TR-diss-0001460(1)

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hydrodynamics of a bubble column loop reactor

7/22/2019 TR-diss-0001460(1)


HYDRODYNAMICSOF A

BUBBLE COLUMN LOOP REACTOR

PROEFSCHRIFT ter verkrijging van

de graad van doctor in de

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de Rector Magnificus,

prof.dr. J .M. Dirken,

in het openbaar te

verdedigen ten overstaan

van het College van Dekanen op

dinsdag 26 november 1985

te 14.00 uur door

ROBERT GERARDUS JAC OBUS MA RIA VAN DER LANS

geboren te Voorburg

natuurkundig ingenieur

Gebotekst Zoetermeer/1985

TR diss1460

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Dit proefschrift is goe dgeke urd door de promotor

prof. J .M . Smith M.Sc.

CO N TEN TS

Summar y VI I I

Samenvatt i ng XCur r i c ul um v i t ae XI I

Chapters

01234567

89

General i nt roduct i onGeomet r yGas in j ect i onSl i p ve l o ci t yDownwar d t wo phase f l owSteady s tate c i r cul at i onTi me dependent ci rcul at i onM x i ng

Mass t ransferConcl usi on

Paragraphs and sections

0 General introduction

0 I nt roduct i on 11 Reactor s w t h energy i nput by gas compressi on 12 Bubble col umn l oop r eactor 43 Bubbl e col umn l oop w t h gas i nj ecti on i n the downcomer 64 Deep shaft 75 Previ ous work 106 Ai m 117 A si mple descr i pt i on of the operati on 11

8 Outl i ne of the thesi s 139 Notes on thesi s l ayout 14

1 Geometry

10 I ntr oducti on 1511 Geometr y of the bubble col umn l oop react or 1512 Experi mental col umn 1613 Measur i ng equi pment 1814 Fr i c t i on number : l i terature 2015 Fr i c t i on number : ca l cu l at i on 2116 I nfl uence of second phase on f ri ct i on number 2217 Fri ct i on number: measurement 2218 Opti mal choi ce of the di ameter rat i o 2319 Concl usi on 24

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VI

2 Gas i n j e c t i o n

2 0 I n t r o d uc t i o n 252 1 P r e v i o u s l i t e r a t u r e c o nc e r n i n g i n j e c t i o n i n d own war d f l ow 252 2 E xp er i me nt a l : d i f f e r e nt I n j e c t o r t y p es f or d o wn f l o w 26

2 3 E x p e r i me n t a l : p r e s s u r e d r o p 282 4 Re s u l t s : d i f f e r e nt t y p es of d o wn f l o w i n j e c t o r s 292 5 Op t i ma l i s a t i o n of t hev e nt u r i i n j e c t o r 3126 The downc omer ai r s p a r g e r s y s t e m 33

2 7 Dy n am c a s p e c t s of the d o wn f l o w i n j e c t o r 3428 The r i s er ai r s p a r g e r s y s t e m 35

2 9 C o nc l u s i o n 36

3 S l i p v el o c i t y

3 0 I n t r o d uc t i o n 37

3 1 B u b b l e s w a r m p h e n o me n a 41

32 Zuber and F i n dl a y - t y p e mo d el s 45

3 3 S l i p v e l o c i t y b a s e d mo d e l s 51

3 4 V oi d f r a c t i o n c o r r e l a t i o ns 53

3 5 S l i p v e l o c i t y . A si mpl e model 543 6 E x p er i men t a l 603 7 Re s u l t s : r a d i a l d i s t r i b ut i o ns 653 8 R e s u l t s : me a n d a t a 71

39 C on c l u s i o n 78

4 D o w n w a r d two p h a s e f l o w

4 0 I n t r o d uc t i o n 794 1 L i t e r a t u r e 794 2 E x p er i me n t a l 814 3 Ov e r a l l d at a ( v o i d f r a c t i o n and v el o c i t y ) 87

4 4 Ra di a l p r o f i l e s of v oi d f r a c t i o n and v el o ci t y 884 5 B u bb l e s i z e 944 6 S l i p v el o c i t y 964 7 T ur b u l e n ce : i n t r o d uc t i o n 984 8 T u r b u l e n c e : r e s u l t s 994 9 C on c l u s i o n 103

5 Steady state circu latio n

5 0 I n t r o d uc t i o n 1055 1 E x p e r i me n t a l and r e s ul t s 1055 2 E x i s t i n g mo d e l s for ai r l i f t t o we r s 1085 3 E x i s t i n g mo d e l s f or d e e p s h af t t y p e c o l u mn s 1115 4 C o mp a r i s o n wi t h t he S b d er b e r g mod e l 113

55 The p r op os ed mod e l 1145 6 V al i d at i o n of t he pr opos ed model 118

5 7 C or r e c t i o n s to the pr opos ed model 120

5 8 P r e di c t i o ns wi t h t he mode l 1235 9 Co n c l u s i o n 125

6 T i me d e pe n de n t c i r c u l a t i o n

60 I ntroduct i on 127

6 1 S t ab i l i t y 127

6 2 Os c i l l a t o r y b e ha v i o u r 1296 3 L i t e r a t u r e 13264 Theory 1336 5 S o l u t i o n 135

6 6 E x pe r i me nt a l v e r i f i c a t i o n 1386 7 Os c i l l a t i o ns 143

6 8 I n st a bi l i t y and o p er a t i o n bo un da r i e s 1466 9 C on c l u s i o n 150

7 Mi x i n g


71 The a x i a l d i s p e r s i o n mo de l 1527 2 Bu b b l e co l u mn s 153

73 The r e c i r c u l a t i o n mo d el 1557 4 L o o p r e a c t o r s 1577 5 E x p e r i me n t a l 158

7 6 I n t e r p r e t a t i o n of measu r emen t s7 7 Ov e r a l l r e s u l t s 1627 8 Downco mer and r i s e r m x i n g 166

7 9 C on c l u s i o n s 168

8 Ma s s t r a n s f e r


8 1 Mod e l s f or mass t r an s f e r measu r emen t 1698 2 B u bb l e c o l u mn l o o p r e a c t o r s 1718 3 I s o b ar i c d y na m c me t h o d s : e x p er i me n t a l 1728 4 I s o b ar i c d y na m c me t h o d s : r e s u l t s 17685 A n o n - i s o b a r i c mo d e l 1788 6 N on - i s o b a r i c me t h o d s : e x p e r i me n t a l 181

8 7 No n - i s o b a r i c dy n am c me t h o d : r e s u l t s 18188 Non - I s o b a r i c s t a t i o na r y me t h o d: r e s ul t s 1838 9 C o nc l u s i o n 186

9 Co n c l u s i o n


9 1 Ge n er a l i z a t i o n of t he s t e a d y s t a t e mo d e l 1879 2 Ge o me t r i c s c a l e up 188

9 3 I n f l u e nc e of l i qu i d p r op er t i e s 1939 4 C o nc l u s i o n s on s c al e up 1969 5 Ge n e r a l c o n c l u s i o n s 197

9 6 Ackn owl ed g emen t s 1999 7 Ap p en d i ces 1999 8 L i s t of r e f e r e nc e s 2479 9 L i s t of s y mb o l s 256

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V I I I

SUMMARY

Bi otechnol ogi ca l p rocesses r equ i r e apparatus of a re la t i vel y l a rge

scal e i n compar i son w th chem cal p rocesses. S ingl e cel l p rote in i s p roduced

f or i nstance i n the 60 met r es hi gh and 7 met r es di ameter col umn of the I CIpressure cycl e ferment er at Bi l l i ngham Waste water i s oft en treated i n very

l arge scale pl ant s. Deep shaft geometr i es, w t h col umns more than 100 metr es

deep and 2 metr es wi de, have been successf ul l y used. Low concent r ated aqueous

sol uti ons of f eed and product usual l y f i l l these col umns.

The subj ect of thi s t hesi s i s t he f l ow phenomena i n l arge scal e

bubbl e col umn l oops w t h gas i nj ecti on i nto t he downcoraer. The l arge hydro

stat i c ef f ects l i mt the usef u lness of convent i ona l t wo-phase - f l ow—model s,

hence an adapt ed concept i s i ntr oduced. Downward t wo phase f l ow, whi ch has

re cei ved l i t t l e at t e nt i o n i n l i t e ra tu re , i s s tu di ed i n mo re de tai l h ere . T h i s

study al so cons iders the f l ow i nstabi l i t i es t hat can devel op i n thi s

equi pment .

The general i ntr oducti on descr i bes a col umn l oop. An excess of

k i ne t i c ( c i r cu l a t i o n) e nerg y r esul t s i f g as i s i nj ec te d i n to the r i ser of a

l arge scale ai r l i f t t ower. I n a deep shaf t type of column thi s energy i s used

to compress gas i nject ed i nto and ent rai ned by the donwf l ow ng l i qui d i n the

downcomer. Desi gn is opt i mal when the hi ghest possi bl e downcomer i nj ect i on

l e ve l i s at t a i n ed w th out r i ser g as i nj e ct i o n, whi l e suf f i c i e nt c i r cu l a t i o n i s

mai nt ai ned t o ent r ai n t he gas i n the downcomer downwar ds.

The fo l l ow ng f i ve chapters dea l w th the steady st ate behavi our .

Chapter 1 tr eat s the geometr i c parameters and t he corr espondi ng f r i ct i onal

pressure l o ss . Thi s l eads to an opt i mal r i ser/ downcomer di ameter rat i o. The10 m high exper i menta l r i g i s a l so descr i bed . Large hydrost at i c e f f ects have

been si mul ated by appl yi ng a r educed t op pressur e ( 10 kPa ) .

Gas i n ject i on and di str i but i on i s the sub ject o f Chapter 2 . The

so lu t i ons t o the probl ems o f gas i n ject i on in to a downf l ow ng l i qu id are

di scussed i n some detai l .

Chapter 3 revi ews the convent i onal t wo- phase- model s f or the

predi ct i on o f vo id f ract i on and t he i r usef u lness f or upward bubbl y f l ow i n the

presence of l a rge hydrostat i c e f f ects . The s l i p ve loc i ty concept i s used and

adapted s i mpl y t o i ncorporate these poss i b le hydrostat i c e f f ects . Radi a l and

axi a l pro f i l es of voi d f ract i on and vel oc i t y were measured i n the r i ser o f the

experi mental col umn usi ng a t wo_poi n t - conducti vi t y bubbl e probe and the

pressure drop method. Sl i p vel oci t y was f ound to be i nverse w t h pressur e fr om

t he commonl y used 0, 30 ms to about 0,40 m s at a gas expansi on of f actor 5

( t hi s i s w th r e duced pressure) .

Downward two phase f l ow was st udi ed i n a speci al t est ri g of 15 cm

di ameter and i s t he subject of chapter 4. Radi al prof i l es of gas vol ume

f ract i on, bubbl y vel oc i ty , bubbl e s i ze and l i qui d ve loc i ty were measured w t ha speci al l y desi gned f i ve-poi nt- opt i cal bubbl e probe and a l aser- doppl er

vel oci t y meter . I nformati on on turbul ence was al so obtai ned t his way. Despi t e

some l i m tat i ons i t has been poss ib l e to show that s l i p vel oc i t y i s l ess than

i n upward f l ow and t hat i t i s approxi matel y equal to t he si ngl e bubbl e

te rm na l vel o ci t y .

Usi ng these resul t s a ( comput er) model was devel oped f or t he s t eady

state c i rcu l a t i on. Th is i s presented i n chapter 5 . Hydrostat i c e f f ects are

i ncorporated i n so far as a) sl i p vel oci ty depends on hei ght (and f l ow

d i r e c t i o n) a nd b ) t he ax i a l pre ssure d i s t r i b ut i o n i s n o n- l i near due to the

presence and expansi on of the gas. Compari son w t h the behavi our i n theexperi ment al col umn shows t hat i n that case the ci rcul ati on rat e is l argel y

det erm ned by the pressure l oss at the downcomer gas i nj ector . The predi cted

l i m ts f or the operat i ng cond i t i ons are w der than those found i n p ract i ce

s i nce the model does not cons i der the poss i b i l i ty o f ve loc i ty f l uctuat i ons.

The si mple model was t heref ore extended w t h a quasi - st ati onary

anal ysi s ( chapter 6) i n whi ch a gas r ate change is al l owed to t ransl ate

through t he system The i ner t i a terms are cons i dered non- s i gni f i cant . Thi s

t i me-dependent model predi ct s osci l l ati on ti mes reasonabl y and st abi l i ty

boundari es qui te wel l i f i t i s assumed t hat spontaneous di st urbances may be

si mulat ed by making st eps i n the gas i nject i on r ate.

Mass t ransfer does not i nf l uence the ci rcul ati on in the experi ment al

col umn, but may be i mport ant i n ful l scal e equi pment. I n chapter s 7 and 8 a

possi ble basi s for a mass tr ansf er model i s gi ven based on presented

measurements of m xi ng and mass t r ansfer . Thi s model , w t h a hei ght dependent

aerat i on const ant (k a ) , may be used t o ext end t he model s pr esented earl i er.

F i nal l y i n chapter 9 the l i t t l e avai l abl e data on deep shaf t systems

are compared w t h a general i zed s teady s t ate model . Furt hermore the i mport ance

of the phys i ca l p ropert i es of the medi um (coa lescent / non coal escent) i s

demonst r at ed on the basi s of some experi ment al measur ement s.

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X

SAMENVATTING

Bi otechnol ogi sche processen vragen in vergel i j ki ng met chem sche

processen apparat en met grote af meti ngen. De " I CI pressure cycl e f erment or" i n

Bi l l i ngham bi j voorbee ld , waar i n e iw t wordt geproduceerd , i s 60 meter hoog enheeft een di ameter van ci rca 7 meter . Afval water wordt gezui verd i n groot

schal i ge apparatuur zoal s de "deep shaft " syst emen di e meer dan 100 met er di ep

kunnen zi j n met di ameter s van meer dan 2 met er. De gebr ui kt e medi a zi j n i n het

al gemeen water i ge opl ossi ngen met l age concentr ati es voedi ngs- en pr odukt -

s to f f e n.

De st rom ngsverschi j nsel en i n bel l enkol om ussen met gastoevoer naar

de daal bul s ( downcomer) vormen het onderwerp van di t proefsc hri f t . Sterke

hydrostat i sche eff ecten beperken de brui kbaarhei d van convent i onel e t weef asen-

model l en, r eden waarom een aangepast model werd ontw kkel d. Neerwaart se t wee-

f asenst rom ng is vr i j gedet ai l l eerd bestudeerd omdat daarover nog ni et veel

was gepubl i ceerd. Verder i s het onderz oek t oegespi t st op de beschri j vi ng van

de ci rcul ati e en de daarbi j i n deze apparatuur optr edende str om ngs -

i ns t a bi l i t e i t en.

I n de al gemene i nl ei di ng wordt het begri p bel l enkol om us geï ntr odu

cee rd. Ga sto evoe r o nde r i n zo ' n l us re ac to r ( l u cht~ i f t t o re n) l ei dt b i j gro te

afmeti ngen tot een onnodig hoge ci rcul ati esnel hei d. Gasi nj ecti e i n de daalbui s

i s dan aant rekkel i j k. Het surpl us aan ki net i sche energi e wordt i n dat geval

gebr ui kt voor de compressi e van het aan de neer waart s s t r omende vl oei st of t oe-

gevoerde en daar door meegesl eurde gas. Bi j het ont werp wordt gest r eefd naar

een zo hoog mogel i j ke l okat i e van de gast oevoer, onder behoud van vol doende

ci rcu l a t i esne lhe id omhet gas mee te s l euren en de c i rcu l a t i e i n stand te

houden.

De volgende vi j f hoofdst ukken behandel en de st ati onai re ci r cul ati e.

I n hoofdst uk 1 komen de geomet r i sc he paramet er s aan de orde en het daar mee

verband houdende drukverl l es door wri j vi ng. Hi erui t wordt een opt i mal e verhou

ding van de di ameters van de sti j gbuis ( r i ser) en de daal bui s af gel ei d. Tevens

wordt i n di t hoofdst uk de 10 m hoge experi ment el e opstel l i ng besproken. De

st erke hydrostat i sche eff ecten worden daarbi j nagebootst door een verl aagde

druk ( 10 kPa) boveni n de kol om

De t oevoer en di spersi e van gas i s het onderwerp van hoofdst uk 2. Er

wordt voor al aandacht best eed aan de probl emen di e opt r eden bi j gastoevoer i neen neerwaart s st r omende vl oei st of en aan de gebr ui kt e gasver del er.

XI

Hoofdst uk 3 geef t een overzi cht van de convent i onel e t weefasen-

model l en voor de voors pel l i ng van de gasfr acti e en bespreekt hun brui kbaarheid

i n het geval van een opwaart se bel l enstr om ng i n aanwezi ghei d van st erke hydro

st ati sche eff ecten. Een eenvoudig te hant eren model op basi s van de sl i psnel -

hei d als par ameter wordt gepr esenteerd waari n deze ef f ecten eenvoudi g i nge

bouwd kunnen wor den. I n de st i j gbui s van de experi mentel e kol om werden radi al e

en axi al e verdel i ngen van de gasf r act i e en de gassnel hei d bepaal d met behul p

van een twee-punts- gel ei dbaarhei dsmeter en met de drukval - methode. De sl i p-

snel hei d bl i j kt omgekeerd evenredi g t e zi j n met de druk en t oe te nemen van

0, 30 m s onder i n de kol om tot ongeveer 0 ,40 m s b i j een vi j f maal l agere druk.

Neerwaart se t weefasenst r om ng (hoofdst uk 4) werd onderzocht i n een

speci aal aangepaste proefopstel l i ng met een meet bui s van 15 cm di ameter . Hi er

i n werden r adi al e verdel i ngen gemeten van de gasvol umefr acti e, de bel snel hei d,

de bel groot te en de vl oeis t ofs nel hei d. Hi erbi j werd een speci aal ont worpen

vi j f punts- opt i sche-bel l enmeter gebrui kt en een l aser - doppl er - snel hei dsmeter .

Ook t urbul ent i e kon zo bepaal d wor den. Ondanks enige beperki ngen kon geconcl u

deerd worden dat de sl i psnel hei d l ager i s dan in een opwaart se st rom ng enongeveer gel i j k i s aan de sti j gsnel hei d van een enkel e bel .

Aan de hand van de gevonden r esul t at en werd een computer model ont w k

kel d voor de stat i ona i r e c i rcu l a t i e i n de kol om Di t wordt beschreven i n hoof d

stuk 5. Een s l i psnel hei d d i e a fhanke l i j k i s van de hoogte (en strom ngsr i ch

t i ng) en een ni et~ i neai re axi al e drukverdel i ng i n verband met de expansi e van

het gas zi j n de i ngebouwde hydrost ati sche eff ecten. Vergel i j ki ng van de model

voors pel l i ngen met experi ment en toont aan dat i n de kolom de ci rc ul ati esnel

he id st erk af hanke l i j k i s van het d rukver l l es bi j de gastoevoer i n de

daal bui s .

Het voors pel de wer kgebi ed i s grot er dan gemet en omdat het model geen

r ekeni ng houdt met mogel i j ke snel hei dsschommel i ngen. Daarom werd het model zo

ui tgebrei d dat een veranderi ng i n de t oegevoerde gasst room zi ch door het

systeem kan verpl aatsen. Deze quasi - st ati onai re aanpak di e de tr aaghei d bui t en

beschouw ng laat i s besproken i n hoofdstuk 6 en bl i j kt t r i l l i ngst i j den rede

l i j k en st abi l i t ei t sgrenzen goed te voorspel l en indi en wordt aangenomen dat

spontaan opt redende f l uctuati es vergel i j kbaar zi j n met st apsgew j ze verande

r i ngen i n de t oegevoerde gasst r oom

Stofoverdracht speel t nauwel i j ks een ro l b i j de ci rcu l a t i e i n de

exper i mente l e kol om B i j i ndustr i ë le toepass ingen staat d i t aspect echt er

cent raal en zal ook de ci rcul ati e beï nvl oeden, reden waarom i n hoofdst uk 7 en

8 een aanzet gegeven wordt tot een st of overdracht smodel waarbi j de

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XI I

bel uchti ngsconstante kLa met de hoogte var i eer t . Hi er toe z i j n i n de kol om

mengti j d- en stof overdr acht smeti ngen ui t gevoerd.

Tensl ott e worden in hoofdst uk 9 de schaarse beschi kbare prakti j k-

gegevens vergel eken met een al gemene vers i e van het st at i onai re model . Ook

wordt met enkel e meti ngen aangetoond hoe groot de i nvl oed van de ei genschappen

van het medi um (bi j voorbeeld coal escerend/ n iet - coal escerend) kan z i j n .

CURRICULUM VITAE

Gebor en 490625 te Voorbur g.

Di pl oma h. b.s. - b i n 1969.

M l i t a i r e di ens t .

I ngeni eursdi pl oma techni sche natuurkunde, f ysi sche t echnol ogi e, i n 1978.

Wetenschappel i j k medewerker 1978- 1983 bi j de st i chti ng Fundamenteel Onderzoek

der Mater i e ( FOM) , gestat i oneerd b i j het l aborator i um voor Fys i sche

Technol ogi e t e Del f t .

Wetenschapel i j k medewerker s i nds 1984 bi j de TH Del f t , Laborat or i umApparat enbouw voor de Procesi ndustr i e.

1

General introduction

Introduction

There i s cons i derable i nterest i n bubble co lumns i n which both l i qui d

and gas c i r cu lat e cocur rent l y, o f ten wi t h l arge hydrost at i c head d i f f erences .

The 1975 devel opment by I CI o f the ai r l i f t f ermentor f or synthet i c protei n

manuf actur e has been fol l owed by the use of thi s t echnology as deep shaf t

waste water t reat ment . Shaft s can be of consi derabl e di mensi ons, a

character i s t i c s i ze being 2 m i n di ameter and 100 m deep. The changes i n

dr i v i ng force f or t he mass t r ans f er processes of so lut i on and desorpt i onbetween l i qui d and gas phases as t he gas ci rcul ates can br i ng l arge benef i t s

to t he process . I n these c i r cu l at i ng l oop reactor s , l i qui d movement can be

i nduced, or a t l eas t as s i s t ed, by t he a i r l i f t wo r k i ng i n t he r i s e r . I n

co lumns of great hei ght i t i s a l so poss i b le to reduce t he supply a i r energy

demands by expl oi t i ng t he l i qui d f l ow i n the downcomer to achi eve some of the

compressi on. Thus ai r i nj ecti on i n the downcomer at an int ermedi ate depth can

cont i nue to provi de a net pos i t i ve ci r cu l at i ng for ce, provid i ng that the gas

f ract i ons and re l at i ve aerated hei ghts i n r i ser and downcomer are sui tab l e.

Devel opment of st eady st ate and dynam c model s f or t he ci rcul at i on in

these systems r equi res dat a on several aspects of two phase f l ow whi ch are not

readi l y avai l able i n l i t erat ure. Data on downward t wo phase fl ow and gas

i n ject i on i n downward l i qui d f l ow are par t i cul ar l y scarce.

Be fo r e gi v i ng t he a i m and s c ope of t h i s t hes i s i n r e l a t i on t o i t s

subj ect , the apparatus , a bubble co lumn l oop w t h downcomer a i r i n ject i on, i s

p laced i n i t s context as an aerator and reactor . The s i mple bas i c f orm of the

equat i on of mot i on i s g i ven to expl a i n the s t r ucture of the thes i s .

1 Reactors with energy input by gas compression

Reactor s i n whi ch the power f or m xi ng and mass t ransf er i s

i nt roduced by i n ject i ng gas are w del y used both i n the chemcal and

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1 3

bi otechnol ogy i ndustr i es. Si mpl e bubbl e col umns are i n common use. There are

a l s o s ever al s peci a l des i gns w t h s t a t i c i n t er nal s t o di r ec t t he f l ow o r

redi sperse t he gas. Lar ge bubbl e dr i ven col umns are al so f ound i n some waste

water tr eatment pl ants. The most i mport ant argument s f or the use of these

reactors i nstead of the more t rad i t i onal mechani cal l y agi tated ones are:

- The constr ucti on is s i mpler and cheaper.

- Someti mes compressors are al ready avai l abl e.- The compressors can easi l y be mai ntai ned or removed w t hout the need t o

empty or even stop the process.

- Mai ntenance i s s i mpl er s i nce there are no rot ati ng part s bel ow the water

l e ve l .

- The maxi mum l ocal shear st ress experi enced by the process f l uid i s much

l ower . Thi s can be i mpor t ant for m cro- organi sms or w th cer t a i n pol ymer

s o l u t i ons .

- Gas di s per s i on e f f i c i enc y i s bet t e r ( Sc hüger l , 1982).

A deta i l ed survey of gas - l i qui d r eactors i s g iven by Schüger l ( 1982). He

c l as s i f i es t he r eac t o r s

a) by t he method of energy i nput :

- by i nternal mechani cal means , such as s t i r r ed tank reactors (s t udi ed in

detai l by Van 't Ri et ( 1975) and Warmoeskerken (1986) ) , brush aerators

and propel l er dr i ven l oop reactor s (B l enke, 1979);

- by external l y pumped l i qui d , such as pl unging j ets ( s tudi ed in deta i l by

Van de Sande ( 1974) and Van de Donk (1981)) , and ( l oop) - r eactors w th two

phase i n jector nozz les ;

- by gas compressi on, such as bubbl e col umns, and

b) i n t erms of geometr y:

- tank reactors wi th an aspect rat i o hei ght to di ameter l ess than three;

- co lumn reactors w th H/d greater than three.One may di f f erenti ate between types of gas dr i ven react ors:

1 . Unconf i ned bubbl e co l umns. I n th i s case w del y separated a i r spargers are

used i n a l arge bas i n . There i s l i t t l e or no int eract i on between the bubbl e

pl umes. Bubbl e curt ai ns are a two-di mensi onal var i ant. Unconfi ned bubbl e

columns are used i n some ef f l uent t reatment p lants ( f i gure 1. 1) . Deta i l ed

research on t he hydrodynamcs of a bubbl e pl ume and i ts surr oundi ng has

been done by Goossens ( 1979).

2. The ( conf i ned) bubbl e col umn w t hout i nter nals . Thi s i s t he most common and

s i mple f orm of contact i ng a i r and water ( f i gure 1. 2) . Extens i ve research

has produced extensi ve l i t erat ure. Schügerl (1977) has present ed a

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.J V

il

Bubbl e colum

external tubul ar doi

sure Cycle Fermentor (ICI); Gou et al. (1975), figure 1.

comprehensi ve t r eatment of bubbl e col umn bi oreact ors and Shah et al . ( 1982)

an extens i ve l i terature r evi ew.

3 . Bubbl e col umns w th i n terna ls to red isperse the gas. Genera l l y th is i s done

wi t h per forated pl a tes ( mu l t i s tage co l umns) ( f i gure 1. 3a) but a lso w t h

s t a t i c m x er s ( f i gur e 1.3b). Such arr angements are necessary i f the

coalescence rate i s hi gh.

4. Bubb le co l umns w th i n ternal s to di rect the f l ow. I n col umns w thout

i nternal s homogeneous bubbl i ng is easi l y di st urbed, and lar ge l i qui d

ci rcu l a t i on patterns can devel op as a resul t o f rad ia l var i a t i ons i n the

gas concentr ati on. The upward f l ow ng l i qui d i n the cent re of the col umncan be stabi l i zed over the whol e hei ght by i nsert i ng a draught t ube ( f i gure

1.4a) . Thi s t ype of col umn i s cal l ed a bubbl e col umn l oop reactor and i s

tr eat ed by Bl enke ( 1979) i n a revi ew whi ch al so consi ders mechani cal l y and

l i qu id d r i ven loop reactors . There i s no d i f f erence i n pr i nci p le between

co lumn l oops w t h i nterna l o r external downf l ow sect i ons ( f i gure 1.4b) .

2 Bubble column loop reactor

Bubbl e col umn l oops are commonl y known as ai r- l i f t react ors. Ai r i s

i n j ected through a sparger at or near the bot t om of t he upf l ow sect i on, the so

cal l ed r i ser. The reduced densi ty of the two phase m xture i n t he r i ser causes

a ci rcul ati on in t he l oop I n the same way as w t h an ai r l i f t pump. Hi gh gas

f l ow rates are possi bl e. I mport ant work on air l i f t reactors has been done by

Hat ch ( 1975) , Hi l l s ( 1976) , L i n et a l . ( 1976) , Wei l and ( severa l pub l i cat i ons,

t hesi s 1978, et al . 1980, 1981, 1982) and Merchuk and Stei n ( 1981a). At t e nt i o n

i s genera l l y d i r ected t owards app l i cat i ons f or the product i on of s i ngl e cel l

pr ot e i n s. L i t t l e l i te rat ure appeared before 1972 a l though an a i r l i f t

f erment or was alr eady pat ented by LeFrancoi s i n 1955. Thi s i s not surpri si ng

si nce appl i cati ons to biot echnol ogy did not recei ve much att ent i on bef ore

about 1968, t he same year that I CI star ted w t h l aborator y work on si ngl e cel lpro te i n p ro duc t i o n po ssi bi l i t i e s ( Ho we l l s , 1982) .

Tower l oop reactor s are r eport ed to be much more sui tabl e f or si ngl e

ce l l p rote i n manefacture than cont i nuous st i r r ed tank reactors (Adl er et a l . ,

1982) . These r eactor s work w t h smal l di ameter downcomers to short en t he

m xtur e resi dence ti me i n the non-aereated part . Anot her way to achi eve this

i s t o i nj ect a i r i n the downcomcr as wel l as i n the r i ser . Al t hough not

ment i oned by Gow et al . ( 1975) on t he occasi on of the i ntr oducti on of the I CI

p ressure cycl e ferment or , th i s poss i b i l i ty i s shown on thei r f i gure 1

( f i gure 2. 1) and was t ested by Li n et al . ( 1976) .

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C I C I ) a t BU l i n g h ;

Use as f ermentor a l so demands suf f i c i ent m xi ng of subst r ates t o

avoid nut r i ent - l i m ted growt h, ef f i c i ent mass t r ans fer f or oxygen and the

desorpt i on of carbon di oxi de to t he bubbl es, the separat i on of gas and removal

of heat . Al though these requi r ements become more d i f f i cu l t w t h i ncreas ing

s c al e , recent appl i cat i ons i nvolve very l arge equi pment . The I CI pressure3

c yc l e fe r ment o r i n Bi l l i ngham ( f i gur e 2. 2 ) c ont a i ns 2000 m of l i qui d ( 60 m

hei ght , 7 m d i amet e r ) ( Schi i ger l , 1983). W th ef f l uent t reatment the aerated

vol umes are of the same order of magni t ude or l arger ( 3000 m = 100000

popul at i on equival ent ) .

Bubbl e col umn l oop w t h gas i nj ecti on i n the downcomer

La r ge a i r l i f t l oop r eac t o r s have i n fact a s u r pl us of ve l oc i t y . On l y

wi t h ver y l ow gas r at es or i f t her e ar e r es t r i c t i ons i n t he l oop w l l l i qu i d

veloci t y be bel ow about 0.75 m s i n the downcomer. For i nstance Wei l and ( 1978)

report ed t his vel oci t y i n the 50 mm di ameter downcomer of a 10 m hi gh column

wi t h a 100 mm di ameter r i ser when the mean r i ser voi d f ract i on was onl y 2%

Ri ser veloc i t i es do not need to exceed about 0 .30 m s t o keep so l i ds i n

c i r c u l a t i on. Low vel oci t i es ar e a t t r act i ve f r om t he vi ewpoi n t o f gas r esi dence

t i me.

One may use the excess of ki neti c energy, whi ch woul d ot herw se be

c onver t ed i nt o hea t by wa l l f r i c t i on, t o

3 7

- red i sperse t he bubbl es , as i s done in t he I CI pressure cyc le f ermentor by

per f orated pl ates ( f i gure 2. 2) or

- compress at l east par t of the a i r by us i ng an i n ject i on point i n the

downcomer. The vel oci t y i n the downcomer shoul d al ways exceed about 1 m s

(dependent on the type of i n ject or , as w l l be d i scussed l ater ) i n order t o

ensure t hat the gas i s carr i ed downwards.

One may als o i ncrease the r i ser- downcomer di ameter rat i o t o decrease r i servel oci t y , but t h i s i s not us ua l l y an opt i ma l c hoi s e .

The I CI deep shaft sys tem for ef f l uent t r eatment uses a l oop des igned

to operate w th a i r i n ject i on in the downcomer on ly but employs r i ser a i r

i n ject i on to s tar t up the system

4 Deep shaf t

The deep shaft process was evolved by t he agr i cu l t ura l d i v i s i on of

I CI i n the ear l y 1970s dur i ng research into synthet i c protei n product i on

( Hi nes et a l . , 1975) . The concept of a deep shaft w t h aerated downcomer I s

not new and was f i r s t used i n the form of a cont i nuous l y d i scharg i ng U- tube i n

Scheveni ngen ( Brui j n and Tui nzaad, 1958) . The I CI rec i rcu l at i ng var i ant was

i ntr oduced soon aft er the energy cr i s i s i n 1973 and recei ved much att enti on

because of the hi gh power economy cl ai med ( up to 1.5 kg 0„/M = 6 kg 0? /

kWh) . Sever al pub l i c at i ons f o l l owed t he f i r s t : Bol t on et a l . ( 1975, 1976) ;

Hemm ng et al . (1977); Hemm ng (1979), whi ch give much general i nfor maLi on. A

number of pi l ot and f ul l scal e pl ant s have been constr ucted si nce then. The

f i r s t pi l o t p l an t was bu i l t nea r t he I CI est abl i s hment a t Bi l l i ngham i n 1974,

soon fo l l owed in 1975 by the f i r s t f u l l scal e co lumn at the potato s t arch

f actor y of Ensl andst arke, GmbH ( KSH) at Em i chhei m BRD, near t he Dutch

boundary. Repor ted p l ants are l i s ted i n tabl e ( 4) .The deep shaf t i s 50 to 150 m deep and i s di vi ded i nto downcomer and

r i ser by a cent r a l draught tube or ver t i ca l baf f l e as shown in f i gure ( 4 . 1 ) .

At t he top there i s a gas di sengagement tank. The shaf t can be 0. 5 to 10 m i n

diameter . The ef f l uent to be t r eated i s f ed to t he top of the downcomer . Ai r

i s i n jected i n the downcomer at a suf f i c i ent depth to cause the l i qui d to

c i r c ul a t e at a vel oci t y o f 1 t o 2 m s . Thi s i s gr eat e r t han t he r e l a t i ve r i s e

veloc i ty of t he bubbl es , whi ch are car r i ed downwards . The a i r l i f t act i on of

t he a i r i n t he r i s e r w l l k eep t he l i qui d ci r c u l a t i ng as l ong as t he

di f f erence i n vo i dage between the upf l ow and downf l ow par t s i s suf f i c i ent to

overcome f r i c t i on and other l osses . The void f ract i on in t he l ower par t of t he

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- r o c k ( T l l b u r y ) ( E s

I n g h a m

i n g h a m

h o e ( N o r t h u m b e r l a

: s b u r y

. c h h e l m

s e x ) ( 1 9 7 8 )

( 1 9 7 4 )

ai)

( 1 9 7 8 )

( 1 9 7 5 )

F 1 .8 6 x 1 3 0

P 0 .4 0 C x 1 3 0

F 1.3 x 50

F G x 33

F a x l o s

P Q .1 3 U x 6 1

F 1.1 x 100

(D

CD

1

0

. 0 0 ) i

. 2 5 ) t

c l x e d ( H O k p e )

■mlelpü; i n g l e c e l l p t o t e

« s e c p a p e r r e c y c

: e r e p h t a l i c a c i d

m u n i c i p a l

I

I n ( 3 0 k p e )

l i n g ( 1 0 0 l ep .

p l a n t ( 1 0 0 k,

( 4 0 k P ü )

t e n ( M a n i t o b a )

F 3

F o.e

( 1 9 7 7 ) P 0 . 0 7 U

P 0 . 4 5 C

( 1 9 7 6 ) P 0 . 4 6 C

F 1 .3 7

6 5 ( 3

1 5 S ( Ï 2 2 0 )

f t e ) m ix ed ( : 0 0 k p u '

1 ( 1 0 I

1 5 3 (2 s h a f t s ) I

a) ( 1 9 7 8 ") P 0 .0 8 U x 1 5 0

! L P ( M a n i t o b a ) F 1 .3 7 K 1 5 3 ( 2 s h a f t s ) I

( H i t a c h i P l a n t K e r n

P 0.1 ■

P 0.1 !

P 0.45C x 150

F = F ul l S c al e P l a n t , P » P i l o t P l a n t .

C ■ C onc e n t r i c ( i n t e rna l ) dom c ome r , U = U - t ube ( e x t e rna l dou i

Dimensi ons i n metr es.

k pe ■ k i l o p o pu l a t i o n e qu i v a l e n t .

References: H = Hemm ng (1979)

C - C ox e t a l . ( 1980) L - Loc k ( 1382)

G - G a l l o a n d S t a n f o r d ( 1 9 7 9 ) Z = Z l o k a r n l k ( 1 9 8 2 )

Downcomer

e n i ( H l n e a ,

downcomer i s higher t han that i n the r i ser at the same l evel because of the

d i r ect i on of t he r e l a t i ve vel oci t y o f t he gas . The dept h o f a i r i n j ec t i on i n

t he downcomer must be l arge enough to compensat e f or t hi s ef f ect as wel l

( f i gur e 4 . 2 ) . F or s t a r t - up, l i qui d c i r c u l a t i o n i s i nduc ed by i nj ec t i ng a i r i n

the r i ser f i r s t and then by gradual l y t ransf er r i ng to the a i r sparger i n the

downcomer. Treated l i qui d i s drawn of f f rom the di sengagement basi n.

As m ght be expected, the l i t eratur e ci ted emphasi zes the advang tages

of the process. Hemm ng ( 1979) summari zes t he aearati on charact er i st i cs as

fo l l ows :

1. A h igh i ntens i ty of oxygen uptake, due to the abi l i ty to t r ans f er l arge

amounts of a i r qu ick l y i nto so lut i on.

2. A h igh ut i l i zat i on of t r ans f er red oxygen, as a resul t o f t he rapi d anduni f o r m ava i l abi l i t y of di s s ol ved oxygen fo r b i o l ogi c al p r oces s es .

3. A high energy economy of oxygen tr ansf er, due to t he hi gh tr ansf er and

m x i n g ef f i c i e nc y.

The fo l l ow ng processes are c l a imed to be respons i b le f or t hese resu l ts :

- Hi gh turbul ence l evel s because of the large vel oc i t i es and d i mens i ons

l eading t o a hi gh rate of bubbl e sur f ace renewal , smal l bubbl es and

ef f i c i e nt m x i n g.

- Hi gh oxygen t rans fer dr i v i ng force i n the l ower r egi ons of the shaft because

of the h igher so lub i l i ty of oxygen at h igher pressures , achi eved w thout

hi gh energy consumpt i on.

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- Long bubbl e cont act ti me ( 3 to 5 mnutes i nstead of 15 s i n a convent i onal

s y s t e m .

Ot her advant ages are ( Hemm ng et al . , 1977):

- Reduced t reat ment vol ume ow ng to t he hi gh tr eatment r at e.

- Low producti on of waste sl udge w t h good processi ng qual i t y.

- Stabl e bi ol ogy w t h good resi st ance to shock l oads.

- Smal l er l and ar ea r equi r ement.Maj or di sadvantages have been r eport ed by users :

- Hi gh i nvestment cost s.

- Hi gh f oam ng r ate.

- Hydrodynamc metast abi l i t y.

The metastabi l i ty f o l l ows f r om the in teract i on between t he voi dages i n r i ser

and downcomer. As soon as t he r i ser dri vi ng for ce, f or what ever r eason, i s no

l onger suff i ci ent to overcome t he opposi ng f orces, i ncludi ng t he downcomer

voi dage, the l i qui d c i rcu l a t i on w l l s ta l l and reverse . One may thi nk of a

d ist urbance i n the downcomer a i r i n ject i on. I f extr a a i r i s p resent i n the

downf l ow sect i on, the l i qui d c i rcu l a t i on ra te w l l decrease. I n consequence

bot h r i ser and downcomer voi d f ract i ons w l l i ncrease. I f the downcomer voi d

f ract i on i ncreases more than that i n the upf l ow secti on or i f the di st urbance

i s too l a rge, t he hi gher gas f ract i on w l l be too s low i n reachi ng the r i ser

to retr i eve the f l ow.

I n p ract i ce thi s prob lem i s overcome by i nj ect i ng a i r i n to the r i ser

(Cox et a l . , 1980) , t hough t hi s af f ects t he power economy adversel y. Another

"so l u t i on" i s f i t t i ng a cap to the shaf t to avert the d isast rous ef f ects of a

poss i b le r eversa l (Lock, 1982). A compl ete modi f i cat i on of the deep shaft

process was i ntr oduced i n Canada by Eco Technol ogy t o provi de bett er

st abi l i t y, l ess foamng and higher power economy ( see Lock, 1982; Zl okarni k,

1982) .

5 Previ ous work

The c i t ed l i terature on the deep shaf t system i s genera l l y occup ied

wi t h ef f l uent t r eatment per f ormance r esul ts . A s i mu lat i on on l aboratory scal e

of t he bi ochem cal process was made by Si ebers and Vel dkamp (1980). The

hydrodynamcs have however recei ved hardly any att ent i on. Hi nes et al . ( 1975)

gave a si mple model at t he i ntr oducti on of the deep shaf t . S i m l ar but f urt her

el aborat ed model s were pr esent ed by Kubota et al . ( 1978) and Sbderberg

(1980). Sbderberg was the f i rst to veri f y his model experi ment al l y.

5 11

Measurement s were made i n a 10 m hi gh pers pex l oop wi t h bot h r i ser and

downcomer of 0.24 m di amet er. Al t hough t he column could oper ate w t h downcomer

a i r onl y , hydrostat i c e f f ects were l i m ted and thei r e f fects cou ld not be

test ed. Söderberg al so st udi ed t he dynam c behaviour of the system

Work on a pumped ver si on i s r eport ed by Hosono et al . (1979), and

there are al so publ i cati ons about the deep U- tube aerat or by Speece et al .

( 1980) and Tak amatsu et al . (1981), but agai n these do not cont ai nexperi ment al resul ts on the ci rcul ati on hydrodynam cs.

The purpose of the research r eport ed i n thi s t hesi s i s t he st udy of

t he tr ansport phenomena of a bubbl e col umn l oop wi t h gas i nj ecti on i n the

downcomer and l arge hydrost ati c ef f ect s, wi t h emphasi s on bubbl e swarm

behavi our and t i me dependent f l ow charact eri st i cs.

7 A si mpl e descr i pti on of the operati on

I t i s conven ient to i ntr oduce now a s i mp le descr i p t i on. The st ructure

of th e th es i s i s l arg el y de term ne d by i t and i t w l l be po ss i bl e to r e f e r t o

i t i n the subsequent chapt ers.

I L i s poss i b le t o l ook at the c i rcu l a t i ng system as be ing equi val ent

to one i nstant dur i ng t he osci l l a t i on of l i qui d i n a U- t ube. By in tr oduci ng

gas i n the col umn, more l i qui d i s present I n t he downcomer l eg than in t he

r i ser l eg. T he l i qui d w l l t h ere f o re f l o w but th e sys te m i s cont i nua l l y kept

out o f equ i l i br i um by the i n ject i on of gas and the over f l ow f r om r i ser to

downcomer. Fri ct i on compensat es t he dri vi ng f orce, resul t i ng i n a uni f orm

mot i on.The most common descr i pti on i s t he momentum bal ance i n i t s s i mpl e

f orm of equat i ng pressure drops

f or a (one di mensi onal ) s t eady f l ow t hrough a duct ( Wal l i s, 1969) . The t otal

pressure drop over a duct i s t he summati on of the cont ri but i ons due to

f r i c t i on ( f ) , accel e rat i o n (a ) and gra vi t y ( g ) . I f t h e duc t i s a cl osed l o op

t he t otal pressure drop and the net accel erati on terms are zero. ( Accordi ng to

Prei ssl er ( 1983) accel erat i on and decel erati on of a two phase mxt ure do not

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compensat e each other , but the t erms are i n our case negl i gi bl e anyhow) . The

bal ance fur t her s i mpl i f i es i n the macroscopi c f orm

AP g - - A p£ ( 2)

For a ver t i ca l l oop the gravi t at i onal pressure drop i s s i mply the d i f f erence

between the cont r i but i ons of r i ser and downcomer:

i pg - ( 1 - ö R) p L gL e - ( 1 - V L g L £ ( 3)

where t he contr i but i on of the gas i s neglect ed and where

a i s the axi al mean r i ser (R) or downcomer (D) vol umetr i c gas f ract i on,

p i s t he l i qu i d dens i t y ,

g the gravi t at i onal accel erat i on and

L the ef f ect i ve hei ght over which the gravi t at i onal f orces are

r e l evant .

The f r i ct i on may be descr i bed si mpl y by means of a f r i ct i on number

A pf

also ca l l ed the number of vel oc i t y heads l os t i n the system

Duri ng the research i t became apparent that an addi t i onal l oss has to

be c ons i der ed. I n j ec t i on of a i r i n a down f l ow ng l i qui d l eads t o l o s s es , due

to c l i ngi ng ai r pockets (cavi t i es ) among other th i ngs , whi ch cannot be

descr i bed i n the same way as t he gravi tat i onal or f r i c t i onal ter ms. For the

pr es ent we w l l c al l t h i s l o ss t he I n j ect i on Head Los s ( I HL ) . W t h equat i ons

( 2 ) , ( 3) and ( 4) we have

(^ - a D) L e - KF * - + I HL (5)

where v i s an appropr i ate veloc i ty to represent c i r cu l at i on.

The downcomer vo id f ract i on i s i n fact a res i s t i ng f orce and on ly t he

r i ser voi d cont r i butes t owards dr i v i ng the system

Ve " Ve +KF h + I HL <6)

Once a co lumn has been bui l t , the on ly var i abl es t hat are avai l abl e to ensure

that the r i ser voi d term i s l arger than the r i ght hand si de res i s t i ng terms

1 13

are the gas f l ow rate and the in j ect i on depth. To predi c t c i r cu l at i on vel oc i t y

as a f uncti on of these i nput parameter s and of desi gn parameter s one needs a

know edge of voi d f r act i on, f r i c t i on and in j ect i on head l osses as f unct i ons of

t hese paramet ers .

8 Out l i ne of the thes i s

E quat i on ( 7 - 5 ) i s t r ans l a t ed s c hemat i c al l y i n f i gur e ( 8 ) w t h bo l d

l i nes : the geometr y, through the f r i c t i on and void f ract i on deter m nes t he

c i r c ul a t i on vel oci t y . I n t he f i r s t f our c hap t er s t hes e var i abl es ar e t r eat ed.

Chapter 1 deals w th the geometr y, especi al l y of the experi mental col umn, and

the way i n whi ch thi s det erm nes t he f r i ct i on number. I n Chapter 2 the

character i s t i cs of gas i n ject i on in a downward f l ow ng l i qui d are cons i dered

together w th our so lut i on to the prob l em Voi d f ract i on i s the subj ect of

chapter s 3 ( upward f l ow) and 4 ( downward f l o w) . F i gure (8) i ndi cates that

r adi a l d i s t r i but i ons o f l i qui d vel oci t y and gas c on t ent t oge t her w t h bubbl e

s i z e det e r m ne t he r e l evant s l i p ve l oc i t y . The sl i p ve l oc i t y i s t he r e l a t i ve

vel oci t y of t he gas compared t o the l i qui d and determ nes i n how f ar voi d

f r ac t i on devi a t es f r om t he vol umet r i c gas f l ow r a t e f r ac t i on. The l i t e r a t u r e

cir cul at i on vel ocit y

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14

concerning the predi ct i on of void f ract i on i s consi dered i n chapt er 3 wher e a

si mpl e sl i p vel oci t y based model i s presented and compared w t h measurements

made i n the r i ser of t he experi ment al col umn. Speci al att ent i on i s paid i n

chapter A to ext ensi ve addi t i onal s tudies of gas- l i qui d bubbl y downfl ow.

Based on the result s descri bed i n the previ ous f our chapt ers, a

st eady st ate model was devel oped and i s present ed i n chapter 5. I t was

possi bl e t o model the ti me dependent behavi our of t he col umn w t h aquasi - st ati onary approach, based on the steady st ate model ( chapter 6) .

Als o i ncl uded in f i gure ( 8) are mxi ng and mass t ransf er, i mport ant

f actors f or desi gn and model l i ng. I n our experi ment al col umn t hese processes

were not l arge enough to i nfl uence t he hydr odynam cs. Some exper i mental

i nvesti gat i ons on t hese aspects are report ed i n chapter 7 and 8. The di agram

i s not f ul l y comprehensi ve, among other aspects t urbul ence and medi um

propert i es are om tt ed. These f actors w l l i nf l uence al most any of the

vari abl es i n the f i gure and t hei r i nclusi on woul d make it unr eadabl e.

The t hes is concl udes i n chapter 9 , w th d i scuss ion o f sca l e up ,

general concl usi ons and appendi ces.

9 Notes on t hesi s l ayout

Each chapt er i s subdi vi ded i nto secti ons whi ch ar e number ed i n such

way that t he chapt er numbers f orm t he tens. Equat i ons ar e numbered per

secti on, w t h t he secti on number onl y shown when cross- ref erenced f rom another

secti on. Fi gures and t abl es are numbered per secti on and ref l ect t h i s .

Appendi ces f ol l ow an appropri ate numberi ng. Refer ences are l i st ed

al phabeti cal l y f or the whol e thesis and have their f orm i n accor dance wi t h

t h at us ed i n A. I . Ch- E . J l .

Unl ess otherw se st ated the dimensi ons used i n the f ormul ae i n thisthes is are i n the S. I . - system On ly f or equat i ons t hat are not d i mensi ona l l y

consi st ent or f or not commonl y used quanti t i es are the dimensi ons gi ven.

Geomet r y

I nt r o duc t i o n

The f i rst par t of the thes i s dea ls w th t he separate terms o f the

equat i on of moti on of the system ( 7 - 5 ) :

"~ F 2g ' " "*

I n this chapt er t he fr i cti on number K i s di scussed i n gener al t erms as wel l

as speci f i cal l y f or t he experi ment al col umn used. Cal culat ed val ues of t he

f ri ct i on number are compared w t h measur ement s. The i mpl i cati ons of thef ormulae used to cal cul ate K the col umn desi gn, i n part i cul ar t he opt i mal

choi se of the rati o of di ameters of r i ser and downcomer, are consi dered. I n

the l ast sect i on the pract i ca l opt i mal i sat i on necessary to make the

experi mental ri g operat e i n the downcomer ai r onl y mode ( no r i ser ai r

i n j e c t i o n) i s d i s cu ssed.

11 Geometry of the bubble column loop reactor

Vari ous conf i gurat i ons of the bubble col umn l oop t ype reactors have

been i ntroduced recent l y. The ori gi nal f orm was the bubble col umn w t h acentr a l s t r a ight concentr i c d raught tube. Design may di f f er i n the c l earance

bel ow and above the i nsert ed t ube, di ameter r ati o, l ocat i on of the sparger and

speci a l f eatures at the top to mn i m ze gas carr yunder . Genera l l y t he r i ng

sect i on for ms t he downcomer but t he deep shaf t vers i on was presented w t h a

centr al downcomer. Thi s i mpl i es an annul ar upward t wo phase fl ow of whi ch

l i t t l e i s kno wn .

A vert i cal part i t i on wal l may also be used to divi de up and down

f l ows . I n th is case the two phase mxt ure f l ows i n a sem ci rcu l ar duct . The

l ack of axi al symmetr y i nduces compl ex fl ow patt erns near t he bends ( Or azem et

a l . , 1 97 9) .

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16 12

The t hird general f orm that has been developed has an external

downcomer. Desi gn of thi s U- t ube l oop reactor may di f f er i n di ameter rat i o

r i ser/ downcomer, l ocati on of the gas sparger( s) , the gas di sengagement

arr angement s and the i nterconnecti on of r i ser and downcomer. Cal cul at i on of

the f l ow hydraul i cs i s re l at i vel y s t ra i ght f orward when the f l ow channel s are

o f c i r c ul a r c r os s s ec t i on.

12 Experi mental col umn

I n this secti on the col umn confi gurat i on used to ver i f y t he developed

model s i s descr i bed. The apparatus i s more or l ess to scale 1:10, w th 10. 5 m

eff ecti ve hei ght , 9" di ameter r i ser and 4" di ameter domcoraer, representi ng

f or i nstance a col umn 105 mdeep, 2 .5 m diameter and a i m diameter concentr i c

downcomer. The exper i mental setup i s shown i n fi gure ( 12.1) and consi st s of

glass (QVF) sect i ons. For detai l ed i nformati on see appendi x ( 12) , but some

rel evant character i st i cs are menti oned here:

F i f i " " 12-l s Exper: a l col umn ( f or

l l f l 12.3i Downcomer bott om sect i on of experi menta l col umn designed to prevent phase separat i on and BhOW n]devi ce to i nsert the oxygen probe.

cross pieces i n the r i ser to i nsert measuri ng probes or to t ake photographs

w t hout opt i c al d i s t o r t i on;

tee-p ieces i n the downcomer at several l ocat i ons t o i n ject the ai r ;

bel l ows to accomodate osc i l l at i ons and d i f f erent i a l expans ion;

a speci a l corner p i ece at the bot t om of the r i ser w th a i r sparger , tap

water supply and drai nage (f i gure 1 2. 2 ) ;

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- a special corner pi ece at t he bot tom of t he downcomer to prevent gas l i qui d

separati on (Hoang and Davi s, 1980) ( f i gure 12. 3) ;

- a heater, whi ch kept l i qui d t emperat ure at 20 (+ 0.2) °C;

- axi a l l y mounted f l ow baf f l es to el i m nate secondary hel i ca l f l ows af ter the

bends.

A spec ia l f eature of the co lumn is the poss i b i l i ty o f l ower i ng the

t op pressure. The ai r di sengagement part i s connected to t he laborator yvacuum I t i s possi bl e to mai ntai n a pressure of about 0.12 bar duri ng the

experi ment s, gi v i ng a pressure r ati o (bott omt op) of about 9.5 comparabl e to

that o f a 90 m deep column open to t he at mosphere. Thi s i s i n accordance w t h

scal e . Because there i s no provi si on to contr ol the vacuum i t depends on the

gas f l ow rate. Thi s devi at i on i s at wors t 20%of the absol ute pressure which

coul d be reached at the top w thout gas f l ow. The act ual operati ng pressure

coul d be descri bed accuratel y under al l c i rcumst ances encountered w th

Ah = 174 W (+ 0.025) ( m ( 1 )

where Ah = p./ p,g i s t he devi ati ng t op pressur e ( m and Wg t he gas massf l ow rate ( k g / s ) .

An overf l ow i n t he r i ser di sengagement pi pe was connected to a

0.11 m r eservoi r some 5 met r es l ower mai ntai ned at t he same top press ure as

t he col umn. Excess wat er was pumped back to t he col umn t o keep the wat er l evel

i n the col umn constant . However t he response was t oo sl ow to be used i n the

dynamcal experi ments.

13 Measuring equipment

Ci r c u l a t i on v el oci t y

I n t he 9" di ameter t op secti on of the downcomer a 0. 2 r a di ameter

i nduct i ve f l owmeter was i ns ta l l ed. I t was not possi b le t o cal i brate the

f l owmeter i n the col umn but i t was tested i n anot her r i g against a cal i brated

or i f i ce meter . I n the re levant r ange of 0. 5-1. 5 m s ( i n a 0-10 m diameter

p i pe) the f l owmeter i ndi cated about 0.02 ms l ower . I t i s probabl e t hat i n the

experi ment al l oop the accuracy was w t hi n 5% of the measured val ue w t h a hi gh

reproduci b i l i ty . The i n f l uence of voi d f ract i on i s known to be l i near , but was

never hi gher i n thi s unaerat ed downcomer sect i on than 1% The dynam c

b-. ihaviour of the meter was cal i brated separatel y (s ee appendi x 1 3. 1 ) .

13 19

Gas f l ow rat e

The ai r s uppl y was t aken fr om the 6 bar compressed air l i ne of the

l abor ator y. The pr essure i n the r otameter s was mai ntai ned at 3 bar by means of

a pressure r educing val ve. A 3 bar rel i eve val ve was i ncorporated i nto the

rotameter l i ne. For both r i ser and downcomer i n ject ors , a smal l or l arge f l ow

rot ameter could be sel ected. The t otal f l ow rat e provi ded was fr om 0. 1 t o2 g /s . A i r f l ow was contro l l ed by need le va lves di rect l y downstream of t he

rot ameter s.

Absol ute pr essures

Pressures above l i qui d l evel and in the mddl e of the bott om

connect i on tube (z = 0 l evel ) coul d be measured w th a Texas I nstr uments

d i f f erent i a l preci s i on gauge (g l ass Bourdon sp i r a l ) against barometr i c

pressure w th a maximum err or of 60 Pa. Barometr i c pressure was determ ned

w th a mercury barometer . The preci si on gauge was al so used t o establ i sh t he

l i qui d l evel i n the col umn ( at z = 10. 50 m) when worki ng w t hout overf l ow. Theaccur acy was + 0.01 m

Pressure drops and void f ract i on

At several places between the gl ass secti ons these are PVC f l anges

w th pressure ho l es , as shown i n f i gure (12 .1) . Ten pressure poi nts at a t i me

coul d be connect ed to t he i nvert ed air water manometer . I n thi s way the

pressur e drop over t he gas i nj ect or and ot her col umn part s coul d be measured

w th an accuracy of about 3 mm water col umn. I n the r i ser wit h a maxi mum

l i qui d veloc i ty of 0 .3 m s t he f r i c t i onal pressure drop i s of the same order

of magni t ude, whi l s t a r i ser voi d f r act i on of 1% r es ul t s i n a d i f f e r ent i a lpressur e 15 ti mes hi gher. However pressur e drops i n t he downcomer are not

eas i l y conver t ed to voi d f ract i on va l ues . The i n terpretat i on of measured

pressure drops i n to vo id f r act i ons i s cons i dered i n appendi x ( 1 3. 2 ) .

Two pressure point s coul d be sel ected arbi t rar i l y and connect ed to a

di aph r agm d i f f e r ent i a l pr es s ur e c e l l , whi ch was cal i brated agai nst the ai r

water manometer . This was used f or the dynam cal experi ments w t h a f i rst

order f i l ter t o suppress t urbu lent no i se.

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20

14 F r i c t i o n n umb er : l i t e ra t u re

The f r i ct i on number can be cal cul a ted w th t he hel p o f t he var i ous

s tandard r el at i ons. I t i s i nterest i ng to compare the methods used by sever al

workers f or si m l ar c ol umns. Söderberg (1980) used a col umn al most 10 m hi gh

i n the fo rm of a U- tube o f 241 r amin terna l d iameter . Kubota et a l . (1978)

t r eated t he deep shaf t theoret i ca l l y wh i l st Wei l and (1978) worked w th aU- t ube t ype col umn about 9 m hi gh w t h a 100 mm di ameter r i ser and 50 mm

di amet er downcomer .

The f r i ct i on number i s general l y consi dered as the summati on of

v ar i o us f a c t o r s . T h e mo s t i mp or t a nt cont r i b ut i o n i s t h e pi pe wa l l f r i c t i o n.

The Fanni ng equat i on:

KF = 4fL / dt ( 1)

depends on an appropr i a te choi ce o f the f r i ct i on factor f fo r two phase f l ow.

Wei l and and Kubota et al . use expr essi ons f or t he Reynol ds number dependence

of t he f r i ct i on factor as i f f o r s i ngl e phase f l ow. Sb' derberg uses a constant

v al ue f o r f of 0.005 as r ecommended by Wal l i s (1969), but assumes t hi s i s an

underest i mate.

The second i mport ant cont r i but i on ar i ses f r om the f l ow reversa l a t

the bott om Thi s may be vi ewed as a 180°-bend or a doubl e 90°- bend. Ei t her an

equ iva l ent p ipe l ength ( about 75 di ameters) o r an equ iva l ent K -va l ue (e . g .

0.36 t he r ather l ow val ue used by Wei l and) may be used f or t h i s .

Cont r act i ons or expansions g ive a lso r i se to l osses. These are

present a t d i ameter t r ans i t i ons i n the co l umn and at the ai r i n j ectors . The

l osses at t he spargers are dependent on gas f l ow rat e and tr eated i n chapter 2

( I n ject i on head l o ss ) . Wei l and assumes t hat al l the moment um i s l ost at theenl argement and nothi ng recover ed at the cont ract i on.

Al l authors assume that t he l oss at the top i s smal l due to the

r el at i ve l arge di ameter s of the di sengagement par t . Sbderberg cal cul ates an

addi t i onal l oss due to a standi ng bubbl e at t he i nner si de of the downcomer

bot t om bend. The ( s i ngl e phase) l osses due to the i nsert i on of i n j ect i on o r

measurement devi ces have general l y be i gnor ed.

21

15 Fr i ct i on number: ca lcu l a t i on

Fi rst we must def i ne an appropr i a te vel oc i t y i n the def i n i t i on of the

f ri ct i on number ( 7 - 4 ) . As we have seen a mni mum downcomer l i qui d vel oci t y i s

necessary to ensure that the gas i s carr i ed downwards. On t he other hand fr om

the vi ewpoi nt of energy consumpti on thi s vel oci t y should be m ni mal . Thi s

means that a col umn w l l be desi gned t o operate w t h a cert ai n downcomer

l i qui d ve l oc i t y and thi s w l l be used as our r e ference. Thus

IL = £— ( l )

For s cal e up purposes a general expressi on is needed. I n vi ew of t he

precedi ng paragraph and inf ormati on fr om Smt h et al . (1981) t he most

i mport ant cont r i but i ons t o K a re :

f r i ct i on downcomer pi pe wa l l : 4fL / oL

dR2

f r i c t i on r i s er pi pe wal l : 4f m L/ d

V \ ( 2)

corrects f o r t he d i f f e rence i n vel oc i t i es between r i ser and downcomer

- equi val ent l engUi of 180°-bend wi t h downcomer di ameter : 75d2

- equi valent l ength of 180 - bend w t h ri ser d i ameter: 75 d m

- cont r a ct i o n: ( 1 -m

- downcomer i n jector dev i ce ( s i ngl e phase): 0. 5

Thi s l ast val ue is based on si ngl e phase measur ement s w t h the i n j ector used

i n our experi mental col umn.

I f we use f = 0.005, whi ch i s reasonabl e f or t he rest ri cted r ange we

K „ = 4f( - £- + ^ m2 + 75 + 75 m2 + 25 + 25( 1 - m) ) ( 3)F dD dR

w t h e qua t i o n ( 2 )

L•) ( 4)

i f the di f f erences i n di ameter of t he downcomer are accounted f or: K = 4. 3) .

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For the other confi gurat i ons ment i oned i n paragraph 11 other

expr essi ons are necessary ( see appendi x). Equat i on ( 4) adapted f or the col umns

used by Söderberg and Wei l and pr edi ct s somewhat conser vati ve val ues.

16 I nfl uence of second phase on f ri ct i on number

Two phase pressure drop has recei ved extensi ve at tent i on in avai l abl el i te ra t u re on mu lt i phase f l ow (Govi er and Az i z , 1972; Wa l l i s , 1969; Berg l es e t

a l . , 19 81 ) . One import ant aspect i s d is cussed here. Hi gh l i qui d and gas f l ow

rat es are char acteri st i c of bubbl e col umn l oops. I n our col umn the rat es are

typ ica l l y : gas 1 g/ s and l i qui d 10 kg/ s- The vo lumet r i c gas f l ow ra t i o i s t hen

about 0.05. I n thi s case t he equat i on of Armand predi cts an i ncrease of

f r i ct i ona l p ressure drop of on ly 7 .5% (see Nakoryakov et a l . , 1981). However

Nakoryakov et a l . measure w th th i s i nput r a t i o a p ressure d rop t hat i s more

than tw ce the s ingl e phase va l ue. Moreover f o r l ow l i qu id ve l oc i t i es

(0. 22 m s) the measured t wo phase pr essur e drop was up to t en ti mes t he si ngl e

phase val ue. Thi s l a rge i ncrease i s re l a ted to a hi gher voi d f r act i on near the

wa l l , wh i ch i s r eported by severa l workers (Ma l nes, 1966; Ser i zawa et a l . ,1975I I ; Nakoryakov et a l . , 1981) fo r two phase up f l ow i n re l a t i vel y narrow

t ubes ( smal l er t han 100 mm di ameter ) . Because of the uncer tai nty of the

i nfl uence of the second phase i t has not been i ncorporated i n equat i on ( 1 5 - 4 ) .

17 Fr i ct i on number: measurement

Equat i on (7 .5 ) f o r r i ser a i r on l y becomes:

""R e "F vL sD' " a ^

Thi s i s the equat i on of mot i on fo r an ai r l i f t l oop . Measurement o f mean r i ser

voi d f r act i on and ci rcu l a t i on vel oc i t y l ead to the s i mp le deter m nat i on of the

f r i ct i on number K . The resul t i s shown i n f i gure ( 17) , wh ich shows that

- the val ue of K i s i ndeed i ndependent o f ve l oc i ty ,

- the K - val ue of the co lumn i s 4.5 f o r t hi s conf i gurat i on, wh i ch agrees w th

the val ue predi cted by equat i on ( 1 5 - 4 ) ,

- coal escence ra te i s i r r e l evant : resu l t s do not d i f f e r between pure water

(coa l esc i ng) o r f o r water w th 20 ppm octano l (non- coal esc i ng) ,

- there i s a cr i t i ca l va l ue fo r the gas i n ject i on ra te above wh i ch bubbl es are

t aken down i n the downcomer f r om the di sengagement sect i on (carryunder) .

23

Thi s va lue i s dependent on the medi um and corr esponds w th a l i qui d ve l oc i ty

i n the top of the downcomer of 0.29 m s (w th t ap-water ) and 0. 27 m s f or

the f i ne di spers i on in a 20 ppm octano l so l u t i on.

18 Opt i mal chois e of d i ameter rat i o

The downcomer/ r i ser cross- sect i ona l a rea ra t i o m i s a parameter i n

equat i on ( 1 5 - 4 ) . The f ri ct i on number may be consi dered as t he i nverse r at i o of

the resul t i ng ki net i c energy and t he d r i v i ng potent i a l energy. The l ower t he

f r i ct i on number the l ower i s t he d r i v i ng f o rce needed f o r a certa i n vel oc i ty .

Thi s means that the downcomer gas i n j ect i on can be hi gher up.

I t i s poss i b le to opt i m ze the downcomer/ r i ser a rea ra t i o by i t s

e f f ect on f r i ct i on number. Equat i on (15- 4 ) is p lo t t ed in f i gu re ( 18) . The

opt i mal val ue i s around 0.5 or a di ameter rat i o of 0. 7. Gi ven a 9" di ameter

ri ser t he downcomer s houl d be 6" di ameter . For other confi gurat i ons t his

o pt i mum w l l be d i f f e re nt ( se e appendi x).

From the v i ewpoi n t o f oxygen t ransf er t he opt i ma l ra t i o wi l l be

l ower, because t hi s means a short er r esi dence t i me i n the unaer ated downcomer

part and a l onger gas r esi dence t i me. Fr om the vi ewpoi nt of the experi mental

mode l a smal l e r r a t i o resu l t s i n more rea l i s t i c gas r es idence t i mes fo r the

ti me dependent behavi our and for the expandi ng bubbl es. The experi mental

col umn has a val ue m = 0.20 wi t h the ri ser aspect r at i o L/d_ = 46.

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For t he other conf i gurat i ons menti oned i n paragraph 11 other

expressi ons are necessary ( see appendi x) . Equati on ( 4) adapt ed f or the col umns

used by Söderberg and Wei l and predi ct s somewhat conser vati ve val ues.

16 I nfl uence of second phase on f r i ct i on number

Two phase pressure drop has recei ved extensi ve at t enti on in avai l able

l i terature on mul t i phase f l ow (Govi er and Az i z , 1972; Wal l i s , 1969; Berg l es et

a l . , 1 9 81) . One import ant aspect i s discussed here. Hi gh l i qui d and gas f l ow

rates are character i s t i c of bubbl e co lumn l oops . I n our co l umn t he rates are

typ ica l l y: gas 1 g/s and l i qui d 10 kg/ s . The volumetr i c gas f l ow rat i o i s then

about 0.05- I n thi s case the equati on of Armand predi cts an i ncrease of

f r i c t i onal pressure drop of onl y 7. 5% (see Nakoryakov et a l . , 1981). However

Nakoryakov et a l . measure w th t h i s i nput rat i o a pressure drop t hat i s more

than tw ce the s i ngl e phase val ue. Moreover f or l ow l i qui d vel oc i t i es

(0. 22 m s) t he measured two phase pressure drop was up to t en t i mes the si ngl e

phase val ue. Th i s l arge i ncrease i s re l ated to a h igher vo id f ract i on near t ne

wa l l , whi ch i s r eport ed by several workers ( Mal nes, 1966; Ser i zawa et al . ,1975I I ; Nakoryakov et a l . , 1981) f or two phase upf l ow i n re l at i vely nar r ow

t ubes ( smal l er t han 100 mm di ameter ) . Because of t he uncert ai nty of t he

i nfl uence of the second phase i t has not been i ncorporated i n equati on ( 1 5 - 4 ) .

17 Fr i ct i on number: measurement

Equat i on (7 . 5) for r i ser a i r on ly becomes:

"R Le " KF VL D

/ 2«

( 1 )

Thi s i s t he equat i on of mot i on for an a i r l i f t l oop. Measurement of mean r i ser

void f ract i on and c i r cul at i on veloc i ty l ead to the si mple determ nat i on of t he

f r i c t i on number K . The resu l t i s shown i n f i gure ( 17) , which shows t hat

- the val ue of K i s i ndeed i ndependent of vel oc i t y,

- the K - va lue of the co lumn i s 4 .5 f or t h i s conf i gurat i on, which agrees w th

the val ue predi cted by equati on ( 1 5 - 4 ) ,

- coal escence rate i s i r r e levant : resu l t s do not d i f f er between pure water

(coal esc ing) or f or water w th 20 ppm octanol (non- coal esc i ng) ,

- there i s a c r i t i ca l va l ue for t he gas i n ject i on rate above whi ch bubbles are

t aken down i n the downcomer f rom t he di sengagement sect i on ( carryunder) .

17 23

Thi s va lue i s dependent on the medi um and cor responds w th a l i qui d vel oc i t y

i n the top of the downcomer of 0.29 m s ( w th t ap-water ) and 0. 27 m s f or

the f i ne d i spers i on in a 20 ppmoctanol so lut i on.

18 Opti mal choi se of di ameter rat i o

The downcomer / r i ser c ross -sect i onal area rat i o m i s a parameter i n

equat i on ( 1 5 - 4 ) . The f r i ct i on number may be consi dered as the i nvers e rat i o of

the resu l t i ng ki net i c energy and t he dr i v i ng potent i a l energy. The lower t he

f r i c t i on number the lower i s t he dr i v i ng force needed f or a cer t a i n vel oc i t y.

Thi s means t hat the downcomer gas i nj ect i on can be hi gher up.I t i s poss ib l e to opt i m ze the downcomer / r i ser area rat i o by i t s

e f f ect on f r i c t i on number . E quat i on ( 15- 4 ) i s p l o t t ed i n f i gur e ( 18) . The

opti mal value i s around 0.5 or a diameter rat i o of 0.7. Gi ven a 9" di ameter

r i ser t he downcomer should be 6" di ameter . For ot her conf i gurat i ons thi s

opt i mum w l l be d i f f e r ent ( s ee appendi x) .

From the vi ewpoi nt of oxygen t rans fer t he opt i mal rat i o w l l be

l ower, because t hi s means a short er resi dence t i me i n the unaerat ed downcomer

part and a longer gas resi dence t i me. From the vi ewpoi nt of the experi mental

model a smal l er r at i o resu l t s i n more real i s t i c gas r es idence t i mes f or t he

t i me dependent behavi our and for the expandi ng bubbl es. The experi mental

col umn has a value m = 0.20 wi t h t he r i ser aspect rat i o L/dL = 46.

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26 21

Blenke ( 1979) descr i bes a l oop react or i n whi ch a downward di r ected

l i qu id j et i s used for the d i spers i on of t he gas . Wt hout t he jet l arge

c l usters of gas r i se against the l i qui d downf l ow and destab i l i ze the

c i r c ul a t i on. P ar t of t he i ns t abi l i t y o f t he c i r c ul a t i on i n t he c ol umn l oop

used by Söderberg ( 1980) i s expl ai ned by the si m l ar escape of l arge bubbl es

f rom the downcomer ai r sparger.

Söderberg' s (1980) experi ences are rel evant and are deal t w t h some

det a i l her e . He f i r s t t r i ed t o i n j ect a i r f r om an aer o fo i l s hape ac r oss a

di ameter (0. 241 m) of the pi pe. Ai r was suppl i ed near the cent rel i ne of the

ta i l ( t welve hol es of 2 mm di ameter ) . Any i nj ected ai r shoul d be swept away

f rom the sur f ace of the aerof o i l s i nce the sur face pressure at t h i s poi nt i s

higher than the str eam pressure ( Gol dstei n, 1938) . However l ow press ure ai r

pockets f ormed at t he tube wal l s beneat h t he aerof oi l . These pockets were

present w t h al l desi gns whi ch presented an obstr ucti on to t he downfl ow. For

thi s reason Söderberg used f i f ty equal l y spaced hol es ( 2 mm di ameter ) i n the

tube wal l and around i t s ci rcum erence. Ai r pocket s were st i l l f ormed however,

now cl i ngi ng to t he hol es. These l ong "st reaks" of ai r (as much as one pi pe

di ameter l ong) usual l y remai ned separat e, unl ess the l i qui d vel oci ty was bel ow

about 1.1 m s. Coalescence may t hen occur and bubbl es w l l escape upwards i nto

the downcomer. When the l i qui d vel oci t y was hi gh enough t o st abi l i ze t he

s t reaks, which was at ta i ned by the use of addi t i onal r i ser a i r , the c l i ngi ng

ai r pockets d i spersed at the ta i l i n to equi l i br i um s i zed bubbl es .

22 Experim ental: different injector types for downflow

From a ser i es of exper i ments us i ng ai r i n ject i on through a s i ngl e

hol e i n the wal l o f a 0. 15 d i ameter water t unnel , the fo l l ow ng observat i ons

were made. Behind each obstr ucti on, even a smal l r i m on the pi pe wa l l , l ow

pressur e pocket s ar e i mmedi atel y occupied by any gas present . Thi s may occur

anywher e i n a downwar d two phase f l ow. Gas emerg ing f rom a wal l or i f i ce acts

i n i t sel f as an obst r uct i on and a gas pocket w l l c l i ng to i t . Th i s may be

very st abl e and reach l engths of several col umn di ameter s, especial l y i n

smal l er pi pes.

The gas pockets have di f f erent f orms whi ch depend on l i qui d vel oci t y.

Waves devel op on the pocket surf ace as a result of var i ous Taylor i nstabi

l i t i e s ( s ee Cl i f t et a l . , 1978). Gas i s di spersed as bubbl es of more or l ess

the equi l i br i um s i ze f r om the lower end of the sl ug, whi ch tends to at t a in a

s tab le l ength. Some i nterest i ng resu l t s on sl ug l ength and s tab i l i ty are gi ven

i n appendix 22.

22 27

i tin

\

i a i r j

vznLuri l Lype2) unstabl e slug plungi ng J zL

Var i ous i nj ecti on geometr i es have been tested i n the downcomer of theexperi mental col umn.

a) A s i ngl e hol e i n the wal l ( f l ange between gl ass p ieces ) . A l arge ai r pocket

woul d al ways develop f r om the or i f i ce at typ ica l l i qui d veloc i t i es of about

1 m s. Rai s i ng the a i r rate t h i s s l ug woul d grow unt i l extendi ng several

pi pe di ameter s downst ream

b) A honeycomb di str i but or f i l l i ng the arm of a tee piece (see secti on 12)

l eads to the unstab le s l ug s i tuat i on envi saged i n f i gure ( 22) . Large

i r r egular s l ugs c l i ng to the lower s ide of the s t ructure, but may eas i l y

escape upwards. W t hout the honeycomb packing the si t uati on becomes even

more chaoti c as l arge bubbl es come f rom the upper si de of the arm are

carr i ed downward and coal esce w t h the sl ug whi ch cl i ngs errat i cal l y t o thel ower si de of the arm Once sl ugs have escaped upwar ds, t hese may break up

and i ni t i at e f ur t h er i ns t a bi l i t i e s.

c) A smal l r i m of about one m l l i meter above the tee pi ece al l owed a plungi ng

j et to develop. R i s i ng s lugs do not escape above th i s r i m but c l i ng to i t

to f orm an annul ar a i r pocket beneath i t . The l i qui d t heref ore p lunges

through the ai r i n a way comparabl e to a fr ee j et, cont ract i ng as i t

accel erat es ( f i gure 22) . Gardner and Nel l er (1979) observed a si mul ar

annul ar air pocket above an injection ring in upf l ow. The pl ungi ng jet

i nj ect o r i s ver y s t abl e .

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28 22

d) The speci a l l y des i gned converg i ng-d i verg i ng or ventur i - i n jector ( f i gure

22) . Ai r i s i nt roduced i nto t he accel erat i ng f l ow i n the converg ing par t

through hol es i n the wa l l . At moderate a i r rates t he ai r i s di spersed i n

the convergi ng part , but w t h hi gh gas r ates ai r pockets may develop i n the

diver gi ng secti on. No escape of s l ugs above the throat i s observed for

l i qui d vel oci t i es above 0.5 m s .

The presence of ai r pockets l eads t o a pressure l oss that i s

character i s t i c of the i n jector used. Al though i n fact a l oss of dr i v i ng f orce

by i ncreasi ng the res i s t i ng term o L (equati on 7 - 6 ) , t he head l oss cannot

be d i s t i ngui shed f rom the fr i c t i onal l osses i n the turbul ent wakes behind the

gas pocket . A speci f i c pressure drop express ion for the i n jector i s t heref ore

needed i n order t o model ci rcul at i on.

23 Experi mental : pressure drop

I n model l i ng the c i r cul at i on in t he co lumn we w l l assume t hat voi d

f r acti on i n t he downcomer becomes homogeneous i mmedi atel y downst ream of the

i n ject i on poi nts . The pressure drop equat i on for the i n jector must then

contai n the ef f ect of any non-homogeneous di spersi on. The int erpret ati on of

pressur e drop measurements must take thi s i nto account. W t h an ai r—water

manomet er we det er m ne:

Apm = Ap + pLgAz (1)

where Az i s t he di st ance between the pressure points and Ap is the t otal

(dynam c) pressure drop (sect i on 7) :

Ap = APf + Ap a + AP g ( 2)

The gravi tat i onal cont r i but i on in two phase f l ow i s ( z upward)

AP g = - ( apG + (1 - a) pL)gdz (3)

but w th i n ject i on f rom a d i s t ance L above the lower pressure point i n

downfl ow, assum ng a homogeneous voi d f ract i on and negl ecti ng t he gas densi t y

t erm thi s becomes

29

APg = " 0LgAz + apLgL (4)

hence

APm = APf + A P a+ * P a ( 5)

%L gL <6)

We want an expressi on for the extra l oss due to i nj ecti on, corr ected f or the

i deal i sed void d i s t r i but i on. Kence

V (x)A P i = A Pm " A P f - J £ PLg L ( 7 )

G

where Ap« is the pressure di f f erence for the si ngl e phase case and determ ned

separatel y. The l ocal veloci t y rat i o est i mates t he homogeneous void f ract i on

(see chapter 3 and 5) . The i n ject i on head l oss def i ned i n equat i on ( 7-5) i s

Ap

I HL = —i (8)PLg

The si ngl e phace cont r i buti on to t he measured pressure dr op i s about 13%of

the tota l . The voi d f ract i on d i s t r i but i on cor rect i on i s general l y between 15

and 20% S ing le phase f r i c t i on loss i s ef f ect i vel y i ncorporated i n the general

K val ue. Pr essure drop was measured over t he i nject or pl us part of the duct

be fo r e and/ o r a f t e r i t .

24 Resul t s : d i f f erent types of downf l ow i n jectors

The three ways of i ntroduci ng ai r i n downfl ow presented i n fi gure

( 22) are compared i n f i gure (24. 1 ) . The plott ed data are not corr ected f or theaxi a l vo i d f r ac t i on d i s t r i bu t i on si nc e t he I nj ect i on he i ght i s no t we l l

def i ned i n some cases . The ventur i i n jector used I s t he f i nal vers i on

descr i bed l ater i n th i s chapter .

The fol l ow ng shoul d be noted.

a) There i s a maxi mum measurabl e pressure l oss corr espondi ng t o the di st ance

L . ( e . g . 380 mm fo r t he ven t u r i , s ee a l s o f i gu r e 22 ) .

b) The pl unging j et l oss shows t wo tr ansi t i ons. The upper one corr esponds to a

j et l ength roughl y equal to the pi pe d iameter ( f i gure 2 4. 2 ) . The f i r s t pa r t

of the loss i s probabl y dom nated by the increase i n j et veloc i ty due to

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4TCT 2 MW^MO""!©'

F ig u r e g 4 . l l Extra head loss due to air injection measured for the

extra pressure dropi i Apm - f i p s p l ' P L g

//'

to p

' I I

// ■

/ :

/

3 of jellenghLps diameter

i

the plunging jet sys i t lo n o f j e t l e i

contr act i on w th the cor respondi ng i ncrease i n head l o s s. The second part

i s t hen domnated by the i ncrease in ai r vol ume and t he corr espondi ng l oss

o f g r a vi t a t i o na l ( d r i v i n g) f o r ce .

S) The ventur i l oss shows a t r ans i t i on too. Based on v isual observat i ons the

f i r s t par t cor responds t o the di spers i on of a i r i n the converg i ng sect i on.

The second part rel ates t o the di spersi on and t he subsequent growt h of ai r

p oc ke t s i n t h e d i v er g en t s ect i on. Thi s l a t t e r e f f ect app ea r s t o b e si m l a r

24 31

to t he other devi ces. The venturi may be thought of as an eff ecti ve means

of delayi ng this undesi rabl e phenomenon. The tr ansit i on it sel f may be

rel ated to t he detachment of the tur bul ent boundary l ayer.

d) The tota l pressure drop over the pl ung ing j et i n jector i s smal l er than that

over t he vent uri ow ng to the very smal l s i ngl e phase cont r i but i on.

Furthermore the extr a pressure drop was st rongl y dependent on l i qui d

vel oc i t y , wh ich i s the reason that a veloc i ty rat i o i s used as an

i ndependent var i abl e. I t makes t he i n jector l ess at t ract i ve for t he s tudy

of t i me dependent l i qui d veloc i t i es .

On these grounds t he ventur i i n ject i on system was sel ected for opt i m sat i on.

25 Op t i m s at i on o f t h e v en tur i i n j ec to r

I n the f i r s t prototype ( 4" , no. 1 see tab le 25) a i r d i spers i on was

good but head l oss hi gh. I n a separate l oop (2" di ameter t ubes) several

conf i gurat i ons were t ested and based on these result s anot her t wo 4" vent uri ' s

were desi gned and const r ucted ( see tabl e 25 and fi gure 2 5. 1 ) .

TABLE 25

Ventur i conf i gurat i ons

2" di ameter 4" di ameter

1 2 3

100 100 100

60 80 80

. 36 .64 .64

4 3 2x2

24 38 40

10 24633 0

1 2 3

100 100 100 mm

60 80 80 mm

. 36 . 64 .64

number

pi pe di ameter

throat di ameter

area rat i o m

50

30

. 36

o

2

50

40

. 64

or 24

I I I I

mm

mm bel ow begi n

convergi ng

par t i n throat

( f i gur e 2 5 . 1 )

hole di ameter

number and

pos i t i on of hol es

2

12

4.5

2

6 12 24

6 6 6

4

24

10

3

38

24633

2x2

40

0

2

12

4. 5

r 19

http://g4.ll/

http://g4.ll/

http://g4.ll/

7/22/2019 TR-diss-0001460(1)


The fol l ow ng desi gn parameters were t ested i n the 2" r i g:

a) t hroat / tube cross sect i onal area rat i o m

b) number of hol es n

c) l ocat i on of t he holes

The resu l t s are presented i n f i gure 25.2:

a ) W t h i n t he r ange t es t ed t he r e i s no s i gni f i c ant e f f ect of l i qui d vel oci t y .

The veloci t i es are somewhat l ow because of the l arger f l ow rest r i ct i ons i n

t he t es t r i g .

b) The area rati os tested ( . 36 and . 64) have no i nfl uence on the corr ected

ext r a pressure drop, but the s ingl e phase l oss of the lat t er i s 2.2 t i mes

l e ss .

c ) The head l o s s i s l i near w t h i nj ect i on ai r r a t e .

d) There i s a st rong dependence on the number of hol es. This cannot be uni f i ed

by usi ng ai r rate per hol e as var i abl e.

Furt hermore t he test s showed that l ocati ng the holes i n the

converging secti on near t he throat was benef i ci al . I t was observed that t he

work i ng d i s t r i butor has an ai r pocket c l i ngi ng to each hol e. For l ow a i r rates

these are broken up in the throat ( f i rst mode) . W th h igh ai r r ates the s l ugs

el ongate i nto the di verging secti on (second mode) . I f more hol es are present

t he smal l er and sl i mmer ai r pocket s break up more easi l y ( sl ugs of t he same

s i ze are exposed t o a l arger l i qu id v el o ci t y ) . A h igher a i r rate i s then

needed t o get i nto the second mode.

corrizclsd extrapressure drop

local superficial gas~~ ve locity (mis)

F l R u r e 2 5 . 3 : C o r r e c t e d e x t r a h e a d l o s s for t h e *" V e n t u r i s . C o m p a r i s o n w i t h th fl 2" r e s u l t s .

On the grounds of these resul ts a 4" i nject or (no. 2) was desi gned tor epl ace t he i n i t i a l p r ot o t ype:

a) w th a l arge number of hol es near the throat

b) w th an area rat i o of 0. 64

A thi rd ( no. 3) ventur i was des igned to tes t the poss ib i l i ty t oseparate in j ect i on and ventur i by i nt roduc i ng ai r j us t upst ream of t heconverging secti on.

Al l types are compared i n fi gure (25 .3 ) . The opt i mal i sat i on i s

demonstr ated cl ear l y. Furt her i mprovements, e.g. on hol e diameter and f orm

are possi ble but a much smal l er l oss i s not to be expected. Hence ventur i

no. 2 was used i n the l ater experi ments.

26 The downcomer ai r sparger syst em

Const r uct i on deta i l s of the 4" ventur i no. 2 are g i ven in appendix

(A26) . Measured head l oss i s presented i n fi gure ( 26) as a functi on of l ocal

s uper f i c i a l gas vel oci t y . W t h i n t he r ange of l i qui d ve l oci t y a t t a i nabl e i n

the experi mental col umn no dependence could be detect ed.

The data may be represent ed w t h two str ai ght l i nes, drawn i n the

Ap , .

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34

4 00

20 0

lOO

BO

60

4 0

20

1000 1 o 002 0 C 006 008 01 0 2

Figure 2b: Corrected ex t ra head loss for the downcomer ai r s p a rg e r (v e n t u r i 4" no 2). Draun Line is equat ion

( 2 Ê - I ) .

and (1)

■ 29-3

Wx )

"

whi ch i n tersect for v n( x) = 0. 082 ms. (Left hand ter m i n met re) . The

accuracy of the r e s ul t s i s l i m t e d by the f o l l ow ng p oss i bl e sy s t e ma t i c

e r r o r s :

a) The pressure at the measuri ng poi nt d i rect l y beneath the d iv e rgi ng se c t i o n

i s i n f l uenced by the wake behi nd the ai r pockets .

b) The a xi a l v oi d f r a c t i o n di s t r i bu t i o n cor r e ct i o n i s based on a s l i p vel oc i t y

of 0 15 ms whi ch i s probabl y too l ow

27 Dynamic aspects of the downflow injector

The t i me dependent ci rcul at i on of the col umn i s studi ed i n chapt er 6

The behavi our of the downcomer i n j ect i on system i s rel evant i n two ways.

1) The di spersi on process must be consi dered to be a po ss i bl e o r i gi n of

di s t urbances. Breakaway of ai r f r o m the c l i ngi ng ai r po ck et s w l l be

f o l l owed by a decrease i n the d i s r u pt i o n rat e wh i l e the sl ug grows to i ts

i ni t i a l v ol u me . T hi s p ro cess may wel l be p er i o di c and dependent on l i q u i d

v

corrccLad cxtr=

p \_q, r n r n

-"-op prcssur,

_J i l_

35

2) The dynam c r esponse of the d ispers i on process on a (gr adual ) change i n

l i qui d v el oc i t y w l l be smal l i n v ie w of the smal l dependence on i t. A

change or di st urbance i n gas i n j e c t i o n rat e h owe ve r w l l ha ve a s i gni f i c ant

e f f e c t . A posi t i ve change resu l t s i n a t emporar y enl ar gement of the ai r

s l ug, wh ich g i ves an i mmedi ate extr a l oss. A negati ve change on the other

hand l eads Co a del ayed syst em r esponse because the ai r pocket vol ume onl y

changes gradual l y.

28 The r i ser ai r sparger syst em

The ai r sparger was constructed of 71 s t a i n l e ss s t e e l t u be s of 1 mm

i nternal d i ameter d placed at the bottom of the r i s er i n such way t hat

coal escence woul d be m ni mal (see sec t i o n 12) . The o bs t r u ct i o n to the f l o w and

hence the head l oss i s a l so m n ima l .

Accor di ng to Hei j nen et al . (1982) bubbl e si ze for t he a ir / water casei s

A - i n 0. 40. 8 - 0. 2d = 1 17 v d eb o o s ( 1)

wher e the ai r v el o ci t y i n the nozz l es (v ) f o l l o ws f r o m

( 2)

Figure 28: Predicted Ini tia l bubble size from riser air sparge;and 3). diameter nozzles (equations 28-1

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28

The mn imumbubbl e s i ze i s ( He i j nen e t a l . ) ( a i r / water )

dL = 0.0033 d l / 3 ( 3)b o

or 3 .3 mm for 1 mm nozz l es . W th a mass f l ow ra te W rangi ng f r om0 t o about

1.2 g/s i t fo l l ows t hat the bubble s i ze i s between 3.3 and 9 mm (f i gure 28) .

I n th i s way the new y f o rmed bubb les w l l have a s i ze i n equi l i b r i umw t h t he f l ow or t he bubbl es comng f rom the downcomer.

29 Conclusion

Whi l e a s imp le des i gn cou ld be used fo r the r i ser a i r i n jec to r , th i s

was not possi bl e for t he downcomer. I n thi s chapt er t he devel opment of the

downcomer ai r venturi - i nj ector has been tr eated. Furt her i mprovements are

probabl y poss i b le and may be att ract i ve for l oop reactor s of l i m ted hei ght .

One such possi bl e i mprovement i s a str eami ned body at the centr e of

the downf l ow and des igned to i n jec t a i r f r omi t . Th i s " i nver ted ven tu r i " was

exam ned, because of the vi s i b i l i ty of the d ispers i on process. Some resul tsare present ed i n appendi x 29.

I n ject i on into downf l ow was shown to l ead to c l i ngi ng venti l ated ai r

pockets, which a re respons i b le f o r l oss o f d r i v ing f o rce , ext r a head loss and

poss i b ly a l so f l ow i ns tabi l i t i es . The deve loped ven tu r i - i n jec to r de layed the

onset of t he s l ugs consi derabl y. The desi gn str ongly i nf l uences i ts operat i ng

character i s t i c s and l osses and i s no t ( yet ) we l l def i ned . I t i s no t c l ear to

what ext ent more vi scous, or non- newt oni an fl ui ds - such as may be present i n

p rac t i ce - w l l i n f l uence the opera t i on o f the i n jec to r . Non-coa lesc i ng f l ui ds

were not found t o af f ect operat i on.

37

3 Slip veloci ty

30 I ntr oducti on

The predi ct i on of voi d f r act i on in r i ser and downcomer i s an

essenti al part of model l i ng the bubble column l oop. Thi s chapter revi ews t he

several model s t hat ar e used i n l i t erat ure and i ntr oduces t he si mpl e model

used for the present research. Measurements of voi d f ract i ons, s l i p

vel oc i t i es , and radi a l p ro f i l es of gas vel oci t i es and concent r a t i ons i n the

ri ser of the exper i mental l oop are report ed and compared w t h the model s.

Downward t wo phase f l ow has been the subj ect of a s eparat e experi ment

and i s t reated as an enti ty i n the next chapter .

Three t ypes of mode ls p reva i l i n l i te ra t u re :1 . S l i p ve loc i ty based mode l s , where a co r rec t va lue of the s l i p vel oc i t y

parameter r esul ts i n the correct voi d f r act i on val ue,

2. The extended dr i f t f l ux model s , or Zuber and F i ndl ay t ype model s , whi ch

predi ct gas vel oc it y w th a str ai ght l i ne dependent on i nput parameter s,

3. S imple correl at i ons w th i nput parameters .

Lapi dus and E l g i n ( 1957) i ntroduced a general tr eatment of d ispersed

f l ow systems i n whi ch the rel ati ve vel oci t y between phases was t he essenti al

parameter . As there i s a di f f e rence between the real l ocal re l a t i ve vel oci ty

and t hat def i ned by:

Vs = Q G / A ° ' " Q L / A £ * C L )

we w l l cal l the parameter def i ned by equat i on (1 ) t he s l i p ve l oc i t y . (Q =

vol umetr i c f l owrat e, A = cross- sect i onal area of col umn, a = void or

vol umetr i c gas f rac t i on, e " X " a * l i qu id hol dup).

A def i n i t i on f o r the voi d f rac t i on a t a def i ned ax i a l l ocat i on i s

given by the f ract i on of t he total cross- sect i onal area occupi ed by gas, t hus

a = AG/ A ( 2)

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38

We def i ne a one- di mensi onal veloci ty, f or the gas phase:

and so

( 3)

VG1 " \ l ( 4 )

For a real one-d i mensi ona l f l ow, w thout radi a l var i a t i ons, th i s becomes

v s = VG ' VL = Vr ( 5 )

I n th is case s l i p ve loc i ty and the re l a t i ve ve loc i ty between gas and l i qui d

are equal .

W t h t he usua l de f i ni t i o n of supe r f i c i a l v el o ci t y , f o r t he ga s pha se :

v Gs = QG/ A, ( 6)

equat i on ( 1) becomes

Vs = V Gs / u - T L s / e <7 )

I t i s now poss i b le to rearr ange thi s to

a = « 1 + — - ( d + - 2 )2 - - ^) %) (8)

( 9)

i s the so def i ned m xture vel oc i t y .

Equat i on (8 ) expresses the voi d f r act i on in t he i nput var i abl es v

and v f w th t he sl i p vel oci ty as an unknown parameter. I f the corr ect

val ue o f the sl i p vel oc i t y i s known, i t i s poss ib l e to predi ct the voi d

f ract i on accurate ly . On the o ther hand, determ nat i on of the s l i p vel oc i t y

f rom measurements of the voi d f ract i on is si mple by usi ng equat i on ( 7) .

Di scussi ng the l i ter ature we must be careful to di f f erent i ate between

two general one-di mensi onal model s. One i s based on the sl i p vel oci ty as

parameter , equat i on ( 8) . The other model , whi ch we wi l l cal l the Zuber and

Fi ndl ay-t ype model , i s based on the dri f t f l ux model descr i bed by Wal l i s

(1969) and extended by Zuber and Fi ndl ay ( 1965) t o i ncl ude radi al vari ati ons

of concentrat i on and vel oci ty. The model equat i on is :

\ 1 ' Co"H

+ VGd l ' ( 1 0 )

where C i s cal l ed the di st ri but i on parameter contai ning the i nfl uence of theo

radial vari at i ons and v i s the weighted mean dri f t vel oci ty contai ni ng

the e f fect o f the local re l a t i ve vel oc i t y . The voi d f ract i on is t hen p redi cted

by

" " VGS / VG1 <" >

We w l l t r eat t hi s mode l i n deta i l l a ter i n sect i on 32.

The dri f t f l ux model i ntroduced some new quant i t i es w th t he

dimensi ons of a vel oci ty. The extensi on to radi al vari at i ons i ntroduced the

averaged and the wei ght ed averaged val ues t hereof, of t en expr essed i n i nput

v el oc i t i es . Some extr a att ent i on w l l be paid to the nomencl ature si nce there

i s some inconsi st ency i n the l i ter ature. For basi c defi n i t i ons see the work of

Wa l l i s ( 1969) .

Wal l i s uses the symbol j fo r a vol umet r i c f l ux o r vo lumet r i c f l ow

rat e per ar ea, and def i nes:

J G = r,:vG ( 12)

h. ° "'l <13)

T he se are al l l o ca l q ua nt i t i e s , j f o r t h e f l ux , v f o r t h e i n di v id ua l v el o ci t y

and a and E for the l ocal (but t i me- averaged) vol ume f r acti ons of gas and

l i qui d . The connect i on w th t he vol umet r i c f l ow ra te fo l l ows f rom the

cross- sect i onal average

' V = <«V = V G s = V A ( 14)

i n wh ich the f o l l ow ng notat i on i s used f o r t he sur f ace i n tegra l :

< > = ( | AdA)/A ( 15)

The anal ogous r el at i on for the l i qui d phase i s def i ned i n t he same way: such

si m l ar i ty app l i es th roughout th is d i scuss ion . The to ta l f l ux becomes

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(16)

40

The dr i f t vel oci ty i s de f i ned by:

vGd " VG " J ' Ev r ( 1 7 )

The dr i f t f l ux i s t hen ana logous to the f l ux:

The sl i p vel oci ty was def i ned by

A<a> A<E>

From equat i on (14) i t f o l l ows that

(18)

(19)

( 20)

W t h ( 3)

VG1 = <c t vG

>/ <a > ( 2 1 )

Thi s i s t he one- d imensi ona l vel oci ty o r t he apparent vel oc i t y i n a

one-di mensi onal model , whi ch i s cal l ed by Zuber and Fi ndl ay ( 1965) t he

wei ghted mean vel oci ty. One may then def i ne the one- di mensi onal dri f t veloci ty

s i m l a r l y :

' Ml " <««>/ <*»> ( 2 2>

Some workers use the charact eri st i c vel oci t y defi ned anal ogous to the dri f tf l ux (equat i on ( 1 8 ) ) , but f o r t he overa l l val ues:

J rtJ1 = v„ <£> - vT <o> = <a><£> 7 ( 23)J Gdl Gs Ls s

(We used here t he subscri pt 1 for one-di mensi onal as t he quant i ty i s based on

overal l parameters i n the same way as t he sl i p vel oci ty. Locket t and

Ki rkpatr i ck ( 1975) used the symbol U Wal l i s (1969) j Q L , Fr eedman and

I f there are any radi a l var i a t i ons VG( 1^ t < vGd>' VG1 ^ <V G>

and j d l £ <J Gd >' T h e l a s t i n e( l u al i t y mean s t hat <ct ><E>vs i <acvr>'

31 Bubbl e swarm phenomena

The di f fe rent mode ls to pred ict voi d f r act i on are a l l more o r l ess

concerned w t h the phenomenon of sl i p. For the i nterpret at i on of these model s

i t i s necessary to know somethi ng about the physi cal processes t hat deter m ne

the apparent or one- di mensi onal rel at i ve vel oci ty of a bubbl e swarm Theref ore

these are t r eated i n th is sect i on, be fore the deta i l ed di scuss ion o f t he

model s.

The sl i p veloci ty of a bubbl e swarm i s not equal to the si ngl e bubbl e

r i se vel oci ty v f o r severa l r easons. The d i f f e rence resu l t s both f rom the

mutual i nter acti on between bubbl es and that between the bubbl es and the l i qui d

f l ow. The present thesi s i s general l y restr i cted to the bubbl y f l ows t hough

there is some consi derat i on of sl ug f l ow. The bubbly r egi me i s f urt her

subdivi ded by Zuber and Fi ndl ay (1965) into i deal bubbl y f l ow, cal l ed l amnar

by these authors, where the re la t i ve ve loc i ty i s l ower than that o f a s i ngl e

bubbl e, and churn turbulent bubbl y f l ow where the rel at i ve vel oci ty i s equal

to o r l a rger t han v . .

I deal f l ow i s i n fact charact eri zed by the absence of phenomena

renderi ng the f l ow nonhomogeneous. Thi s i mpli es t hat the bubbl es are uni f orm y

di st r i buted and a l l have the same re l a t i ve ve l oc i ty . The system i s comparab le

to one i n whi ch sol i d part i cl es are sedementi ng in a l i qui d. There i s general

agreement i n the def i n i t i on of the re la t i ve ve loc i ty i n such a system as

v r - f <e ) V ( 1)

The dependence on the voi d f r acti on is of t en descri bed by

He) = en

(2)

wher e n is gr eater than 0, i n anal ogy t o the rel at i on that Ri chardson and Zaki

(1954) proposed fo r so l i ds in l i qui d . Thi s e f f ect of the reduced re la t i ve

vel oci ty can be descri bed i n terms of the i nteract i on between bubbl es or

hi ndered set t l i ng.

I f except i ona l care i s t aken to excl ude heterogeneous e f f ects i t i s

possi ble to mai ntai n an i deal bubbl y f l ow regi me up to voi d f r act i ons of about

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42 31

40% However i deal bubbl y f l ow can general l y only be produced under caref ul l y

cont r o l l ed laboratory cond i t i ons and devi a t i ons w l l be usua l under i ndust r i a l

ci r cumst ances.

As stated by Locket t and Ki r kpatr i ck ( 1975) such devi at i ons f rom

i dea l f l ow may resu l t f r om

- F l o o di ng

- L i q ui d c i r c ul at i on

- The presence of l arge bubbl es.

F l ood ing w l l onl y occur a t very h igh voi d f r act i ons (h i gher t han

50% and i s there fore not re l evant to the p resent study. L i qui d c i rcu l a t i on i s

i mpor tant but the l arge scal e ci r cul at i on patt erns i n bubbl e columns that are

predi cted i n l i t erat ure, and revi ewed by Joshi and Shah (1981) , ar e di f f i c ul t

t o di s t i n gui sh f r o m ra di a l v ar i a t i o ns . Ed di e s o n a sma l l e r s cal e ( sma l l e r t ha n

hal f the di ameter of the col umn) w l l be more i mport ant as bubbl es

encounteri ng a vert i cal eddy f rom bel ow w l l move to the upward f l ow ng part

o f t he eddy and tend to fo rm cl usters o f bubb l es. These c lusters w l l

re i n fo rce t he ro ta t i on of the vort ex. On the other hand stochast i ca l l y f o rmed

b ubb le c l us t e r s wi l l a l so i ni t i a t e l i qui d c i r cul a t i o n. T h e bu bbl e s i n a

c l uster w l l have h igher upward vel oc i t i es as a resul t both o f the smal l e r

drag of the bubbl e group and the extr a upward l ocal l i qui d vel oci ty. Lockett

and Ki rkpat r i ck ( 1975) not i ced r i se ve l oc i t i es t hat were about tw ce those of

the i ndi vi dual bubbl es. The number of bubbl es i n a cl uster depended on the gas

rat e and i ncreased f rom about 6 at a superf i c i a l gas ve l oc i t y o f 33 mm s up to

about 25 at 50 mm s. Coal escence, i f no t h i ndered by i mpur i t i es o r addi t i ves,

i s most l i kel y to occur i n these c l usters , p roduci ng la rger bubbl es. I n

p ract i ce there w l l a lso be o ther reasons fo r l a rge bubbl es, wh ich can be

produced at the gas i n jector o r devel op as a r esul t of l ocal re tent i on and

coal escence in the apparatus. I n typi ca l f l ow f i e lds i ns ide equ ipment , i t may

al so happen that severe pressure gradi ent s, i n bends or vort ex f l ows, canencour age coal escence. The pr esence of l arge bubbl es has an enor mous ef f ect on

the s l i p ve loc i ty s i nce they are equi val ent to a re la t i vel y l a rge number o f

normal s i zed bubbl es movi ng w th a h igh r i se vel oc i ty . I n a bubbl e swarm th i s

vel oc i t y i s even h igher t han the s ingl e bubb le t erm nal vel oc i t y (Hi l l s and

Dart on, 1976). Al l these ef f ects o f I nhomogene it y r esul t i n an i ncrease in the

re l a t i v e ve lo c i t y w t h re spe ct t o t hat o f h i nde red se t t l i ng .

Fur t h ermo re , a s a re sul t o f r a di a l v a r i a t i o ns t he s l i p ve lo c i t y

d ev ia t e s f r o m t he l o cal r e l a t i v e v el o ci t y . Be ca use t h e s l i p ve lo c i t y i s a

one-di mensi onal par ameter , t wo- and/or t hree-di mensi onal ef f ect s, f or exampl e

31

- 20 -10 O 10

pa rabo l i c f l o w and concent r a t i o n pro f i l e s , w l l i nc re ase t h e v al ue of t h e s l i p

vel oci t y i n an upward cocur r ent t wo phase f l ow. More bubbl es move i n the

f aster centr e of the duct and hence the apparent rel at i ve vel oci ty bet ween the

two phases i s l arger than the l ocal one. I n downward cocurr ent bubbl y f l ow the

same prof i l es may lead to a l ower val ue of t he apparent r el at i ve vel oci ty i n

the same way. Thi s ef f ect i s compl etel y dependent on the f orm of t he profi l es

and t he f l ow di rect i on and can theref ore work both ways but as parabol i c- typepro f i l e s pre v ai l i n bubbl y f l ow t h e t rend w l l be ge ne ra l .

Radi a l vo i d pro f i l es have recei ved much at t ent i on i n l i t e ra t u re s i nce

Mal nes ( 1966) measured non- paraboli c- type profi l es i n 46 mm vert i cal p i pes.

These di st r i but i ons show voi d peaki ng c l ose to the wa l l s ( f i gure 3 1. 1 ) . Thi s

phenomenon i s conf i r med by numerous author s, such as t he i mport ant work of

Seri zawa et al . (1975II ) f or a 60 mm pi pe and by Nakor yakov et al . (1981) i n a

86. 4 mm pipe ( al l upward f l o w) . The l att er authors al so measured t he

phenomenon of st rongl y i ncreased wal l f r i ct i on (sect i on 1 6) , i ndi cat i ng a

r el ati on bet ween the t wo phenomena.

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F i g u r e 3 1 . 2: R i s i n g b ub bl e i n a n ( u p wa r d ) v e l o c i t y g r a d i en t . T h e

c o wa r d s t h e a i d e wh e r e t h e ab s o l u t e l i q ui d v e l o c i t y i s s ma l l e s t .

Several total l y d i f f erent model s have been proposed to expl ai n the

tr ansverse m grat i on of bubbl es towards the wa l l , of t en i n compar i son w th t he

same phenomenon w th sol i d part i cl es i n a f l ow ng f l u i d. However , the Magnus

f orce on a rotat i ng body travel i ng through a f l u id ( Swanson, 1961) , cannot be

used as expl anat i on i n the case of bubbl es si nce there i s no spi nni ng bubbl e

i nterf ace. A Zhukovski f orce caused by the di f f erence i n vel oci t y encountered

by the s l i pp ing bubb le i n a l i qui d vel oc i t y g radi ent f i e ld ( f i gu re 31 .2 ) may

be i mport ant f or t he tr ansverse moti on. I bragi mov et al . (1974) counteract s

the resu l t i ng t ransverse f l ow to the wa l l (upward f l ow) w th a di f f us ion or

d ispers i on f l ow:

dv- 1 2 L

K, v d. c . v - r -b L b b r dy ( 3)

where c, i s the bubbl e concent ra t i on (m ) , y the d ist ance f r om the wa l l

and EL a bubb le t r ansverse l i f t coef f i c i ent , def i ned by the equat i on. Themodel p redi ct s voi d p ro f i l es wh ich f o rm i s i n reasonabl e agreement w th

measur ements f or bot h upwar d and downward gas l i qui d f l ow.

However ot her more compl i cated model s are al so proposed and f i t t ed to

measurements w t h good resul ts . These are based f or i nstance on the hypothesi s

o f r o l l i ng vort i ces generat i on near t he wa l l s ( Rouhan i , 1976), on the

i n f l uence of the gas vel oc i t y g radi ent (R i quart s , 1977), t h e d i s t r i b ut i o n of

the l i qui d phase turbulence and i ts ani sotr opi c nature (Drew and Lahey, 1981)

or the c i rcu l a t i on o f l i qu id a round t he bubb le caused by the l i qu id ve l oc i t y

gradi ent (Zun, 1980).

Al l these model s concentr ate on the predict i on of the peaki ng

phenomenon at t he wa l l . However i n l arge ducts ( di ameter great er t han about

0. 15 m) t his peaki ng is not observed ( thi s t h es i s ) . I t may be t hat

redi st r i but i on of l ong i t udi nal tu rbul ence by the bubbl es ( Ser i zawa et a l . ,

1 975 I I ) , and the correspond ing f l a t teni ng of the l i qu id vel oc i t y p ro f i l es in

combinat i on w th l a rge pi pe d i ameters , resu l t s i n a re l a t i vel y smal l a rea w th

s i gni f i cant l i qui d vel oc i t y g radi ents . Thi s a rea may be too smal l to cause any

measurable tr ansverse m grat i on of the bubbl es to t he wa l l .Another expl anat i on may be the i nfl uence of bubbl e si ze. Measurements

of Sekoguchi et al . (1979) show that t he bubbl e l ayer near the wal l consi st s

l argel y of bubbl es smal l er than A - 6 mm and when no peaki ng i s observed

bubbl es l arger t han t hi s were concentr ated near t he centr e of the 35 mm

diameter p i pe. Thi s t r end i s conf i rmed by Va l uki na et a l . (1979) who , f o r a

gi ven si tuat i on, measured peaki ng w t h 0.5 mm bubbl es but not w t h 1 mm

bubbl es ( 15 mm di amet er p ipe) . Ser i zawa et a l . (1975 I I ) r eport s bubbl e s i zes

l ess t han 4 - 6 mm di ameter and Nakoryakov et al . ( 1981) even l ess than 3 mm

i n the non- coal esc ing sa l t so l u t i on they worked wi th . Bubb les i n th is thes i s

are greater than 4 - 6 mm (sect i on 28) .

Radi a l var i a t i ons are a lso r eported i n bubbl e co lumns w thout l i qui d

t hroughput , but al ways w t h maxi ma at t he col umn cent r e and backfl ow al ong the

wal l ( e. g . Hi l l s , 1974) . One may expect t hat th i s "gulf st ream ng" ef f ect

caused by l i qui d ci r cul at i on (Fr eedman and Davi dson, 1969; J oshi and Shah,

1981) i s s t i l l pr e s ent w t h l ow l i qui d vel o ci t i e s.

The combi nat i on of these vari ous eff ects expl ai ns why voi d peaking

near t he wal l s does not al ways occur i n upward f l ow. I n downward cocurr ent

f l ow the d i f f e rent e f f ects work i n the same di rect i on - towards the cent re -

re i nf o rc i ng each other , wh i ch resul t s i n parabo l i c- t ype or even be l l - shaped

voi d prof i l es ( Chu and J ones, 1980, p i pe di ameter 26. 7 mm (see also

chapter 4 ) .

32 The Zuber and Findlay-type models

I n vi ew o f the def i n i t i ons g i ven i n sect i on 30 , t he Zuber and F i ndl ay

model can now easi l y be deri ved. Combi ni ng equat i ons ( 30-17) and (30- 21)

gives :

VG1 = <a J >/ <a > + <av ><a> ( I )

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32 47

Zuber and Findl ay ( 1965) def i ne the di st ri but i on parameter C aso

C - <° 3 >

o <a>< > ( 2)

hence equati on (30-10) :

Gl o M Gdl ( 3)

or i n terms to predi ct <a>

— Wc + ^

° VM

One needs expressi ons f or t he di st r i but i on parameter C and the

one-di mensi onal (or wei ghted mean) dri f t veloci ty to be able to predict the

cross-sect i ona l average vo lumet r i c concent r a t i on of gas.

Zuber and Fi ndl ay i nvest i gated t he di st ri but i on par ameter C by

assumng power l aw d ist r i but i ons o f the fo rm

j / j c = 1 - P ( 5 )

= 1 - p ( 6)a - a

c w

where t he subscr i pts c and w ref er to t he val ues at the centr e l i ne and at t he

wal l of a ci rcul ar tube and wher e

P = r/ R ( 7)

i s the d imensi onl ess radi a l coord i nate . ( For the f o rm of these p ro f i l es seethe f i gures 32 . 1 , 2 and 3 ) . F rom t h ese di s t r i but i o ns i t f o l l o ws t h at

C = 1 f o r uni f o rm pro f i l e s

C > 1 f or a > ao c w

C = 1.5 at most , f or m = k = 1 and a =0o * w

I t i s i nt e re s t i ng t o no t e t hat t h ese di s t r i but i o ns p erm t a f i ni t e

voi d f r ac t i o n at t h e wa l l , but n ot f o r t he l o ca l v o lu me t r i c f l u x, wh i l e i n

vi ew of equat i ons (30-12, - 13 and - 1 6 ) , i = 0 w l l onl y be pos s i b l e i f a =

0 or f or a ze ro re l a t i v e ve lo c i t y a t t he wal l . I n bubbly f l ow, where the

l i qui d i s cont i nuous and compl etel y wett i ng the wa l l , onl y t he f i r s t condi t i o n

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48 32

TABLE 32

L i tera tu re val ues of t he di st ri but i on parameter C

Zuber and Fl ndl ay model . Data

Co1.60

1.2

1.45

1.09

1.07

2.28

1.08

1.07

1.07

0.86

0.90

0.93

1. 2

1.18

1.16

1.03

1.16

1.165

<a>

0.07- 0.35

> 0. 30

> 0. 27

< 0. 10

< 0. 14

?

> 0. 25

< 0. 20

< 0. 20

< 0. 20

0.03- 0.20

0.03- 0.20

\ ,0.45-0.82

0 -0 . 3

0

0. 03- 0. 3

0

- 0. 017—0. 04

- 0 .09 —0. 94

- 0 .30 —2. 00

-0 . 50 —1. 00

-0 .6 —2. 6

- 1.6 < v

0—0. 75

0

< \< 0. 8

0. 3-2. 7

< 1. 0

0. 75 - 3.1

0. 75 - 3.1

uni t s i n S I .

VGS0. 10-1. 0

0 -9 . 0

0 -2 . 5

0.003- 0.025

0.003- 0.025

0.012- 0.025

- 0 . 1 —0 . 9

- 0 . 01—0. 4

- 0 . 003—0. 12

- 0 . 1 —0 . 9

M < 0

0—0. 03

< 2. 60

< 50

0. 07-3. 5

< 0. 20

?

?

d t0.14

0.07

0.15

0.037

0.037

0.037

0.046

0.044

0.090

0.14

0.10

0.026

0.026

0.0455

0.15

0.14

0.052

0.052

(equat i on 32- 3) of the

system & re ference

a ir water bubbl y f l ow

P et r i c k ( 1962) *

a ir water s l ug f l ow

Smssaert ( 1963)*

bubbl e col umn, sl ug f l ow

Ba i l e y et a l . ( 1956)*

bubbl y a i r water f l ow

Bhaga and Weber ( 1972)

bubbl e col umn, bubbl y f l ow


countercurr ent bubbl y f l ow


downward bubbl y f l ow

Cur tet et Dj oni n (1967)


Lorenzi and Sotgi a ( 1976)


Lorenzi and Sotgi a ( 1976)

downward sl ug f l ow

Mart i n (1976)

downward l i qui d f l ow w th

s l u gs , Mart i n (1976)

downward l i qui d f l ow w th

s l u gs , Mart i n ( 1976)

upward l i qui d f l ow wi th

s l u gs , Mart i n ( 1976)

upwar d, Van Thanh Nguyen

et a l . (1977)

bubbl y f l ow i n r i ser

Hi l l s ( 1 976 )

bubbl y f l ow i n r i ser

Mer chuk and St ei n ( 1981a)

up f l o w

downf l ow

Cl ar k and Fl emmer ( 1984)

* ci ted by Zuber and Fi ndl ay ( 1965)

il 49

w l l p reva i l . Thi s means that bubbl es w l l not t ouch the wa l l and thus have af i ni t e v el oc i t y .

Zuber and Fl ndl ay show that f or a gi ven ful l y establi shed f l ow regi me

Cg i s a const ant . However values report ed i n l i ter ature f or a gi ven type of

f l ow are not consi sten t ( see tab le 32) .

Zuber and Fi ndl ay also s uggest that the second t erm i n equat i on ( 3) ,

the one-di mensi onal dri f t veloci t y, shoul d be a const ant f or both t he sl ug

f l ow and churn tur bul ent regi mes and si mply equal to t he ri se vel oci ty of asi ngl e bubbl e:

vGdl " vb» (8)

However f or t he l am nar bubbl y f l ow r egi me there i s a dependence on

concentr at i on:

n+1VGd =V E O)

( n+1, as thi s resu l t s i n the exponent n fo r the re la t i ve ve l oc i t y :

r " vGd" - - V - ( 1 0) )

Zuber and Fi ndl ay st ate t hat

n = 0. 5 f or bubbl e di ameter s d, l arger than 5 mmo r n = 2 f or dL < 5 mmb

Know edge of the concentr at i on prof i l e i s agai n necessary t o deter m ne the

wei ghted mean dri f t vel oci ty i n these cases. A constant val ue for t he

one-di mensi onal dri f t vel oci ty i s reasonabl y support ed by values i n the

l i terat ure obtai ned f or churn- tur bul ent bubbl y and sl ug upf l ows. Thi s means

that i f data are p lo t t ed in the v^v, . , p lane fo r a g i ven f l ow reg i me, i tshoul d be possi bl e t o draw a str ai ght l i ne through the poi nts, the sl ope of

wh ich w l l g ive the d ist r i but i on parameter C Q and the in tercept w th thev

G1- a x i s w l l gi v e t he one -di me ns i o na l dr i f t v el o ci t y v , . A t y pi cal

resul t i s shown in f i gure ( 3 2. 4 ) .

I t must be noted t hat the l i ter ature data used by Zuber and Fi ndl ay

f or compari son were r el evant to hi gh gas f l ow rat es and voi d f ract i ons

general l y above 15% Moreover the l i qui d f l ow rat es were kept const ant or

var i ed w th i n fa r narr ower r anges t hen the gas f l ow ra t es. I f the d r i f t

vel oci ty depends on the concentr at i on then thi s may resul t i n a change i n the

50 32 12 51

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s l ope of the f i t t ed l i ne, s i nce the concent ra t i on depends on the gas f l ow

ra t e . The cal culat ed val ues f rom i ntercept and sl ope are t hen not very

accur ate. I ndeed Bhaga and Weber ( 1972) who i ncorporat ed equat i on (9) i nto t he

model of Zuber and Fi ndl ay (1965), showed that f or cocur r ent upward f l ow the

val ue of n may vary between -1 and 2 w t hout a si gni f i cant change i n the

corre l a t i on coef f i c i ent when f i t t ed to the i r mode l . Moreover W sman (1979)

po in t e d out t h at e qua t i o n ( 3 ) i n f a c t d esc r i be s a set o f s t ra i ght l i ne s w t h a

constant v as parameter ( appendi x 32) . Thi s ef f ect may be import ant f or

the bubb ly f l ow i n th is r esearch in wh i ch superf i c i a l gas vel oc i t y var i ed

consi derab ly as a resu l t of expansion at the same l i qui d vel oc i t i es .

Conclusions on the Zuber and Findlay-type models

- The Zuber and Fi ndl ay model or extensi ons thereof can be used t o report

measurements on f u l l y establ i shed f l ow pat t erns, especi a l l y s l ug f l ow

(whatever f u l l y establ i shed may mean f or vert i cal two phase f l ow w t h normal

gas expansi on).

- The method makes use of the re l a t i vel y i nsensi t i ve var i abl e v , and so

di sgui ses measurement s er r ors .

The mode l i s onl y o f l i m ted use fo r p redi ct i ng vo id f ract i on in an unknown

si tuat i on, as there are no re l i abl e re l a t i ons fo r the mode l parameters

avai l abl e. Among other thi ngs t his i s a consequence of d i f f erences i n syst em

geometr i es such as pi pe di ameters and gas i n j ector s. Ther e i s no gener al

agreement between the val ues predi cted f rom di f f erent model s and those

measur ed.

33 Sl i p vel oci t y based model s

For the ca lcu l a t i on of vo id f ract i on w th equat i on (30- 8 ) an

express i on i s needed fo r the s l i p vel oc i t y . Ow ng to the d i f f e rent i n tensi t i es

w t h whi ch t he i nhomogeneous phenomena may occur , due among other t hi ngs to

di f f e rences i n ge ome t r i e s , e x pre ss i o ns f o r t h e s l i p v el o c i t y i n churn

tu rbul ent f l ow that a re reported i n the l i t e ra tu re show a con fus i ng var i a t i on.

No att empts have been made to separate the eff ects of radi al vari at i ons and

other i nhomogene i t i es . There fore one must r eal i ze t hat nei ther these ef f ects

nor hi ndrance have been consi dered i n devel opi ng exi st i ng corr el at i ons.

Genera l l y these re la t i ons pred ict a s l i p vel oc i t y that I s l a rger than the

s i ngl e bubbl e r i se v el o ci t y .From al l the gi ven equat i ons a genera l fo rm comes out :

en

where the added funct i on f i s

- ge ne ral l y p os i t i v e ( cent re v oi d f r a ct i o n maxi mal ) ,

- dependent on v or a, as a r esul t of hi ndrance and/ or i nhomogeneous

ef f ec t s l i k e cl us t e r i ng,

- dependent on v , probabl y chi e f l y as a resu l t o f rad i a l var i a t i ons,

- dependent on col umn di ameter , al so due to radi al vari at i ons caused by l i qui d

ci rcu l a t i on and perhaps al so re la ted to the t r ans i t i on to s l ug f l ow.

Compar i son o f t he var i ous t ypes o f r e l a t i ons i s on ly poss i b le f o r cert a in

de f i ne d s i t u at i o ns . T h i s i s done i n t a bl e ( 3 3) f o r o ne s i t u at i o n i n t he r i ser

o f our own exper i menta l r i g and fo r two o ther s i tuat i ons f rom the reported

l i te ra tu re (onl y bubbl y f l o w) . These are charact eri zed by l arge val ues f or

voi d f r act i on and di ameter (Towel l e t a l . (1965) ) o r f o r l i qui d and gas

vel oc i t i es ( Hi l l s (1976)) . The measured s l i p vel oci t i es are much hi gher thanvb» (Cwce a n d thr ee t i mes re spe ct i v el y ) .

33 53

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TABLE 33

Compar i son of s l i p ve

s ax>

- a v b »

- 0 . 73 ( 1 - E 2 - 8 ^

+« v b « / e

+0. 065 ms. . 1 72 .+4a ms

+v. / ( c f 1/ 3- l )b" s

+0. 065 + 1. 35a( g<L)

-0 . 1 l v,L s 5 R

- 0. 73(1 V ^ v +

+v 1 ' K

L s (K - a)e

1 3

+0 . 9 ( g d t v Gs )1/ J

t l l ï / r + 0.1vM/ e

^ b »/ E + ° -2V e

VGS a

s t a t e 1 0. 020 0.045

s t a t e 2 0. 068 0 16

s t a t e 3 0.27 0.10

** g roup 1 v l es s

group 2 v l es s

group 3 v grea

group G

*** Towel l et al . f ou

( I n the expressi ons f

t he phase f ract i ons a

oci ty equat i ons

eq

2

3

4

5

6

7

8

9

3

10

11

state*

1

- 0.011

- 0.021

0.011

0.065

0.019

0. 13

0.155

- 0 .016

0.011

0. 04

0. 320.028

0.045

s t a t e *

2

- 0 .038

- 0 .066

0.045

0.065

0 17

0. 28

0. 49

0

- 0.066

0. 14

0. 58

0.054

0.061

s t a t e *

3

—

- 0 .044

0.026

0.065

—

0. 20

- 0 . 2 0

- 0 .025

(0 . 54)

-0. 250. 48

**

2

?

G?

1

2

3

7

2

2

G

G

v v. d v -vLS bo° t s b^0.144 0.235 0.224 0.065

<0. 01 0. 235*** 0 39 0.192

1 78 0.235 0.150 0. 49

t han 0 3 ms l ow l i q u i d

re ference

Wal l i s ( 1969)

( l amnar bubbl y f l ow)

Gomezpl at a et a l . (1972)

( l a mn ar bub bl y f l o w)

Zuber et al . ( 1965)

Hi nes et al . ( 1975)

Hi l l s ( 1 97 6)

I ordache et al . ( 1981)

Kubota et al . ( 1978)

Mal nes ( 1966)

Gomezpl at e et a l . ( 1 97 2)

( appendi x 33, e q. l OSl l )

Towel l et al . ( 1965)

He i j nen et al . ( 1982)

Zuber et al . ( 1965)

( appendi x 33, eq-A33- 15)

** re ference

1 thi s experi ment

2 Towel l et al . (1965

3 Hi l l s (1976)

v el o ci t y

than 0. 05 ms bubbl e col umn

:er than 0 3 ms h ig h l i qui c

generali d an i n tercept of 0 27 ms

t hroughput s

3r the s l i p v el oc i t y the use of the symbol s a and e for

r e ne er l oc; l y, but g en era l l y the vol umetr i c mean

val ues a and £. Only Hi l l s ( 1976) and in pr i n ci pl e t h eo ret i cal r e l a t i ons use

the l ocal cross- sect i ona l averaged va lues <a> and <e> For conveni ence t hese

i n di cat i o ns are l e f t out, as i s u s ua l ) .

The compar i son made i n t a bl e ( 33) i s c l e ar : the r andomness of the

f o rm of the equati ons i s re f l e ct e d i n the predi cted val ues for the three

pra c t i cal ca ses . I t seems t hat the s i mpl e re l a t i o n of Towel l et al . (1965),

equat i on ( 10) , i s qui t e good even for the hi gh l i qui d ve lo c i t y s i t u at i o n, t hat

was not present i n thei r exper i ments. Al so equat i on (A33- 15) , r educed fr om the

model of Zuber and Fi ndl ay , gi ves bet ter r esul t s t han any of the other

re l a t i o ns . Bot h equati ons are att ract i ve si nce these depend onl y or l a rge l y

onl y on i nput var i abl es i n a si mpl e f orm However the physi cal meani ng of the

parameter s i s not very cl ear and there f ore thei r va l ues are d i f f i c u l t to

predi ct i n pr a ct i c al s i t u at i ons . T hi s may be i mpr oved, as wi l l be shown i n

sect i on 35.

34 Voi d f ract i on corre l a t i ons

There have been many cor r el ati ons to descr i be voi d f ract i on. Shah et

al . ( 1982) have recentl y r evi ewed t hese and concl ude that no uni que equat i on

i s appl i cabl e. However gas volume fr act i on i s a l ways re l a ted to the gas

v el o ci t y , g e ne ra l l y i n the f ol l ow ng way

a ~ v P B ( 1)

For the heter ogeneous ai r/ water bubbl e col umn Hei j nen and van ' t Ri et ( 1982)

propose the f o l l o w ng va lu es for the parameter s:

" ' " ' Gs

An even more si mpl e f orm i s gi ven by Botton et al . ( 1978)

0. 75C = vGs

( 2)

( 3)

However the exponent p i n var i ous corre l a t i ons can l i e anywhere between 0 4

and 1. 2. The r e l a t i on i s someti mes made di mensi onal l y c onsi st ent by the

i n t roduct i on of medi um propert i es or col umn di ameter . The f i t t ed parameters

are sel dom based on suf f i c i e nt d at a to j u s t i f y t h i s , and agai n show

considerab le scat ter . An i nt e re s t i ng al t e rn at i v e was proposed by Oel s et al .

( 1976)

35 55

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0 . 9 1 ( v G s / ( g d 3 2 ) Va 9 (A )

i nf l uence of sparger and medi um i s i ncluded i n this way.

I t w l l be cl ear that the gi ven equat i ons can probab ly onl y be used

f or a ( f i rst ) est i mate o f t he vo id f ract i on val ue fo r bubb le co l umns that a re

0. 3 m s ) .

35 Sl i p vel oci ty A si mpl e model

o r An a l t e rna t i v e f o rmu la t i o n f o r s l i p v el o ci t y

S l i p vel oc i t y i s an art i f i c i a l te rm used i n the one-d i mensi ona l mode l

t o pre di c t v oi d f r a ct i o n. I t i s as such t he resul t o f t h e l o ca l r e l a t i v e

vel oc i t y and o f non uni f o rm d ist r i but i ons of ve loc i ty and voi d f r act i on. To

separate t hese two e f f ects one can rewr i te t he def i n i t i on (30 -20)

s <ot>

w i t h ( 3 0 - 5 )

VL " VG " V r

S <£> «X> <£> <a><£>( 1 )

I f the re la t i ve vel oc i t y i s constant , as Brown et a l . (1969) assumed, one may

wr i t e

v = v + v ej . ( 2 )s r pr of i l e

i n whi ch now

(2)prof i l e <a><£>

One may est i mate t his val ue easi l y by measuri ng l ocal voi d f ract i on and gas

vel oc i t y w th a bubbl e probe.

Even i f v i s not a constant, equat i on (2) can be used as a

approxi mati on. For t he same t ype of d i st r i but i on prof i l es as proposed by Zuber

and Fi ndl ay ( 1965)

( 4)

( 5)

- v ( 1 ( 6)

Even for t he tr i angul ar pr ofi l es ( in = k = 1) and a voi d f ract i on of 10% the

factor a t t he r i ght hand s i de i s c l ose to 1 ( 0 . 9 4) . For more rea l i s t i c va lues

(k greater than 2 and m greater than 7, see among others Seri zawa et al . ,

197511) t he corr ect i on is 2% or l e ss . I n f i gure ( 3 5. 1 ) t h e re l a t i v el y sma l l

var i a t i on in e is compared t o that i n a. I n f i gure ( 35, 2) i s shown how the

profi l es f or ev and v compar e.

I t i s i mport ant to real i ze that the p ro f i l e i n f l uence i s dependent onvel oci t y . Equat i on (3 ) can be wr i t t en as

p r o f i l e ( 7)

0 4

'

\ . ,afctc

\

■

F igu re 35. U Voi d f ra c t i on p ro f i l e i n t he Ü - C, k) f o rm a nd t he i

« - 0 . 1 ) .di n e " q u i d ho l d u p p r o f i l e ( k = 3 ;

35 57

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X E V

lm/alt

1

E/E c^ ~

"

P1

x \

= 7AE \ _

0

Figure 35.2: Inf luence of a r a d i a l di s tr i but i o n of the rel at iv e veloc i ty on E

the relat ive v eloci ty (v ^ * 0.25 m/s) . Power prof i le (1 - p3 ) for the void 1

parameter « e v^> /<£>< v> - 0.958 (m = 50) and 0.716 (m = 7).

' prof i le (1 - p ) fo

Hence for gi ven const ant di mensi onl ess profi l es

pr o f i l e " G <e> ( 8)

where C„« i s a di st r i but i on parameter , def i ned i n the same way as C, f or

the f l ux j ( equati ons A33-13 and 32-2 ) . Equat i on ( 3) may be rearr anged w t h

c ons t ant r e l a t i ve vel oci t y i nt o

<2W

ET<< l )prof i l e <e> N<a><v >

or , f or gi ven constant d imens i onl ess prof i l es

( 9)

- " < " « -t (10)

where K is the di st r i but i on parameter used by Brown et al . (1969), equati on

(A33-3) . I n fact equat i ons (2) and ( 10) are i dent i ca l to t hat model as

descr i bed i n equat i on ( A33-10). For power l aw d i s t r i but i ons f or a ( equat i on 4)

and v (exponent m) t he di st r i but i on parameter CT , i s :

LI F T T T T ( 11)

General l y one may expect a dependence of v _ _ on a rel evant vel oci t y.

I t f o l l ows t hat for bubbl y f l ow the s l i p veloc i ty can be expressed

w th equat i on (2) , where v , though i ndependent of r adi a l var i at i ons» s t i l l

i ncorporates cl uster i ng and other i nhomogeneous ef f ects . Al so hinderi ng by the

presence of other bubbl es as occurs i n a bubbl e swarm and resul t i ng i n a l ower

rel at i ve veloc i ty i n the l amnar bubbl y f ow reg ime i s enc l osed i n v . Both

eff ects are di scussed by Lockett and Ki rkpatr i ck ( 1975) and they concl ude t hat

for vo id f ract i ons up to about 20%t he f o l l ow ng dependence for i deal bubbl y

f l ow i s acceptable

v - vv £ (12)r b™

Thi s i s conf i rmed by Bhaga and Weber ( 1972) al t hough they not i ce t hat f or

cocurr ent upf l ow and voi d f ract i ons l ower t hat 10%t he val ue of n i n the

general expressi on (31-2)

i s not a very sensi t i ve parameter . I n the countercur rent s i t uat i on however

they fi nd that n i s much mor e sensi t i ve and must be equal to 1.

Agai n equati on ( 12) can be wri t t en as a summati on

r b^ b™

So for t he l amnar bubbl y f l ow regi me equat i on (2) can be extended to

v = v, + v , . . + v, . . ( 14)s b00 pr of i l e hi nder i ng

i n whi ch

v , = -ctv. (15)hi nderi ng b™

A s di s c us s ed , l i t t l e c onsi s t ent i nf o r mat i on i s avai l abl e on t he

i nfl uence of i nhomogeneous ef f ects . Zuber and Fi ndl ay ( 1965) st ate t hat f or

the churn- t urbulent bubbly f l ow the dr i f t vel oc i t y i s i ndependent of vo i d

f r a c t i o n

vGd v b»

whi ch means that i n equat i on ( 33- 1) n = - 1.

(16)

35 5 9

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\ = VGd/ £ * V ^ (17)

Al though Bhaga and Weber (1972) f i nd that the parameter n i s i nsensi t i ve

enough to even a l l ow a val ue of m nus one wi th equat i on (31 -2 ) st i l l l ead ing

to r easonabl e predi ct i ons, equat i on (17) does account f or i nhomogeneous

eff ects that were probably not present wi t hi n the experi ments of Bhaga and

Weber .Equat i on (17) can agai n be wr i t t en as a summati on

( 18)

Compari ng wi t h equati on (13) t hi s means that t he i nhomogeneous eff ect s have an

i n f l uence that i s about t w ce as l a rge and opposi te to the hi nder i ng e f f ect .

One may expect that measur ement s i n systems t hat are not f u ll y i deal w l l g i ve

resul ts i n between. Equat i on ( 18) does compri se i nhomogeneous as wel l as

hi nderi ng ef f ects . Taki ng these together we may rewr i t e equat i on (14)

v + v ( 19)prof i l e i nhomogeneous + hi nderi ng v

Equat i on (19) f orms t he si mpl e model i ntr oduced here. I t has the

advantage that t he s l i p ve loc i ty parameter i s b roken up i n part s w th a c l ear

physi cal background. To use t he model one must det erm ne t he val ues of these

ter ms f or t he speci f i c si tuat i on encountered. Thi s may be done fr om the

l i t e ra t u re ( e spe ci a l l y v i s we l l d es c r i b ed ) , or f r om measurements or even

i n tu i t i on and observat i on, wh ich w l l be bet t er i n much cases than

corr el at i ons for t he voi d fr act i on based on other s ystems and geometr i es.

For i nstance the inhomogeneous t erm v. m ght be approxi mated i n

the fo l l ow ng way. As we do not want the express i on fo r the s l i p ve loc i ty to

be descr i bed i n terms o f a , we reca l l the express i on fo r the vo id f ract i on o f

Oe l s e t a l . (1976) equat i on (34- 4 )

whi ch i n our c ase coul d be appr oxi mated by

a = %Fr ( 20)

i n wh ich

Fr = v / (gd, ) '5 ( 21)

Gs b

as w l l be d iscussed l a ter on. The express i on fo r v . i n equat i on (18) i s an

approxi mat i on i t s e l f , so i t seems acceptab le to use

V « <sV% ( 2 2 )

(These do not di f f er more t han 0.05 m s f or the range of bubbl e di ameter s

between 3 and 8 mm) . Then i t f ol l ows

v. = av, = \v n (23)i n b™ Gs

or more general l y

v. - vp ( 24)I n Gs

I n thi s way one gets some f eel i ng of t he tr end and t he or der of magni t ude of

th i s t e rm and may even t r y to est i mate i t s va l ue i f no add i t i ona l i n fo rmat i on

i s av ai l a bl e .

We wi l l test the model i n the same way as was done i n t abl e ( 33) . The

val ue of v - v . was compared fo r th ree arb i t r a r i l y chosen s i t uat i ons. W th

equat i ons ( 10) , ( 11) , ( 19) and ( 23) we have

^VGs (25)

For k=2 and m=6 the di st r i buti on par ameter becomes 0. 2, what agrees w t h the

of t en r ecommended val ue of 1.2 f or C (s ee equati on A33- 15). Hence

% - v "• ° -2 Vjf + -\B

(26)

For t he s i t ua t i o ns w t h out l i qui d f l o w ( s t a t e 2 ) we wi l l u se t h e ci r cu la t i o n

vel oci t y as expressed i n equat i on (33- 11) as re l evant vel oc i ty and est i mate

£ w t h e qu at i o n ( 3 4 - 3 ) . W t h h igh l i qui d th roughput ( st a te 3) t he l i qui d ho ld

up can be approxi mated w t h

( 27)

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The resul ts are:

TABLE 35

s ta te

a:

approxi mat ed

cal cul ated

measur ed

V calcul ated (s

cal cul ated

measur ed

eq

( 27 or

(26)

m s)

(30-8)

34 - 3)

1

0 053

0.042

0.065

0 277

0.047

0.045

2

0 133

0 187

0.192

0.422

0.161

0. 16

3

w t h v =0. 3

0 13

0.60

0 49

0.835

0.096

0. 10

3

0. 835 ms

0. 10

0. 58

0 49

0.815

0.097

0. 10

I n compar i son w th t abl e (33) t hi s shows t hat the proposed model l eads to very

good resul ts for al l three largel y di f f er i ng cases , even w th the si mpl e

assumpt i ons made.

36 Experi ment al

To determ ne the t er ms i n equati on (35-19) v and v. for the71 pr i n

experi mental col umn, measurements were requi red of the l ocal and mean voi d

f r act i ons and the v el oci t y pr o f i l e s . The downcomer sect i on was not e as i l y

access ib l e for l ocal measurements , nor could pr essur e drop measur ements be

convert ed i nto mean voi d f racti on val ues, because the f r i c t i ona l pr es s ur e d r op

was nei t her neg l i g ib l e nor known. Theref ore downfl ow charact eri st i cs have been

deter m ned separat el y ( chapter 4) . Measur ement s i n the r i s er of the

experi mental col umn are reported i n th is chapter ,Local measurements were done w t h a two-poi nt conducti v i t y bubble

probe ( f i gure 36, 1 ) . The s i gnal f r om such a probe, when tr avers ed by a bubble

( f i gur e 36. 2 ) , may be i n terpreted to give loca l voi d f r act i on, bubbl e vel oc i t y

and si ze. I n thi s work we were onl y i nterested i n the r adi a l p r o f i l e s . Loca l

void f racti ons were deter m ned by averagi ng the processed si gnal of the f i r s t

p oi n t , and bu bb l e v el oci t y f r om c r oss - c o r r e l a t i ng the si gnal of both po int s .

The probe was made f r om 0 1 mmpl a t i n um wi r es , i n s ul a ted e l ec t r i c al l y

and str engthened w t h a l ayer of 0 1 - 0 2 mmepoxy resi n around thewi r es ,

l eavi ng about 0 3 mmof w r e f r e e ( f i gur e 36. 3 ) . Sta in l ess s t eel tubes were

used to r e i nf o r c e the probe f urther ( see f i gur e 36. 1 ) .

on and bubb le velo,

L ro 3b-1: S i g n al s f r o m e e. en 1 and Z of

The s i gnal f r om the probe i s of course not an i deal b l ock but can be

characteri zed by a f i n i t e r i s e and f a l l t i me ( f i gur e 36. 2 ) . The r i se t i me i s

r e l a t i v el y l ong as a resul t of the l i qu id f i l m remain i ng beh ind af ter the

probe enters the ai r of the bubbl e. For the void f r acti on measurements a

squar e wave si gnal must be of fered to the dig i ta l l y averagi ng vol tmeter .

; U =V u. A t / T ( 1)

wher e U± i s tne block vol tage level cor responding w th the gas or l i q u i d

62 36 63

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Figure 3b.3: The bubble probe.

2

voltag

1 /^ A

i ri

!f

uULTlnL 71 1 1 .

02m/

ï\\i r1

i/j

/ i l

V[T \±i

Figure '0 .4; Typical traces from the bubble probe

phase, U the output of the averagi ng volt meter aft er t i me T and (At ) t n e

sample f requency (10 kHz) . The bl ock i s s i mply based on a threshol d l evel

( f i gure 36.2) whi ch l eads to an est i mate t , o f t he probabl e a i r res i dence

t i me t , . The threshold must be suf f i c i ent l y h i gh to exc l ude no i se and dr i f t .b

The accuracy of the esti mate depends on the exact f orm of t he det ected si gnal

( f i gur e 36. 4) . The t hreshol d l evel was regul arl y checked wi t h a wave form

recorder . There i s some di scuss i on in t he l i ter ature ( see e. g . Hewi t t , 1978)

as to the poss ib l e er r ors that ar i se wi t h voi d f ract i on measurements made wi t hsensors of thi s t ype. The most i mport ant are

a. d i s tor t i on of the bubbl e on i mpact ;

b. t endency of the bubbl e to centr e i t sel f around t he probe poi nt ( Buchhol z et

a l . , 1 979 b) ;

c . def l ect i on, espec ia l l y o f smal l bubbl es , caused by the di s turbance to t he

f l ow f i e l d ; and

d. decel arat i on of the bubble as a resul t o f t he hydraul i c r es i s tance the

probe of f ers .

Ser i zawa et a l . ( 1975I ) es t i mate the tota l er r or o f t he voi d f ract i on

measurement usi ng thi s t ype of pr obe to be l ess that 3% Cal i brat i on of the

probe i s not rea l l y poss i b l e , but t he resu l ts can be compared wi t h thevolumetr i c measurements aft er i ntegrat i on of t he measured voi d prof i l es (see

appendi x). The resu l t s w th t he probe were about 30%l ower . Thi s i s probabl y a

resul t o f t he l i m tat i ons of t he averag ing vol tmeter coup led w t h a sampl i ng

f r equency that i s rather l ow i n compar i son w th t he typi ca l smal l es t s i gnal

durat i on (about 2 ms agai nst At = 0.1 ms) . I t i s bel i eved t hat th i s er r or has

no i n f l uence on the f orm of the prof i l es der i ved. The er rors i n bubble

vel oci t y measurements ar e

a. a d i f f erence i n the di s tor t i on of the bubble on i mpact of the two probe

p oi n t s , the second poi nt gi v i ng more r esi st ance on pi erci ng (Seri zawa et

a l . , 1975I ) ,

b. decelerat i on of the bubbl e.Both ef f ects l ead to measurements that ar e l ow, but can be al l owed for by

c al i br a t i o n ( appendi x).

The cross - cor re l at i on techn ique may lead to addi t i onal er rors whi ch

can ar i se f r om

a. the averagi ng method. The l arger bubbl es have more i nf l uence, t hough thi s

i s not suf f i c i ent to cor rect t he est i mate of t he rea l mean gas vel oc i t y

(For more deta i l s on the cross -cor r e l at i on techni que see Ser i zawa et a l . ,

19751)

64 3È

b. d ef l e c t i o n of s ma l l e r b u bb l e s T h e d e t e r m n a t i o n of t h e v o l u me t r i c me a n v o i d f r a c t i o n s n a nd t h e

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c . i n ac c ur a c y i n r e ad i n g t he e s t i ma t e d t i me t f r o m t h e c o r r e l e l l o gr a m wh os e

ma x i mu m i s n or ma l l y r a t h e r f l a t .

T h es e e f f e c t s c a n n ot s i mp l y b e c a l i b r a t e d , b ut a g a i n i t i s be l i e v e d t h at t h e

f o r m o f t h e p r o f i l e s d er i v e d i s n ot s e r i o us l y i n e r r o r .

T h e pr o b e wa s mo u n t e d o n a t r a v e r s i n g un i t , wh i c h c o v e r e d r a d i a l

l o c a t i o n s be t w ee n 5 a nd 14 0 mm f r o m t h e wa l l i n t h e 2 29 mm d i a me t e r r i s e r ( a n

a x i s y mme t r i c d i s t r i b ut i o n i s a s s u me d ) . A x i a l l o c a t i o n s we r e a t z = 2 . 1 3 m a nd

8 . 7 4 m i n t h e r i s e r .

u

1C

OS

0 5

\ x

normal top prsssur? \

I WG vG s V L S 5 •

V 87", 101 0O19 021 0O37 'o 6 7" 165 0029 02 6 0 050

■ 213 165 0019 0 2B O 023

1 \s\ \ o 9

z WG V ,^ yL S a \ \Imllgfcllm/») Iml. . U

o B74024 0017 Q19 OO02 EA ,, 060 0039 026 0063 \• , , 066 OOS3 031 0075 0A „ 1 16 0069 034 0107

-6*

. Nz wG vGS v LS a V

Iml Ig/jl Im/sl Imftl &o 6 7« 030 OO20 017 0043 IA „ 060 0039 0B4OO71 \• „ 0 66 0 05 3 0 27 O0 90 1

z wG v G £ v L S a \ u

1ml tg/*I ImfcJ (m/»l \* \□ 2 13 O 24 0OO6 0 19 0 010 IA 060 0O13 02BOO 24 \■ ,, 066 0019 031 0031 \ |A ,. 1 16 0O25 034 0O37

■ * — * = | _ _ _ _ -

t w G v G S v L S a o \

o 213 O30 0O07 017 0011 \A ,, 06 0 0013 02400 24 °\

• ,, 066 0019 027 0O31

i n t e r p r e t a t i o n a s l o c al c r o s s - s e c t i o n al me a n v a l u e s h av e b e en d i s c u s s e d i n

a p p en d i x ( 1 3 . 2 ) .

37 Re s ul t s : r a di a l d i s t r i b ut i o ns

I n t h e f i g u r e s ( 3 7. 1 a nd 2 ) t h e me a s u r e d v o i d f r a c t i o n s an d bu b bl e

v e l o c i t i e s ar e p l o t t e d, ma de di me ns i o nl e s s by t h e i r c e nt r e l i n e v a l u e s . P o i n t s

wi t h a b a r r e f e r t o me a s u r e me n t s o n t h e f a r s i d e o f t h e c e n t r e l i n e .

66 37 37 67

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void fraction distribution

An el l i pt i c curve was found to gi ve a good f i t to the data.

a 2 h

(1 - k f>y ( 1)

The centr e l i ne val ue and the f i t t i ng parameter k were deri ved f rom a l east

square l i near r egressi on procedure. Other r el at i onshi ps were consi dered e.g.

the power curve (1- p) descri bes the data as wel l but i s less si mple to

i ntegrate. The power curve ( l - pm) , proposed by Zuber and Fi ndl ay (1965) , has

a d i f f e rent fo rm ( f i gure 37 .3 ) and i s genera l l y not i n good agreement wi t h

meas ur ement s.

For most of t he data t he f i t ti ng parameter k was about one. At

barometr i c t op pressure and the upper measuri ng locat i on the di st ri but i ons

were fa i r l y f l a t . At the lower measur i ng l ocat i on the ef f ect o f the bot t om

bend on two phase f l ow resul ted i n dev i a t i ons f rom the e l l i p t i c p ro f i l e . I twas a lso observed v isua l l y t hat i t took a t l east two met res f o r t he p rof i l es

to devel op ful l y f rom the asymmetr i c condi t i ons at the lower bend. The

i n f l uence of the superf i c i a l ve l oc i t i es i s smal l . However t h is coul d not be

tested i ndependent l y , and the var i a t i on i n l i qui d vel oc i t y i s re l a t i vel y

smal l . The ve loc i t i es tested are typ ica l fo r the exper i menta l l oop in

operat i on.

bubbl e v el o ci t y di s t r i but i o n

The data are aga in f i t t ed by an e l l i pt i c cu rve:

be

(For compari son some of the best power f i ts are added i n f i gure (37. 2) . ) The

f i t t i ng constant k i s smal l e r t han one as a resul t o f the f i n i t e val ue o f

the bubbl e vel oci ty expected cl ose to the wa l l . The di f f erence i n k betweenv

the two axi al measuri ng l ocat i ons must be seen in ter ms of the absol ute

values. The centr e l i ne values at t he upper l ocat i on are about t w ce those of

the l ower one, whi ch means that t he wal l val ues are l ess t han 60%above those

f or the gi ven f i t t i ng parameter s. Bubbl es ' hoppi ng' al ong the wal l (Sekoguchi

et al . j 1979) a re expected to have a vel oci ty that i s c l ose to thei r f ree r i se

vel oci ty. The f i t t ed wal l values vary between 0. 22 m s (normal top press ure, z

= 2.13) and 0. 60 ms ( l ow top press ure, z = 8 .74) . (See appendi x f or a survey

of these data ) .

The vel oci ty pr ofi l es do not show entr y or exi t ef f ects and no

i nf l uence of t h e supe r f i c i a l v e lo c i t i e s .

i nt e rpre t at i o n of the results

To determ ne the cont r i but i on o f t he rad ia l d ist r i but i ons t o the s l i p

vel oc i t y , v , w th equat i on (35 -3 ) or (35- 9 ) we need to determ ne the

cross-sect i ona l averages of vo id f ract i ons, ve loc i ty and vol umet r i c f l ux. We

assume that t he measured bubbl e vel oci t y i s a good est i mate f or t he l ocal gas

veloci ty. He rewr i te equat i ons ( 35- 3 and - 9 ) :

V = ( VG1 " <v

G>) / <£> C4)

( 5)

( 6)

f av a = «>*v* >/ <a*> (7)

68 37 37 69

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are t he di mensi onl ess cr oss- sect i onal average and one- di mensi onal (wei ghted

mean) ve l oc i t i es . The var i ous f o rmu lae fo r t he el l i p t i c p ro f i l es are p resented

i n the appendi x t ogether w th t he resul t s. To gi ve an i mpressi on of the

sensi t i v i t y , the d i f f e rent d imensi onl ess means are presented i n f i gu re ( 37. 4 )

as funct i ons of the parameter k (equat i on (3) ) and f or k =1 .

The di f f erence between t he one- di mensi onal gas vel oci ty and thecross- sect i ona l average gas vel oci ty ( f - f ) i s the essent i a l parameter

that determ nes t he cont r i but i on to the s l i p ve l oc i ty . The i n f l uence of the

f o rm of t h e v oi d pro f i l e s ( k ) on i t i s shown i n f i gure (37 .5 ) . I t shou ld be

noted however t hat fo r the p red ict i on of voi d f r act i on i t s sensi t i v i t y as a

parameter i s l i m ted . The cal cul a ted val ue o f v shows considerabl e scat t er

as a resul t of t h is sensi t i vi ty and the restr i cted accuracy of t he measurement

method. The presentat i on of the data i n f i gure (37. 6) shows t h i s . Moreover i t

show'_. that the data are di vi ded i n two gr oups above and bel ow a mean val ue of

o ^ L

F iB Ut

m e a s u

3 7 . 6

euea: The con rlb uti on of t

(37-5)h e r a d i a l d l s t r

Based on e l l it o t h e s l i p v e l o c i t y v c a l c u l a t ed f r

f * p rCo mp a r i s o n w t h eq u a t i o n ( 3 5 - 1 0 ) .

about 0. 04 m s. These groups cor r espond roughl y to t he upper and l ower

measuri ng l ocat i ons. The str i k i ng di f f erence between the two top pressure

modes shoul d be noted. Examnati on of the voi d and vel oci ty profi l es ( f i gure

37. 1 and 2) shows t hat f or the l ow top pressur e mode the di f f erence between

the two measuri ng locat i ons i s general l y smal l f or t he voi d prof i l es but

s i gn i f i cant f o r the vel oci ty p ro f i l es . For t he normal p ressure mode th is i s

opposi te.—2

I n f i gure ( 37. 6 ) the ' i ndependent ' var i abl e v (Z was used, t o

compare the result s w th equat i on (35 -26) .

T he el l i p t i c p rof i l e s f o r t h e g as v el o ci t y i n comb in at i o n w t h t h e

constant r e la t i ve vel oc i t y assumpt i on resu l t s i n the fo l l ow ng l i qu id ve loc i ty

pro f i l e s:

Gw ( 8)

i s t h e r e l a t i v el es ar e compared

the power prof i l es proposed by Zuber and Fi ndl ay f or t he f l ux (see al so f i gure

32. 3) . The di f f e rence i s smal l fo r an exponent m = 2 .5 . The d ist r i but i on

paramet er

( 9)

was cal cul a ted wi th e l l i p t i c voi d p ro f i l es ( k ) and equat i on (8 ) ( k )

70 37 3771

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Figure- 37.7 : Compari son of the form of Eh . d imens l on lcs s (1 -

l i q ui d v el o c i t y , equ at i o n ( 3 7 - 8 ) , « r i c h f ol l o u f r o m e l l i p t i c

) p o ue r p r o f i l e

s velocity p t o f i

F i g u r e 3 7 . 3 : Distribution p a r a met e r t i , ( eq u at i o n 3 7- 9 ) a s f u nc t i o n of t h e el l i p t i c p r o f i l e p ai

(void) and k (gas velocity).

( f i gure 37. 8) . F r om t he f i t t ed k^ and k y val ues ( f i gure 37.1 and 2) the

f o l l o w ng resul t e d:

cal c. meas.

l ow press ure mode k k c C

z - 8.74

z » 2. 13normal press ure mode

z = 8.74 0. 5 0. 87 0.35 0. 22

z » 2.13 0.94 0.88 0.11 0.10

"I A "L I0.95 0.79 0.09 0. 20

0.95 0.65 0.06 0.06

The agr eement w t h the measur ed val ues i s reasonabl e except f or t he

combi nat i on of l ow top pressure and upper measuri ng l ocat i on. As wi l l be seen

later the voi d f ract i on is t hen and there above 10% Thi s gi ves a severel y

i nhorcgeneous regi on whi ch would i nval i date the above reasoni ng. On t he other

*(mle? ' .

v0

21 3

V

L.

A

•""<=' lm

- ^

-

/// /

/ /Z

v

a. r

ae j mc

.*1

//

/ /

t o p

p r«

no r

l o w

/

//

1

sura-Ti 01

,/ -

60"'"

y.v // • /

• / // ° // /

4 y

/A A

/" "/ A

' 9

a Irmnanturi

F i g u r e 3 7 . 9 : C o m p a r i s o n o f t h e o n e d i r J en 3 1 o n a I g a s v e l o c i t y M a a a r e d b y p r o b e a n d b y m a n o m e t e r .

hand i t must be noted that t he gas vel oci t y measured w t h a pr obe i n thi s case

i s about 60%hi ger t han by t he manometr i c method ( f i gure 3 7. 9 ) . Correct i ons

f or t h is woul d bri ng measured and cal culat ed di st ri but i on parameters c l oser.

Because the manometr i c method does not show thi s di f f erence (f i gur e A36. 1) , i t

must be concl uded t hat the probe bubbl e vel oci t y measurements are too hi gh.

There i s no evi dent explanat i on for t h i s .

I t has not been poss ib l e to der i ve an accurate re l a t i on fo r the

cont r i but i o n of t he ra di al di s t r i but i o ns t o t he s l i p v el o ci t y . Ho we ve r , i t i s

not greater than 0. 07 m s and a mean est i mate of 0.04 m s seems acceptabl e.

There i s a smal l dependence on the ( l i qui d) vel oci ty but t his i s not

s i gn i f i cant f o r the smal l range o f vel oc i t i es encountered in the r i ser .

Hydrost at i c eff ects are also present but two measuri ng l evel s are not

suf f i c i ent to g ive a he ight dependent re l a t i on. Furt hermore the ef f ect i s

dif f erent for each of the two t op pressure modes. These concl usi ons are

restr i cted to t he ri ser of the experi ment al col umn i n the worki ng area of t he

l oop.

38 Results: meao data

In thi s paragr aph the vol umetr i c measurements w t h t he manometr i c

method are presented. The volumetr i c mean val ues ar e i nterpreted as I dent i cal

73

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f:

-; ^

A-°"-°-

<*

y.

+/t

"

,¥'

f *

m ■

. „ - ( *

* -'■_

—ssssx,

F igure 38.1: R i ser vo id f ract i on data (manometr i c method) agai nst super f i c i a l gas ve loc i t y 01

a ; proposed cor r e la t i on fo r bubbl e co lumns by He i j nen at a l . (1562) , equati on ( 3 4 - 2 ) , 0 - 0. (

ve loc i ty r anges between 0.1 and 0.34 h/s. R i ser d i aa-= er 9" .

t o c ross - sect i ona l ave rages at g i ven l eve l s . F i r s t a compar i son i s made w th

the corr e lat i ons presented i n sect i on 34. The data are al so presented i n thef orm of the one- dimensi onal gas vel oci ty accordi ng to the Zuber and

F i ndl ay- t ype mode l s ( sect i on 33) and i n the form of s l i p ve loc i t i e s . Ax i a l

profi l es are shown to gi ve more i nformat i on on the vari ous contr i but i ons to

t h e sl i p v el o ci t y .

The re l at i on between vo id f ract i on and supe r f i c i a l gas vel oc i t y i s

shown i n f i gure ( 3 8. 1 ) . W i t t en i n t h e f o r m o f e qu at i o n ( 3 4 - 1 ) ,

- 0 .85a = VGs ( 1)

descr i bes most o f the data w th i n + 10% The r i se r vo id f ract i on a i s based on

the manometr i c method and compared w t h the corr espondi ng super f i ci al gasvel oc i t y ( characte r i s t i c va lue about hal f way the pressure po in t s , append ix

13. 2) . These val ues are l ower than the top val ues whi ch are normal l y used,

whi ch i s i mportant i n the compari son w th equat i on (34-2) f or bubble col umns

and shown i n the f i gure.

I t w l l b e c l e ar t h at c o r r e l a t i o n ( 1 ) i s g oo d but w l l p r ob ab l y no t

ho l d f or o t he r c o nf i g ur a t i o ns a nd l i qui d v el o c i t i e s . I n an ea r l i e r

conf i gurat i on of the col umn w th a h igher f r i c t i on number (A , sect i on 19) t he

( e q ua t i o n 3 8 - 1 0 ) . Compar

su p er f i c i a l g a s v el o c i t y n a de d i

equati on (38-2) (sl ope 1) and eq

. i onless I n te rms o f bubble

on (34-4) ( sl ope 1 . 1 9 ) .

data coul d be f i t t ed w th a power 0 .83 . Th i s corr esponds to vo id f ract i ons

5- 1 0% h i g he r . T hi s i s a n i l l u st r a t i o n o f t h e l i m t e d v al u e o f t he c or r e l a t i o n

f o r v oi d f r a ct i o n p r ed i c t i o n.

For compar i son w th corre l at i on ( 3 4 - 4 ) , we have assumed an i ni ti al

bubbl e d iamete r o f 6 mm (at t he r i ser gas i n j ector , sect i on 28) and app l i ed

i dea l gas equat i ons ( f i gure 3 8. 2 ) . Our l ocal one-di mensi onal data may be

descri bed w th t he si mple expressi on

,* ( 2)

The spread i n the data i s t he same as i n f i gure ( 3 8 - 1 ) .

I t must be noted that l i qui d vel oci ty i n the col umn i s dependent on

gas f l ow rat e and could not be vari ed i ndependent l y. Dependi ng on the top

pressure , the re l at i ve Increase i n supe r f i c i a l gas vel oc i t y ove r the he ight

was a f actor two or e i ght poi n t f i ve.

The one- di mensi onal gas ve loci ty plot ted accordi ng to t he Zuber and

F ind lay mode l agai nst the sum of the supe r f i c i a l ve loc i t i e s v ( f i gure 38. 3)

wi l l n ot g i v e t he pr e di c t e d s t r a i g ht l i n e. W t h e qu at i o n ( 1 ) :

Gs 0. 15— = VGs

( 3)

A s a n i l l u st r a t i o n t hi s e qu at i o n i s p l o t t e d i n f i g ur e ( 3 8. 4 ) f o r f i v e

i s not use fu l i n our case.

38 75

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norms

•*V

°

lo w

o

A

0V

°

J t» P

j p r s s s u r c

1 7 73 555 607 649 40

<

*&a

vL s , ml s ,

O 1760 2 4 4

O 272Q 2 7 9

W GH tg/s )

0 1

0 6 *

0 6

0 6

0 9

"normal top pressure

i p l o t t e d f o l l o w n g Zu be i 1 F i ndlay: one dim<

F i g u r e 3 B. 4 : L i k e f i g ur e 3 8. 3 b u t w th o n l y se l e c te d d ata

(S u pe r f i c i a l g as v e l o c i t y i n c r e ase s b y e xp an s i o n a t d l f f e i

and Findl ay model cannot be used i n our case.

tsur i ng leve l s) and com

on ( 3 8 - 3 ) . T h i s a g a i n ;

At f i r s t s i gh t ( f i gure 38. 5 ) the s l i p ve loc i ty as parameter i s no t

very usef ul ei t her . But f i gure (38. 5) shows r aw data and moreover the s l i p

vel oci ty i s a sensi t i ve parameter . F rom the f i gure s t rong hydros ta t i c e f f ects

may be deduced. The r esul ts for the upper part of the r i ser ar e considerabl y

hi gher than for the l ower part and for the barometr i c t op pressure mode.

Superposed on th i s i s a decrease w t h superf i c i a l gas vel oc i t y.I t was concl uded t hat the data m ght be reduced by averagi ng the

resul ts f or t he same measur i ng l evel w t h gas f l ow rat e as parameter. Thi s has

been accompl i shed as shown i n fi gur es ( 38. 6 and 3 8. 7 ) . The two top pressure

modes are presented separat el y.

We w l l make the fol l ow ng remarks:

a. The spread i n the data f or l ow gas f l ow rates i s i nherent to t he manometr i c

measuri ng i i ethod.

F i g u r e 3 8 . 5 : Loc a l ( o n e- d i me n s i o n al ) s l i p ve l o c i t y c a l c u l a te d w th th e d e f i n i t i o n (3 0- 7 ) f r o m th e r i se r v oi d

f r a c t i o n d ata ( ma no me t r i c me th o d ) a ga i n s t c o r r e sp on di n g l o c al su pe r f i c i a l g as v e l o c i t y . L i q u i d v e l o c i t y r a n ge

0 . 1 - 0 . 3 4 m s -

b. A va lue f o r t he s l i p ve loc i t y o f 0 .3 ms (equat i on 33- 11) i s genera l l y a

good, somewhat conservati ve est i mate, except at the hi ghest l evel (z =

9. 4 m) .

c . The s l i p ve loc i ty i s f a i r l y i ndependent of hei ght excep t t hat i t i ncreases

st rongl y w th l ow top pressure at the upper two measur i ng l ocati ons. Thi s

corresponds roughly w th void f ract i ons h i gher than about 8%

d. S l i p ve loc i ty does not decrease w th super f i c i a l gas vel oci ty (wh ich i ss t r ong ly dependent on hei ght ) but w th overa l l gas f l ow ra te ( f i gu re 3 8. 8 ) .

Sl i p vel oc i ty may therefore be dependent on l i qui d vel oc i ty.

To compare t he two pressure modes t he resul ts are al so presented i n

f i gure (38 .8 ) as func t i on o f t he pressure r a t i o

P* " 1 = p (0 ) / p (z ) (4 )

which gi ves the expansi on, and i s therefor e also equal to t he rat i o

78

Because t he devi at i ons are s mal l , r ather t han a cal culat ed agai nst measured

79

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voi d f r ac t i on p lo t , the dev i a t i ons i t se l f (en la rged) a re shown as a func t i on

of v oi d f r a c t i o n ( f i gur e 38. 9 ) . These show t hat t he "good guess" of equati on

(35-26) i s surpr i s i ngl y accurat e, al though some syst emati c err or i s present

and the hydrostat i c ef f ects are not accounted f or . St i l l Che devi at i ons r emai n

wel l bel ow 10%r el at i ve and 0. 01 absol ute. The ot her model , equati on ( 6) ,

reduces t he spread i n the data t o the measuri ng err or of about + 3 mm

ai r - water —manometer readi ng ( = 0.002 absol ute voi d f r a ct i o n) . Thi s e r ro r

resul ts mai nly f rom the strong f l uctuati ons i n water l evel i n the manometerdue to t u rbu lent p ressu re d i f f e rences . Equat i on (6 ) wi l l be used in mode l l i ng

the c ir culat i on of the exper i ment al col umn. Equati on ( 1) mght be used as wel l

but i s thought to be l ess accurat e for ot her conf i gurat i ons. Scal e up aspects

w l l be t r eated in the l ast chap ter o f th i s thes i s .

39 Concl us i on

I n th i s chapter the p redi c t i on of voi d f r ac t i on has been t reated w th

speci a l a t ten t i on to the l a rge hydros ta t i c e f f ects whi ch are p resent i n hi gh

or deep col umns. The survey of t he l i t erat ure shows t wo general appr oaches:

the Zuber and Fi ndl ay-t ype model s and s l i p vel oc i ty based models . The f i rst

use the super f i c i a l m xtu re ve loc i ty v as var i abl e wh ich i s obvi ous l y

i ns uf f i c i e nt t o i nc l u de hy dr o s t at i c e f f e ct s ( f i gur e 38. 3 ) . The l att er model s

coul d be modi f i ed s i mply to g ive a more i ntui t i ve model i n whi ch hydrostat i c

ef f ects coul d be easi l y i ncorporated. Al t hough the l ocal one-di mensi onal void

f r ac t i ons cou ld be we l l cor re l a ted w th super f i c i a l gas ve loc i ty , the

parameters have no physi cal f oundati on and w l l probabl y change unpredict abl y

f o r o t h er c o nf i g ur a t i o ns , e spec i a l l y w t h h i g he r l i qui d v el o ci t i e s . T he s l i p

veloc i ty was t heref ore used as parameter , and spl i t i nto components f or

buoyancy ( vht o) , concent r a t i on and vel oci ty p ro f i l es (v ) and other ef f ects

( v. ) , l i ke hi ndrance, coa lescence and c l uster i ng of bubbl es . The pro f i l es

were measured and shown to have a smal l contr i buti on of about 0. 04 m sresul t i ng f rom the re l a t i vel y l ow l i qu id ve loc i ty . The ' i nhomogeneous '

cont r i but i on i s i mpor t ant i n the upper sec t i on of the r i ser w th l a rge

hydrosta t i c e f f ec ts and a re la t i vel y h igh vo id f rac t i on (> 8% p resen t . A very

s i mp le emp i r i cal re l a t i on f o r t hi s cont r i but i on resul t s i n a p redi c t i on f o r

the void f ract i on that i s w thi n the accuracy of the measurements.

Dowrvar d t wo phase f l ow has not been consi dered i n thi s chapt er.

Downward two phase flow

40 I n t roduc t i on

I n thi s chapt er the separat e experi ments on downward two phase fl ow

are descri bed. The advanced techni ques devel oped and used i ncl uded

l aser- doppl er vel oc i t y measurement s and a f i ve poi nt opti cal probe for bubbl e

detec t i on and charac ter i sa t i on. L i qui d vel oc i t y , bubbl e vel oc i t y and vo id

f r ac t i on pro f i l es were measured t o determ ne the downf l ow s l i p vel oc i t y . Data

on bubb le s i ze and fo rm and on tu rbu lent vel oc i t y f l uc tuat i ons were a l so

produced. The progr amme was mot i vat ed by t he l ack of data on downward f l ow i n

the open l i t erat ure, and that whi ch exi sts shows many di screpancies. The f l ow

rat es st udi ed were of the same order as t he downcomer f l ows i n the bubble

co lumn loop , but measur i ng d i f f i cul t i es l i m ted voi d f rac t i ons t o be low 5%

41 L i te ra t u re

Report ed experi ments on downward two phase fl ow are r est ri ct ed t o a

f ew publ i cat i ons. Several f l ow maps are avai l abl e ( Golan and Stenni ng, 1970;

Oshi nowo and Char l es, 1974; Speddi ng et al . , 1980 and Barnea et al . , 1982) .

Alt hough these are al l for p i pes 1 - 2 I nch diameter there are maj or

di f f erences between them bot h in the f l ow patt ern def i n i t i ons and predict ed

areas. Mart i n ( 1973) presents work on downward sl ug f l ow i n a 0.14 m di ameter

p ipe (8 m l ong) and predi c ts a t rans i t i on to bubb ly f l ow fo r

WVL s " °"3 ™

W t h a mn i mum l i qui d vel oci ty o f about 0 .20 ms th i s means t hat the super

f i c i a l gas f l ow ra te mus t be be low 0.06 m s i f the f l ow i s t o be cocur ren t .

Mart i n ( 1976) p l ott ed hi s dat a on s l ug f l ow accordi ng to t he Zuber and F indl ay

mode l ( see sec t i on 32) and achi eved good f i t s w th d i s t r i but i on coef f i c i ents

C l ess than 1. He showed that gas s l ugs i n downward f l ow ' r i de' the wal l

80 41

(Söderberg, 1980) and thus di f f er essenti al l y fr om sl ugs i n upward f l ow. One

41 81

St i l l s ma l l e r s l i p ve l oc i t y v al u es a r e r epo r ted by F uj i e et a l . ( 1980): 0. 175

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may even di st i ngui sh a fl ow patt ern between bubbl y and sl ug fl ow i n whi ch

l arge bubbles c l i ng to the wal l w thout f i l l i ng the whol e tube as occurs w th

upf l ow. I f a suf f i c i ent par t o f t he a i r f r act i on i s entrapped i n these s l ow

moving bubbles a di st r i but i on coef f i ci ent l ess than 1 is expected.

The most i mport ant r eport on downward gas l i qui d bubbl y f l ow i s the

work of Lorenzi and Sotgi a (1976) . These authors measured voi d f ract i ons i n

44 mm and 90 mm pipes over a w de range of i nput parameter s (v f rom 0.5 to

2 m s , v f r om 5 to 4 00 mm s w th v / vM rangi ng f rom 0. 01 to 0 . 2) .Their resul t s coul d be represented wel l enough by the Zuber and Fi ndlay model :

v = 1. 07 v M - 0.283 ( 1)

where t he parameter s agree w t h t hose f ound by Curt et and Dj oni n ( 1967)

ear l i er (d = 46 mm v f r om0.45 m s to 1 m s) but l ess so w th recent

r esul t s of Cl ark and Fl emmer ( 1984) (C = 1. 16, d = 52 mm v L g f r o m 0. 8

to 2.3 m s ) . Thi s cor r esponds w th s l i p vel oc i t i es wel l bel ow 0.3 m s:

v f« 0.28 - 0.3a

Fifiura 41 : Downward bubbly f low void fracti on pro fil es . Reported by Curte t and Djonin (1967) (a) , Ibraglmovet a i. (1973) (b-e) and Chu and Jones (I960) (£,g ) .

to 0.19 m s i n a 0 .45 mdi ameter co l umn w th l ow l i qu id veloc i t i es (below

0.23 m s) and by Ohba and Yuhara ( 1979b): 0.12 t o 0. 22 m s. These l at ter data

however are probabl y not r el evant si nce they are obt ai ned i n a square duct

w t h a w dth (11. 5 mm of the same order of magni t ude as t he bubbl e di ameter s

(Local l y determ ned l aser - doppl er data. The s l i p ve loc i ty r epor ted for upf l ow

was even l ower t han f or downfl ow) . I nhomogeneous ef f ects l i ke bubbl e cl usters

and l i qu id c i rcu l at i on are hard l y poss i b le i n such a smal l duct .

Comparati ve val ues f or s l i p vel oci t i es i n downward t wo phase f l ow areexpected t o be smal l er. Radi al for ces that woul d expl ai n void f ract i on peaking

at the wal l i n upward f l ow are al l dir ected t owards t he centr e i n downfl ow

(sect i on 31) . I n f i gure (41) t he cor r esponding bel l - shaped prof i l es are shown

whi ch were r eport ed by I bragi mov et al . ( 1975) and Chu and Jones ( 1980) f or

bubbl y f l ow. The t ransi t i on to parabol i c - t ype or power - t ype prof i l es appears

to occur sooner ( i . e . w th smal l er p i pes) than wi th upward f l ow. Thi s

f l a t t eni n g i s c l ear l y a f u nc t i on of ( i n cr easi n g) v oi d f r act i on and

(decreas i ng) ve l oc i t y .

42 Experi mental

The resu l t s presented in t h is chapter are obta i ned i n a spec i a l r i g

i n whi ch a pumped ci rcul ati on can gi ve downward water veloci t i es up to 4 ms

( f i gur e 4 2. 1 ) . The test secti on consi sted of a 2 m l ong, 0.15 m di ameter

perspex pi pe. A convergi ng- di verging pi pe segment w t h ai r i nject ed

peri pheral l y i n the convergi ng part was agai n used as dist r i but or (chapter 2) .

The overal l parameters were deter m ned as f ol l ows:

a. Li qui d r ate by a magneti c i nducti ve f l ow meter . There were some probl ems

resu l t i ng f rom the bu i l d up of t rapped a i r i n the f l owmeter sect i on which

l ed to di screpanci es between the apparent overal l and l ocal vel oci ty

measurements. A speci al cal i brati on for t he f l owmeter ( appendi x 42. 1)

al l owed adequat e corr ecti on to t he measurements .b. Vol umetr i c void f ract i on by the pressure poi nt method. Dynam c pressure

drop was cor r ected w th s i ngl e phase f r i c t i on data; a procedure that i s

s uf f i c i en t l y ac cu r ate w th t he vo i d p r o f i l e s and l ow v oi d f r ac t i ons

present. Press ure drop was det erm ned w t h both an air - water - manometer and

a di f f e r ent i a l p r es su r e c e l l .

The mai n a im of t h is s t udy was the determ nat i on of the s l i p vel oc i t y

parameter f or downward two phase f l ow i n a rel ati vely l arge duct , w t h i nput

42 83

parameters i n the operat i ng range of t he column l oop. I t was al so desi rabl e to

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acquir e data on the phase and veloc i ty d is tr i but i ons i n such a system so as to

be able t o model s l i p veloc i ty as di scussed i n sect i on 35.

I nstr umentat i on f or l ocal measurements was therefor e used. Thi s

compr i sed a l aser - doppl er vel oci meter (LDV) f o r determ nat i ons o f l ocal l i qui d

veloc i ty and a f i ve point opti cal probe whi ch was developed i n the laboratory

fo r the determ nat i on of l oca l vo id f rac t i on and bubbl e vel oc i t y .

Laser Doppl er Vel oc i metr y

The arr angement used is shown schemati cal l y i n fi gure ( 4 2 . 2 ) . I t i s

based on the ref erence-beam method where scattered l i ght f rom moving part i c l es

i n the measur i ng vol ume ( l ength 2.3 mm w dth 0.1 mm) m xes wi t h t hat of a

ref erence beam at the photo d i ode, g ivi ng a doppler f requency and t hus

veloc i t y. S ince the tap water used was part i cular l y c l ean some seedi ng w th

speci a l par t i c l es ( 3 pm (Texi cote 03 -040) was used . The l ocat i on of the

measuri ng vol ume coul d be t raver sed t hrough the whol e t ube by movi ng the f ront

l ens . The preshi f ted doppler f requency was f i l tered and convert ed i nto a

volt age by a ' tr acker ' (TPD 1077) operat i ng in md r ange. Thi s volt age was

t hen off ered t o an average vol t age meter ( developed i n the laborat ory) andcor r e la to r ( HP3721A). The speci al precauti ons needed f or LDV measurements i n a

t wo phase f l ow are revi ewed by Mari e ( 1983) . The arr angement used i s

i nsens i t i ve to scat t e red l i ght f r ombubbl es whether ou ts i de or i n the

measur i ng volume. However s i gnal - presence i s l ow, wel l bel ow 5% f or void

f ract i ons of about 3% and s i gnal durat i on extremel y short ( f i gure 4 2. 3 ) .

Under t hese c i rcumst ances the operat i on of t he tr acker i s not f u l l y correct

and thi s had to be al l owed for ( appendi x 4 2. 2 ) . I t shoul d a l so be real i zed

11

0 100

1 n

gas fractionliquid vel

20 0

=57

3%

1m/s

"

30 0

Li meys)

F i g u r e 42 . 3 : D r o p- i n d ur a t i o n t i n e d i s t r i b u t i o n . (D r o p- i n t i me l a e q ua l t o s i g n a l d u r a t i o n t i me m n u s a

v a l i d a t i o n t i me o f 1 2 0 u s ) .

42 85

The f i ve poi nt opt i ca l probe

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2 1 34 5

F i g u r e 4 2 . 4 : T h e 5 - p o i n t b u b b l e

F i g u r e 4 2 . 5 : O p t i c a l p r o b e o p er ;

■ e me nC a f t e r B u r g '

optical coating

F l f i u r e 4 2 . 6 : H P b a r c o d e s c a n n e r c o u p l e d t o o p t i c a l p r (

s i g n a l .

i 4 2 . 7 : O p t i r a l p r o b e a r r a n g e m e n t .

l i g h t a n d r e C1 r e f l e c t e d l i j

. . 92 m m ; r - 0 . 82 m m . T he f i h r i

that l aser doppl er determ nat i ons r equ i r e unobstructed paths across t he ent i re

channel , and t heref ore t end to gi ve undue wei ght i ng to peri ods w t h l ow voi d

f r act i ons ( negat i ve voi d p lugs) .

Thi s probe is a devel opment of t he pri ncipl es presented by Perei ra and

Cal derbank (1976) . P r i n ci p l e of ope r at i on i s i l l us t r a ted i n f i gu r es ( 4 2. 4 and

4 2. 5 ) . Detai l ed i n format i on on the pr i nc ip l e of operat i on of an opt i ca l probe

i s g i ven by Abuaf et a l . (1978) and J ones et a l . ( 1979) . The great est

adyant ages over t he conducti v i ty probe are t he short s i gnal r i se and f al l

t i mes, but i ts f rag i l i ty and s i ze l i m tat i ons ( Onken and Buchhol z , 1982) are

di sadvantages.The probe used f or t he experi ments i s made of 0.2 mm di ameter f i bers

coupl ed to HP sensors ( HEDS-1000) (f i gure 4 2. 6 ) . These sensors ar e desi gned to

read bar- codes ( combi nati on of emssi on and sensi ng of l i g ht ) . The probes are

assembled as a f i ve poi nt sensor (f i gures 42.7 and 4 2. 8 ) .

The s i gnal f rom the bar - code scanner i s conver ted by di f fer ent i at i ng

e l ec t r oni c s i n to a b l oc kf o r m ( f i gur e 4 2. 9 ) . The passage of a bubbl e may t hen

r es ul t i n t he s i g na l o f f i gur e (42 .10) . The passage ti mes i ndi cated are

Fi gure 42 .9 : Convers i on of s i gnal of bar code sensor/ probe combinat i on in to b l ock form

F i g u r e 42,10: Typ ical converted s i gnal f rom the opt i cal p robe (compare f i gure 42 .4) and determ ned t i mes.

86 42

deter m ned by counter s and condi ti onal l y tr ansf err ed t o a programmabl e

42 87

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calcu l ator . The programagai n checks the su i t abi l i ty o f the data, ca lcu l ates

i ndi v idual bubbl e character i s t i cs and determ nes the di s t r i but i ons t hereof .

W t h the descri bed arrangement i t i s possi bl e to deter m ne:

a. i f the bubbl e i s penetr ated centra l l y ( c o nd i t i o n) ,

b. i f t he bubbl e i s penetr ated by an axi al movement ( c o nd i t i o n) ,

c . the d i f f erence i n penetrat i on s tar t t i mes of f r ont poi nt (1) and back

poi nts ( 2 , 3, 4) (bubbl e v el o ci t y ) ,

d. penetrat i on t i me (bubble s i ze, ver t i ca l h ei g ht ) ,

e. total t i me averaged penetr ati on ti me (uncondi t i onal l y) ( voi d f r a c t i on) ,

f . i f the bubbl es are assumed to have an el l i psoi dal f orm the data can be

used t o determ ne the eccentr i ci t y and t hus t he volume-equival ent diameter

of the bubbl e.

The bubbl e veloc i ty determ ned w th t he probe i s ca l i brated i n the

same way as w t h t he conducti v i t y probe (see appendi x 36) and the result s are

presented i n f i gure (42 .11) . W thi n the l i m ts of the exper i menta l scat t er and

the vel oci ty range of the experi ments (bel ow 1.5 r a/ s) agreement i s perf ect. At

hi gh bubbl e veloci t i es t he probe gives val ues that ar e somewhat l ow The probe

resul ts f or bubbl e si ze and form have been compared wi t h photographs ( f i gures

42. 12 and 42. 13) . Since bubbl e t rai ns are used, the photographs arerepresent ati ve for the bubbl e si ze di st r i buti ons measured by the probe.

Agreement i s good, al t hough one may noti ce that the smal l est spheri cal bubbl e

that can be detected i s about 1.9 mm di ameter . I n practi ce bubbl es as smal l as

1.5 mm equival ent di ameter could be regi ster ed due to their f l att er f orm

P l g u f e 42,11: Compari son of babb le vel oc i t y by probe and by c rossed beam I nterr upt ion (append ix 36 J .

F i g u r e 42.12: Compari son of bubble di ameter by probe (r aw) and by the photograf l c method.

F i g u r e 42.13: Compari son of bubbl e excentr l ci ty by probe and by the photogr afl c method.

43 Overall data (void fraction and velocity)

To obtai n the volumetr i c voi d fr acti on the singl e phase pressure drop

was subtract ed f rom the measured pressure drop over 1 m of pi pe. The single

phase pressure drop was deter m ned s eparatel y (f i gure 43. 1 ) . The f i gure

i l l ustr ates the r ange and accuracy of the pressure dr op measurements.

The vol umetr i c voi d f r act i on i s presented i n f i gure (43.2) where i t

i s compared w th the data of Lorenzi and Sot gi a ( 1976) and w t h l i nes based ona constant sl i p vel oci ty model (equati on 30-8 f or downward f l o w) . The

agreement i s good i n both cases. The f i tt ed sl i p vel oci t i es however are l arger

than expected especi a l l y f or h igh l i qui d vel oc i t i es . Th is i s probabl y due to

the two phase f l ow not bei ng ful l y establ i shed i n t he sense that some

i nf l uenc e of t h e i n i t i a l g as d i s t r i bu t i on r ema i n s . I n t h e f o l l ow ng s ec t i on

thi s i s di scussed i n more detai l . For t he same reason the devi ati on at hi gher

l i qui d vel oc i t i es l eads to a di s t r i but i on parameter l ess than 1 (0 . 92) f or a

Zuber and Findl ay type plot (f i gure 43. 3 ) . Thi s again i l l us t rates t he dangers

of plot ti ng data t his way, t his t i me because the di st r i buti on parameter C i s

a ( weak) f unct i on of f l ow rat e.

Figure 43.1: Single phase pressure drop data.

our data | pred icted voidlllrws)

44 89

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V v Ls =059mfso ; 07 ' l

o =098A -196

v s =0 27 m/s.0 27. 030' O Ï 5

data of Lorenzl. Sotgla 1976 : j.

0003 0.O05

Figure: A3.2: Volumetr ic mean void fraction a inlines and data from Lorenzi and Sotgla (1976).

■p velocity

Figure A3.3: The data of f igure 43.2 plotl and Fin dlay (1963) and compared

with the f it found by Loren zi and Sotgla (1976) (C Q - 1.07; Vgj/0) - -0 .2b ra/s) .

44 Radial profiles of void fractions and velocity

Radial Gas distributions

Di mens i onl ess voi d f ract i on prof i l es are presented i n f i gure (44 .1) .

Thi s i l l ust r ates the res i dual i n f l uence of the d is t r i butor even when th is i s

two metr es above t he measuri ng l ocati on. As a consequence of the i nj ecti on of

F i g u r e 44 . 1 : R e l a t i v e r a d i a l v o i d f r a c t i o n p r o f i l e s a t t wo l i q ui d f l o w r a t e& ma de d i n ens i o nl e s s b y t h ei r

f i t t ed c en t r a l v a l u e a -

air around the peri phery of the di st r i butor bubbl es are concentr ated near t he

wa l l i n s pi t e o f t he r edi s t r i bu t i on c ha r ac te r i s t i c s o f t h e d i v er g i n g s ec t i on

( W sman, 1979). W th a bubble vel oc i t y of about 1.2 m s at the hi ghest l i qu id

f l ow rat e, on ly 1.7 s i s avai l abl e for t he estab l i shment of t he prof i l e . Thi s

i s obv i ous l y t oo br i e f , at l eas t w th l ow vo i d f r act i ons .

Nevert hel ess t he resul t s show (see al so f i gures 44.2 and 44. 3) :

a. The gas d i s t r i but i on i s not Gauss ian but i s c l oser to a power - t ype form

F i g u r e 44 . 2 ; A s f i g ur e 4 4. 1 f o r t h e o l d d l e l i q ui d f l o w r a t e .

90 44

as report ed by Nakoryakov et al . were r el ated t o voi d peaki ng near thewal l s and hence are not present.

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O 06 -

b. W th i ncreas ing vo id the d is t r i but i ons f l at t en becomng a lmost uni f orm The

same tendency was f ound w t h upward bubbl y f l ow ( chapter 3 and Nakor yakov

et a l . 1980).

c. Void f racti on becomes zero at about 2 - 3 mm fr om the wa l l . Obvi ousl y

bubbl es do not sl i de al ong the wal l surf ace as i s t he case in upward f l ow

(Sekoguchi et al . 1979) .

I t i s now possi bl e to compare t he probe dat a w t h the vol umetr i cmethod by averaging t he l ocal dat a ( appendi x 4 4 ) . Fi gure ( 44. 4) shows t hat t he

probe data may be as much as 25% below the vol umet r i c data. Some possi bl e

reasons for t his di screpancy are:

a . The cor rect i on for f r i c t i onal pressure drop i s too l ow. I t may be noted

that t his corr ecti on is of the same order of magni tude as the gravi tat i onal

term However for the smal l voi d fr acti ons encountered t his ef f ect i s not

expected t o account f or more t han about 10% The strong f r i cti onal eff ects

b. W th t he actual vel oci t i es ( and bubble di ameter s) the durat i on ti me of a

bubbl e passi ng the probe is very short (ca. 2 ms) , however a sampl i ng rat e

of 100 kHz should be suff i c i ent for an accuracy of 1% or l e ss .

c. Systemati c err ors i n the operati on of t he probe i t sel f . These may i nclude

i n f l uence on the f l ow, the def l ect i on of smal l bubbl es , delay of bubbl e

penetr ati on or the di sregard of contacted but not penetr ated bubbl es by the

di f f e r en t i a t i ng e l ect r oni c s , a s a r es ul t of t he gr adu al r i s e and f a l l oft h e s i g na l s ( M l l er and M tc h i e , 1969). These errors ar e general l y bel i eved

to be less t han 10%i n tot a l .

Other er r ors must be present , but the form of the prof i l es as such i s

consi dered re l i abl e.

Radi a l v el oci t y d i s t r i b ut i ons

S ince the entry l ength of the tes t sect i on i s onl y about th i r t een

di ameters a ful l y developed turbul ent vel oci ty prof i l e cannot be expected. On

the other hand t he special desi gn of the water tunnel suppresses l arge scal e

turbu lence, whi l e the t r ans i t i on f rom square t o c i r cul ar duct and t he presenceof the converging-di vergi ng i nject or pi ece may qui cken the devel opment of t he

boundary l ayer.

The si ngl e phase water vel oci ty prof i l e was measured t o check t h i s .

The longi tud ina l turbul ence i n tens i ty f o l l ows f rom the s tandard devi at i on of

the probabi l i ty curve produced by the cor r e lator af ter cor rect i on for noi se.

The resul ts are compared w th the wel l known data of Laufer (1954) (f i gure

44.5 and 4 4. 6 ) . The agreement i s, despi te some scatt er, surpri s i ngl y good,

al though the t urbul ence i ntensit y at the centr e i s somewhat l ow. This means

that t he boundary l ayer i s probably not yet f ul l y developed and that t he l ower

entr y i n tens i ty prevai l s ( Di spl acement i n teract i on, see Johnston, 1978). I t

must be noted that t hese resul ts ar e of l i m ted value f or t he two phase f l owcase, s i n c e the i n j ec t i on of a i r di s tu r bs t h i s p at t e r n r adi c al l y .

Both bubbl e veloc i t y (probe) and water vel oc i t y ( LDV) prof i l es are

shown i n fi gures ( 44. 7 and 4 4. 8 ) .

a. On ly f or vo id f ract i ons above 1% do the l i qui d vel oc i t y prof i l es devi ate

s i gni f i cant l y f rom the si ngl e phase s i tuat i on and become more un i f orm w th

i ncreas ing voi d f r act i on. The lat t er ef f ect i s al so found i n upward two

phase f l ow (Nakoryakov et al . , 1981).

93

c. Al l prof i l es can be f i tt ed by a power l aw equat i on of t he t ype:

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Re c . 150 000 f \V L )i 0& 6m/ i \

b. Form and tr end of bubbl e and l i qui d vel oci ty pr of i l es are the same,

al though bubbl e vel ocit y prof i l es are somewhat more unif orm and

i ncreas ingl y so w th decreas ing l i qui d f l ow rat e. Bubble vel oci ty prof i l es

seem to be more pronounced ( i . e. both f l att er or sharper) than comparabl e

upward bubbl y f l ow (Seri zawa et al . , 1 97 5I I ) .

Z- - a - ~>1 /n

env K

W th except i on of t he uni form dis t r i but i on at 3% voi d f r act i on and vLs

0.22 m s, the val ue of n is between 7 (a typical s i ngl e phase val ue) and

22. Prof i l es were not of Gaussi an t ype, nor di d t hey show a maxi mum near

t he wa l l .

I n two of the cases sharper l i qui d vel oci ty prof i l es are found i n

compari son w t h si ngl e phase. One possi ble expl anat i on is t hat the i nject i on

of ai r may qui cken the devel opment of t urbul ence. The fl ow coul d t hen have

developed to the poi nt t hat shear l ayer i nteracti on is i mport ant. I n that case

the centra l l i qu id veloc i ty I s known to be l arger than that o f t he fu l l y

devel oped t urbul ent f l ow ( J ohnston, 1978). I t w l l be c l e ar t hat

i nterpret ati on of two phase data i s very di f f i cult where such compli cated

eff ects can be present.

As st ated earl i er both measuri ng methods of l i qui d f l ow rat e, LDV and

i nducti ve f l owmeter, were sensi t i ve to the presence of ai r and corr ecti ons

94 44

wer e needed. Thi s l essens accuracy. A di rect compari son of the t wo methods

cannot be gi ven si nce the corr ect i on of the LDV was based on the f l owmeter

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data.

Loc al r e l a t i v e ve l oc i t y f o l l ows d i r ect l y f r om th e l i qui d and bub bl e

vel oci ty measurements . Relat i ve vel oc i t y i s par t i cu lar l y sens i t i ve to

I naccuracy s i nce i t i s t he d i f ference between two l arger quant i t i es , each of

l i m ted accuracy . Bear i ng i n m nd th is l i m tat i on, t he resu l t s are presented

i n f i gur e (44 .9 ) . The resul ts are s i m l ar to t hose of Ser i zawa et a l . (1975I I )

f or upward f l ow. Dependenci es cannot be cl ar i f i ed cl earl y alt hough rel ati vevel oc i t y i s l ower nearer t he wa l l . There i s a tendency of the rel ati ve

vel oc i t y to i ncrease w th f l ow rat e, but not w th qual i ty s i nce th is was kept

near l y constant per vo id f ract i on. Th is contrasts w th t he resul ts o f Ser i zawa

et a l . who found an i ncrease w th qual i ty . F i nal l y t he val ues get l arger t han

v (0 .235 m s) for h i gher f l ow rates ; th is must be a resu l t o f

i nhomogenei t y.

45 Bubbl e si ze

The ti me that el apses between i nject i on and detect i on of the bubbl es

i s t oo short for break up and coal esci ng processes to produce an equi l i br i umbubbl e di st r i but i on si ze. This means t hat the mean bubbl e si ze is l argel y

deter m ned by the i nj ecti on and is thus normal l y dependent on gas and l i qui d

f l ow rat es (chapter 2) . W th excess i ve gas i n ject i on, a bubbl e s i ze, dependent

onl y on l i qui d vel oci ty and determ ned by the sl ug break up process m ght be

expected.

For the resul ts presented i n th is chapter i t i s i mpor t ant to be

i n formed about bubbl e s i ze and i ts radi a l d is t r i but i on but i t shoul d be noted

that t hi s i s not necessar i l y r epresentat i ve for t he genera l case. S ince the

bubbl es are created near t he wal l and a uni f orm voi d di st r i but i on has

devel oped in most cases , i t i s i n terest i ng to see i f the necessary m grat i on

of the bubbl es t o the centr e i s s i ze-dependent.The (number) mean equi val ent bubbl e di ameter i s i n general

i ndependent of rad ia l pos i t i on ( f i gure 45. 1 ) . Thi s i s a l so repor ted by

Ruchhol z et al . ( 1979a) i n a bubbl e column. The bubbl e si ze dis tr i but i ons

(f i gures 45.2 and 45. 3) i nf l uence the vol ume to surf ace equival ent di ameter

( f i gure 45.1) somewhat . Thi s i s a wal l e f f ect . Some other typ ica l bubbl e s i ze

di st ri buti on dat a are presented i n appendi x 45. Some comments are al so made

t' iere on the rel ati on between bubbl e si ze and detect i on probabi l i t y .

96 45

The resul ts i ndi cate t hat a l though there are mnor d i f f erences i n the

deta i l ed bubble s i ze di s t r i but i ons across t he pi pe di ameter , these di f f erences

46 97

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are so smal l that they are un l i kel y to af fect the voi d di s t r i but i on, ve l oci ty

prof i l e and other presented two phase f l ow propert i es si gni f i cant l y .

46 Sl i p v el oci t y

Bef ore di scussing t he concl usi ons t o be drawn fr om the l ocal

measurements about sl i p vel oci t y, whi ch was t he mai n ai m of thesemeasurements , the f ol l ow ng poi nts s houl d be made.

F i r s t ; the resul ts are of l i mt ed val ue as a bas is for extr apolat i on

to pract i ca l sys tems. Genera l l y the entr y l ength needed to at t a in f ul l y

establ i shed f l ow w l l be re l at i vel y shor t i n compar i son w th t ota l co l umn

l ength. I n the experi ment al l oop t he aerated downcomer l ength (2.9 m i s about

th i r t y d i ameters , a l r eady tw ce that o f the downf l ow exper i ments . I t I s not

possi bl e t o determ ne how complet el y t he f l ow had devel oped, al though onl y at

the hi ghest ve loc i ty s tudi ed (1 .47 m s) was a typ ica l i n jector dependent voi d

f r act i on found.

Second; t he somewhat l i m t ed accuracy of bot h the devi ces used f or

l i qui d veloci t y measurement, combined w th the necessary assumpti on t hat thebubbl e vel ocit y measured by probe is the same as the local gas veloci t y, l i m t

the accuracy of the s l i p ve loc i ty determ nat i ons .

Thi rd ; the i n i t i a l bubbl e s ize d is t r i but i on devel oped In the

experi mental system i s determ ned by the i nject or desi gn and operati on rat her

than by the bu lk l i qui d f l ow rate i n the pi pe sect i on.

I n order to determ ne s l i p vel oc i t y the fo l l ow ng equat i ons are

recal l ed and rewri tt en for cocurr ent downward f l ow. Equati on ( 30- 7)

VLs VGs , Nv = —; —• ( I )B <e> <o> *■ '

was used w th the overa l l data ( sect i on 41) . Equati on ( 30- 20)

<£> ( 2)

wi l l be us ed t o d ete r m n e the s l i p ve l oc i t y d i r ec t l y f r om th e l o c al d ata ,

whi l e equat i on (35-3)

may be used t o det erm ne t he prof i l e i nfl uence i n the same way as was done for

upward f l ow. Fi nal l y the inf l uence of assum ng v cons tant i n the deri vati on

of equati on ( 3) may be assessed di rect l y .

To determne these quant i t i es f r om the l ocal data a curve f i t t i ng

procedure was used together w t h numeri cal i ntegrati on. The result s are

presented in f i gure (46. 1) .

The s l i p vel oci ty based on l oca l data agrees l argely w th t hat based

on ove r al l d ata , f i gur e ( 4 3. 2 ) . Even the t rend towards higher values w t h

i ncreasi ng vo id f ract i on can be seen i n the l at ter p l ot .The prof i l e i n f l uence cal cul ated by equat i on (3) i s very smal l , l ess

than 0.03 m s, and negl i g ib l e f or h igher voi d f ract i ons as a resul t o f more

uni form prof i l es• The non-parabol i c vo id prof i l es at v = 1.47 m s are

i nsuf f i c i ent l y pronounced to produce pos i t i ve ef f ects , a l t hough the ef f ect i s

far l ess than i t woul d have been w th parabol i c - t ype prof i l es .

Wh en c or r ec ted f o r t hi s p r of i l e i n f l u en ce , t he s l i p ve l oc i t y i s

al most t he same as t he mean rel ati ve veloci t y cal culat ed f rom the l ocal dat a

98 46

( f i gur e 44. 9 ) . I n vi ew of t he smal l bubbl e s i ze ( ca. 3 mm di ameter ) a t erm nal

ve loc i ty v = 0.2L m s i s expected and found fo r t he l owes t l i qui d f l ow ra te

47 99

eddies are af f ected, depends on the system consi dered and the operat i on

cond i t i ons .

7/22/2019 TR-diss-0001460(1)


(0 . 62 m s ) . Clear l y i nhomogeneous ef f ects (chapter 3) are responsi b l e for t he

i nc r ea se of t h e s l i p ve l o c i t y w t h hi ghe r f l o w r a t e s . T hi s l a t t e r e f f e ct i s

cont r ar y t o t hat measured by Ohba and Yuhara (1979b); , who however wor ked w t h

a 11. 5 mm square duct where i nhomogenei t i es are suppress ed and pr of i l es are

re l a t i vel y peaked .

F i n al l y , f or t he se s i t u a t i o ns t he r e i s l i t t l e or n o d i f f e r e nc e

between the mean and wei ght ed mean rel at i ve vel oc i t i es , thus conf i rm ng theval i d i t y of t he assumpt i on o f a rad ia l l y constan t re l a t i ve vel oc i t y i n the

der i vat i on of equat i on ( 3 ) .

I t can be concl uded that i nhomogeneous ef f ects and prof i l e i nf l uence

counteract each other. I n downward bubbly f l ow i n l arge ducts there i s a

tendency to uni f o rm rad ia l p ro f i l es reduc i ng the p ro f i l e i n f l uence. S t rong

i nhomogeneous ef f ect s, i n compari son w t h upward f l ow, wer e encount ered but

extensi on to other syst ems i s not poss i b le s i nce the f l ow was not I ndependent

of the gas i n j ect i on system More data on downward two phase f l ow are c l ear l y

needed.

As a ru l e of thumb the s i ngl e bubbl e t erm nal veloc i ty may be t aken

as a reasonabl e est i mate f or t he s l i p vel oc i ty i n downward bubbl y f l ow.

47 Turbulence: introduction

The p resence o f bubbl es modi f i ca tes t he l i qui d f l ow s t r uc tu re , wh i l e

t he turbul ence of t he f l ui d may i nfl uence bot h si ze and movement of t he

bubbl es . H inze ( 1972) s t a tes that there i s an i ncreased e f f ec t i ve shear r a te ,

whi ch w l l modi f y t he energy spectr um of t he f l u id i n the wave number range

corr espondi ng w th t he average di st ance between part i c l es. Moreover t here i s

an e f f ect due to the wakes o f par t i c l es havi ng a re la t i ve ve l oc i t y w th

respect t o the f l u id bul k, t hus modi f yi ng the energy-s pect rum i n the wave

number r ange corr espondi ng t o part i cl e di mensi ons. When bubbl es move i n groupsor c l us ter s t here a re addi t i ona l e f f ec ts co r respond ing w th g roup separa t i on

and s i ze . I t may be noted that f o r l ow voi d f rac t i ons (above 0. 5% bubbl e

separat i on and si ze are of the same order of magni t ude.

The inf l uence of t he presence of part i c l es on the turbul ence i s not

yet c l ear , a l t hough I t i s genera l l y accep ted that t he di s s i pat i on ra te

i ncreases w th the par t i c l e concent r a t i on (H i nze, 1972). But the e f f ec t on the

re l a t i ve i n tens i ty o f the f l u id t u rbul ence , onl y no t i ceab le when the b igger

Si nce the determ nat i on of t urbul ence was not t he pr i mary obj ect i ve

of t hi s st udy of downward two phase f l ow, use was made of the avai l abl e

equ ipment on ly t o study the l ong i t udi nal vel oc i t y f l uc tuat i ons . Th i s r es t r i c ts

the in f o rmat i on to ax i a l tu rbul ence I n tens i t y .

I n bubbly upward f l ow Ser i zawa et al . ( 1975I I ) f ound a uni f orm

d is t r i but i on of tu rbul ence i n tens i ty . I t was however no t a l ways i ncreased

above t hat I n s ing le phase f l ow. They ment i on the f o l l ow ng poss ib l e e f f ec ts .a. Decrease i n the l i qui d volume i n whi ch the energy di ss i pati on must take

p lace , i ncreases t u rbu lence in t ens i ty .

b. The work done i n f l oat i ng the bubbl es decreases t urbul ence i ntensi ty.

c . There w l l be a dec rease of axi a l i n tens i ty to supp ly energy f o r t he

l a t e r al v el o ci t y f l uc t u at i o ns o f l i qui d a nd bu bb l e s .

d. The bubbl es can absorb energy as a r esul t of defor mati on and gas movement

wi th i n bubbl es .

e. One may add here t he poss i b le d i ss i pat i on of bubbl e-s i zed t urbul ent eddi es,

by t he bubbl es t hemsel ves.

General l y however a str ong I ncrease i n t urbul ence i ntensi ty i s found

w th gas concent r a t i on (Lance and Bata i l l e , 1983; Theof anus and Sul l i van,1982; Ohba and Yuhar a, 1979a). Sat o and Sekoguchi ( 1975) i ntr oduced the

concept of the superposit i on of t wo k i nds of t urbulence. One i ndependent of

bubbl e presence ("wa l or g r i d -genera ted") , the other due to t he bubb le

a gi t a t i o n ( "buoyancy-dri ven" or "excess" tur bul ence i n t en si t y ) • Thi s concept

was used success fu l l y by Sato e t a l . (1981) to pred ic t l i qu id ve l oci ty

p ro f i l es and by Van der We l l e ( 1981) t o predi c t f r i c t i ona l p ressu re drop .

The resul t s on longi tudi nal tu rbul ence i n tens i ty repor ted i n the

f ol l ow ng sect i on are unique in so f ar t hat they ref er t o downward f l ow and to

f l ow i n a pi pe of l arge di ameter. Al l the other r esult s ment i oned were

obta i ned i n pi pes o f 60 mm or sma l l e r , o r downs t ream of a g r i d ( Lance e t a l . ) .

On ly Buchhol z e t a l . (1979a) and l a ter Zakrzewsk i e t a l . (1981) havedeter m ned t urbul ence i ntensi ty i n a bubble col umn of 0.14 m di ameter . Their

very low l i qui d f l ow rat es however ( v < 0.02 m s) prevent val i d compari son

o f t he r e su l t s .

48 Turbulence: results

The s tandard dev i a t i on o f t he ax i a l ve l oc i t y p robabi l i ty cu rves ,

co r rec ted f o r no i se , i s equal to the RMS val ue of the longi tud ina l vel oc i t y

100

fluctuations vl

48 48 101

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ih< (o

( 2)

( 3)

Resul ts are presented as f o l l ows :

a . r ad i a l d i s t r i b ut i o n of t he r el

b. r a di a l di s t r i b ut i o n of t he d i me n s i o n l e s s t u r b u l e n ce i n t e n s i t y v ' / v f

c . p i p e c e nt r e l i n e v a l u e s v' / v as a f u n c t i o n of v oi d f r a c t i o n.

H e r e v . i s t he s o- c al l e d f r i c t i o n v el o ci t y f or the s i n gl e p ha s e

s i t u at i on or

Yf v L s ' \t ( 4)

I n f i g ur e s ( 4 8 . 1 , 'I and 3) al l d a t a are p l o t t e d a c c or d i n g t o a) and

c o mpa r e d wi t h t he s i n g l e p h as e c a s e r e p r e s e n t e d wi t h a b r o k e n l i n e . T he s ec l e ar l y s h ow t he i n c r e as e i n i n t en si t y due to the p r e s e n c e of ai r . The s l i g ht

de c r e a s e ne ar t he wa l l i s a r e s ul t of the i n c r e as e of the t u r b u l e nt c o r e a r e a

01 6

oo a

-

ï

" I

" ^

, - -

IV Lï »0.63

A 05a 1 0V 30

„ . - -^, ~-~ single phas.

i

„ ,.

=

',

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o

-^

11

I

.—- — /

„ * „

x £ > -

-

— ,m

1

F i R u r e 4 8, 1 : R a di a l d i s t r i b ut l o i e l a t i v e l o n gi t u d i n a l i

parameter {broken l i ne - s i ng le phase) {v - 0.62 m s ) .

F l s u r e 4 B . 2 : As f i g ur e 48. 1 (v - 0. 86 m a ) .

0.16

oo s

V L

vr

- I

I I——

0-*~

A

1

-9—

i

1

Ls ' m'

s

A OSa 10V 2 2

°

—-§~~~~z

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1

1

-~z$?

,

^ A * '

1

^

11

/

-

fH

L S S J » ! " „ t , f u n ct i o n of q ua l i t y w t

and t he d e c r e a s e of t he wa l l s u b l a y er t h i c k n e s s a s s o c i a t e d wi t h t he measur edf l a t t e r ve l o c i t y pr o f i l e s ( Oh ba and Y u h a r a , 1 9 7 9 a ) .

T h e i n c r e a s e i n r e l a t i v e t u r b ul e nc e i n t e ns i t y i s mo s t s p e c t a c u l a r f or

l o w l i q ui d f l o w r a t e s and at t he c e n t r e of t he d u c t . T h e r e i s some dependenc e

o n v o i d f r a c t i o n , but b as i c a l l y at l ow v a l u e s . A l r e a dy at 1%v oi d f r a c t i o n an

a l mo s t c o n s t a n t l e v e l of t u r b u l e n c e i n t e n s i t y s e ems to ha v e be e n r e a c he d .

T h e s e t e n de n c i e s a g r e e wi t h t h o s e f o u nd by S e r i z a wa et a l . ( 1 9 75 I I )

f o r u p wa r d f l o w. The f a c t t h at t h e i r d at a d ev i a t e f ar l e ss f r o m t he s i n gl e

p ha s e s i t u a t i o n i s a r e s ul t of b ot h t h ei r h i g h i n i t i a l l e ve l of i n t en si t y

( c e nt r e l i n e v al u e 5% and our l ow l e v el i n the c e n t r e pa r t ( d e v e l o p ed

t u r b u l e nt f l o w c e nt r e l i n e v a l u e 3% L a uf e r , 1 9 5 4 ) .

T he s e ns i t i v i t y of t he c en t r e l i n e a xi a l r e l a t i v e t u r b u l e n cei n t e n s i t y ma k es i t a t t r a c t i v e t o p l o t i t s e pa r a t e l y i n o r d e r t o s ho w t he

i n f l u en ce of v oi d f r a c t i o n or q ua l i t y ( gas ma s s f l o w r a t e f r a c t i o n ) . I n f i g ur e

( 4 8 . 4 ) our d a t a are c o mpa r e d wi t h t ho s e of T h e o f a n u s and S ul l i v an ( 1 9 8 2 ) .

T he s e do not c o nt r a di c t o u r s , u nl i k e t h o s e of L a n c e and B at a i l l e ( 1 98 3)

( f i g ur e 48 . 5 ) who d et e r m n e d t he i n f l u en c e of b u b bl e s on g r i d t u r b u l e n c e .

Cl e ar l y t he i n f l u en ce of the wa l l c a n n ot be i g n or e d e v e n at the c en t r e l i n e of

a 0. 15 m di amet er downwar d t wo p h as e p i p e f l o w.

T h e q u e s t i o n has a r i s e n wha t ha ppe ns to the l o ga r i t h m c ve l o c i t y

d i s t r i b ut i o n, c ha r ac t e r i s t i c f or t u r b u l e n t s i n gl e p h as e f l o w. The answer

4 3 103

f o r d ec r e a s i n g l i q ui d f l o w r a t e . T he s t r o n g r e s e mb l e n c e s h o wn i n f i g ur e ( 4 8 . 6 )

i s h o wev e r mo r e t h a n ex p ec t ed .

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Fi g u r e 4 B .5 : C e n t r e l i n e a x i a l r e l a t ll iqu id f low rate as parameter . Coopai

f u n c t i o n o f i

i upward gr i d t urbul e i

F i g u r e 4 S . 6 : C omp a r i so n b e twe en l i q u i d v e l o c i t y a nd b ub b l e v e l o c i t y f l u c tu a t i o n s .

i l l u s t r a t e d i n a p pe n di x 4 8 , i s c o mp l i c a t e d , b u t t h e p r o f i l e s k e e p t h ei r

l o ga r i t h m c c h ar a c t e r .

F i n a l l y a co mp a r i s o n ca n b e ma d e b e t ween t h e r o o t mea n s q u a r e

d e vi a t o r i c l i q ui d an d b ub bl e v el o c i t i e s . I n v i e w o f t h e f o r e g o i n g d at a o n e ma y

e x pe c t t h a t t h e r e i s a s t r o ng r e l a t i o n s hi p be t w ee n t he f l u c t u a t i o n s i n

vel oci ty o f t he bubb les and the to tal tu rbul ence i n tens i ty , i ncreas ingl y so

Do wn wa r d f l o w v e l o c i t y f l u c t u a t i o n s ar e o f t h e s a me or d e r o f

ma g n i t u d e a s f o r u pwa r d f l o w, wh i c h i s c o n s i s t e n t wi t h t h e c o n c l u s i o n s o f

S e r i z a wa et a l . ( 1 97 5I I ) . T h e p he n ome na a r e l o c a l l y , d e t e r m n e d by r e l a t i v e

v e l o c i t y . A s a g e ne r a l c o n c l u s i o n i t ma y b e s t a t e d t h a t t h e r e i s l i t t l e

d i f f e r e n c e i n t h e i n f l u e nc e of t h e p r e s e nc e of b u b bl e s o n t h e l i q ui d

t u r b u l e n c e f o r u pwa r d a nd do wn wa r d f l o w. F o r l o we r l i q u i d v e l o c i t i e s b u o y a nt

t u r b u l e n c e ma y e a s i l y d o m n a t e wa l l g e n er a t e d t u r b u l e n c e . W t h z e r o l i q ui dv el o c i t i e s pa r a bo l i c t y p e d i s t r i b ut i o ns a r e f o un d ( B uc h ho l z e t a l . , 1 9 7 9 a ) .

4 9 C o nc l u s i o n

T h e s ep a r a t e ex p e r i men t a l p r o g r a mme o n d o wn wa r d t wo p ha s e f l o w h a s

p r o du c ed s e ve r a l i n t e r e s t i n g r e s u l t s . U nf o r t u n at e l y t h e s l i p v e l o c i t y r e s u l t s ,

wh i c h we r e t h e ma i n o b j e c t i v e , a r e no t v er y r e l i a b l e b e c a us e of e n t r y e f f e c t s .

S c a l e u p t o pr a c t i c a l s i t u a t i o n s o r e v en t h e e x p er i me n t a l l o o p i s d ou b t f u l .

S o me of t h e r e s u l t s a r e u s e f u l h o w e ve r . F i r s t , i n d o wn wa r d f l o w t h e r e

i s a c l e ar t e n de nc y t o wa r d s f l a t pr o f i l e s f o r v oi d f r a c t i o n a nd ve l o c i t i e s

wi t h i n c r e as i n g, b u t l o w v oi d f r a c t i o ns . T he c on t r i b ut i o n of t h e pr o f i l ee f f e c t t o t h e s l i p v e l o c i t y i s t h e r e f o r e s ma l l , a n d mu c h l e s s t h a n f o r u p wa r d

f l o w, d e s pi t e t h e h i g he r l i q ui d ve l o c i t i e s i n t h i s e xp er i me nt . Ne v er t h el e s s i n

a ho mo g en e ou s di s p e r s i o n s l i p v e l o c i t i e s l e s s t h a n t h e f r e e b ub bl e t e r m n a l

v e l o c i t y wi l l b e f o u n d. I n ho mo g en e i t i e s l e ad t o l a r g e r v a l u e s a nd t h e s e oc c u r

mo r e f r e qu en t l y a t h i g he r l i q ui d ve l o c i t i e s . Me a su r e d s l i p ve l o c i t i e s we r e

a l wa y s be l o w 0 . 3 5 m s an d a r e g e n e r a l l y l e s s t h a n t h o s e f o u nd f o r u p wa r d f l o w.

T h e t u r b u l en ce d a t a a r e u n i q u e a n d emp h a s i s e t h e n eed f o r mo r e d a t a

o n t u r b u l e n c e i n t wo p h a s e f l o ws .

104105

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5 Steady state circulation

5 0 I n t r o d u c t i o n

S teady s tate c i rcu l at i on i s def i ned as the s t abl e operat i on mode i n

which the t i me averaged c i rcu l at i on vel oc i t y i s constant . I n fact t he

c i rcu l at i on vel oc i t y i s observed to f l uctuate about a mean val ue. I n th i s

chapter experi mental dat a on veloci t y and mean r i ser voi d fr acti on are

presented. Some exi st i ng model s t o descri be the st ati onary behavi our f or bot h

ai r l i f t t owers and deep shaft type col umns are di scussed. A model i s

presented whi ch i ncl udes t he l arge hydrostat i c ef f ects i n ta l l co lumns or

col umns under reduced pressure. Downcomer i nj ecti on head l oss i s al so

i ncluded.

Operati on w t h downcomer ai r i nj ecti on may be metast abl e. The col umn

has to be s tar t ed w th a i r i n ject i on in the r i ser t o get a downcomer l i qui d

vel oc i t y that i s suf f i c i ent to entra i n a i r i n jected in to t he downcomer . Th is

vel oc i t y must c l ear l y be l arger than the i ndi v i dual bubbl e r i se veloc i ty over

al most the whol e cross sect i on of the downcomer. I n practi ce a superf i ci al

water veloci t y of about 1 m s i s needed. Once reached, downcomer air i nj ecti on

rat e may be st art ed and than i ncreased, whi l e the r i ser ai r i s reduced.

Operat i ng l i m ts are reached which depend on conf i gurat i on ( i n ject i on hei ght ,

col umn hei ght ) . Ou ts i d e th es e l i m t s t h e dr i v i n g f o r c es a r e i n s uf f i c i ent t o

overcome the res is t i ng for ces leadi ng to f l ow reversa l . S tab i l i ty boundar i es

have been det erm ned both by experi ment and cal cul ati on.

51 Experimental and results

Steady state measurements are very si mpl e. Cir cul ati on vel oci t y

( v n ) and mean r i ser voi d f ract i on (ct. ) were recorded and t i me-averaged for

d i f f erent combinat i ons of r i ser and downcomer a i r mass f l ow i n ject i on rat es .

Resu l t s are shown i n f i gure ( 5 1. 1 ) . Ci r c ul a t i o n v el o ci t y ( v ) i s

p lot t ed agai nst t ota l r i ser a i r mass f l ow rate to show the in f l uence of

106 5 107

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( WGR1>"

downcomer ai r i nject i on on the perf ormance. I n f i gure ( 51. 2) the i nf l uence of

r i s er a i r i nj ect i on r a te (W ) i s s h own. The f i gur es al s o show th e

f o l l ow ng:

- There are st abi l i ty boundari es i n the downcomer ai r onl y mode (no r i ser ai r

i nj e ct i on) ( I ndi c a t ed w t h * ) . The col umn operat es onl y i n a restr i cted gas

f l ow rat e range.

- W th a r e l a t i v el y s ma l l r i s er a i r i nj ect i on r a te t h e l ower s t abi l i t y

boundary di sappears .

- The i nf l uence of downcomer i nject i on hei ght x and t op pressure.

- The data for r i ser ai r onl y may be f i tt ed w t h a power cur ve, ( exponents0.378 l ow top pr essure; 0.461 normal top pressure).

The spread i n the data, especi al l y near t he l ower st abi l i ty boundary

resu l t s f rom the d i f f i cul ty of mainta i n ing a constant t op pressure i n the

l oop , and the sens i t i v i t y o f t he c i rcu l at i on to th is parameter .

The resul ts may be int erpol ated to for m a perf ormance chart , (f i gure

51. 3 and 4) whi ch shows more cl earl y t he operati ng area of the experi mental

co lumn f or a g i ven conf i gurat i on ( i n ject i on hei ght ) and t op pressure. These

f i gures al so i l l ustr ate how the col umn can be start ed up. Note that t he

For a clear underst anding t he stabi l i ty boundari es must be def i ned.

The c i rcu l at i on i s unstab le when i t does not recover f r om a smal l d is t urbance.

Genera l l y t he di s turbance ar i ses f rom a change i n ai r i n ject i on rate. S i nce

the column i s al ways metast abl e w th downcomer ai r i nj ecti on onl y, each

d is t urbance that i s l arge enough w l l d isor i ent c i rcu l at i on vel oc i t y . Near t he

stabi l i ty boundar i es the smal l es t poss i b le pract i ca l change i n in j ect i on rate

(by hand and needle val ve) i s of t he same order as s pont aneous di st urbances.

One is never sure t hat an operati on mode is real l y st able, because i t may take

a l ong t i me, someti mes hours, bef ore a l arge spont aneous dis tur bance occurs .

Duri ng that t i me other vari abl es may have changed.

I n the present work t he operati on is considered t o be stable i f t he

c i r c ul a t i on v el oci t y r ecove r s f r om th e s ma l l est p r ac t i c ab l e a i r i n j ec t i on

change. This def i ni t i on is l ess arbi tr ary than the demand that t he col umn

should be stabl e f or some ti me aft er t he change. The smal l est practi cabl e air

i nject i on change i n our case was 0.01 g/ s.

The s tab i l i ty boundar i es found are di f ferent f r om those determ ned by

S' óderberg. He def i nes t wo types of st abi l i ty boundari es. One, cal l ed

downcomer sl ug breakt hrough w t h conti nued cir cul ati on", i s characteri zed by

108 51

TABLE 52

Geomet r i es of c ol u mns f r om l i t e r a tur e

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F i g u r e 5 1. 5 : S t a bi l i t y b o un da r i e s f r o m S ö d er b e r g (1980) . (See tabl e 52 for geometr y col umn) . T he f o r m i s

comparabl e w th the boundari es found in th is uork but def i ned d i f fer ent l y ( see t e x t ) .

the escape of s l ugs upwards t hrough the downcomer at l i qui d vel oci t i es smal l er

th an about 1. 1 m s . I t happ ens onl y w th s u f f i c i ent r i s er a i r i nj ect i on t o

k eep the s ys tem c i r c u l a t i ng . I n our c as e th i s t y pe of i ns tabi l i t y i s

eff ecti vely prevented by the downcomer venturi ai r sparger (A vel oci ty of

1.1 m s i n the throat cor responding to a c i rcu l at i on vel oc i t y of 0. 70 m s) .

However the l ocat i on of the corr espondi ng unstabl e area i n the operat i on chart

i s r oughl y t he same ( f i gure 51.5, broken l i ne s) . Probabl y t h is area i s

ext ended by t he phenomenon.

Söderberg descri bes t he other st abi l i ty boundary as a cessati on of

the ci rcul ati on or f l ow reversal aft er downcomer sl ug breakthrough. I n our

case sl ug breakt hrough only occurr ed aft er t he ci rcul ati on veloci ty was

decreased bel ow a c r i t i cal val ue and retr i eval was no l onger poss i b le . S t i l l

thi s part of t he boundary r esembl es our upper st abi l i ty boundary (f i gure 51. 5,

dr awn l i n es ) . I t shoul d be not ed t hat the boundary f ound by Söderberg i s

cont i nuous and does not i ntersect the absci ssa (Compare wi t h t he operati on

c ha r t gi v en i n s ec t i on 93 f o r ope r at i on w th d i s t i l l ed wate r) .

52 Ex i s t i ng models for air lift towers

Mode l s t o pr edi c t c i r c ul a t i on v el oci t y i n an ( ai r l i f t ) l oop r eac to r

vary f r oms i mple cor re l at i ons to models based on equat i on (7- 6) w th

appropri ate expressi ons f or t he voi d fr acti on and the fr i cti on number. Merchuk

and Stei n ( 1981a) ( see tabl e 52 for the di mensions of the col umns of ci ted

authors ) g ive

Ref erence

Mer chuk

et a l . (1981a)

Hi l l s ( 19 76)

Bl enke (1979)

Wei l and

( 1978)

Söderberg

Thi s work

Hei ght

( m

4.05

12.

2.

3. 2

3. 2

9.

9.85

10. 3

Di amet er

Ri s er ( m

0.14

0.149

0.17

0.17

0.194

0.1

0.241

0.225

Downco mer

0.14

0.149

0.29

0.29

0.29

0.05

0.241

0.10

Downcomer

External

External

Annul ar

Annul ar

Annul ar

External

External

External

Sparger(s)

A: 25 mm hol es

B: 90 mm pi pe

51 mm pi pe

r i ng n oz z l e ( j e t )

2 mm hol es

2 mm hol es

porous pl ate ( 0.2 mm

per f orated pl ate

( 0. 5 mm

spec ia l (chapter 2)

1 mm pi pes/speci al

(chapter 2)

Gsc ( 1)

as a gener al f o r m f o r a c o r r e l a t i on, and f i nd f or un r es t r i c t ed c i r c ul a t i on:

a = 34; b = 0.41. I t i s i nteresti ng to see that Wei l and ( 1978) f ound the same

exponent:

0.414VGsb ( 2)

The proport i onal i t y f actor a depends on the geometr y of t he col umns

and the def i n i t i ons of t he vel oc i t i es . As s tated ear l i er we wi l l use the

downcomer l i qu id veloc i ty as the character i s t i c ve loc i ty for a deep shaf t - type

system For the super f i c i a l gas veloc i ty the l ogar i thm c mean veloc i ty i s

representat i ve for operat i on w thout downcomer a i r I n ject i on ( r i ser a i r onl y

mode) ( appendi x 77. 2 ) . However when downcomer ai r i s i nj ected thi s mean i s not

charact eri st i c and a hei ght i ndependent val ue l i ke the gas mass f l ow rat e or

the barometr i c superf i ci al vel ocit y ( based on 1 bar and exist i ng temperatur e)

must be used.

] 10 J Z

Hei j nen et a l . (1982) compare the c i r cu l a t i on ve loc i ty w th t hat in a

bubbl e col umn w thout i nternal s . This wel l d i scussed phenomenon (see al so

chapter 7) may be descr i bed by ( Hei j nen et al . )

52 111

dependency on col umn di ameter rat her than col umn hei ght. Not e t hat the

exponent s f ound i n the previ ous paragraph ( 0. 378 and 0.461) di f f er f r o m t he

o ve r al l v al ue i n ( 7 ) .

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v c ■ 0.9 ( gd t v Gs ) 1/ 3 ( 3)

Here , o f course , the super f i c i a l gas vel oci ty i s based on the ( to ta l )

cross- sect i onal area of the col umn. The rel at i on w th the col umn di ameter d

f o l l o ws f r o m t he c ha r ac t e r i s t i c c i r c ul a t i o n l o op l e ng t h. H ei j n en et a l .

suggest that equati on (3) i s appl i cabl e for bubbl e col umns w t h draught t ube

and negl i g ib l e f r i c t i on ( l a rge co lumn d iameters , 0 .25 mm nimum i f d i s

repl aced by the col umn hei ght . As such, equati on ( 3) i s expected t o predi ct

the maximum vel ocit y att ai nable i n l oop react ors .

Thi s sugges t i on i s c lear l y i nva l i d s i nce equat i on (3 ) i n that f o rm

woul d predi c t an inc reas ing ci r cu l a t i on vel oci ty w th i ncreasi ng hei ght ,

despi t e the fact that the mean superf i c i a l gas vel oc i ty woul d be reduced as a

resul t of hydros ta t i c p ressure e f f ec ts .

The ci ted equat i ons may be compared w t h equat i on ( 7 - 6 ) , rearr anging

( r i s er a i r o nl y ) :

2gL e t r k

With equation ( 38-1) follows

H -w «n>(5)

where f (p* ) i s a f unc t i on of the pressure r a t i o between t op and bot t omand

thus dependent on (dec reas i ng wi th ) co lumn hei ght . W th (4 )

2gL f (p* ) t 0 A2C-v = ( S L . ) 5 v 7 " ( 6 )

Ls D v K J Gsb K J

The exponent agrees w t h those of Wei l and and Merchuk et al . The

proportionality factor in our case and p » p i s 5. 9. It may be not ed here

that equation (6) resembl es (3) :

, „ , , , , * , v0. 5 0.425 , . ,VL s D " 10 <8 d tf ( p * » v ^ ( 7)

(with the simple approximation Kp = 0.02 L ^ / d ^ ) . This shows agai n the

Merchuk and Stei n ( 1981a) noted t hat the exponent 0-41 decreased if

there i s s i gni f i cant res i s tance i n the downcomer to t he f l ow. I n f ac t the

agreement of the exponent between t he c i ted authors i s coi nc i dental . Thi s can

be seen f rom the least square f i ts of the data of Hi l l s ( 1976) ( exponent

0.265) , and Bl enke (1979) ( t hree val ues 0. 225, 0. 320, 0.395 f o r d i f f e r e nt

geometr i es). Cl ear l y t he genera l equat i on (1 ) i s usef u l as a descr i p t i on of

t h e c i r c ul a t i o n v el o ci t y but of l i m t e d v al u e f or p r ed i c t i o n.

Wei l and ( 1978) also uses equat i on (4) to model c i rcul at i on vel oc i t y.

However the void f ract i on i s assumed t o be known and was not modell ed.

Furt hermore t he compl i cated express i on f or K cont ai ns an empir i cal f actor .

Merchuk and Stei n ( 1981a) use an expressi on for K that i s weakl y dependent

o n c i r c ul a t i o n v el o ci t y w t h t h e e qui v al e nt l e ngt h t e r m f i t t e d t o the

measurements. For t he voi d f ract i on the Zuber and Fi ndlay model i s

recommended, which descr i bes the data w thi n 10% (sparger 25 mm holes, C ■

1 .03, v Gd l m 0. 3 3 m s ) .

53 Existing models for deep shaft type columns

The deep sha f t system a l oop reac to r w th a i r i n jec t i on in the

downcomer, has not received much att enti on. S i nce the i ntr oducti on by Hi nes et

al . ( 1975) , Kubota et al . ( 1978) and Soderber g ( 1980) have di scussed the

sys tem The i n fo rmat i on gi ven by Hi nes e t a l . i s i n te res t i ng s ince th i s f o rms

the sta r t i ng poi n t f o r our r esearch f o r wh ich t he f o l l ow ng featu res a re

e ss ent i a l :

- Voi d f ract i on should be bel ow 0.2 to prevent i mport ant bubbl e coal escence.

As a resul t voi d f rac t i on i s genera l l y l ow and the vo lumetr i c f rac t i on o f

l i qui d c l ose to one. L i qui d vel oc it y may be consi dered t o be constant.

- A s l i p ve loc i ty o f 0.3 m s , i n r i ser and downcomer a l i ke i s chosen as a

conservat i ve des i gn ru le .

~ Li qui d veloc i t y i s between 1 and 2 m s.

- The same cross- sect i on for r i ser and downcomer i s " c l ose to t he opt i mum".

- Ma ss t r a ns f e r i s s uf f i c i e nt l y l i m t ed as not to affect estimates of the

c i r c ul a t i o n.

- I nf l uence of voi dage on l ocal pressure I s negl ected.

- The mechani ca l l y opt i ma l super f i c i a l a i r vel oc i t y l i es in the range f rom 0.1

t o 0.2 m s ( 1 ba r , t o t a l c r o s s - s ec t i o n) .

1 12

A more s ophi st i cated model was presented by Kubota et al . ( 1978) who

al so i ncorporated mass t ransfer . The model was not ver i f i ed w t h measurements.

For t he f ri ct i on number basi c wel l known for mul ae were used (see als o

2gc P (0) p (0)1 s

— — i n 7—s ^ e s

( x )

( 5)

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chapter 1) and void f ract i on was cal culat ed w t h a s l i p vel oc it y based model,

equati on (33 -8 ) . The sl i p vel oci t y used was t he same f or r i ser and downcomer.

The det ai l ed r esearch on the perf ormance of a deep shaf t model

perf ormed by Soderberg w l l be tr eated here i n more detai l . Hi s approach is

somewhat d i f f erent f rom the above model s because he i ncorporates the f r i ct i on

number K as a f ree parameter i n the s l i p vel oc i ty based model . The s l i p

veloc i ty i s a constant and equal i n r i ser and downcomer. Equat i on ( 7-6)

w thout i n jec t i on head l oss

( o

i p

S "

hen be comes

h "

VL s D

2SLe 1 W »2 lvT „ + v

VGS D( X)

VLsD - Vs( 2)

( V, „ T V V T T , — V /

Ls D

where al so expansi on i s negl ected. The model equati ons gi ven by Soderberg

have been modi f i ed f or our exper i ment al col umn, x i s t he i n ject i on hei ght and

R and D i ndi cate t he r i ser and downcomer sect i ons. W t h correct values f or K^F

and v equati on (2) shoul d predi ct t he c i rcul at i on veloc i ty. K and v were

deter m ned f rom measurements (superf i c i a l vel oc i t i es) by varyi ng v to get a

constant K - val ue. Soderberg showed t hat i t was not necessary to i ncorporate

hydrostat i c i nf l uences for h i s col umn of 10 metr e h i gh and under barometr i c

pressure.

The extensi on for gas expansi on fol l owed f r om the i nc lus i on of t he

pressure dependence of the superf i c i a l gas veloc i t y w t h the assumpti on of a

l i near p ressure -he ight re l a t i on (neg lec t i ng the in f l uence of voi dage) .

Equati on ( 2) becomes

^ " / i V ^ - V W vLsD-Vs pLglnU sWj (

(3)

Ls D

Here p denotes t he hydrostat i c pressure

P0( z ) = pc( 0 ) - PTgz ( 4)

equati on ( 3) may be rewri t ten i n terms of gas mass f l ow rate

( m / k g) ( 6)

Mass t ransfer i s negl i g i b l e i n a 10 meter h igh column ( see al so chapter 8) and

was not i ncorporat ed.

54 Compari son w t h t he Soderber g model

Measured c i r cu la t i on vel oci t i es were used to cal cul a te the f r i c t i on

number w t h equati on (53 -3 ) . Fol l ow ng Soderberg v was chosen to g i ve the

most constant K_ ( m ni mum st andard d ev i a t i o n) . The bott om pressure p (0) was

cor rec ted w th equat i on (12 -1 ) . For a d i mensi onl ess i n ject i on hei ght x* = 0. 28

the resul t s a re :

P s ( 0)— — ( wr - o)

11. 7

20. 8

m

V

s0.47

0.27

m s

KF

3.76

4.99

-

£

0.6

0. 4

-

I n sect i on 17 the f ri ct i on number was deter m ned by measurement of

the re l a t i on between c i r cul a t i on ve loc i ty and r i ser voi d f rac t i on. The val ue

was 4.5 i n agreement w th cal cul at i ons, for both pressure modes. Thi s

i l l u st r a tes the l i m tat i on o f t he Soderberg mode l i n i t s p resented fo rm

Soderberg a l so f ound d i f f e r i ng resul t s f o r var i ous i n jec t i on he i ghts , and

thought the inef f i c i ent gas i n ject i on in both r i ser and downcomer r esponsi b l e.

Lower K val ues were found i f the in j ector was l ocated l ower i n the

r i s er as a r e sul t o f ai r j e t t i ng a t t he r i s er i nj e ct o r . I ne f f i c i e nt i nj e ct i o n

producing c l i ngi ng ai r pocket s i s counteracted i n the downcomer by the

probabl e i ncrease in f r i c t i on w th t wo phase f l ow. The resul t i ng ef f ec t was

not sys temat i c . I n al l cases Soderberg f ound that a s l i p vel oci ty o f 0.5 m s

ga ve t h e be st f i t . T hi s a ga i n r e f l e ct s t h e e f f e c t o f i ne f f i c i e n t a i r

i n j e c t i o n.

The resul t s of our own measurements ar e reasonabl y i n agreement w t h

the val ues f ound f or K_ and v ( secti on 17 and 38) f or t he barometr i c topF s v L

pressure mode. I n t h is case the i nf l uence of the downcomer i n jector (chapter

i n compar i son w th the con t r i but i on of the vo id f r ac t i on, p ressu re d rop as a

r e s ul t of f r i c t i o n and ac c el e r at i o n i s n eg l i gi b l e i n t he r i s e r ;

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2) i s s t i l l sma l l and the r i ser s l i p vel oci ty f ound to be cons tan t . W th a

reduced top pressure t he s l i p veloc i ty i s constant over t he hei ght and i s al so

genera l l y h igher wh i l e the downcomer i n jec t i on i s f a r l ess ef f i c i ent . Thi s i s

re f l ected i n the h igher es t i mate f o r t he s l i p vel oc i t y and t he l ower

55 The pr oposed model

Based on t hese resul t s a steady st ate model i s proposed for the

predi c t i on of the t i me average c i r cu la t i on ve loc i ty . The bas i c equat i on o f

mot i on (7 -5 ) i s

2

The mean voi d f r act i ons f ol l ow f rom i ntegrat i on of t he l ocal one-di mensi onal

va lues , equat i on (30 -8 )

( 2)

( where the upper + or - i s f or upfl ow and t he l ower f or downfl ow).

The s l i p veloc i ty i s based on equati ons ( 38-6 to 8)

v n = v, + 0. 04 + 0. 02( P( V° \ - 1) ( 3 )sR b*> PR^Z'

whi l e i n the downcomer the s l i p vel oc i ty i s assumed to be constant ( l ower part

of the downcomer, short gas resi dence t i me) and expect ed t o be about equal to

v. (chapter 4 ) . The i nj ecti on head l oss i n the downcomer was di scussed i n

chapter 2. Equati ons ( 26- 1) are taken together:

^ - ( (0. 7 v Gs D ( x ) ) 1 0+ (29. 3 v ^ W2 - 5 ) 1 0 ) 0 - 1 ( 4)

Expansion of the gas is incorporated wi th

V Gs ( z ) p(0)( 5)

dp - ~P M g dz ( 6)

PM= üP G + (1 " a )p L ( 7)

Negl ec t i ng gas dens i ty p re l a t i ve to p

dp = - p L g dz + ap L g dz ( 8)

= dp + dp

s *a

Here p denotes the hydrostat i c pressure ( i . e. w t hout gas) and p the

dev ia t i on f rom th i s p ressure as a resul t of the p resence of gas . Fo l l ow ng

Hi ne s et a l . ( L 9 75) ( c ons t a nt s l i p ve l o c i t y a nd l i qui d v el o ci t y ) i t f o l l o ws

that the gas vel oci ty i s cons tan t and t hus w th ( 5 )

a ( z ) p( 0 ) , 0.

Thi s approxi mati on i s suf f i c i ent l y accurat e as a voi dage dependent express i on

fo r the pressu re . An anal y t i cal so lu t i on of (8 ) and (9 ) however l eads to an

i mp l i c i t f unct i on f o r t he p ressure . I t i s pre f erabl e to use the express i on

a ( z ) P„(0 )(10)a ( 0 ) ps( z )

wh ich i s o f s i m l a r accu racy . I n tegra t i on o f ( 8 ) then y i e lds

p(z) = p GO + a( 0) p( 0) In -E i °) ( 1 1 )

s p s u ;

or , di v i d i n g b y p ( 0 ) ,

p* = p* - a (0 ) I n (p* ) (12 )

Since p i s exactl y the pressure determ ned w t h the ai r - water - manometer , i t

i s poss ib l e to compare t h i s ca lcu la ted dev i a t i on f rom hydros ta t i c p ressure

atolo 0.036. 0029A 0.020

» M «.2, 5 "06

55 ] 17

c o r r e c t v a l u e . W t h s ma l l c o l u mn s ( 10 m h i g h , b a r o me t r i c t op p r e s s ur e ) t h i s

p r ed i c t i o n may s t i l l be 20%too h i g h.

A s b ou n da r y c o n d i t i o n for the i n t eg r at i o n of e qu a t i o n ( 8) t he

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V 0.012"norm,

0.3

PP I

F i £ u ; 5 5 , ? : C omp a r i so n of i

w th measurements. F i gure (55. 1) shows p rel ati ve to the constant bottom

apressure ( p* = p / p(0)), where using sem- l og coordi nates the theoreti cal l i ne

a ai s s t r a i g ht and of s l o pe a ( 0 ) . Fr o m t he f i g ur e i t f o l l o ws t h at wi t h a r e d u c e d

t o p p r e s s u r e p may be as h i g h as 10%of t he h i g he s t p r e s s u r e ( p ( 0 ) ) , wh i c h

i mp l i e s t h a t i t i s a l mo s t eq u a l to the l o c al h y dr o s t a t i c p r e s s ur e p at t he

t op of the co l u mn . The d r i v i n g f o r c e f or c i r c ul a t i o n ( r i s er v oi d f r a c t i o n ) i s

l a r g e l y g e ne r a t e d i n t h i s t op p a r t of the co l u mn and t he d ev i a t i o n f r o m

h yd r o s t a t i c p r e s s u r e may c l e ar l y not be n e gl e c t e d . F i g u r e ( 5 5. 1 ) s ho ws t h at

e q ua t i o n ( 1 2 ) i s s u f f i c i e nt l y a c cu r a t e t o c o r r e c t f or t h i s ef f e c t .

To compar e t he i n f l u e nc e of t he v a r i o u s a s s u mp t i o n s on the p r e di c t e d

e x pa n s i o n of the v oi d f r a c t i o n, ( a / a ( 0 ) ) i s shown i n f i g ur e ( 5 5 . 2 ) . E qu at i o n

( 1 0) ( l i n e a) i s s h o wn t o g e t h e r w i t h t he e x pa n s i o n p r e d i c t e d by the p r o p o s e d

mo d e l i n c l u d i n g t he e f f e c t of t he c h an gi n g s l i p v el o c i t y ( l i n e b ) . As an

i l l u s t r a t i o n a l s o s o me me a s u r e me n t s ar e a d d ed , but i t mus t be e mp h a s i z e d t h a t ,

du e to the i n ac c u r a c y of the e s t i ma t e d v a l u e f or a ( U ) , t he p os s i b l e e r r o r s i n

t h e pl o t t e d d a t a may be l a r g e. The f i g u r e s h o ws t h a t i n the mo s t ex t r eme ca s e

t h e l i n e ar mo d e l ( e q ua t i o n 10) p r e d i c t s l o c a l v o i d v a l u e s a b ou t t w i c e t he

c o n s t a n t p r e s s u r e at the b ot t o m ( z =0 ) was u s e d. T h i s i s i n a g r e e me n t w i t h t he

us e of the e x pe r i me n t a l c o l u mn , wh i c h o pe r a t e d wi t h o u t o v e r f l o w ( c o n s t a n t

l i q u i d v o l u me ) . I n p r a c t i c e de e p s h af t c o l u mn s wi l l o p e r a t e wi t h o v er f l o w or

wi t h a l a r g e gas d i s en g a g emen t t a n k at t he top. In t h e s e c a s e s t he top

p r e s s u r e ( at the o v er f l o w l e v el ) i s c o n s t a n t and t he mo d e l s o me wh a t d i f f e r e n t

( s e e a p p en d i x 55. 1 f or c o mp a r i s o n ) .

I n t he d o wn c o me r f r i c t i o n i s not n eg l i g i b l e and a c c o u n t e d f or wi t h

Ap ,f 1 7 5- = O 00907z v

D

PTg L sD( 13)

which fol l ows f rom the Fanning and B asius equat i ons.

Computer progr am

The proposed model i s not anal yti cal l y sol ubl e and a FORTRAN program

was wr i t ten (appendix 55. 2). Start i ng the cal cul ati on w th an arbi tr ary val ue

00 2

o n

a R - a D '

-f ƒ

/// /stable

/m\f(

W G D lg ^

stable

able

/

' 2 / .0 9 /

/ 06

-

KFFr/2

ng void (RHS of eq. (14)) ca lcu l i

t the computer program STS. (No i

i of velocity (LHS of eq. (14))

Ion, x* ■ 0.28, reduced top

p r es s u r e : p° - 0 . 1 1 6 ) .

118 55

fo r the c i r cu la t i on vel oci ty o f 1 m s , the program i tera tes and converges

towards t he sol u t i on. I f the i te ra t i on d iverges t here i s no ( s tab l e ) sol u t i on

(F i gure 55. 3 g ives a f l ow sheet) .

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One may rewri te equat i on ( 1) i n di mensi onl ess f orm

Ls DAP„

h^-^-^-^ir

iKFFr " °R " °D - K ' \ " "I, < 1 5 )

to ta l mean re la t i ve ai r con tents i n the downcomer ) . The sol u t i on o f t hi s

equat i on i s s t abl e i f an i ncrease (o r dec rease) o f t he vel oc i t y ( l e f t hand

ter m cor r esponds w th an inc rease ( dec rease) i n dr i v ing f o rce ( r i ght hand

t e r m . Fi gure ( 55. 4) shows t hat the upper i ntersect i ons of these terms are

st abl e, and i l l ustrat es the solvi ng procedure of t he program

56 Val i dati on of the proposed model

Ai r l i f t ope r at i on

The proposed model i s expected to be val i d f or t he r i ser ai r only

mode, si nce the parameter s ( not abl y t he f ri ct i on number K = 4. 5) are based

on measurements i n that mode. However several assumpti ons and general i sat i ons

are made. The hydrost ati c ef f ects may be compared w t h measurements w t hout

t he added uncert ai nti es of t he downcomer parameter s ( sl i p vel oci t y and

i n jec t i on head l o ss ) .

I n f i gure ( 56. 1) the measurement s w t h r i ser ai r onl y are compared

w t h the calcul ated vel oc i t i es . The agreement i s very good. I t must be noted

that t he calcul at i ons are based on a s i ngl e selected t op pressure r el at i ve to

the col umn hei ght :

The spread i n pressure between t he measurements i s not l arge however.

A more sensit i ve ver i f i cat i on of the model i s a compari son of the

calcul ated and measured mean r i ser void f ract i ons (f i gure 56. 2) ( the program

cal cul ates t he r i ser voi d f ract i on between the hi ghest and lowest pressure

po in t and th i s val ue i s used) . Pl ott ed t h is way the actual top pressure can be

used f or t he cal cul at i ons, whi ch i mproves t he compari son.

i 5 6 . 1 ; Computed and measured ci rc ul ati on vel oi

. i o n ) . Compare f i gure 51.1 fo r s ymbo ls .i n j e ct i o n ( a i r l i f t

fr act i on compared, t l o i • ai r i n j e c t i o n .

1

- 1

^

.,;■[ r . , -^ , , . , - ,

**

/

a t /

' , - - - ' ■ »

■d dis tr i buti on compared. Ho downcomer a i r i n ject i on.

120 56

F i nal l y l oca l vo id f rac t i ons a re compared in f i gure ( 5 6. 3) . Thi s

shows cl earl y t hat the accuracy of t he model i s w t hi n the measurement err or

range. Thi s accuracy of th i s par t o f the mode l i s su f f i c i ent to use i t as a

basi s for model l i ng the operat i on w th downcomer ai r i n j ect i on.

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Operati on w th downcomer ai r i n j ect i on

To veri f y the model we have concentr ated on t he measurements i n the

downcomer ai r onl y mode (no ri ser ai r i nj ec t i o n ) . Because thi s mode devi ates

mos t f rom that di scussed i n the prev ious sect i on, i t i s t he mos t sens i t i ve f or

compari son.

I n f i gure ( 56. 4 ) cal cul a ted and measured c i r cu l a t i on ve loc i t i es are

compared on t he basi s of a s l i p vel oc it y i n the downcomer of 0.2 m s. These

show to di f f e r i n var i ous ways:

- The predict ed operat i on range is l arger. Thi s i s not surpr i s i ng because the

steady state model d is regards vel oc it y f l uctuati ons. These were measured to

be pl us or m nus 5% or more and w l l be d i scussed l ater . Moreover no other

dynamcal behavi our i s i ncorporated.

- The cal culat ed curves go through a maxi mum val ue. Operat i on t o the r i ght

hand s ide of th i s maximum i s onl y poss i b le i f the increase i n energy i nput

(gas i nput) i s compensated by a l arger i ncrease of potenti al energy, s i nce

at the same ti me ki neti c energy decreases. Such a case mght happen i n

p ract i ce w th i ncreas i ng car ryover of a i r f r om the a i r d i sengagement sect i on

but such eff ects ar e not i ncorporat ed i n the model . Theref ore t hese model

sol ut i ons i n th i s r ange may be consi dered to be uns tabl e. Thi s woul d i mply

however that t he upper st abi l i ty boundary i s predi cted f or l ower gas i nputs

t han measur ed.

- General l y the cal culated veloc i t i es are less t han those measured.

Obvi ousl y t he parameter s used for the downcomer sect i on, t he sl i p

vel oc i ty and the i n ject i on head l o ss , a re not exac t l y co r r ec t . I n the next

sect i on poss i b le corr ect i ons to t he proposed model are d i scussed.

57 Corrections to the proposed model

S ince opera t i on w th r i ser ai r i n jec t i on was ca lcu l a ted cor rec t l y ,

; he devi at i ons f ound for operat i on w t h downcomer ai r must ar i se f r om that

part of t he model . Basi c el ements here are t he downcomer sl i p vel oci t y and the

i n jec t i on head l oss .

UB2IS " ■ ! : E f f e c t ° E c o r r e c t i o n s t o c . e mo d el . Lo we r Un a s i d e n t i c a l t o th ose I n f i g ur e 5 6 . 4 . Upp e r l i n e s

Downcomer s l i p vel oc it y was discussed i n chapter 4. Val ues f ound in

l i ter ature were as l ow as 0. 15 ms . Substi tut i ng th i s val ue in the model l eads

to an overal l i ncrease i n predict ed vel oc it y. I n chapter 2 the i naccuracy of

the measurements and the corr el ati on for the i nj ecti on head l oss have been

consi dered. The val ues of I HL der i ved in chapter 2 were f ound not to sat i s fy

the requi rements of the model s for the measured c i rcul at i on r ates, and t hese

coul d onl y be predi cted us i ng somewhat l ower i n ject i on head l osses. The

fo l l ow ng emp i r i cal equat i on desc r i bes t he ci r cu la t i on resul t s r easonabl y .

P X " (°-6 ^ n ^ 1 0 + C8 v rx)

2)10)0-1

( 1)

Substi t ut i ng th i s i n the model l eads t o the calcul ated curves shown in f i gure

( 5 7. 1) . On t his bas i s cal culat ed and measured vel oc i t i es agree r easonabl y, so

that t he model can be used as a basi s f or t i me dependent cal cul ati ons (next

chapte r ) . The maxi mum i n the calcul ated cur ves now corr esponds t o great er

downcomer ai r i n j ect i on rates than the measured upper stabi l i ty boundary, as

woul d be expect ed.

Validation of the corrected model

To compare t he whol e range of vel oc i t i es and air i nputs, the

operati on chart s previ ousl y presented i n sect i on 51 are used. Bot h cases are

repeated here and compared w t h t he model i n fi gures (57. 2 and 3) . General l y

57 123

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- 57.2: Computed ( ed. Compare f l gui

Fl Bure 57.3: Computed and measured operat i on chart c ompared. Compare t l gure 31.4,

agreement i s good, but t he predi cted operat i on range i s l arger . The measured

l ower st abi l i ty boundary encl oses an area that i s at l east three t i mes that i n

whi ch no solut i on is f ound.

Ri ser voi d f ract i on may be al ter ed dur i ng downcomer air i n j ect i on

because the d i s t r i but i on o f a i r a t the bot t omof the r i ser i s di f f e r ent . As a

resul t o f coal escence i n the hor i zonta l bot t om sect i on la rge bubb les enter the

r i ser . These break up agai n on their way up. Thi s ef fect was vi sual l y observed

to be s iga i f i can t onl y f o r l ow c i r cu l a t i on ve loc i t i es . The cal cul a ted and

58 Predict i ons w t h the model

I n t h is sect i on some ext rapolat i ons ar e made w th the model , These

w l l show the sensi t i v i t y o f the mode l and thus of the sys tem to cer t a in

parameters and gi ve an i ndi cat i on about scale up ef f ects . Onl y t he most

sensi t i ve case , t ha t o f r educed top pressure w th a re l a t i ve i n jec t i on he ight

o f 0 .28, i s di s cussed.

When operat i ng w th r educed top pressure, the i nf l uence of var i at i ons

i n the barometr i c p ressu re are re l a t i vel y l a rge ( f i gure 5 8 . i ) . I n f a ct t hi s

i nf l uences the pressure r at i o between bott om and t op, whi ch af f ects t he

equi val ent hei ght of the col umn represented by t he model .

The i nf l uence of t he downcomer s l i p vel oc it y i s rel at i vely smal l

( f i gur e 58. 2) , but the rel evant i n jec t i on he ight i s l i m ted . However th i s

parameter i s not very sensit i ve and the chosen opt i mum of 0.15 m s not very

r e l i abl e.

)1 1P</P|_glm)

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/ # ^ ;«nj downcomerlg/i)

58-1: Computed influ i i (us ual ly p t /p , 8 - 1 .2 ■) ;

Flf iu re 56 .2 : Computed Inf luence of douncomer sl ip velocity (n

i n j e c t i o n ) .

n 1 J _ . 1 1 1 1

: 5 8 . 3 : C o mp u te d I n f l u e nc e o f i n j e i n he i g h t ( r

Fi gure ( 58. 3) shows the inf l uence of t he i n ject i on hei ght . Of course

th i s i s the mos t i n teres t i ng var i abl e f o r t he sys tem studi ed. An i n jec t i on

poi nt onl y 0.6 m higher than the used height of 2.9 m i s, a l though

theoret i cal l y poss ib l e , c l ear l y not p ract i cabl e . The measured m n imum vel oci ty

o f 0.9 m s w th the in j ect i on hei ght 2 .9 m i s about equal to the ca lcu la ted

maxi mumvel oci ty w th t he hi gher i n jec t i on l eve l .

F i n al l y s t a bi l i ty i s consi dered. Alt hough nothi ng conclus i ve can be

said on basi s of a steady-st ate model , exam nati on of the equati on of moti on

reveal s the group

5 ; ■ _ C'R e n

t o be i n t e r e s t i ng . I t s v al ue i s i nf i ni t e f or t h e a i r l i f t mo de , whe n t he

col umn i s def i n i t e ly s tab le . I f d r i v ing voi d QL and res i s t i ng vo id cc

( i ncl usi ve o f I n jec t i on head l oss ) are equal ( r a t i o ( I ) = 1 ) t here i s no net t

dr i vi ng f orce to overcome fr i ct i on and operat i on is not poss i b l e. Operat i on

may be considered to be most st abl e i f , for a g i ven set of ai r i nputs, the

rat i o ( 1) is maxi mal . The rat i o i s cal culat ed and shown in f i gure (58 .A) . Thi s

re f l ects the st r ong s tabi l i z i ng ef f ects o f a sma l l amount o f r i ser a i r and

also shows a maximum i n th is r at i o when very l ow r i ser ai r i n ject i on rates are

used.

59 Concl us i on

I n t h is chapter a steady st ate c i rcul at i on model has been presented,

wh ich , i n compar i son w th o ther mode ls f r oml i te ra t u re ,

- accounts for hydrost at i c ef f ects i n gas rat e, voi dage and pressure,

" a l l ows f or d i f f e rences i n the s l i p ve loc i ty i n r i se r and dow i comer f l ows ,

" accounts for l osses i n dr i vi ng force at t he downcomer air sparger.

126 59

Al t hough empi r i ca l r e la t i ons are i ncorporated , i t i s be l i eved that

these do not i nhi bi t the use of the model f or s cal e-up and other

conf i gurat i ons. I nstabi l i ty cannot be pred ict ed w th a steady stat e model , but

th e l i m ts of po ss i bl e s ta bl e o pe ra t i o n ca n be cal cul a te d. As we l l a s

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predi ct i ng these l i m ts the model p redi cts the genera l a rea of operat i ng

cond i t i ons wh ich woul d be the most st abl e . The model i s su f f i c i ent l y accurate

t o be extended to i nclude ti me dependent ci r cul ati on, and make rel i abl e

compari sons w t h measurements .

Time dependent circulation

60 I n tr oduct i on

The steady state model i n chapter 5 pred ict s t he l i m t s o f st abl e

ci rc ul ati on i n the absence of any i mposed di st urbance. The measured operat i on

area however i s f ar smal l er t han t h i s .

I n thi s chapter the steady st ate model i s ext ended t o descr i be ti me

dependent behavi our of the col umn. Thi s s i mul ati on model i s compared w t h t he

response of the c i r cu la t i on ve loc i ty to a step change i n the gas i nput .

Concl usi ons ar e dr awn about mechani sms that can lead to i nstabi l i t y, one of

whi ch i s an undamped osci l l ati on.

61 St a bi l i t y

I n t hi s sect i on the poss i b le mechani sms l ead ing to f l ow reversa l i n a

bubbl e co l umn l oop w t h downcomer gas i n ject i on are d i scussed. S i nce, as wi l l

be see n, i n er t i a l e f f e ct s a re of l i t t l e s i gni f i cance f l o w re ver sa l ca n be

consi dered to occur at the moment s that Che posi t i ve dri vi ng for ces cease to

be l arger t han the opposi ng for ces. Thi s may be reached i n di f f erent ways.

I nstabi l i ty usua l l y resu l t s f rom a change in the gas i nput ra te made

t o get to another operat i on poi n t ( e .g . a t st ar t u p ) . I t may be t hat such a

change i nvokes an i nstabi l i ty or that i t generates a l a rge secundary

di st urbance. I n the experi ment al col umn a f l ow reversal someti mes occurr edspont aneousl y some ti me af t er an al t erat i on In gas i nput r ate.

Based on experi ment al observat i ons t he fol l ow ng mechani sms ar e

t hought to be r el evant.

a ) I noperab l e i n ject i on ra tes. Thi s case i s s i mp le and ment i oned f or the sake

of compl eteness. I t happens when one att empts to i mpose condi t i ons outsi de

th e l i m ts o f s ta bl e op era t i o n.

b ) B locki ng of the downcomer by a l a rge ai r f i l l ed cav i t y . Thi s was

exper i enced i n one o f t he ear l i e r con f i gurat i ons o f t he co l umn

128 61

( s ec t i on 19) . Air col l ected on the i nner s i de of the smooth 90°-bend at the

bott om of the downcomer. Thi s ai r bubbl e can grow to such an extent that

any moment ary r educti on i n the l i qui d f l ow rat e al l ows i t to move t owards

the downcomer. Thi s then causes a f urt her reducti on in the l i qui d vel oci t y

whi ch exacerbates the si t uati on and al l ows the bubbl e to move more qui ckl y

61 129

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agai nst t he l i qui d , l ead ing al most i mmedi atel y t o f l ow reversal . A i r

pockets may f orm anywhere i n a downf l ow ng t wo phase f l ow behi nd an

obstr ucti on. SÖderberg ' s (1980) co lumn al so operated w th a quas i - s t abl e

ai r pocket i n the downcomer bott om bend. Stabl e operat i on w t h such a l arge

c av i t y i s onl y pos s i b l e i f gas l o s s f r om t he r ear of t he c av i t y bal anc es

t he steady accumul ati on fr om the bubbl y downf l ow.

c) Sl ug breakt hrough. Thi s phenomenon was not observed i n our col umn but was

report ed by Söderberg. I t i s t he escape of s l ugs i n the downcomer f rom

obs t r uct i ons , s uc h as i n t he a i r i n j ec t o r , t o a hi gher l eve l . The

convergi ng sect i on of our ai r sparger prevent ed t hi s. The escaped sl ugs are

gener al l y soon broken up and washed down agai n, but the i ncrease i n the

downcomer gas content w l l general l y but not necessar i l y l ead to f l ow

reversal . A l arge and thus unat t ract i ve r i ser a i r rate may prevent t h i s .

d) A very l arge change i n the a i r i n ject i on rate. St abl e operat i on i s based on

an equi l i br i um between r i ser dr i v i ng f orce and res i s t i ng f orces togetherw t h a sel f - cont ro l l i ng mechani sm by whi ch a smal l (pos i t i ve) change i n

dr i v i ng force i s compensated by an increase i n res i s t i ng force due t o the

hi gher vel oc i t y. However there i s an essent i a l t i me l ag between t he change

i n downcomer ai r i nput rat e and t he moment that the correspondi ng var i at i on

i n mass dens i t y of the a i r - water m xture reaches the r i ser . Gi ven t i me,

both r i ser and downcomer gas content w l l i ncrease. I f th i s does not happen

qui ckl y enough the system may have al ready reached an i rr ecoverabl e st ate.

I n f i gur e ( 6 1) t h i s I ns t a bi l i t y i s i l l us t r a t e d. F or a gi v en s t abl e

mean ci rcu l at i on veloc i ty a s i m l ar pos i t i ve s tep change i n the downcomer

i nj ecti on gas rat e has been appl i ed several t i mes. The syst em someti mes

revers es and someti mes not. The ci rcul at i on dynam cs depend on the exact

value of t he l i qui d veloc i ty and the a i r d i s t r i but i on around the loop at

t he moment of i mposit i on of the st ep change i n gas rat e.

e) A lesser change i n the a i r i n ject i on rat e can al so l ead to f l ow reversal

though th i s se l dom happened I n pract i ce. The voi dage pl ug resul t i ng f r om

the ext r a gas reaches the r i ser i n a shor t enough t i me for c i r cu l at i on to

c ont i nue , but an os c i l l a t i on w l l devel op. I f t he ampl i t ude of t he

os ci l l a t i on i ncr eas es t he s yst em w l l be uns t ab l e .

06-

04 -

k. 3 0 S B 18 24 3 0 3 6 * 2 4 8 5 4 6 0

F J - B" " 61; p l ow re v e rsa l due W a l a rge cha nge In downc ome r a i r i n j e c t i on ra c e . De pend ing on t he ac t ua l s t a r t

c ondi t i on ( but u i t h e qua l a i r i n j e c t i on ra t e s ) a g i v e n s t e p c ha nge i n In j e c t i on ra t e ca y l e a d t o re ve rsa l o r

not . ( U nc or re c t od r e sponse s o f f l ow met e r ) .

f ) Spontaneous d i s turbances may a l so c r eate i ns tab i l i t i es . One may t h ink of as t ochast i c axi a l mal d i s t r i but i on of gas i n the downcomer . As th i s passes

f rom the downcomer to the r i ser a change i n overa l l dr i v i ng f orce occurs

whi ch, i n pr i nc ip l e, can l ead to an undamped osc i l l at i on. These

di st urbances are general l y smal l er than the ai r r ate change necessary to

r eac h t he I ni t i a l oper at i on po i n t . Di s t u r bances wh i c h i n i t i a t e at t he

downcomer ai r i nj ector ar e an except i on to t h i s . The breakof f o f a i r

bubbl es f r om t he vent i l a t ed a i r c av i t i es i n t he i nj ect o r i s a ver y

turbul ent process . Occas i onal l y re l at i vel y l arge amounts of a i r may break

o f f , f o l l ow ng which t here i s a per i od i n whi ch the creat i on of bubbl es i s

t emporar i l y r educed. I f such a sequence happens at t he moment that t he

c i rcu l at i on vel oc i t y Is ext reme fo l l ow ng some previ ous d i s t urbance, t he

eff ects may be addi t i ve and l ead to i ns tabi l i ty .

62 Oscillatory behaviour

The c i rcu l at i on veloc i ty f l uctuat i ons have al ready been ment i oned. I n

f i gure (62. 1) a t r ace f rom the i nduct i ve f l owmeter i s shown and compared w th

t he f i l t e r ed r espons e of t he d i f f e r ent i a l p r es s ur e c e l l wh i c h i s

_L !

130 62

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Figure 62 .1 : Typical (uncorrected) t races o f veloci ty and ri ser vo id fract ion In esperl raencal co lumn d l lou i flgcomparison of the fluctuat ions.

re pre sent a t i v e f o r t h e r i ser dr i v i n g f o r ce. Cl e ar l y t he re i s a di s t u rbe d

per i odi c i ty w th a t r i angul ar , ra ther than s i nuso i dal , fo rm The mean per i od

of the vel oci ty tr ace shown i s 43 s ( st andard devi at i on 3. 5 s ) , whi ch may be

compared w th t he ci rcul at i on t i me of about 66 s or a gas resi dence t i me ( f rom

downcomer a i r i n jector to r i ser l i qui d l evel ) o f about 23 s-

The osci l l a t i ons were studi ed fu rt her w th a corr e la tor and f requency

ana l yzer w thout much success. Thi s was a resu l t o f the very l ong i n tegrat i on

t i mes needed (more t han 15 mnutes) together w t h t he pecul i ar behavi our of

the f l uctuat i ons. These are sel dom i n phase fo r l ong and are r egul ar l y

"doubl ed" or i nversed, as may be seen fr om the fam l y of t r aces shown i n

f i gur e ( 6 2. 2 ) .

From th is f i gu re certa i n character i st i cs may be recogni zed:

a ) The ampl i tude o f t he f l uctuat i ons i ncreases w th decreasing a i r i nput

( downcomer o nl y ) . The probab i l i t y o f a " doubl e" di stu rbance i s l a rger .

b ) The f l uctuat i ons are smal l e r w th a smal l r i ser a i r ra t e , a l t hough the

g i ven t r ace is f a i r l y cl ose to the unstabl e area ( f i gu re 5 1. 4 ) .

c) The per i od of the osc i l l a t i ons decreases w th i ncreas ing gas i nput (and

t h us c i r cul a t i o n v el o ci t y ) . Thi s i s not very ev ident f r om the t r aces.

Theref ore a peri od was defi ned as t he di st ance between two adj acent

ext remes ( top to t op or m n imum to m n i mum w thout an in terveni ng

di st urbance. I nversed or "doubl ed" f l uctuat i ons were int erpreted as such.

I n f i gure (62 .3 ) the mean such def i ned per i od i s p l o t ted agai nst a i r ra te

Fi gure 62 .2 : F l u c t u at i o ns i n ci r c ul a t i o n v el o c i t i e s w t h d i f f e r e nt o p e r at i o n co nd i t i o ns : a ) n o r i s e r a i r

f l o w r e ve r s al i s shown). A r b i t r a r i l y c h o s en u nc o r r ec t ed t r a c t s s o o n a f t e r a c ha n ge i n do wn c omer a i r i n j e c t i o n

5 0

4 0

3 0

C

™ » n P » rl °d of_<MclllaUon(*l

\

"

\ l 1

IX I

"

S iL V- OB

i10

1

\

tnpu

r

s ,

Ig/sl

i

-

-

-

V. "

'

( n o r i s e r a i r I n j e c t i o n ) . V e r t i c a l b ar e d e no t e t h e si

i n f i gure 62 .2 and as def i ned In the

_A

132 62

show ng a cl ear decrease. No data are presented f or operat i on w th s mal l

r i ser a i r i n ject i on ra tes, s i nce the f l uctuat i ons do not then show a

per i odi c character .

The phenomenon can be expl ai ned as f ol l ows: S ta rt i ng f r oma sudden

63 133

a s i ngl e constant s l i p vel oc i t y . Thi s t r eatment i gnores the typi cal

i nteract i on between the voidages i n the upper ri ser sect i on and at t he

downcomer i n j ector , and over- emphasi zes the i nf l uence of the tr ansi t i on of a

voi dage pl ug fr om downcomer to ri ser.

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breakaway of ai r f rom the downcomer i n j ector the amount of ai r i n the

downcomer does not change much. Af t er a cert ai n t i me t his i ncreased voi dage

reaches the upper part of t he ri ser. Because of the l arge expansi on it s

cont r i but i on to the dr i v ing fo rce is now s i gni f i cant and thus c i rcu l a t i on

v el o ci t y h ig h. As so on as t he pl ug l e av es t h e sys t e m t h e i n i t i a l c i r cu la t i o nvel oci ty i s rest ored. Duri ng decel erat i on a new di st urbance may form at the

i n jector and i n i t i a te a new sequence. From the t races i t appears that a

reduced voi dage pl ug i s al so possi bl e and even more l i kel y. I t i s not cl ear

how such a negat i ve ef f ect m ght ari se.

Another mechan ism i s t he i n teract i on between i n jector and c i rcu l a t i on

vel oci t y . At a moment o f h igh c i r cul a t i on ve loc i ty a g iven a i r supp ly r a te is

i n jected i nto l ess l i qui d , and vi ce versa , thus creat i ng a pl ug of hi gher gys

f ract i on. At the moment of i t s g reatest e f f ect i n the upper r i ser sect i on thi s

posi t i ve voi dage pl ug creates a new plug of l ower gas fr act i on at the

i n jector . I f th is new pl ug is as l a rge or smal l e r than the fo rmer, an

o sci l l a t i o n de ve lo ps . I f i t i s l a rg er , unda mpe d o sc i l l a t i o n o r i n s t a bi l i t y i st h e resul t .

The l ast descri bed mechani sm may be cal culat ed si nce i t i s

i ndependent of t he di st urbance mechani sm

63 Literature

The occurr ence of osci l l at i ons i n bubbl e dri ven systems has been

di scussed by other wor kers. Beek ( 1966) star ted a di scussi on on a bubbl e

dri ven osci l l at i on i n a U- t ube, f ol l owed by Whal l ey and Davidson ( 1972) and

Gar l and and Davi dson (1975). Thi s mechani sm was t hought to be representat i ve

for the se l f -exci t ed osci l l a t i ons on s i eve t r ays. These osci l l a t i ons however

di f f e r i n t y pe , s i nce t here i s no l i qui d c i r cu la t i o n a nd t he i nt e ra c t i o n

bet ween two coupl ed gas i n ject i on systems i s essent i al . Hj al mars (1973)

descr i bes an a i r l i f t pump wh i ch start s to osc i l l a te when the l i qui d l eve l o f

the dra i ned basin r eaches a cert a in cr i t i ca l l evel . A l though comparab le i t

om ts the i nteract i on w th t he downcomer gas content- The l att er was

i ncorporated i n the di scussi on presented by Sbderberg (1980). However he

s i mp l i f i ed the mode l consi derab ly , assum ng, i n ter a l i a , no a i r expansion and

Sbderberg' s deri vat i on is based on his s teady st ate model , d i scussed

ear l i e r ( sect i ons 53 and 54) . A smal l pert u rbat i on v' of the c i rcu l a t i on

veloci t y i s assumed and t he ef f ect on devi at i ons i n the voi dage ori g i nat i ng at

the i n ject ors (and t he ri ser/ downcomer connect i on) was deri ved. A standard

f orm for the osci l l a t i on was proposed:

v' = D exp( b + ir,.,) t ( 1)

(D cons t an t, b dampi ng, u) per i od) , and t he const ants cal cul ated.

The di f f erent i al equat i on was compl i cated and solved numeri cal l y. The

calcul ated peri ods of osci l l at i on were smal l er than those measured by a f actor

of about two. I t must be noted t hat the cal cul ati on was based on the

assumpti on of a smal l pert urbat i on whi l e t he measured f l uctuat i ons can easi l y

be as l arge as t wenty percent of the stat i onary val ue.

I n thi s work i t was decided to si mulat e t he system numeri cal l y on the

basi s of the steady state model . Thi s el i m nated t he need f or t he severes i mpl i f i c at i ons .

64 Theor y

The general t i me dependent f orm of the equat i on of mot i on (55-1) i s:

L

dt A

0

where M i s an eff ect i ve mass (kg) and A an eff ect i ve (cr oss- sect i onal )

a rea of the co lumn. The fo l l ow ng s i mp l i f i ca t i ons w l l be considered here :

a) Ti me dependence of mass cont ents i s negl ect ed.

b ) The consequences of l eav ing out t he i nert i a - t erm ent i re l y .

c) The assumpti on that the gas i n j ect i on resi st ance i s i ndependent of t i me.

Notes

a) The eff ect i ve i nert i al mass depends on the gas content i n the syst em si nce

the water d i sp laced by the gas e i t her overf l ows or i s separated f rom the

ci r cul at i on. The vol ume f ract i on of the gas i n the col umn under operat i ng

cond i t i ons i s however smal l , bel ow f i ve percent , and t he departu re f rom an

operat i on po in t i s fa r smal l e r and negl i g i b le .

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65 137

The previ ousl y i n jected i ncrements of gas ar e

m i , j ) = m i - l . j - l ) ( 2)

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Of f

^s^fLkl^^:o ti 2A1 3ii lj,-i)At j . At

.merical solution

-ation of tho E a:

Shown from sti

a) i ts spat i al l ocat i on (upper and l ower boundar i es ) ,

b) i ts vol ume, whi ch depends on

c) i ts press ure, a mean val ue between the boundar i es,d ) i t s ve l oc i t y , o r more preci se ly the vel oc i t i es o f upper and l ower

boundar i es.

Si nce the movi ng el ement i s r el ated to a gi ven mass of gas, both i ts

vol ume and the associ ated l i qui d mass change as i t i s f ol l owed thr ough the

system I t i s assumed that except at t he tr ansi t i on f rom downcomer t o ri ser,

t h ere i s no ax ia l di sper s i o n of gas r e l a t i v e t o ea r l i e r o r l at e r i nj ec t ed

p or t i ons .

The t otal gas content ( volume) i s t he summati on of al l consi dered gas

mass es, and determ nes t he l i qui d c i rcu l a t i on ra t e .

Taki ng a constant t i me step At a consi dered gas mass m resul ts f rom

the gas i n ject i on ra te W . Dur i ng a t i me between ( j - l )At and j A t aft er t=0the i n jected gas mass i s

( 1)

where t he suf f i xes o f m i , j ) re f er t o the part i cul a r mass of gas downst ream of

the i nj ec t or and the t i me step respect i vel y . W ( j - 1 ) i s t he ( mean) gas ra te .

The creat i on and t r ans l a t i on o f t hese gas masses i s i l l ust r a ted i n f i gure (65)

and governed by t he fol l ow ng equat i ons.

Transl at i on of upper boundary

z ( i , j ) = z ( i - l . j - l ) + v" G( i , j ) At ( 3)

Mean gas vel oci ty

Actual gas ve loc i ty

( 5)

The s l i p ve loc i ty v i s g iven by the steady sta te mode l . I n the r i ser

(equat i on 55-3) there i s a dependence on l ocal pressure.Ac t u al l i qui d ve lo c i t y

v L( i , j ) = vLs ( j ) / ( l - a( i , j ) ) ( 6)

Voi d f r a ct i o n

v ^ c m i , j ). . _ G _ m = p _lZl (j-\

a t >V * V PGAAz " p A( z ( i , j ) - z ( i - l , j ) ) { ;

where c i s a gas const ant (p/P,-, ) and A the cross- sect i onal area of l ocalP »

tube sect i on.Pressure

P = Mp Ci . j ) + P( i - l > ) ) ( 8)

p( i , j ) = P( i - l , j ) - ( z( i , j ) - z( I - l , j ) ) ( l - a ( i j ) ) ( 9)

( i n the program p is shorthand f or p / P . g ) .

__:

138 65

An ana lyt i cal so lu t i on o f the equat i ons i s not poss i bl e but onl y a

f ew i ter at i ons ( commonl y 2) are needed i f good i n i t i a l est i mates are used.

These are avai l abl e fr om the preceding t i me step.

The comput ed voi d fr acti ons are summed and l i qui d ci r cul ati on

66 139

Exper i mental

Both the s igna ls f r om the f l owmeter and the di f f e rent i a l p ressure

cel l (void f ract i on) are sampled by a processor (HP 2240A) w t h a sampli ng

ti me of 200 to 400 ms. Some of the tur bul ent noi se of the pressure si gnal was

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iy max L

v L s D( j ) = C g ( £ ( z( i , j ) - z( i - l , j ) ) a ( i , j ) - AP v ) r ( 1 0)F i

Agai n an i ter at i on i s perf ormed and tested for accuracy.Up t i l l now the di scussi on has been i ndependent of col umn sect i on

( r i se r o r downcomer ) . The tr ansi t i ons f rom downcomer t o hori zontal bott om

sect i on, f r om there to r i ser and f r om ri ser t o di sengagement sect i on pl us

other detai l s are now di scussed.

a) Downcomer gas vel oci ty i s checked f or a posi t i ve val ue l arger than zero.

b) I f the upper ( = downstr eam si de) boundary gets bel ow z=0 i n the downcomer

or above z - L i n the ri ser t he gas mass i s di vi ded proport i onal l y and

the part s red ist r i buted.

c) I n the hor i zontal bot t om sect i on the redi st r i buted gas mass enters f r om the

downcomer. S li p veloci ty i s assumed zero. Gas i s r edi st ri buted i n the same

way as under b) between the hor i zontal sect i on and the ri ser. The l att er i sadded to the ri ser gas i n j ect i on H . .

d) The downcomer sl i p vel oci t y and i nj ecti on head l oss are t he same as t hose

used for the steady stat e model .

e ) Ga s re di s t r i but i o n ( d i sp er s i o n) r e sul t i ng f r o mco ncent r a t i o n di f f e re nces

was t est ed and f ound to be uni mport ant ( appendi x 6 5. 2 ) .

f ) To l i m t comput i ng t i me the t i me step used fo r the r i ser was taken as t w ce

t hat used f or the downcomer. The FORTRAN program i s pr esented i n appendi x

( 6 5. 1 ) , together w th f l ow sheet and descr i pt i on.

66 Experimental verification

In th i s sect i on the quasi - st at i onary model I s compared w th

experi ment s. Fi rst the experi mental procedure t o determ ne the response t o a

step i n the gas i nput i s descr i bed . The responses t o step changes i n r i ser a i r

i n ject i on (a i r l i f t mode) and downcomer a i r i n ject i on ra tes are t reated.

pre f i l t e red ( f i rst o rder , t i me constant 1.25 s) . Sampl i ng was cont ro l l ed f rom

the ter m nal of the HP 2100 comput er w t h a sampl i ng program ( appendi x 6 6. 1 ) .

To recover the physi cal data on veloci ty and voi d fr act i on f rom the

si gnal s ( see al so appendi x 13.1) a numeri cal d i f f erent i al procedure was

devel oped w t h pr esmoothi ng of t he r emai ni ng noi se ( appendi x 66. 2) .

S i nce the gas i nput could onl y be contr ol l ed by hand a step change In

the gas i nput was chosen for t he experi mental veri f i cat i on. A puls e response

I s al ways s wamped by noi se whi l e other ti me dependent i nputs , e. g . s i n uso id al ,

are di f f i cul t to produce manual l y. Furt hermore operat i ng condi t i ons are of ten

changed st epw ze. Even a step however cannot be ideal l y adm ni st ered and i s

si mulat ed i n the model program by a l i near change over a short t i me. To

separate the response f rom spontaneous f l uctuat i ons rel at i vely l arge steps

together w th the in j ect i on of some r i ser a i r were necessary .

Ri ser a i r only

Compari son of t he dynam c model w t h t he steady st ate model showed

very smal l d i f f e rences ( l ess t han four percent ) i n the p red ict ed vo id f ract i on

stalfc prassL1 P'PL9l

ra do v

"1 low5t«que

^™ r . l r r . u 10 .

Lop pressure *idy state mode! /si stationary modal / y

/ ydm*

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' ifiure 66.2; Computed and o es Following

Figure 66.3: Compute

of t he upper ri ser sect i on. S i nce the two model s di f f er onl y i n the expressi onf or the pressure (equat i ons 55-11 and 65-9, of whi ch the f ormer i s an

approxi mati on, see f i gure 55. 1) th i s d if f erence was tested by pl ott i ng the

dev i a t i on f r omhydrostat i c pressure (no a i r ) p aga inst hei ght ( f i gu re 6 6. 1 ) .a

The f i gure s hows t he di f f erence bet ween the two model s and - i n compari son

w t h given measurements - that the dynamc model i s corr ect .

The response of both vel oci ty and ri ser voi d f ract i on to both

posi t i ve and negati ve step changes ar e presented i n f i gures (66.2 and

/

, _

-»- helghtlml

-

Figure 66 . k: Aa f i gure 66 .1 for downcomer ai r I n jec t i on on ly .

Fi gure 66. 5: Computi

v el o c i t y ) .

i n d o wn c omer a i r i n j e c t i o n r a t e ( h i g h

66. 3) . Agr eement i s ver y good. Note t he smal l overs hoot r esul t i ng fr om the

h igher t han sta t i onary gas i nput d i r ect l y a f t er t he step.

Dovncomer air injection

Agai n the model i s compared w t h the st eady st ate model wi t h a pl ot

o f t he di st r i but i on o f t he pressure dev i a t i on f rom the hydrosta t i c case (The

model s were s l i ght l y i mproved i n a way not pr esent ed i n chapt er 5: The

pressure drop over the hori zontal sect i on was i ncluded l eading t o a hi gher

i

J P ' i »

66 143

s t r at i f y ( f i gur e 6 6. 7 ) . The coal esced bubbl es enter the ri ser and contr i bute

l ess to the mean r i ser voi d f r act i on as a resu l t of thei r hi gher term nal

v el o ci t i e s. Thi s a lso means that the m n imum vel oci t y i s l ower . Th i s e f f ect i s

not re l evant fo r real col umns w thout hor i zonta l sect i ons.

One may concl ude t hat the model agrees r easonabl y w t h t he

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riser air injection 0.25 g/sdow noo m«r air injec tion from O to o.2 g/sp\ =0103

Flpure 66.6; Computed I

v e l o c i t y ) .

ured re6ponaoB on a posltivü step change In dowi

P l gl H i o f b u bb l e s I n t h e

pressure and a l ower voi d f ract i on i n the downcomer. Thi s ef f ect was

compensated by usi ng a hi gher val ue of 0.2 m s f or t he downcomer sl i p vel oci ty

parameter , whi ch i s t hought to be more r e a l i s t i c ) . The d ist r i but i ons f o r the

downcomer agr ee ( f i gur e 6 6. 4 ) .

The response to a posi t i ve step change i s presented in f i gur e ( 66. 5)

f o r both c i rcu l a t i on vel oc i t y and r i ser voi d f ract i on. Agreement fo r these

cases i s good, al though there i s some i naccuracy i n the det erm nat i on of themoment of the step. Note that the start of the change i n ri ser voi d

cor r e sp ond s w t h t h e f i r s t m ni mum i n c i r cu la t i o n v el o ci t y .

I n f i gure ( 66. 6) another r esponse t o a posi t i ve step change is shown

whi ch does not agree as wel l w t h t he measurements . The measured r esponse i s

s l o we r i n r e co ve r i ng f r o m t h e m n imu mv e lo c i t y . T hi s d i f f e re nce, wh i ch i s

typ ica l fo r l ower vel oc i t i es (be low about 0.8 m s ) , may be expl ai ned by the

coal escence o f the a i r bubb les i n the hor i zonta l bot t om sect i on wh i ch tends to

measurements , except f or s pontaneous f l uct uati ons, and may be used t o study

i nstabi l i t y and osci l l a to ry behavi our i ndependent l y o f t hese f l uctuat i ons.

67 Osc i l l a t i o ns

The osci l l atory behavi our of t he system may now be studied usi ng the

model . I t i s noted once more that s pont aneous di st urbances make comprehensi ve

experi mental i nvest i gat i on of t h is i mpossi bl e. The response to step changes i n

downcomer ai r r ate was tested for d i f f erent r i ser ai r r ates. Step changes as

great as 0.1 g/s were f ound to gi ve sim l ar ef f ects to the behavi our of the

experi mental col umn. Peri od and ampl i t ude were cal cul ated.

Peri ods are presented i n f i gure (67. 1) and compared to t he data

presented ear l i e r i n f i gure (62 .3) . The t ests were genera l l y st a rted w th a

downcomer i n j ect i on rat e of 1.0 g/ s. Steps were posi t i ve for h i gher and

F l R u r e 67 . 1 : Co mp ut ed p er i o d s f o r d i f f e r en t s ma l l r i s e r a i r I n j e c t i o n r a t es as f u n c t i o n o f d o w

I n j ec t i o n r a t e . Co mp a r i s o n w t h mea s ur ed v a l u es ( f i g . 6 2 . 3 ) f o r z e r o r i s e r air I n j e ct i o n r a t e .

67 145

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FlRure 67.2: Simulated .the same start condition

t ive and negat lv .

Figure 67 .3 : As figure 67 .2 . Step size 0 .05 g /s

1) — *

0 KX3

I'°

20 0

1mj

3O 0

11n 193io194q/s

tlmsls

4O0 5- JO

F i g u r e 6 7 . 4 : C o m p ut e d o s c i l l a t i o n a f t e r s t e p c h a n g i

F l g u r o 6 7 . 5 : C o u p u t , s t r l b u t l o n d e v e l o p m en t d u r i n g o s c i l l a t i o n .

f lG u i- e 6 7 . ; . At u p p e r i n s t a b i l i t y U n i t ,

negat i ve fo r l ower i nj ect i on ra tes. Data are presented fo r the in j ect i on ra te

a f ter the step .

Consi deri ng the possi bl e i naccuracy of the measurements , agr eement

between computed and actual f requenci es i s r easonabl e f or i n ject i on rates

bel ow 1.0 g/ s ( the l ower i nstabi l i ty boundary) . Above this there i s a

dif f erence both i n peri od and operat i ng envel ope,As an i l l ustrat i on two typical computed responses are gi ven i n f i gure

( 6 7. 2 ) . These show that the peri ods ar e wel l def i ned and depend on the fi nal

i nj e ct i o n rat e . T h e ampl i tudes however are not wel l defi ned and depend on

whether the step is posit i ve or negat i ve and on the step si ze ( f i gure 67. 3) .

Fi gure ( 67. A) shows how the comput ati ons suggest an osci l l ati on may be

susta i ned . I t w l l be c l ear that such osci l l a t i ons do not occur i n pract i ce

because of spontaneous di s tur bances w ch can be of the same order as t he st ep

change.

To demonstr ate t he mechani sm of t he osci l l at i on the devi at i on f rom

the steady state val ue of the computed void f ract i on di st ri but i on around t he

l oop at var i ous moments of a per i od (at the extr emes and hal f way) i s pl ott edi n f i gures (67.5 and 67. 6 ) fo r two di f f e rent operat i on condi t i ons. These

cl earl y show the tr ansl at i on of voidage pl ugs t hrough the system and the

i nter acti on between vel oci t y and cr eated downcomer voi dage.

The peri od is determ ned by the t i me i t takes for a voidage pl ug to

tr avel through the system fr om downcomer i n ject or t o the top of t he r i ser. A

total peri od consi st s of t he tr ansl at i on of both a posi t i ve and a negat i ve

voidage pl ug. These di f f er i n resi dence t i me as may be seen fr om the f i gures.

146 67 147

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F l f l u re 67 . 6 : A s f i g u r e 67 . 5 . A t l o we r I n s t ab i l i t y l i m t .

A posi t i ve pl ug i s somewhat f aster si nce i t w l l be i n the ri ser at t he moment

of maximum veloci ty, downcomer resi dence t i me bei ng very s hort . I n th i s

respect t he system response w l l depend on the geometr y, part i cul arl y the

di ameter rat i o and downcomer i n j ect i on l evel .

Tw ce t he steady state gas resi dence t i me i s a good est i mate f or t he

pe r i o d o f t he osc i l l a t i o n.

As shown i n chapter 3, i t should be remembered t hat sl i p vel oci ty i sh igher w th hi gher voi d f r act i ons. Th is i s however not i ncorporated in th i s

model . I t expl a ins the l a rger di f f e rence i n r i se and fa l l t i mes o f t he rea l

f l uctuat i ons shown i n f i gure (62 .2) . Thi s may al so be r eason f or t he

di f f erences between comput ed and measured per i ods (f i gur e 67. 1) f or hi gher

i nj e ct i o n ra t e s .

68 Instability and operation boundaries

The mechani sms t hat l ead to f l ow r eversal whi ch were di scussed i n

sect i on 61 under d) and e) are si mul ated by t he program Gener al l y a st ep may

l ead to ( f i gu re 6 8. 1 ) .a) d i r ect f l ow reversa l

b) undamped osci l l at i on

c) damped osci l l a t i on

The f i gure shows t hat the response fr om a gi ven star t i ng condi t i on depends

onl y on the step si ze. Start i ng condi t i ons are not usual l y the same ( f i gure

51), and thi s l eads to di f f e rences in resul t s ( f i gure 6 8. 2 ) . Furt hermore there

i s a di f f erence dependi ng on the step di rect i on (posi t i ve or negat i ve)

as shown i n f i gure ( 68. 3) for t he same start condi t i on and i n f i gure ( 68. 4)

f o r t h e same f i na l a i r i nj e ct i o n ra t e s . A sma l l r i ser ai r i nj e ct i o n ra t e

s t a bi l i z es t h e sys t e m F i gure ( 6 8. 5 ) , which has a r i ser a i r r a te of 0 .1 g / s ,

can be compared w th f i gure (67. 2) whi ch corr esponds to t he case w t h no

addi t i onal r i s er a i r .

We may now agai n compar e t he measur ed and computed oper at i ng char t s,w t h emphasi s on the oper ati on boundari es, i n the same way as wi t h the st eady

state mode l ( f i gure 57. 4) . To deter m ne these stabi l i ty boundari es, t hey were

approached st epw ze. The program was all owed to stabil i ze for at l east 200

si mulat ed seconds before a new step was i n i t i ated. Di f f erent boundari es were

found fo r d i f f e rent step s i zes. Smal l s teps (0 . 05, 0. 02 g/ s and l ess) l ed to

s i m l ar r esul t s . S teps wh ich were l a rger than 0.1 g/ s devi a ted st r ong ly I n

e f f e c t .

68 149

I n f i gure ( 68.6) measured and computed val ues are compared. The

agreement i n form and place of t he l ower unstabl e area i s very good. The

comput ed boundary f or t he hi gher unstabl e area l i es on the r i ght hand si de of

that measured (whi ch actual l y l i es at the l ocati on comput ed for st eps of

0.2 g/ s) and i s t o the r i ght hand si de of the computed m ni ma of t he i so-

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veloci t y curves. However i t must be noted t hat the l ocati on of thi s boundary

i s very sens i t i ve s ince vel oc i t i es are al most constant i n th i s area. Moreover

the computed vel oc i t i es are somewhat too h igh (s l i p vel oc i t y , see sect i on 67,

f i n al r emar k ) and t h us s t ab i l i t y l a r g er . Neve r th e l es s i t i s an i mp or t ant

resu l t that t he proposed model actual l y s i mulat es both s t abi l i ty boundar i es .

S tep di r ec t i on de te r m n es th e pos si b l e ons e t o f i n s tabi l i t y . Thi s was

shown i n f i gure (68.4) f or t he top of the boundary curve (s t ep 0. 02 g / s ) . I t

must be noted t hat pos i t i ve s teps are more l i ke ly t o lead to i ns tabi l i ty and

thus that the r i ght hand s i de of the lower unstabl e area i s l ocated at h i gher

gas i n ject i on rates for such s teps ( not shown i n f i gure 6 8. 6 ) .

Cons i s t en t wi th t h e s ma l l es t p os s i b l e u s e of r i s e r a i r , t h e c ol u mn

shoul d be operated i n such a way that i t I s most s t ab le. Thi s i s character i zed

by th e g r ea tes t c apaci t y t o r e tu r n t o t he i n i t i a l s t ead y s ta te ve l oc i t y a f t e r

a di s tu r b an ce i . e . w th t h e g r ea test dampi n g o f t he i n i t i a t ed os c i l l a t i on. A

dampi ng fact or may be def i ned as t he rat i o between the f i rst and second

ampl i t udes. The program gi ves the successi ve peak to peak val ues (f rom m ni mum

Figure 68 .7 : Computed damping factor (ra t io o f fi r st to second top-top value of response) fo r d i ff eren t ri s era i r In ject ion ra te s and d i ffer en t s tep size s as funct ion of douncomer a i r In je ct ion r a te (Compare wi th fig ure5 8 . 4 ) .

150 68

t o m nimum. Note , f r om th e f i g ur es , t h at d i f f e r ent d ampi n g f ac to r s a r e

calcu l ated for pos i t i ve and negat i ve s t eps .

General l y however di f f erences are smal l . The dampi ng factor I s

pr es en ted as s i n gl e l i ne i n f i gu r e ( 6 8. 7 ) f o r a l l s t ep s i z es (0 . 01, 0.02, 0.05

and 0.1 g/s) except f or t hose cases where the di f f erence was si gni f i cant. The

151

7 M x i n g

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damping factor i s c losel y re lated to the re l axat i on t i me of the osc i l l at i ons :

the rat i o of per i od to re laxat i on t i me i s - as a f i r s t approxi mat i on - equal

to i ts l ogar i thm c val ue. Th i s means that i f the damping f actor equal s e

( 2.72) ( compare f i gure 68. 7) peri od and rel axati on t i me are about the same,

The f i gure bears cl ose resembl ance t o f i gure ( 58. A) where t he rat i o bet ween

steady s t ate dr i v ing and res i s t i ng vo i dage (0 /Q ) was p l ot ted i n the same

way. I t demonstr ates that the same f actors determ ne both t he maxi mum steady

s tat e operati on area and the maximum resi st ance t o di st urbances:

a ) l ow s l i p vel o ci t i e s

b) ef f i c i ent downcomer a i r i n jector

69 Concl usi on

I n thi s chapt er a t i me dependent si mul ati on model has been present ed

based on the steady st ate model proposed previ ousl y i n chapter 5. The model i s

a quas i - s tat i onary approx i mat i on om tt i ng iner t i a and i s governed by the ax i a l

tr ansl ati on of voidage pl ugs t hrough the col umn l oop. The model predict s t he

operat i ng envel ope fa i r l y accurate l y , show ng that i ner t i a i s i ndeed a second

order ef f ect . Fur t hermore i t i s shown that osc i l l atory behavi our i s

r es pons i b l e f or t h e i ns tabi l i t y w th a pe r i od th at i s app r ox i mate l y tw c e the

gas res i dence t i me. The capac i ty of the system to recover f rom pert urbat i ons

i s shown to be re lated to t he rat i o of dr i v ing and res i s t i ng voi d f r act i ons .

70 I nt roduct i on

Li qui d phase m xi ng has an i mportant i nf l uence on the mass t ransf er

i n gas - l i qui d reactors . Both m cro- and macromx i ng af f ect t he l i qui d

ref reshment r ate of the i nter f ace. I n tal l bubbl e columns where l ocal mass

t ransfer i s enhanced by t he lar ge pressure changes, m xi ng may wel l be a

l i m ti ng f actor . The mass t ransf er i n a deep shaft system may be l arge enough

to i n f l uence the hydrodynam cs by d isso l v i ng the in j ected ai r . Hi nes et al *

( 1975) f or exampl e suggest that most of the air bubbl es are compl etel y

d isso l ved at t he bottom of t he deep shaf t . Know edge of m x ing rate i s

necessary f or t he i nterpr etat i on of the mass t ransfer measurements i n the next

chapt er. Furt hermore good descri pti ons of the macrom xi ng are al so requi red

f or t he desi gn of f ermentors , where cont i nued suppl y of oxygen and nutr i ents

i s es s en t i a l .

M x i n g i n a r eci r c u l a t i ng b ubb l e c ol u mn r es ul t s f r om l i qui d r ec i r -

cul ati on, secondary f l ows i n bends, l arge scale eddi es produced by the

bubbles, l i qui d vel oci ty prof i l es , t urbul ence, etc . One may compare the m x i ng

i n a bubbl e co lumn loop reactor w t h that i n

- a bubble co l umn w thout i n ternals , or

~ a si ngl e phase l oop reactor.

I n the f i r s t case the i n f l uence of the conf i nement of the l i qui d c i rcu l at i on

pat ter ns I s compared, i n the l att er t he inf l uence of the second phase on the

m x i ng process .

I n thi s chapter measurements of l i qui d mxi ng are report ed and

compared w t h l i t erat ure. M xi ng i n the column as a whole has been examned as

wel l as that I n the gas - f r ee downcomer separate l y . Theoret i ca l aspects w l l

onl y be di scussed bri ef l y i n order t o i l l ustr ate t he measurement method and

I n te r pr e ta t i on of t h e r es ul t s . Va r i ous and extens i v e l i t e r a tu r e on thi s

subj ect i s ref er red t o , espec ia l l y t he rev i ews by Shah et a l . (1978) and

( 1982) .

71 The axi al di spersi on model

The axi al di spersi on model i s t he one most w del y used t o charac

ter i ze t he mx i ng per f ormance f rom res i dence t i me d i s t r i but i on measurement s .

The bas ic equat i on i s the s impl e, one-d i mens i onal f orm of F i ck 's l aw w th t he

di f f u si on coef f i c i en t r epl aced by an axi a l d i s p er s i on coef f i c i en t D:

153

- ^ « r f -0

^ )

w t h an average ti me

t * = 1 + 2/Pé ( 5)

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££- D A - v ^ (1)at

u3 z 2 L ?z

KL)

where c i s t he concentr at i on. The model i s at t ract i ve because a s i ngl e

parameter , D , i s used to character i ze mx i ng per f ormance. I t i s essent i a l to

use the same model i n order to make compari sons w t h t he l i t erat ure.

The di mens i onl ess form of equat i on ( 1) i s

3c* = 1 'd 2c* _ 3c*a t * P I 3z A 2 aS* * ;

wher e c* = c/ c cw th c = c ha r ac te r i s t i c c on cent r a t i on ( e . g. i n i t i a l o r f i na l v al u e)

and t* = t/ t c

w th t = c ha r ac te r i s t i c t i me ( e . g . t he t i me f o r one c i r c u l a t i on i n t h el oop)

and z* = z/ L

w th L = c ha r ac te r i s t i c l eng th ( e . g. t he l eng th of t h e p i p e s ec t i on

consi dered)

M - - £ - ( 3)

There i s some i nconsi st ency i n the use of thi s number. Some authors

cal l that def i ned by equati on (3) t he Bodenst ei n number, whi l e i f the diameter

of t he p ipe i s used as character i s t i c l ength , i t i s cal l ed the Pécl et number .

We w l l c al l t h i s l a t t e r g r oup th e d i s p er s i on i n tens i t y a f t e r Levens p i e l

(1972) . The value of the Pécl et number denotes the degree of ( axi al ) m xi ng.

I f Pé = 0 , m x i ng i s compl ete, wh i l s t for very l arge Péc let numbers

condi t i ons approach pl ug f l ow.

For an ideal pul se i nput t o an open system the sol ut i on of equat i on

(?.) I s

and vari ance

( v a r ) * = v a r / t * 2 = 8/ Pé2 + 2/Pé ( 6)

(Levenspi e l and Smt h, 1957).

A r ecord of a tr acer curve may be used t o deter m ne t he Pécl et

number i n vari ous ways. Ar i s (1959) has shown that w th a non- i deal pul se

i nput of t r acer whose concentr at i on i s measured at two l ocat i ons , L apar t , the

fo l l ow ng re lat i on for t he change in var i ance can be used

( 7)

w thout i n t roduc i ng a s i gni f i cant er ror . Thi s s i mpl i f i es the measurement

method consi derabl y.

72 Bubble columns

I n bubbl e columns where t he l i qui d vel oci t y i s general l y much l ess

than that o f t he gas , t he convect i ve term i n the axi a l d i spers i on equat i on

( 71- 1) may be neglect ed. The sol uti on, and hence the determ nati on of the

axi al di spersi on coeff i ci ent i s then much si mpl er. The st ati onary method may

then be used, as i s f requentl y done. Towel l and Acker man ( 1972) used bot h

approaches.

S i n c e the l i qui d ve l oc i t y i s s o s ma l l , a l i qu i d v el oci t y b as ed

Péclet number i s not r epresentat i ve for the m xi ng process . Measured

d ispers i on coef f i c i ents are therefore somet i mes presented i n the form of

Péclet numbers based on the rel ati ve vel oci t y w t h bubbl e or col umn di ameter

as c ha r acte r i s t i c l eng th. F ur t h er mor e t he s u per f i c i a l g as v el oci t y , b ei n g

i ndependent , has been used as character i s t i c ve l oc i ty .

The cor re l at i on of t he d ispers i on coef f i c i ent w th t he most i mpor t ant

parameters , super f i c i a l gas vel oc i t y and co l umn d i ameter , i s more

st r a igh t fo rward . Di mensi ona l S . I . equat i ons that corr e l a te the di spers i on

coef f i ci ent are t hose of Towel l and Ackerman ( 1972) :

D . 1.23 d^5v ° ; 5 ( 1)

and Deckwer et al . ( 1974):

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D = 0.678 <£*4v ! j ' 3 ( 2)

The most I mpor tant process f or m xi ng i n bubbl e col umns are theci rcu l a t i on pat terns. These are a resul t of the h igher gas concent r a t i ons I n

the centr e of the col umn. I f not st abi l i zed by a draught tube the ci rcul at i ng

f l ow i s broken up i n ci rcul at i on cel l s as l arge as the col umn di ameter ( J oshi

and Shar ma, 1979) . The di spers i on coef f i ci ent I s then expressed as

( 3)

wher e t he constant i s f rom J oshi (1980) who based i t on a vast amount of data.

The equat i on i s comparabl e w th t hat f rom Tayl or (1954) for si ngl e phase

turbul ent f l ow:

( 4)

wi t h f - 0.005 ( F i e ld and Davi dson , 1980; Hi l l s , 1977):

( 5)

The veloci ty prof i l e in si ngl e phase tur bul ent f l ow i s comparabl e to that due

to r eci rc ul at i on. Vi al ter and Bl anch ( 1983) have recent l y suggested usi ng the

centr al pl ume vel oci ty i n equati on (3) and f ound, based on measurements of

t h i s v el oc i t y ,

( 6)

He i j nen et a l . ( 1982) s i mpl i f y the express ion fo r t he c i r cul a t i on vel oc i t y

f r o m J o shi (1980):

v c - 0 . 9 ( g d t v G/ ( 7)

L

■

D

_ DETECTION

157

74 Loop reactor s

The only source t hat has been tr aced whi ch report s data f or a si ngl e

phase loop reactor i s that o f Murakan i et a l . ( 1982) , who worked w t h a l oop

of 0.1 m diameter and a total l ength of 2.8 m Thei r resu l t s i ndi cated t hat

axi a l d ispers i on was between a ha l f and a th i r d t hat i n a st r a ight p i pe. Th i s

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7 3 . 2 : P r i n c i p l e o f I

response i s t he summati on of the i ndi v i dual responses, whi l e i n t i me these

f l a t t en out ( s ee f i gur e 7 3. 1 ) .

Voncken et al . ( 1964) devel oped a reci rcul ati on model based on the

axi a l d i spers i on concept . A l ong tube reactor i s di v ided in to sect i ons w th

l ength L . Fo l l ow ng the re lease of a t r acer pul se at the reactor entr y , t he

response at vari ous di st ances j L downstr eam i s expected t o be

2. , . , , Pé .\ , ( j - t * ) Pé,

( 1)

The rec i rcu l at i on i s model l ed so that t he i ns tantaneous output f r om a sect i on

of l ength L forms t he i nput t o that same sect i on ( f i gure 73. 2) . The total

response i s t hen the summati on of the r esponses ( 1) :

c* - X C*(j ) ( 2)

athemat i ca l l y more thorough der i vat i on (Murakani et a l . 1982) r esul ts i n

c*= tsllf^ I e x p ( - M^i + z* " ti)2) (3)

j — ™

whi ch does not d i f fer much f r omequat i on (2) as l ong as Pé i s greater than

10.

Normal i zat i on and equ i l i b r i um condi t i ons are sat i s f i ed for c* = c /c ,

wh er e c i s t h e t r ac er c onc ent r a t i on a f t e r i nf i n i t e t i me ( c ) . I n pr ac t i c als i t ua t i o ns t h e def i ni t i on w l l be

( 4)

I n t he model m x ing i s character i zed by D or Pé, a mean val ue resul t i ng f r om

a si n gl e c i r c u l a t i on.

i s s u r pr i s i n g s i n c e i t d i f f e r s f r om th e r es ul t s o f o th er wo r k er s , wh o have

st udi ed m xi ng in arbi t rary pi pel i ne geometr i es, and revi ewed by Park and

Gomezpl at a (1971). At comparabl e Reynol ds numbers the overal l axi al di spersi on

coeff i ci ent has been found t o be much hi gher for systems w t h bends. Park and

Gomezpl ata ( 1971) model l ed the i nf l uence of bends w t h

1 + K(Re) i T" . + L CDPi pe "pi pe eq

For a loop reactor the equi valent l ength of pi pe f or t he f our 90°-bends( L e q ) i s 1 2 ° dt ( o r 1 5 0 dt f o r t wo i SO' - bends) and assumed t o be the same

as for f r i ct i on. The Reynol ds dependent const ant K(Re) was f ound to be about 2

f or Reynol ds between 4,000 and 12, 000 and a pi pe di amet er of 21 mm One may

quest i on the val i d i t y o f t he model , but at l east i t prov i des a bas is agai nst

whi ch to compare t he present r esul t s.

Ai r l i f t type reactor s have been revi ewed by Bl enke ( 1979) . Wei l and

(1978) measured overa l l m x i ng in hi s a i r l i f t l oop and i n the two phase f l ow

of the r i ser (10m col umn, r i ser 0. 10 and downcomer 0.05 m di ameter ) . Hi l l s

( 1977) has al so measured the axi al di spersi on i n the r i ser of a bubbl e col umn

l oop ( 10.5 m hi gh, 0 .150 m diameter tubes). Wei l and f i nds a d ispers i on

i ntens i ty of about 0.5 which i s t w ce the va l ue found by Hi l l s or predi c ted by

Tay lor (1954) f or s i ngl e phase f l ow (equat i on 72- 5) . The di f f erence bet ween

the resul ts o f Wei l and and Hi l l s can be exp la i ned by the di f f erence i n ( r i ser )

l i qui d ve l oc i t y , wh i c h i s a f acto r 1 0 h i g he r w th Hi l l s ( 1 - 2 m s ) . I n f act

thei r resu l t s do not d i f f er much f r om the repor ted va lues for s i ngl e phase

f l ow (see Levensp i e l , 1972) . On t he other hand t he data of Wei l and al socompare reasonably w t h t hose obtai ned i n bubbl e col umns ( equati on 72- 8) .

Accordi ng to Bl enke ( 1979) an overal l Pécl et val ue of about 200

corr esponds to a medium axi al m xi ng i ntensi t y and Pé = 50 to a hi gh one.

Recent l y Fi el ds and Sl ater ( 1983) r eport ed measurements i n a l aborator y ai r

l i f t r eac to r w th i nte r n al d r aug ht t u be (0 . 1 52 m ex te r na l d i amete r , l i qu i d

hei ght between 1 and 2 m) . Thei r resu l ts were of the same order : Pécl et

between 50 and 150 dependent on gas f l ow r ate and aspect r at i o. They wer e abl e

to predi ct these val ues on the basi s of the model of Bair d and Ri ce ( 1975)

(equat i on 7 2 - 8 ) , extended to i ncl ude the i n f l uence of l i qui d vel oc i t y

D = 0.35 ( gC( vGs 1 Ls1/3

( 2)

75 159

i s below the level at which coalescence ef f ects become si gni f i cant. A

reproduci b i l i t y test conf i rmed t h i s . Potassi um ni tr ate was chosen as a tr acer

si nce i t modi f i es t he water s tr ucture l ess t han most other sal ts (Van der

Lans, 1978) .

The l ocal l i qui d conduct i v i ty was recorded a t appropr i a te l ocat i ons.

Two probes ( i n the centr e and at the wal l of the downcomer) were s i tuated

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2The Pécl et number was obtai ned f rom equati on ( 7 1 - 6 ) , wher e ( v a r ) * fo l l owed

f r om the summati on of the (change i n) vari ances of downcomer, r i ser and head

regi on as est i mated fr ommass bal ances. They show that t he di spers i on w l l

general l y not be homogeneous as i s i mpl i ci t l y assumed i n the axial d i spersi on

model .

75 Experi mental

The overf l ow ci rcui t of t he experi mental col umn ( f i g. A12) was

modi f i ed to all ow a sw tch bet ween pi pe sect i ons. Measurements w th t he

overf l ow operat i ng i mply that t he volume of the t wo phase m xture i s constant.

The hydrodynamc behavi our of bubbl es, especi al l y t hei r coal escence rate, i s

modi f i ed i n a l l but very d i l u te e lect r o ly t e so lu t i ons. For t hi s reason onl y a

smal l amount of t r acer was i n jected ( 250 m of sat urated potassi um ni tr ate at

50 m / s ) . The col umn l i qui d was ref reshed aft er each seri es of t en measure-

r aent s. The mean tr acer concentr at i on i n t he col umn was then 0. 55 kg/ m , whi ch

120 IS O

F i g u r e 7 5 : T y p i c a l r e s p o n s e c

a l o n g t h e d o w n c o m e r ( t - 0 ) .

! C o r d . 1 : t o p p r o b e . 2 : b o t to m p r o b e . I n j e c t i o n o f s a l t t r a c e r h a l f w a y

cl ose to the sal t i n ject i on poi n t f or the overa l l measurements. M x i ng in the

downcomer secti on was det erm ned separat el y by probes at top and bot t om A

typi cal response curve i s s hown i n f i gure ( 75) . Measurements were done w t h

downcomer l i qui d vel oci t i es rangi ng f rom 0. 75 t o 1. 35 m s, at both barometr i cand reduced top pressur es. Because of t he hi gher f r i ct i on number ( sect i on 19)

onl y a f ew measurements were possi bl e w thout some ai r addi t i on to the ri ser

i . e. i n the downcomer ai r onl y mode. M xi ng i n the downcomer sect i on coul d not

be deter m ned when downcomer ai r i nj ecti on was i n use because the si gnal f rom

the bott om probe was great l y di st urbed by the di spersi on.

76 I nter pret ati on of measurements

Reci rcul at i on model

The conducti vi t y r esponse tr ace was di scr eti si zed and made di mensi on-

l ess by usi ng equat i on (73-4) . The Pécl et number was deter m ned by fi t ti ng

the maximum and m ni mum val ues to t he r eci rcul at i on model . The f i rs t maxi mum

was di scarded, bei ng al ways si gni f i cant l y hi gher than expected, probabl y as a

re sul t of i nsuf f i c i e nt r a di al d i sp er s i o n. F i g ure ( 7 6. 1) sho ws t hat f i t t i ng t he

f i rst f l ank woul d l ead to an excessi vel y hi gh val ue.

Fifiu re 76 .1 ; Analysis o f response. Po in ts: meaeured and d iscre t i s ized response (conduct iv i ty p robe d i rect lydownst ream of in ject ion po in t ). Fi t ted curves (reci rcu la t ion model ) based on fi rst : flank (Pé = 106) o r on

15 0

too

0 50

:

^'^.'.*'

.

^.oa^ ^

76 16

For the ax ia l d ispers i on model (equat i on 71-1)

F ( s ) = exp( Pë( l - (1 +^Lfy) ( 2 )

where s* i s t he di mensi onless Lapl ace operat or

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^ '^

st«

?.

-»'

F i p u r e 7 6 . 2 : A na l y s i s o f r e s p o ns e w t h eq ua t i o n ( 7 6 - 4 ) . Measun

r a n ge o f s * f r o m 5 t o 0 . 8 .

F i g ur e 7 6 . 3 : D r

the met hod.

i o f P é w t h .

Downcomer

To deter m ne m xi ng in t he downcomer s ecti on the i nput ( c. ) and

output ( c ) s i gnal s are compared. Fahi m and Wakao (1982) r evi ew the di f f erent

techni ques f or obta i n ing t he Péc let number . We used t he " t r ans fer f i t t i ng

met hod" as used by Wei l and (1978) , whi ch gi ves a reasonable accuracy i n ter ms

of the measurement er ror s ( Fahi m and Wakao) .

The t r ans fer f unct i on i s

dt

F ( s ) = ■ ( 1)

s * = s t D ( 3)

w t h t p t he mean resi dence t i me i n t he downcomer. Rearr angement of equati on

(2 ) g i v es

p s " m 2 F ( s ) l n F<

s>

( 4 )

When p lot t ed th i s i s a s t r a ight l i ne w th i n tercept Pé and s lope t

( f i g u r e 7 6. 2 ) . The choi ce of t he s*- range i s i mpor t ant because i t sh i f t s the

wei ght i ng ef f ect o f t he method. I f s* i s l arge, t he wei ght i s shi f ted to the

f r onta l por t i on of t he s i gnal . However i f s* i s too smal l the t rans fer

f u nc t i on h as l i t t l e di s c r i m na t i n g ef f ect . Ac c or d i n g t o Hopk i n s et a l . ( 1 9 69 )

val ues shoul d be between 2 and 5. Smoothi ng t he tr aces i mproves t he resul t

c on si d er abl y , e s peci a l l y f o r h i g h va l u es o f s * .

The response char t s were d i scret i s i zed, normal i zed and i n tegrat ed

usi ng the t rapezoida l ru le to gi ve the t r ans fer f unct i on for d i f f erent

s*- values. Only that part of t he response curve whi ch corr esponded t o the

f i r s t i ndi v idual response was used. The ta i l was r econstructed by curve

f i t t i ng to avoi d cal cul at i ng a systemat i ca l l y h i gh val ue of t he Pé-number .

Péc l et as f unct i on of s* was cal cul ated w t h equat i on (4) t o gi ve an i dea of

the accuracy ( f i gure 7 6. 3 ) . Péc l et and t were determ ned w th a curve

f i t t i ng procedure.

As a consequence of equat i on ( 71-7) one may st ate

A ( v a r ) c = A ( v a r ) D + A ( v a r ) R ( 5)

The change i n vari ance for one ci rcul ati on ( c) i s t he summati on of the

cont r i but i ons of r i ser and downcomer. Once t he system and downcomer Pécl et

numbers are deter m ned, the r i ser par t (w th the bends) can a lso be

c al c ul a ted. F r om (5 ) and ( 7 1- 7 ) i t f o l l ows th at

ps =_ —^L L _ ( 6)^ t 2PC - c?Pg

c D D cwhere t i s the mean c i rcu l at i on t i me, w th

c

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( 7)

77 Ov er a l l r e s ul t s

The Pé-val ues determ ned f rom the reci rcul ati on model may be pl ott ed

wi t h t he gas f l ow rat e as an i ndependent vari abl e. Al t hough the gas mass f l ow

r a tes di f f e r w d el y i n t he di f f e r i n g t op pr es s ur e modes s i m l a r P é- v a l u es a r e

f ound w th g i v en c i r c ul a t i on v el oc i t i e s . The r es ul t s a r e t h er e f o r e pl o t t ed

agai nst a charact eri st i c Reynol ds number.

( 1)

The charact eri st i c val ues f or a col umn l oop are deter m ned by anal ogy wi t h the

def i ni t i ons used by Bl enke (1979)

L. = 2L„ ( 2)

( i = R, D) (3)2

c

c

1

QL

■ wd 2

(A )

2~ir

For our col umn w t h changi ng cross sect i ons equati ons ( 2) and (3) must be

general i zed (appendi x 77. 1) .

L = T l . ( 5)c ' V il

d L = 2Vd. L J ( 6>c c V i i

i n f i gure ( 77. 1) i s the mean of three determ nat i ons .

■ J±±: O v e r a l l P é - v a l u , l e y n o l d s - n u a b e r . E a c h p o ;

The f i gure shows t hat:

- Di spers i on decreases somewhat w th i ncreas i ng c i r cu lat i on vel oc i t y but i n

the range s tudi ed a Péc l et number of 75 i s t ypi ca l for the operat i on as a i r

l i f t . Thi s i s about the same val ue as f ound by Wei l and ( 1978) w th a

Reynol ds number about one thi rd t he pres ent.

- W t h downcomer ai r added t he di spersi on i ncreases somewhat.

- There i s no s i gni f i cant d i f f erence bet ween operat i on at normal pressure and

at l ow pressure.

One may concl ude that ;

- The m x i ng rate i n the co lumn loop i s re l at i vel y h igh (B l enke 1979).

- The presence of a i r i ncreases t he m x ing r ate, a l though m x i ng decreases

w th i n c r easi ng ci r c u l a t i on v el oci t y .

The Pé- number def i nes the di spers i on coef f i c i ent

( 7)

which may be pl ot t ed agai nst a character i s t i c super f i c i a l gas vel oc i t y . The

barometr i c ( output) va lue or t he i nput va lue of the super f i c i a l gas vel oc i t y

are commonl y used to descri be and compare data but t hese are not rel evant . I t

i s shown i n appendix (77.2) t hat a l ogar i thm c mean is a su i tab l e bas i s f or

th e c ha r acte r i s t i c v el oci t y

( 8)

164 77

rn!

/N

cq 177-

/V 721

B i r 7 b

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where v i s the barometr i c super f i c i a l gas vel oc i t y based on the

character i s t i c d iameter ,

h, = p , / PTg, the barometr i c pressure,

H = the hei ght o f t he l i qui d level (m and

h = p . / PTg, the top pressure.

A corr ecti on i s made for the gas di sengagement part of the column ( see

appendi x 7 7. 2 ) . The data are pl ot ted i n f i gure ( 7 7. 2 ) . A s t r a i g ht l i ne w t h

sl ope 1/3 was drawn t hrough the poi nts

D = 1. 22 <g<v Gs c ) 1 / 3 ( 9)

Thi s may be compared w t h equat i on ( 7 2. 8 ) . The const ant ( 1.22) must then be

cor rected s i nce for bubbl e co lumns the character i s t i c l ength i s t he co lumn

hei ght . Even the cor rected val ue (0 . 63) i s about tw ce that f or bubbl e co l umns

( 0 . 3 5 ) .

The data w th downcomer a i r i n ject i on are a l so shown i n the f i gure.

Al t h oug h the ch ar ac te r i s t i c s u pe r f i c i a l g as v el oci t y i s no t we l l de f i n ed i n

thi s case, t he tr end i s t he same. Data of Wei l and are I ncorporated I n the

f i gure an' 1 show the same tr end but w t h a much l arger constant ( 3. 3) ( d =

0.112 m) . Note that these data f o l l ow f rom a near l y constant Pé-number , wh ich

means t hat t he dependence on the gas fl ow r at e i s t he same as that f or t he

c i r c u l a t i on v el oci t y ( s ect i on 5 2 ) . A Pé- number of 75 means, w t h equati on

(72-7)

0

\ 1A 6

_ L «

a io

77.3; E nv t l o

(10)

and hence t he measur ed dependence on gas f l ow r at e was to be expect ed. However

mor e data are needed t o veri f y t he dependence on t he other paramet ers .

Equat i on (10) predi c ts val ues that are 15 - 45% too low.

I t must be noted t hat the Pécl et- number determ ned i n thi s way is

not very accurat e. As a result of the measuri ng t echni que, the i nadequacy of

the t racer i n ject i on, whi ch i s not rad ia l l y homogeneous , and t he res tr i c tedval i d i t y o f t he reci rcu lat ed ax i a l d i spers i on model , both systemat i c and

c as ua l e r r o r s a r e p r es en t . S ys temat i c e r r o r s a r e d i f f i c ul t t o es t i ma te .

Compari son of the response tr ace extr emes wi t h those of t he reci rcul ati on

model l eads to a mean Pécl et- number w t h a st andard devi ati on of about 7. The

mean Pécl et- number was al so det erm ned i n a reproduci bi l i t y t est (10 measure

ments ) . The st andard devi ati on was 3.7.

M x i ng t i me of f ers a th i rd way of present i ng the resu l ts . Usual l y

m xers are c l ass i f i ed accord i ng to the t i me requ i r ed to ach ieve a cer ta i n

degree of homogeneit y. This i s descri bed i n ter ms of the parameter

(11)

and can be deri ved fr om the envel ope of the maxi ma and m ni ma of t he response

curve ( f i gure 77. 3)

P T(12)

166 77

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F l R u r e 7 7 . 4 ; A b s ol u t e a n d r e l a t i v e m x i n g t i mes ( 5 %l n ho mo g en e l t y > f o r t h e ex p er i men t a l c o l u mn .

wher e

t * = - 3 (13)m t c

i s t he re l at i ve t i me requ i r ed to achi eve a cer ta i n degree of m x i ng. Equat i on( 12) of f ers a reasonably accurate cor r e l at i on of the l ocat i ons of the ex tr emes

calcu l ated w th t he rec i r cu lat i on model . The m x i ng t i mes for 5% i nhomogenei t y

have been cal cul ated using equati ons (12) and ( 13) and pl ott ed i n f i gure

( 7 7 . 4 ) . I t takes about 7 c i r cu lat i ons to get th i s degree of m x i ng and hence

th e m x i n g t i me decr eas es w th i n cr eas i n g ci r c u l a t i on v el oc i t y . Cox e t a l .

(1980) repor t that 15- 20 c i rcu l at i ons were r equ i r ed to d i sperse a t racer peak

i n a 61 m hi gh U- t ube col umn wi t h 126 mm pi pe di ameter . A r el ati vely smal l

contr i but i on of the bott om and head sect i ons and t he very h i gh aspect r at i o

(350) resul t i ng in near p l ug f l ow are respons i b le f or t he l esser m x i ng rat e.

These aspects must be consi dered f or pl ant desi gn and scale up.

78 Downcomer and riser mixing

The resul ts for the downcomer determ ned wi t h the t rans fer f unct i on

f i t t i ng method are g i ven in f i gure (78.1) as a f unct i on of the downcomer

Re-number. Al l these measurements wer e done w t hout downcomer ai r i nj ecti on.

The di f f erence i n Pecl et- number bet ween the nor mal and l ow t op press ure modes

Fé-values. Each po in t i s th ree measurtments.

g -values (equat ion 76-6). Each po in t i s based on th ree measurements.

i s probabl y a resul t of t he carr y-under phenomenon. W t h barometr i c t op

pressur e t he ai r di sengagement at t he head of the col umn was not as good as

w t h low top pressure.

The ca lcu l ated Pé- va lues for the r i ser sect i on (and bends) are gi ven

i n f i g u r e ( 7 8. 2 ) . I t shows - i n compar i son w th f i gure ( 77. 1) - the smal l

i n f l uence of t he downcomer sect i on on the tota l m x ing rat e. Th is i s l argel y a

r es ul t of t h e r e l a t i v el y s ho r t r es i d en ce t i me of t h e l i qu i d i n t h i s s ec t i on:

23% of t he tot a l c i rcu l at i on t i me. On the other hand d ispers i on in the

downcomer sect i on seems t o be much hi gher t han t hat i n the r est of t he col umn,si nce the Pêcl et- numbers are i n both cases about 50.

To ge t a b et t e r i dea o f t h e c ont r i b ut i on of t h e di f f e r ent s ec t i ons ,

t h e s i n gl e p hase d i s p e r s i on i s c a l c ul a ted w th eq ua t i on ( 7 2 - 4 ) f r om Tayl o r

(1954) , as appropri ate f or t he actual Re-number range (Levenspi el , 1972) . The

i nf l uence of bends was accounted f or usi ng the model of Par k and Gomezpl at a

(1971) ( equat i on 74-1) w t h the constant K ex tr apol ated f rom the data g iven by

th es e au th or s . I t pr edi c t s a di s pe r s i on i n tens i t y f o r t he r i s e r s ec t i on o f

about one and a hal f t i mes t hat w t hout bends.

The rati o of t he measured two phase f l ow di spersi on t o that

c al c ul a ted f o r s i n gl e ph as e di s pe r s i on i s p l o t t ed i n f i gu r e ( 7 8 . 3 ) . I t shows

an al most thr ee fol d i ncrease excepti ng the ai r f ree f l ow i n the downcomer.

Even here t he di spers i on i s approxi matel y doubl ed, probabl y as a r esul t o f the

var i ous obstr uct i ons and t ee pi eces i n the downcomer , together w th the

i nf l uence of the head region on the top part of the downcomer sect i on. The

i nc r eas e of t h e d i s p er s i on r a t i o w th Re i s l a r ge l y a r es ul t o f t he ex t r a

polat i on of t he model of Park and Gomezpl ata pr edi ct i ng the ef f ects of the

bends . The Tayl or equat i on (72- 4) which appl i es to s t ra i ght p ipes predi c ts an

al most const ant val ue of Pé, and i n fact t his behavi our was f ound in our

^ n s n a

168 78 169

8 Mass transfer

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Fi gure 78 . 3 : Rat i o of two phase to s ing l e phase di spers i on (eq . 72-4 and 7 4 - 1 ) .

experi ment i n the l oop. The di f f erence i n downcomer r esul t s shows t hat event he smal l amount of ai r carr i ed downwards duri ng at mospheri c pr essure

operat i on i s suff i ci ent to i ncrease the di spers i on al most t o the val ues found

i n the r i ser . However i t i s c lear that t he cont r i but i ons t o the to ta l m xi ng

i n each sect i on are of s i m l ar magn it ude. Thi s i s the most s i gn i f i cant resul t

o f these m xi ng studi es.

79 Conclusions

- The Pécle t - number fo r an a i r - l i f t reactor o f the p resent d i mensi ons i s

around 75.

- Thi s i mpli es that about 7 ci rcul at i ons are needed for a m xi ng t i me»

corr espondi ng to a homogenei t y of 95% to be achi eved.

- The di spersi on i s about 2.5 t i mes that expected f or a si ngl e phase f l ow i n a

pipe on the basi s of the Taylor equat i on modi f i ed wi t h t he model of Park and

Gomezpl ata to account f or bends.

- The dependence of the overal l d i spers i on coeff i ci ent on the gas f l ow rate to

the power 1/3 i s a result of t he same dependence of the ci rcul at i on vel oci ty

on gas f l ow ra te and fo l l ows di rect l y f rom the def i n i t i on o f t he Pécl e t -

number and expressi ons f or t he ci rcul at i on veloci ty i n bubbl e col umns.

80 I n t roduct i on

The deep shaft system i s an aerat or. Theref ore oxygen tr ansf er r ate

i s an essent i a l per f o rmance var i abl e . The t ransfer o f carbon di oxi de f r om the

l i qui d i s als o i mport ant. From the vi ewpoi nt of the hydrodynam c behavi our of

a bubble col umn l oop w th downcomer ai r i n j ect i on, gas t ransfer rate i s

re l evant because i t may i nf l uence t he c i r cul a t i on dr i v i ng fo rce .

I n the experi mental col umn used in t his work mass t ransfer i s too

smal l to i nf l uence the hydrodynamcs and has t heref ore not been i ncorporated

i n the c i rcu l a t i on mode l s . However i n la rge col umns th i s wi l l be di f f e rent ,

al t hough the ef f ect i s assumed to be smal l because the mass t ransf er r ate i s

greatest i n the l ower part of the col umn, where t he cont ri but i on to the

c i r cul a t ory d r i v i n g f o r ce i s sma l l e st .S i nce l i ter ature data on bubbl e column l oops i s l i m ted, and because

their i mport ance as an ai rl i f t reactor , use is made of t he uni que opport uni ty

to det erm ne oxygen t ransf er i n such a devi ce. Moreover a model f or mass

tr ansf er whi ch consi der s the hydrostat i c eff ects has been devel oped and

veri f i ed. Thi s can be used to extend the hydr odynamc model s.

81 Model s for t he mass transfer measurement

Vari ous met hods and model s ar e commonl y used t o deter m ne mass

tr ansf er i n bubbl e col umns. Two gener al appr oaches may be r ecogni zed. Fi r st

the axi al d i sper si on model (Deckwer et al . , 1974) whi ch i n the general case isbased on the f ol l ow ng bal ances f or the two phases:

-T- Ec = - T —( v E C) + - ( £D ~) + k a(mc - c)at d z L ' J z dz L G

( 1)

d 3 9 3c G— ac = - » - ( v a c ) + -x—(aD - =— - k a(mc - c )dt G 3z v G G 3z G 3z L G

170 81

w t h the assumpti ons

a) Mass t r ans fer ra te (k a ) i s con t ro l l ed by l i qui d phase res i s tance.

b ) There a re no radi a l var i a t i ons o f concent r a t i on or o ther p roper t i es .

c ) Mass t rans fer th rough the f ree l i qui d i n ter f ace i s neg l i g ib l e ( Merchuk and

S t e i n , 1980).

These bal ances are of ten f urther s i mpl i f i ed i n two di f f erent ext reme ways.

171

dis pers i on vol ume. The assumpti on of a complet el y m xed gas phase l eads t o

CG = CGi ( 6 )

whi ch i mpl i es t hat the oxygen concent r ati on i n the gas i s assumed const ant and

equal to the concent rat i on in t he in j ected gas ( c , , . ) . To al l ow fo r the

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1. The assumpti on t hat the bubbl e col umn i s compl etel y m xed i s made for t he

anal ys i s of dynam c measurements. Thi s approach wi l l be d iscussed l ater .

2. The NTU method, i n whi ch t he phases are assumed t o f l ow i n pl ug fl ow w t h

no axial d i spers i on (D = D = 0) . For t he st eady st ate measuri ng techni que

based on the determ nat i on of ax ia l concent r a t i on p ro f i l es and us ing the

assumpt i on gas phase (oxyg

equati ons ( 1) t hen become

dc . ,. _ , _ . . . ( 2 )

I f the oxygen concentr at i on i n the gas phase i s not constant , but gas phase

d ispers i on neg l i g i b le , th i s di f f e ren t i a l equat i on i s coupl ed to (Deckwer ,

1976)

dz Cs G L G

Deckwer et al . ( 1974) do not consi der a var i abl e vol umetr i c l i qui d phase

t r ans fer coef f i c i ent l ea . Hi gher k. a-val ues i n short er col umns are expl ai ned

i n terms of i ncreased mass t ransfer dur i ng bubbl e f ormati on (Deckwer et al . ,

1980). Alvar ez-Cuenca et al . (1981) present a t wo-zone model to account for

t h i s e f f e ct .

When dynam c measurements are used car e i s t aken t o keep the hydro

dynam cs constant and hence the voi d f ract i on i ndependent of t i me. W t h a

comp lete l y mxed l i qui d phase i t f o l l ows f r om (1) that

£-5— = k a ( mc - c ) ( 4 )a t L G

but the more general l y used form i s ( Lopes de Fi guei redo and Calderbank, 1979)

3 1 = k L a ( mc G ~ C ) ( 5 )

where l ea i s now based on the volume of aerat ed l i qui d i nstead of the tot al

deplet i on of oxygen i n the gas phase, i f necessary, the fol l ow ng mass balance

i s used ( Lopes de F i guei redo and Cal derbank, 1979) (compare w t h ( 1))

dc v

° "dF = T T ( c G i " CG ) " \ a ( m C G " C ) ( 7 )

f o l l ows f rom the di spers i on vo lume (V / A ) . The compari son made by the ci t ed

authors between model s based on equati on (6) or ( 7) i ndi cates t hat i t i s not

necessary to take gas phase dynamcs i nto account f or l e a val ues l ower t han

0 .02/ s .

Dynam c measurement s i n ai r - sparged syst ems f ol l ow the var i at i on of

the concentrat i on of oxygen di ssolved i n the l i qui d phase ( c) produced by a

sudden change i n the tr ansf er dr i vi ng for ce, of ten by a step change i n the

composi t i on of the gas enter i ng the apparat us. The i nterf ace ni tr ogen

t r a ns por t w l l i nt e r f e r e w t h i t . T he ef f e ct i s ag ai n n eg l i g i b l e f o rk a -va l ues be l ow 0. 02/ s ( L i n ek et a l . , 1981) . Furt hermore, the response of

the oxygen sensor must be much fast er t han the tr ansfer proces s. These ef f ect s

were di scussed by Ruchti , Dunn and Bourne ( 1981) , whi l e Kei t el and Onken

(1981) d i scuss poss i b l e error s i n both the dynam c and steady st ate methods to

deter m ne mass tr ansf er i n gas- l i qui d d is pers i ons. As a concl udi ng remark i t

must be noted t hat the NTU- method based on non- i sobari c equati ons l eads t o

val ues t hat are conservat i ve, whi l e the wel l m xed approach based on i sobar i c

equati ons may l ead t o val ues that are too hi gh.

82 Bubbl e col umn l oop react ors

I n ter ms of model l i ng, bubbl e column l oop reactor s are i ntermedi ate

between bubbl e col umns and st i r r ed tanks. As a result of t he l ow l i qui d

super f i c i a l ve l oc i t y l i qui d di spers i on i s no t negl i g ib l e i n bubbl e co lumns .

Also the gas res i dence t i me i s l ong compared wi t h that i n a loop reactor . On

the o ther hand gas r es idence t i me i s r e la t i vel y shor t i n s t i r r ed tank

reacto rs , wh i l e m xi ng i s a l mos t comp let e .

172 82

The bubbl e col umn l oop may be char acter i zed by

- an appr oach to pl ug fl ow of both phases,

- s ma l l c onc ent r a t i on di f f e r enc es r es ul t i ng f r om th e f a s t l i qui d r ec i r c ul a t i on

( Wei l and, 1978) , and

- short gas r esi dence ti me compared to the character i st i c t i me of the mass

tr ansfer process ( i n the experi mental column t . t f 20 S compar ed t o

(k a) greater t han 50 s ) .

i i .

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L i n et a l . (1976) determned mass t r ans fer i n a 3 metre h i gh ai r l i f t

react or by a dynamc met hod. A mean k a-val ue was based on fi ve i ca- val ues

measured wi t h probes at di f f erent posi t i ons i n the col umn. Wei l and ( 1978) al soused t he dynamc met hod but det erm ned a mean I I a f rom the mean concentr ati on

reg is t ered w th s i x probes at d i f f erent l ocat i ons i n a 10 metre h i gh ai r l i f t

t ower. The resu l t s were comparab le w th t hose f r omLi n et a l . (1976) for

s i m l ar super f i c i a l gas veloc i t i es (based on r i ser c ross sect i on and s tandard

pressure).

Mass t ransfer has al so been det erm ned w t h t he dynam c method in

thi s work. The r esul ts are moreover compared w t h a non- i sobari c model whi ch

takes i nto account the ax ia l pressure and vo id d i s t r i but i ons i n the r i ser .

Thi s model a lso predi c ts the ax ia l oxygen concentr at i on di s t r i but i on, wh ich i s

compared wi t h measur ement s.

83 I sobari c dynam c met hods: experi mental

Oxygen concent r ati on was measured w t h a si ngl e pol arographi c oxygen

probe at the bot t om of the unaerat ed downcomer. Thi s l ocati on ensured a

conti nuous r efr eshment of the l i qui d f i l m adj acent to the membrane and gave a

probe ti me const ant l ess t han 10 s.

Measurements were made f or f i ve di f f erent c i rcu l at i on vel oc i t i es

( v L s D = 0.75; 0.90; 1. 05; 1.20; 1.35 ms) w t h top pressures of about 1.0 or

0.1 bar abso lut e. The co lumn was f i l l ed w th 0 .60 m tap water .

The dynam c met hods are based on measur ement of t he response of a

st ep change i n the mass t ransf er dr i vi ng for ce. Thi s may be done by changi ngthe composi ti on of the i nlet gas ( method I ) or by a change i n the overal l

pressur e ( method I I ) , to produce a dif f erent oxygen saturat i on concent rat i on.

The fi rs t method ( I ) was achi eved by changi ng the gas suppl y fr om

ni t rogen to a i r . Because of the f i n i t e gas res idence t i me and l i mt ed l i qui d

di s pe r s i on th e d i f f e r enc e i n i ni t i a l a i r c on tac t t i me of t h e l i qui d i s c l ea r l y

v is i b le i n the record of oxygen concentr at i on w th t i me ( f i gure 83. 1a) . I t was

Figure 83.1 : Typical form of the re cords of oxygen concent:

(changing overall pressure) (b) . The f lat level In the f ir .iods I (nitrogen) (a) and II

even possi ble to measure the l evel of oxygen concent rat i on aft er onec i rcu l at i on, wh ich a l l owed a th i rd method ( I I I ) to be used to determ ne the

k a. Thi s may be consi dered as a st eady st ate method i n whi ch oxygen fr ee

l i qui d moves past the oxygen probe and t hrough t he aerated r i ser back t o the

probe. The aerat i on constant f ol l ows si mply f rom

( i )

l i qui d vol ume and cm the f i nal concentr at i on, measured wi th the cal i brated

probe. That thi s method can be used conf i rms the l ack of compl ete m xi ng i n

each phase whi ch i s r eal l y assumed i n t he dynam c met hod.A change i n the overal l pressure, method I I , was achi eved by usi ng

the vacuum l i ne or venti ng the t op of the col umn. Such a change al so

i n f l uences the vo id f ract i on and t he hydrodynam cs . The gas i n ject i on f l ow

rate was s i mul t aneous ly ad justed to mai nta i n the l i qui d c i rcu l at i on veloc i ty

as constant as poss ib l e . As i s di scussed previ ous l y t he l i qui d c i rcu l at i on

rate i s determ ned by the voi d fr acti on in each l i mb of the l oop. A gi ven

ft\V \

method I (nitrogen)barometric top pressure

9 \ A

method Ilnitrogenllow top pressure

vLs D0 7 50 910 51.2

m/s)

AeVK

S3 175

l i qui d super f i c i a l vel oci ty cor r esponds to v i r tua l l y the same average vo id

f ract i on in both barometr i c and low top pressure operat i on. The or i g inal

c i rcul at i on rate was ret r i eved w thi n about 20 seconds. The great advantage of

th i s method i s t he overal l change i n mass t ransfer dr i vi ng f orce. The response-

curve i s more st able t han that of method I i n consequence ( f i gure 83. 1b) .

Bot h methods ( I ) and ( I I ) are based on equati on ( 8 1. 5 ) . Wei l and

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\ V % N

\ \ \ v_ ^

x\ V \ \

A N Nl\ \ \ \

\ \ \

\ " \

V

13 5

JC \ N A b

\ \ \ \

■ l\ \ . \

method Hfrom low to barometric

top pressure

£ ê

V\

\. \

\

method IIfrom barometric to low

top pressure

\ \A o

\ \ \ '

\ \x\

v

\

Pteure 63.2: Oxygen c.

p l o t , d esp i t e th e sea

c o nc e nt r a t i o n l e v e l .

. o n d i s t r i b u t i o n s I n t i me a r e ex p on en t i a l a nd sh ow s t r a i g h t l i n e s a s a se m

r e su l t o f t h e r e c l r c u l a t i o n . T h e t i me t - 0 c o r r e sp o nd s t o 2 0 2 o f t h e tin

def i ne an average pressur e:

yp

e

where p i s t he average pressure, y the mol f ract i on of oxygen i n the gas2 2phase and H the constant of Henry (m / s ) .

Equati on ( 81-5) may be sol ved to give (c = c f or t = 0 and c = c

for t = «>

c - c - k a t1 - c* « — — — = e ( 3)

W th a non- l i near r egress i on method based on the l east square cr i ter i um

(Warmoeskerken, 1985) i s i t poss ib l e to determ ne L a w t hout t he need to

know the concentrat i on aft er i nf i n it e t i me. Moreover th i s method provi des awei ghti ng for the measured response curve i n the essenti al part . A st art i ng

concentr at i on of about 0.20 c i s used in order t o avoi d any l i nger i ng ef f ects

of t he deoxygenati on t echni que ( Brown and Bail l od, 1982).

Typi cal response curves are shown as a sem- l og pl ot i n f i gure ( 83. 2)

f or the four d i f f erent cases: methods I and I I and wi t h l ow or barometr i c t op

pressure. Note t hat method I I for l ow top pressure i s a desorpti on process.

The di mensi onless c oncentr ati on c A was calcul ated f rom the measured dissol ved

oxygen values w th the i n it i a l and f i nal concentr at i ons est i mated by the

non- l i near regressi on comput er program The same program showed that the probe

dynam cs could be negl ected. Thi s i s in accordance w t h the cr i t er i on that the

probe t i me cons tant ( i ncl udi ng f i l m d i f f us i on ou ts i de the p robe) mus t be l essthan the inverse k a-val ue (Ruchti et al . , 1981) . The f i gure shows t hat the

response curves are s t ra i gh t but f o r t he osc i l l a to ry var i a t i ons as a resul t o f

the ree l r cul a t i on.

176 83

too hi gh values t hat f o l l ow f r om the m xi ng assumpti on. The resul ts may be

descr i bed w th the f o l l ow ng f i t t ed cu rve, d rawn I n the f i gure ,

V0.83v„Gsc CD

where t he exponent agrees w t h t he val ue of 0,82 menti oned by Shah et al .

( 1982) f or bubbl e col umns.

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F i g u r e 8 4 . 1 ; Measured k,a a s fu n c t i o n o f t h e characteristic superficial gas vel oc i t y ( which i s a measure o f

overall dissipated power) . Comparison of the t h r e e dynamic methods. Drawn l i ne i s aquation ( 8 4 - 1 ) .

F igure B4.2: Compar i son o f measured l a (eq. B4-1) w th data o f o ther l oop reacto rs and w th bubbl t co l umns.

84 Isobaric dynamic methods: results

The measured k a val ues for the t hree dynam c methods ar e pl ott ed

together i n f i gure ( 8 4. 1 ) . I t must be emphasi zed t hat t hese val ues are based

on the tot al vol ume of t he aerated l i qui d. Secondl y t hese are mean val ues

def i ned by the measuri ng method used. Met hod I i s si m l ar to t hat used by

Wei l and ( 1978) and Li n et al . ( 1976) . The result s are presented as f uncti ons

o f t he character i s t i c super f i c i a l gas vel oc i ty based on the total cross-

sect i onal area of the col umn ( see sect i on 77) .

The resul ts of the d i f f erent methods essenti al l y agree, al t hough thespread i n the dat a f or the l argest val ue amounts to some 20% Thi s corresponds

w t h t h e l ar g es t l i qui d v el o ci t y ( 1 . 35 m s i n t he downcomer ) , when t he col umn

no l onger works wel l because of extr eme t urbul ence i n the gas di sengagement

space l eadi ng to carr yunder. Because the mean depl eti on of oxygen i n the ai r

can be esti mated to be as much as 15% i n the wors t case (method I I I ) , the

r - r - su ts must be cons i dered t o be re l a t i vel y conservat i ve , i . e . l ower than the

Compari son w t h the data of Wei l and (1978) for a comparabl e col umn

and measur i ng method ( I ) ( f i gure 84. 2) shows that the resul t s agree very we l l .

The mass t ransfer dat a of L i n et al . (1976) ar e f ar l ower. The agreement notedby Wei l and where t he Independent vari abl e was based on ri ser cr oss sect i on and

s tandard p ressure , i s acc identa l . The data o f L i n e t a l . shoul d be cor rected

for the medi um (f ermentat i on medi umw th 2% ethanol and pH * 4. 6) and

measuri ng met hod. I t may be not ed here t hat the dat a of Li n et al . f or t wo

dif f erent geometr i es f al l together when us ing the charact er i st i c val ues of

v . The same hol ds for t he data f rom J ackson et al . (1975), whose result s

may be presented w t h a s i ngl e l i ne for bubbl e col umn l i qui d heights between 4

and 21 m The correl at i on gi ven by Hei j nen et al . ( 1982)

c l g / m ü l

f

l

7

; r ^ - ^

.

A ^ _ ^

, u o . l . 0 5 m l .

-

) 100 200 300 40O 50

clg(m3 )

|

,

f .00 20 0

—- *-">£ z^^~^~

injection

30 0

I

„ c o m

■» « . r ^ w ^ / v

r »

-AGO KX

Figure 34.3: Examples of chart records . Comparison between r iser air only and ulth downcomer air (0.1 g/sr i s e r a i r ) .

k a = 0. 32 v° - ' ° (2 )L GSD

i s mai nly based on l i ter ature data f or bubbl e col umns about 2 m hi gh, whi ch

has been al l owed fo r i n f i gure (84 .2 ) ( f acto r 1 . 1 ) . Bubb le co lumns g ive h igher

F i gure ( 84. 3) g ives shows oxygen concentr at i on response curves w th

% . ypMO) . ^ ( 6)

e

The speci f i c su r f ace a rea i s

a = 6a/ d32 ( 7)

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the in j ect i on of downcomer ai r . Despi te the consi derabl e noi se i n the s i gnal ,

even w t h the l ow i n ject i on hei ght (x* = 0.28) the col umn l oop att ai ns

k a- val ues that are hi gher t han those of bubbl e col umns.

85 A non- i sobari c model

I n th i s sec t i on a mode l i s presented wh i ch , w th i n cer t a in

assumpti ons, account s for t he axial pressure var i at i on and i ts i nf l uence on

the var i abl es .

Assum ng a cons tan t l i qu id vo l ume, the pressu re var i a t i on i n the

r i s e r i s

( 1)

( 2)

z* m z /L (3 )

and L the unaerated l i qui d hei ght (10. 5 i n) , p the top pr essure and E the

mean l i qui d volume f ract i on ( i n the r i ser) between z = 0 and z. Equat i on (1)

may be wr i t ten as a pressure rat i o

P P ( 0 ) p ( 0 ) W

The satur at i on concentr at i on i s pressure dependent accordi ng to Henry' s l aw:

Assum ng, as i n the f o rego ing sec t i ons , that the mo l f r ac t i on y i s cons tant

For an i deal gas w th v const ant , the void f ract i on may be approxi mated by

a ( z ) = a ( 0 ) / p * ( 8 )

Assum ng no i nteract i on between the bubbl es, i . e. no coal escence, but al so no

d ispers i on, then

d 3 2 ( z ) = d3 2

( 0 ) P * ~1 / 3 O )

descri bes the expansi on of the bubbl es. I f bubbl e shape does not change as the

bubbl es expand, equati on ( 7) becomes

, a( 0 ) , - 2 / 33

■

6

z£m

p

* e")We w l l cons i der a l i qu id vo lume el ement i n the r i ser t hat t ravel s w th no

di s pe r s i o n w t h v el o ci t y

en)l i

VLR dt

and def i ne the di mensi onl ess group

B* = k. 6 a ( 0 ) P ( 0) mi

Then, w th equati ons ( 6) and (10) , equat i on (81-5) becomes

dc* - 2 / 3i f r - B* P* <c* " P * ) (13 )

wher e

c* = c/c (0) (14)

W th e constan t (wh ich i s reasonabl y t rue i f a i s 10%or l e ss ) , t he gr oup B*i s a cons tan t .

180 35

B * =- Ü ! L - P ( 5 M0 ) ( 1 5 )vLs R P l , gd32 ( 0)

To use the anal y t i cal so lu t i on o f t he di f f e ren t i a l equat i on (13) i s about as

compl i cated as a di rect numeri cal appr oach. The numeri cal method descri bes the

oxygen concentr at i on of a vol ume el ement of l i qui d t ravel l i ng through the

whol e col umn w t h al l owance for a resi dence t i me i n the downcomer, where

181

86 Non-i sobari c methods: exper i mental

The non- i sobar i c mode l i s f i r s t f i t t ed to t he dynamc response

(method I ) . The group B* was based on the fol l ow ng val ues. The in i t i a l bubbl e

di ameter f ol l ows f rom secti on 29 and was vi sual l y checked t o be about 6 mm

The void f ract i on for z = 0 and the mean l i qui d vol ume f ract i on fol l owed f r om

the steady st ate model , chapter 5. The mass tr ansf er coef f i c i ent k_ i s t aken

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concent r a t i on i s kept cons tan t . As such i t w l l desc r i be the oxygen

concentr at i on at t he l ocati on of t he oxygen probe w th t i me i nterval s equal t o

the c i r cul a t i on t i me. Fo r l ong t i mes equi l i b r i um i s reached and t he s teadystat e axial concent rat i on di st r i buti on is g i ven. The method may be used to

adapt the constant BA to t he probe measurements or to predi ct the

concent r a t i on d i s t r i but i on w th p lausi b le val ues f o r t he cons tan ts .

100 200 300 400 50O

Figure 86 .1: Example of the comparison of measurements with t he non -isoba ric model. The model Is f lLLed to

the measurements with a f ac to r ( kR) In the k.a. Barometr ic top pressure.

c l g / m 3 l

0 TOO 200 300 'tOO BOO

Flji- jre 86.2: As f igure 86.1.

to be 0.45 mm s ( Hei j nen et al . , 1982). We may defi ne now a corr ecti on fact or

k whi ch fol l ows f r om the f i tt i ng procedure and is a measure f or the

predi ct i on accuracy

_ B* f r o m f i t

Fi gures ( 86. 1) and ( 86. 2) show the compari son bet ween measur ement and model

f o r the m ddl e l i qui d vel oci ty (v =1 . 05 m s) and the two top p ressure

modes. The hor i zontal part s i n the model l i ne correspond to the l ocati ons i n

the downcomer sect i on where the oxygen probe i s si t uated. The measurements are

the same as t hose used w t h t he isobari c dynam c met hod.

The model i s t hen compared i n the stat i onary s i tuati on w th

measurement s of the axi al oxygen concentr at i on di st r i but i on. Because of the

need t o measure smal l absol ute d i f f erences i t was necessary to work w t h a

s i ngl e probe, whi ch could be lowered to any posi t i on i n the r i ser . To ensure a

suf f i c i entl y hi gh l i qui d approach vel oc it y the probe has a smal l propel l er

st i rr er i n fr ont of t he membrane ( Van de Donk, 1981) . The ti me const ant of

th i s system was 10 s or l e ss . The probe was pr essure i ndependent.

The col umn was not desi gned t o al l ow these measurements w t h l ow top

pressure. The model was based on the B*-val ues f i tt ed by the fi rs t met hod.

The model was also extended w th axi al d i spers i on, us i ng the

Pécl et- numbers i n t he previous chapt er . Hi gh Pécl et- numbers however result ed

i n a second o rder d i f f e ren t i a l equat i on that i s too s t i f f f o r the numer i cal

method ( Runge Kut t a) to converge.

87 Non- i sobar i c dynam c method; resul ts

I n tabl e ( 87) t he resu l t i ng val ues f o r k a re gi ven fo r t he 10

di f f e ren t cases .

TABLE 87

VLs D( m s )

0.75

0. 9

1.05

1. 2

The devi at i on f

p = 1 bar

1.04

1.02

0.93

0.87

rom the model , expressed by k„

P t = 0. 12 bar

1.27

1.16

1.07

1.20

87 183

pred ic ted . The mass t ransfer coe f f i c i ent k. w l l a l so be g reater s i nce w th

l ow top pressur e t he greater gas expansi on cr eates more f resh sur f ace area

w t h corresponding much l arger val ues of tc .

The tendency f o r the f acto r k to decrease w th l i qui d vel oci ty

p robab ly resul t s f rom i ncreas i ng coal escence w th i ncreas ing vo id f rac t i on. I n

consequence the speci f i c surf ace area shr i nks, and the ef f ect i s apparentl y

not total l y compensated by the expected i ncrease i n mass tr ansf er coef f i c i ent

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1.35

mean:

1.13

1.00

1.24

1.19

These show

a) When worki ng w t h l ow top pressure a l ea-val ue about 20% above that

predi cted i s measured whi l st at at mospher i c pressure predi ct i on and

perf ormance agr ee.

b) There i s a tendency f or l ower k a- val ues i n the m ddl e of the l i qui d

vel oc i t y r ange.

I t i s di f f i cul t to expl a in t he di f f e rences f rom the mode l because of

t he compl exi t y of the assumpti ons and t he constant B A. However some r emar ks

may be st ated. The t ea-val ues measured w t h the dynamc method were si m l ar

f or the two top pressure modes. The model predi cts l ower speci f i c surface area

wi t h l ow top pressur e, s i nce the bubbl es are l arger and the same gas vol ume

f r ac t i on i s ma in ta i ned .

The hi gher than predi cted mass t ransfer may resul t f rom there being

more di spers i on than coalescence, s i nce the r i s i ng bubbl es become l arger than

the i r equi l i b r i um s i ze . Th i s l eads to a l a rger speci f i c su r face a rea than

i 6 7 ; H e i g h t d e p en d en c e o f I L 3 i n I o rdl ng co the non- I sobar i c mode l .

at l a rger l i qu id vel oc i t i es . Mass t r ans fer as a resul t o f car r yunder i s

thought t o be responsib l e for t he i ncrease i n k f or the hi ghest l i qui d

v el o ci t i e s.As an i l l u st r a t i on i n f i gure ( 87) t he predi c ted changes o f k a w th

posi t i on are shown. One may i ntegrat e t hi s dependence to get a mean val ue.

Thi s was done for the result s g i ven in t abl e ( 87). These mean lea- values were

18 ( + 1) %l ower t han those predi ct ed by the corr el ati on based on the dynam c

methods, equati on ( 84—1) . As al ready stated, t he assumpti on of a t otal l y mxed

l i qui d phase woul d l ead to val ues t hat are too hi gh.

88 Non- i sobar i c stat i onary method: resul ts

Af te r ten to f i f t een c i r cul a t i ons the cal cul a ted oxygen concent r a t i on

i s constant I n t i me but not i n pl ace. The l atter dependency i s compared w th

the measured axi al oxygen concentr at i on dis tr i buti on in f i gure (88 .1 ) .

Alt hough t he agreement i s good, i t shows t hat the real oxygen concent rat i on

dist r i buti on di f f ers somewhat i n the locati on of the maxi mum (where c = c )s

and t he boundary c ondi t i ons. The assumpti on i n the model t hat the

concentr ati ons are equal befor e and aft er t he unaerat ed downcomer ar e viol ated

i n prac t i ce by

a) t he desorpti on process i n the gas di sengagement sect i on of t he col umn

( whi ch i s account ed f or i n a model of Merchuk and St ei n ( 1981b) , but i s not

apparent i n the axi al d i st r i but i on measurement of Wei l and (1978)) ;

b) t he mass t ransfer i n the downcomer as a result of carr yunder of bubbl es;

c) t he rel at i vely h i gh mass t ransfer i n the nei ghbourhood of the ai r sparger,wh i ch i s p roduci ng f resh con tac t su r f ace. Thi s i s not i ncorpora ted i n ou r

measurements (nor i n the model of Merchuk and Stei n) but cl earl y present i n

t he resul t s of Wei l and who measured the concentr ati on bef ore and af t er t he

sparger.

The i nf l uence of sparger mass t ransfer must cert ai n l y be consi dered

si nce Al varez-Cuenca et al . (1980) c l aimed that up to 95%(!) of t otal mass

1,,''\~—

J*

— ^

—

■*-»,,

«,,'.

i

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8 . 1 : Ax i a l o x yg en c o nc e nt r a t i o n d i s t r i b u t i o n i n th e r i se r - C o mp ar i so i

i c t op pressure . The maxima o f the curves co r respond w th c = c .

vLs D[mi l l

1 351050 75

1 fl l07 r,0 4 V

F igure 8S. 2: As f i gure 68.1 but w th t he mode l adapted to I ncorpora te t he measured concentra t i on jump outsi de

th e r i se r , mo de l l e d w th a l o c a l h i g h ma ss t r a ns fe r a t t h e r i se r sp a r g er .

tr ansf er t akes pl ace in the sparger regi on, t hough of course th i s i s dependent

on sparger desi gn and the rel at i ve vol ume of that r egi on. I n our case the

sparger i n f l uence w l l be re l a t i vel y sma l l s i nce the sparger used i s des igned

to c reate bubb les o f equ i l i b r i um s ize w thout the in t r oduct i on of much

turbul ence due to j ett i ng or coal escence and di spers i on processes. A mni mum

val ue for L a at t he sparger may be cal culat ed us i ng the di f f erence i n

concent rat i on bet ween top and bott om The resul t i s around 0.4 s ( i f

rel ated t o 0. 05 m) whi ch agrees w th the measurements of Al varez-Cuenca et al .

(1980) . Thi s cont r i buti on may be i ncorporated i n the model (where i n fact th i s

descr i bes the combined ef f ect of absorpti on at the sparger and desorpti on at

t he col umn head). Thi s t hen resul t s i n the concent r a t i on pro f i l es gi ven in

f i gur e ( 8 8 . 2 ) . I t shows t hat the maxima i n the curves corr espondi ng w t h the

l ocal saturat i on concent rat i on c are at about z* = 0.2. Thi s conf i rms t hats

mass t r ans fer ra te a t a porous p la te sparger i s re l a t i vel y l a rge.

„ ' '

_

■ y ^ \

•

r

- *■ j

e d a x i a l o x y g e n .

C a n t ^ a . ■ t h f o r m a n d l e '

1 kLacs ~ -| t g * » * *

barometr ic! ^ > L top pressure

V Lj D . 1. 05 m/ S _ _ J > - _

F i g u r e SBA: C o m p a r i s o n of t h e p r e d i c t e d m a x i m a l l o c a l o x y g e n t r a n s f e r r a t e f o r v a r i a b l e o r c o n s t a n t k a .

Deckwer et al . (1974) di d not f i nd i t necessary to consi der that k a

m ght vary over the col umn hei ght when f i tt i ng measured prof i l es i n a bubble

column w th a comparabl e model . Thi s i s conf i rmed by the f orm of the curves

given i n f i gure ( 88. 3) appl i ed to the present col umn for barometr i c t op

pressure. However the di f f erent l evel s of t he curves show that d i f f erent

values f or k a w l l be found. Theref ore t he values as determ ned by Deckweret al . ar e - again - def i ned by the model used. For col umns w t h l arge gas

expansi on rat i os the f i gure shows t hat t he prof i l es also change in form

Moreover i t must be noted t hat the assumpti on of the i ndependence of

k ,a p redi c ts a t o ta l l y d i f f e ren t l oca l mass t rans fer ra t e . Assum ng an al mos t

const ant and l ow oxygen concentr ati on proport i onal to c t he mass t ransf er3 s

rate may be expressed as proport i onal to k ac (kg/ sm) .

I n f i gure ( 88. A) th i s i s compared f o r a var i abl e k_a( predi ct ed by the model ,

and a const ant k a ( average value ) . I n par t i cul a r t he t rans fer r a te near t he

bot t omwoul d be less w th a var i abl e k a . Tota l t r ans fer i s a l so l e ss , whi ch

m ght expl ai n why the mass tr ansf er ef f i c i enci es of the deep shaf t I n pract i ce

are l ower than were expected.

89 Conclusion

Conclusion

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Al t hough much more r esearch i s needed i n t he subj ect bef ore resul ts

w l l be concl us i ve, some conclus i ons may be drawn fr om the for egoi ng sect i ons.

- Oxygen t rans fer r a te in an ai r l i f t r eac to r i s l ess than that i n a ( t ower )

bubbl e co lumn, but w th a i r i n jec t i on in t he downcomer the t r ans fer w l l be

at l east comparabl e. Mass tr ansf er w l l then be more ef f i c i ent because l ess

compres si on power i s needed.

- The non- i sobar i c model presented predi cts oxygen tr ansf er r ate l ocal l y and

may be adapt ed t o extend the hydrodynam c model s• However i t has onl y been

ver i f i ed agai nst measurements w t h a barometr i c t op pressure and r i ser ai r

i n j e c t i o n. I n t hi s c as e t her e i s no s i g ni f i c ant di f f e r enc e i n t he f o r m of

the axi a l oxygen concent r a t i on pro f i l es w th o r w thout a var i abl e

k a- val ue, and th i s aspect could t heref ore not be tested.

- The i nf l uence of a l ocal l y h i gh mass t ransfer rat e at the sparger cannot be

negl ect ed. Desorpt i on at the col umn head may al so be i mport ant but was not

st udi ed.

- There was l i t t l e d i f f e rence between the lc^a - v al ue s r e s ul t i ng w t h t h r e e

di f f erent dynam c methods, based on the assumpti on of wel l m xed phases. The

k a- values cal culat ed f r om a model based on l i qui d p l ug f l ow were 18%

l ower . L i qui d p lug f l ow i s a reasonabl e approxi mat i on to rea l i ty .

Further work i s necessary to ver i f y t he non - i sobar i c model . The

v ar i a bl e l e a-val ue may be ver i f i ed w t h measurements of the axial oxygen

concent r a t i on di s t r i but i on w th l ow top pressu re . The i n f l uence o f sparger and

gas di sengagement sect i ons on mass t ransf er need more r esearch. I ncorporat i on

of a gas phase balance w l l probabl y i mprove the model. More i nformati on about

the bubbl e s i ze and bubbl e expansi on i s needed as wel l as some quanti f i cat i on

of the i nf l uence of bubbl e di spers i on and coalescence.

Introduction

This l ast chapter concl udes t he thesi s w t h some r emarks on scal e up

and the i nf l uence of t he l i qui d medi um on the rat e and st abi l i t y of t he

c i r c ul a t i o n.

The most i mport ant r esul t s of t hi s work are summari zed i n secti on 95.

Al l appendi ces are i ncl uded thereaft er .

Genera l i za t i on of the s teady s ta te model

The model presented i n chapter 5 i s specif i c to t he exper i mental

column and i s not general l y appl i cabl e. For scal e up the model can be modi f i ed

i n the f o l l ow ng ways .

1. The f r i ct i on number can be general i zed us i ng the express i ons der i ved in

chapter 1, assum ng negl i g i b l e l oss at the head tank. W t h an exter nal

downcomer :

F - 4 f ( m2 ^ + 125 + L/ ( dRm2) ) ( 1)

a s i m l ar t ype of exp ress i on (A15 -7 ) i s used fo r the case of an i n ternal

concentr i c downcomer .

2. I n jec t i on o f r i ser a i r can be at any arb i t r a ry he i ght .

3. For th i s genera l mode l f r i c t i ona l p ressure d rop i s i ncorpora ted i n the

cal cul a t i ons o f the pressure di s t r i but i on.

4. S l i p vel oc i ty i s based on the s imple model presented i n chapt er 3. For t he

r i ser t h is means

Ls R ( 2)

( compare equati on 35- 26) , wher e

188 9

( 3)

As a basi s for the bubbl e si ze i t i s assumed t han 6 mm bubbl es are

created at the downcomer i nject or whil e gas fr om the r i ser i nject or has

bubbl es of the same si ze as those present i n the l i qui d at t he

corresponding dept h. Rel ati ons f or coal escence and di spersi on and i njector

bubbl e si ze may be incl uded i f necessary. Downcomer sl i p vel oci t y ( see

92 18

condi ti on di d not apply for the experi ment al col umn used i n t he present work.

Cl ear l y t he operat i on of the Ay lesbury p i l ot p l ant i s not r epresentat i ve of

t he hydr odynamc behaviour of deep shaft col umns i n general . For compari son

the cal culat i ons are repeated w th a deeper i nject i on poi nt (30 m) and shown

i n f i gur e ( 9 2. 2 ) .

The axes are expressed both i n mass ai r rat e as i n power per vol ume

accordi ng to

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chapter 4) i s assumed to correspond to t he si ngl e bubbl e t erm nal veloci ty

w th a smal l a l l owance for the i n f l uence of l i qui d vel oc i t y :

( 4)

92 Geomet ri c scal e up

Any col umn may be calcul ated w th thi s general i zed model . For the

present thi s has onl y been done for those f ew systems f or whi ch dat a are

ava i l abl e .

The known detai l s of the pi l ot scal e col umn at Ayl esbury (60 mhi gh

U- tube w th 0 .125 m i nternal d iameter , see Cox et a l . 1980) are as fo l l ows:

a. a s i ngl e i n ject i on depth (20 m i s used.

b. the quanti ty of ai r needed . to cir culat e t he col umn contents was greatl y i n

excess of t hat needed f or t he process .

c. when a normal quanti t y of pr ocess ai r was i nj ected i n the downcomer , as

much as 10 ti mes t hat amount was needed i n the r i ser to ensure a st abl e

c i r c u l a t i o n.

d . the ci rcu l at i on vel oc i t y was around 0.7 m s.

Fur t hermore i n a d iscuss ion on such a p i l ot sca l e i ns ta l l at i on a

rat i o 50/50 of r i ser t o downcomer ai r i nject i on rate was ment i oned f or the

Ti l bury pl ant (130 m high col umn of 1.86 m di ameter w t h 1. 00 m di ameter of

t he int ernal downcomer: see Lock, 1982 (Robi nson, 1984) .

The predi ct i ons of the general i zed model f or t he geometr y and

operati ng condi t i ons of t he Ayl esbury plant are presented i n f i gure ( 9 2 . 1 ) . I t

shows t hat the col umn was probabl y operat i ng i n an area where an i ncrease of

downcomer ai r rate whi l e keepi ng the same r i ser ai r rate r esult s i n lower

c i rcu l at i on vel oc i t i es . ( Operat i on poi nt shoul d be the in tersect i on of the

0.7 m s l i ne and t he broken l i ne whi ch represent s W„ .GRi

. power per vol ume ( W m3)1QO 200

/ velocity (m/s)11 / injec tion height.

downcomer airinjection ratelg/s)

2 00( W / m 3 )

A

Calculate d o per ati n | cha rt for a 60 m high U-tube «1th 0.125 o dl;-er and downcomer (Aylesbury pilot plant).

»- powe r per v olume IW/m 3)100 200 300

and 20 in Injecti on deptl

(W/m3)

W„ p.

A t L PG P t( 1)

wher e

A i s the tot al cross sect i onal area of the col umn

L i s the col umn height

p/ pr i s a const ant

p and p. ar e the pressures at t op and i n ject i on l evel (compare equati on

- power per volu100

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A77.2- 12) ,

Fi gure ( 92. 2) shows t hat operat i on in downcomer ai r onl y mode i sp os s i b l e w t h t hi s l a r ge r i nj e ct i o n de pt h b ut w t h a n ai r r a t e abo ut t h r ee

t i mes and a power about 5 t i mes t hat used i n Ayl esbury. (For compari son a

power per vol ume of 100 W m corr esponds t o 0. 3 g/s ( downcomer) or 0.27 g/s

( r i ser) i n the Del f t exper i ment al column under l ow pressure (f i gure 5 7. 4 ) ) .

The i n j ect i on depth of the Ti l bury f u l l scale col umn is not known.

Hi nes e t a l . (1975) c la i m that i t i s poss ib l e to opera te l a rge scal e col umns

w th an in j ec t i on dep th as sma l l as 18 m F i gure (92 .3 ) i s cal cul a ted f o r an

i n jec t i on dep th of 30 mand i t can be seen that even th i s i s not su f f i c i ent l y

deep to al l ow stabl e operat i on w t h downcomer ai r only. For compari son cal cu

l a t i ons f o r t wo other i n jec t i on dep ths are p resen ted in f i gure ( 92 .4 and . 5 ) ,

wh ich show the rapi d ly i mprov ing s tab i l i ty w th i ncreas ing i n jec t i on dep th .

■power per vol ume (W m )50 100 150

F i g u r e 9 2 . 3 i Cal cula t ed opera t i on chare fo r a 130 m deep col umn o f 1.86 m di ameter and L. 00 r

d our, c o me r ( T i l b ur y f u l l s c a l e p l a n t ) . I n ject i on depth 30 in fo r bo th r i ser and dounconer .

0 250

Figure 92.4: As f igure 92.3 . Inject ion depth 40 B.

e» power per volume (W/m^)

TOO

(W/m3)

The fo l l ow ng i s no ted:

l . W t h i nc r e as i n g s cal e

a. power r equi r ements decrease

b. c i r cu la t i on vel oci ty i ncreases , and thus

c . voi d f rac t i on dec reases .

192

'■ »

1

i

i /T I ' 1

iC

a -■;

Y ■ ■ ■ ■: B,

0

-r

92 ]93

Söderberg f i t t ed hi s mode l to hi s data by ad jus t i ng the s l i p vel oc i t y

parameter. Us i ng 0. 5 m s for both r i ser and downcomer was sat i s f actory. Lower

val ues w l l be needed i f these ef f ec ts a re no t p resen t ( sec t i on 54 ) .

93 Influence of liquid properties

A second e f f ec t associ a ted w th t he scal e of the i ns ta l l a t i on a r i ses

f r om the l i qu id medi um used i n prac t i ce . I n par t i cul a r su r f actan ts wh ich

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.' /

/: 1 _ _ L 1 _

•-

•

. as! H

m

-•- ■-m

a2

g/ i

OerO.rgs daLe

— - » —

""""*""" ~

of the data of Söderberg (1980) for a 9.85 a high U-tube of 0.24

enerallzed mode l. Data give the range of measured val ues.

2. Downcomer i nj ecti on head l oss (whi ch was not scal ed) becomes much l ess

i mport ant as t otal column hei ght i ncreases. The gravi t at i onal pressure drop

term i ncreases al most proport i onal l y to scal e but the downcomer head l oss

i s expected to stay s i m l ar or even to decrease in absol ute t erms, s i nce

s l ugs a re l ess s t abl e i n w der duc ts and because ci r cu l a t i on ve loc i ty

i ncreases w th scal e .The r i ght hand unstabl e area whi ch corr esponds to t h is i n ject i on head l oss

i s t heref ore not encountered i n the f i gures presented here.

3. Operat i on w th downcomer ai r onl y general l y r equi res more ai r than that

needed fo r mass t rans fer , especi a l l y f o r sma l l e r i n jec t i on depths . I t may

be advantageous t o use r i ser a i r o r deeper i n jec t i on poi n ts i n the

downcomer .

A f i nal i l l u s t ra t i ve compar i son i s made w th t he data o f Söderberg

( 1980) ( 9.85 m high U- t ube w th 0.241 m diameter p ipes) : f i gur e ( 9 2. 6 ) .

Agreement i s good i n vi ew of t he general charact er of our model and the

rel at i ve i mport ance of i n ject i on head l oss in short col umns:

a. The downcomer i n j ector of Söderberg i s not desi gned to pr event s l ugs f rom

r i s i ng above the i n jec t i on po in t . Head l oss there fo re i ncreases as l i qui d

vel oc i t i es dec rease to about 1.1 m s , wh i l e f o r even l ower vel oc i t i es so

much ai r escapes t hat stabl e operat i on ceases ( see f i g ur e ) .

b. The r i ser i n ject or of Söderberg cr eat es bubbl es much l arger t han equi l i

b r i ums i ze . Thi s l eads to a sma l l e r r i ser d r i v ing f o rce and a vel oci ty l ess

than predi cted.

i n f l uence coal escence behavi our w l l be p resen t . Term nal r i se ve loc i ty i s

al so Inf l uenced by addit i ves in the water . Vi scous ef f ects are not consi dered

here si nce typi cal wast e water s may be regarded as non-vi scous. However,

f ermentat i on brot hs may be non-newtoni an and/ or vi sco-el ast i c and requi re

speci al tr eatment. To study the poss ib l e ef f ects of contam nat i on in the water

some exper i ment s were conducted w th di st i l l ed water and w t h t ap water

contai n ing 20 ppra octanol . Coal escence Is repressed i n the octanol sol ut i on,

wh i l e coa lescence ra tes i n d i s t i l l ed water a re s l i ght l y f as ter t han those in

tap wat er.

Resul ts are presented i n the f orm of operat i ng chart s , where t he

curves of constant veloc i ty are compared w t h the t ap water dat a presented

ear l i er ( f i gur e 51. 4) . when the r ig was f i l l ed w th d i s t i l l ed water

( f i gu re 93 .1 ) opera t i on w thout r i ser a i r was no l onger poss i b le . The mos t

s i gni f i cant d i f f e rences a re presen t w th h igher downcomer a i r ra t es . Thi s i s

consi stent w t h an i ncrease i n the coalescence rat e, whi ch woul d be expected

to become more r el evant at hi gher voi d f r act i ons. Coal escence phenomena w l l

be most i mport ant i n the upper part of t he r i ser where voi d f ract i on is

h ighes t . The l a rges t bubb les w l l have a h igher r i s i ng ve loc i ty and so there

w l l be a r edu ct i o n i n t h e r i s er d r i v i n g f o r c e r e s ul t i ng i n l o we r c i r c ul a t i o n

veloci t i es and a lar ger unstable operat i ng area.

The l arger bubbl e s i zes and i ncrease of i nhomogeneous ef f ects al so

r e sul t i n l a r ge r s l i p v el o ci t i e s as i s c onf i r me d i n f i g ur e ( 9 3 .2 ) ( c . f . f i g ur e

38. 7) . one case (W _. = 0.1 g/s and W = 1.2 g/ s) i s c l ose to the

operati ng l i m ts and can be seen to devi ate consi derabl y f rom the constant

s l i p veloc i ty which i s usual l y assumed. The average s l i p vel oc i ty i mpl i ed by

the voi d f r ac t i on data i s about 0 .35 m s i n d i s t i l l ed water , compared w th

about 0.27 m s i n tap water ( f i gure 38. 7) .

E f f ec ts i n a non -coal esci ng medi umare to ta l l y di f f e ren t ( f i gure

93. 3) . Ci rcu l a t i on ve loc i t i es are a lways l a rger and cor r espond ing ly there i s a

s mal l er un st a bl e a r e a. T hi s i s c l e ar l y i n l i ne w t h t he e xp ec t a t i o n of a

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O6O0090012O0

010601090112

••'DP.0 6OSI 2

0 1

W G D

O

060.91 2

i n r i se r o f e x pe :

ryt ute 93.3, Measure

■■ ' i eur e 9 3 . 4 : M ea su r e d a xi a l v oi d f r a c t i o n d i s t r i b u t i o n w th p n

c o mp ar e d w c h d i s t i l l e d wa te r . { Br o k e n l i n e b a ae d o n l i n e ar p n

a x i a l p a r a me te r . Oc ta n o l a dd i t i o n

i n de p en d en t v a r i a b l e ) .

22zl- The e ffect o f octano l on the bubbles. Upp< p i c t u r e : t a p i

196 93

reduced s l i p vel oc i ty. Moreover i t was observed that there was al most no

tendency to st r a t i f i ca t i on i n the l ower hor i zonta l sec t i on> whi ch i ncreases

the c i r cul a t i on s tabi l i ty . The i n f l uence o f the addi t i on of a sma l l amount o f

octanol on the voi d di st r i buti on Is shown in f i gure (93. 4) and i s compared

wi t h t hat f o r d i s t i l l e d wa t e r . T h e de vi a t i o n f r om t h e s t r a i g ht l i ne f o r

expansi on onl y,

a ( z ) _ p( 0 ) , . .a ( 0 ) p ( z ) u ;

94 197

d. Columns w t h i nter nal downcomers are t o be pref err ed above t he exter nal

type pr i nci pal l y because coal escence ( s t ra t i f i ca t i on or a t the inner s i de

of a smooth bend) at the bott om secti on i s prevented. This must be weighed

aga ins t the poss ib l e h igher f r i c t i on number ( t a l l co lumns w th d i ameters

bel ow 2 m) due to energy l osses associat ed w t h t he sharp r eversal of

l i qui d d i r ect i on at the bot tom of the co lumns .

95 General concl us i ons

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i s c lear . (For compar i son the resu l t i s al so g iven w th p ( 0 ) / p ( z) as

parameter , see als o f i gure 5 5. 2 ) . W th oc tano l there i s no devi a t i on ow ng to

t he l ack of i nhomogeneous ef f ects r esul t i ng f rom coalescence.

The ef f ect of the presence of octanol on the bubbl es i s demonstr ated

wi t h p i c t u r es ( f i gur e 93- 5) .

94 Concl usi ons on scal e up

a . F rom the l i mt ed data ava i l abl e i n the open l i te ra tu re i t i s conc l uded that

col umns are commonl y operat ed i n the area above t he cal cul at ed maxi mum of

t he unoperabl e boundary ( f i gures 92. 1 to 5) . I n vi ew of the exper i ments,

however , ( f i gure 57. 4 or 68 .6 ) th i s a rea i s r e la t i vel y c l ose to the

b ou nd ar y o f o s ci l l a t or y i n st a bi l i t i e s .

The calcul at i ons show that deeper i n ject i on l evel s may be att ract i ve, s i nce

the consequent i ncreased worki ng range makes i t poss ib l e t o operate w t h

hi gher downcomer/ r i ser air rat e rat i os at the same net power i nput

( f i g . 92. 3 to 5) . Operat i on is t hen poss i b le w t hi n the most stable area.

b. Alt hough t he downcomer i n ject or head l oss i s of m nor i mport ance i n fu l l

scal e pl ants both i n an absol ute sense and because of t he hi gher

c i r cu l a t i on ve loc i t i es , ca re fu l a t ten t i on to the des i gn i s r ecommended . I n

par t i cul a r a r es t r i c ted c ross sect i ona l a rea prevents the possi b le escape

o f l a rge bubb les r i s i ng a long the wa l l . I t i s a l so i mpor t ant to ensure a

smoot h l i n ing of the downcomer not only i n order to r educe f r i ct i on, but

al so to prevent c l i ngi ng ai r pockets, whi ch reduce dr i vi ng force and mayunnecessar i l y r estr i ct the downcomer.

c . Mos t was te waters con ta in su r f ac tants wh ich w l l s tab i l i ze opera t i on by

reduci ng coal escence and the corr espondi ng i nhomogenei t y. Car e must be

taken to avoi d contam nat i on w t h coal escence promoti ng addi t i ves such as

o i l (an t i - f oam agents ) , wh ich may rapi d ly a l t e r t he ci r cu l a t i on to the

e xt e nt of l e adi ng t o i n st a bi l i t y .

The most i mport ant concl us i ons, outs i de those concerned w t h scal e

up, can be summari zed as f ol l ows:

a. I t i s poss i b le t o opti m ze the downcomer/ r i ser d i ameter rat i o in terms of

i t s e f f ect on f r i c t i on. W th an external downcomer t h i s r a t i o i s 0 .7

( sect i on 18) and f or t he concentr i c t ype 0. 36 (appendi x A18) .

b. Based on exper i ence w th gas i n jec t i on I n a downf l ow ng l i qui d a

ventur i - f orm i njector was desi gned based on di spers i on by t he column

l i qui d. Thi s i n jector has a lower head l oss than other ar rangements, and

al so prevents the escape of ai r upwards. I t can be constr ucted i n var i ous

f orms of whi ch the i nverted t ype has the advantage of eas i l y al l ow ng the

i n jec t i on depth to be al te red ( chapter 2 ) .

c . A s i mple model has been presented for the s l i p vel oc i t y i n upf l ow. Thi s i s

bett er for the predi ct i on of voi d f ract i on than the Zuber and F i ndl ay model

when l arge hydrostat i c ef f ects are present . The l att er model was i n fact

f ound to be total l y unsui tabl e for t hese condi t i ons ( sect i ons 35 and 38) .

d. I n upward f l ow radi al d i st r i buti ons of voi d f r act i on have been f ound to

have a di st i nct maximum at the pi pe centr e.

e. A new t ype opti cal f i ve poi nt bubbl e probe was desi gned and constr ucted

( secti on 42) and used i n the downward two phase f l ow program w t h good

r e s ul t s .

f . S l i p vel oc i t i es i n downward f l ow are less t han in upward f l ow. Term nal

vel oci t y i s a good though somewhat conservat i ve est i mat e of t he f ormer

(chapter 4 ) .

g . The rad ia l d i s t r i but i ons o f vo id f rac t i on and bubbl e vel oc i t y i n downward

f l ow are re la t i vel y f l a t , excep t f o r the lowes t voi d f rac t i ons measured

( 0 . 5% . I t i s concl uded that t he cont r i but i on to the s l i p vel oc i t y i s sma l l

( chapter 4 ) .

h . Longi tud ina l tu rbul ence i n tens i ty p rof i l es are f l a t tened i n two phase pi pe

f l ow, except i n a very sma l l r egi on cl ose to the wa l l ( sec t i on 48 ) .

A steady st ate model whi ch i nc ludes hydrost at i c ef f ects and i n jec t i on head

l o ss and a l l ows p l ace dependent s l i p vel oc i t i es has been present ed

(chapter 5 ) . The model was modi f i ed to f o r m an accurat e basi s for a t i me

d ependent model .

I t i s poss i b le to model the t i me dependent behavi our i n quas i - s ta t i onary

t erms. The predi c t i ons are in reasonabl e agreement w t h the unst abl e

boundar i es f ound exper i ment al l y.

95 199

k. I n s t abi l i t y o r i g i n at e s f r om os ci l l at i ons w t h a per i od of approxi matel y

t w ce the gas resi dence t i me.

1. The 95%m xi ng t i me corresponds to about 7 c i r c ul a t i o ns , but axi a l

d i spers i on i s su f f i c i ent l y sma l l to assume plug f l ow i n mass t ransf er

model l i ng.

m Oxygen tr ansfer rat e of an ai r l i f t r e a c t o r i s l es s than I n a bubbl e

col umn, but w t h ai r i nj ec t i o n to the downco mer the t rans fer can be

i ncreased unti l the perf ormances are equ iva l ent .

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?®

Figure A12: Detailed sketch of experimental calm

n. Dynam c met hods for mass t ransf er measurement consi st entl y gi ve val ues that

ar e too hi gh, and in th i s type of equi pment the d i f f e rences are not

n eg l i g i b l e .

96 Acknow edgement s

This work i s part of the research program of the Foundati on for

Fundamental Research on Matt er ( FOM and was made possi bl e by f i nanci a l

support f r om the Nether l ands Organi zat i on for t he Advanc ement of Pure Resear ch

( ZWO), the f a c i l i t i e s of the Laboratory for Physi cal Technol ogy of the Del f t

Un ivers i ty of Technol ogy and the e f f o r t s of i ts personnel and st udents .

97 Appendi ces

The appendi ces f ol l ow the number i ng of the sect i ons they ref er to.

A12 Experimental setup

F or d e t ai l s see f i gur e (A12)

The vari ous vol umes are:

Di schar ge vol ume

Constant l i qui d vol ume

Di spersi on vol ume

Total vol ume c ol umn

A13.1 Dynamic calibration of the inductive flowmeter

The user ' s i nst r uct i ons for the Alt of l ux K300 magneti c- i nducti ve f l ow-

meter gi ve as s pe ci f i c at i o n a ti me const ant of about 7 s (f or i nstr ument

vel oci t i es bel ow 06 m s, a t y pi c al v el o ci t y i n our col umn i s 0 25 m s ) . Mor e

0.570 m3

30.600 m ( w t h ou t o v er f l o w)

30.630 m (w th over f l ow opera t i ng)

0.655 m

97 /A13

a , | „ ( z ) d z . M ^ w lnCp(0) _ ^g2 )

97/A13

i n terpreta t i on o f p ressu re grad ien ts i n te rms o f voi d f rac t i on. For t he

one-d imens i ona l s i tua t i on, neg lec t i ng radi a l var i a t i ons , w th equat i on

(55 -10 ) :

L

(1 )0

where a i s t he vol umetr i c mean void f ract i on between t wo pressure poi nts of

the col umn, connect ed t o t he manomet er .

For the ai r wat er manomet er

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Fifiure A13.1 ; Dynamic cal ibra tio n of Inductiv e f loui

(2U Q) according to a f irs t order model (2 U j) and a i

response of a fas t differential pressure cell measui

r . The velocity calculatednd order model (2u,) coopai

detai l ed i nformati on was necessary for the t i me dependent veloc i ty

meas ur ement s.

The exper i mental setup for the cal i brat i on was s i mpl e: the pressure

drop over about 6 m downcomer pi pe was measur ed w t h t he DP L5 Val i dyne

d i f f e rent i a l p ressure t r ansducer and cal i b rated aga ins t the i nduct i ve

f l owmeter i n steady st ate. The DP- cel l response t i me of about 1 ms i s much

f aster t han that of t he ti me dependent f l ow i n the col umn, about 20 s. The

i nduct i ve f l owmeter response was f ound to be adequat el y descri bed by thef ol l ow ng f i rst order equati on which i ncl udes a dead t i me:

Ls D( t - 2 ) - 2( u ( t ) + 4.5

du ( t )

dt CD

where u = the output vol t age of t he f l owmeter .

The resu l t i s i l l u st r a ted i n f i gure (A13. 1 ) f o r a re l a t i vel y ext r eme

case (start i ng up the col umn). The response curve of the f l owmeter (2u ) i s

compared w th t he recal i brated response of the DP-cel l (v ) and equati on

( 1 ) ( 2u, ) . A second order cal i brat i on ( 2u? ) i s a lso shown. Apparentl y t he

f l owmeter response i s more compl ex t han equati on ( 1) descri bes but the c a l i

brat i on i s suf f i c i ent l y accu rate f o r ou r pu rposes .

A13.2 Interpretation of the manometer pressure drop measureme nts

In chapter 5, equati on (55 -10 ) , a si mpl e model f or t he height

dependence i s i n t roduced, wh ich i s su f f i c i ent l y accura te to a l l ow the

Zp - gLAp - PLgAh ( 2)

where Ap i s t he pressur e dif f erence bet ween two pressure poi nts of t he col umn

due to f r i c t i on and accel e ra t i on,

Ah i s the water l evel d i f f erence i n the manometer ,

Ap i s the di f f erence i n densi t y of water and "medi um" i n the col umn and

L i s the dist ance between the two pressure poi nts .

For a t wo phase f l ow:

( 3)

( 4)

( 5)

Negl ect i ng f r i c t i on and acce le ra t i on:

aL = Ah ( 6)

Thus the ai r wat er manometer i s a very si mpl e devi ce t o measure voi d f r acti on.

I t i s now possi bl e to answer two probl ems:- The measured axi al mean voi d fr acti on between respect i ve pressure poi nts

rep resents the local voi d f r act i on on a cer t a in l evel .

- The measured axi al mean voi d f ract i on bet ween the hi ghest and l owest

p ressu re poi n t dev ia tes f r om the axi a l mean of the to ta l r i se r .

202

The f i rst probl em can be solved by stat i ng:

o:( z) = a

or w t h ( 1 )

97/A13

( 7)

97/A15 203

Cent r a l ba f f l e .

The tube w th d iameter d i s di v ided i n two ha l f cy l i nders . The

hydraul i c d iameter i s

( 1)

The l oss at the l 80°- bend w l l be comparabl e w t h the l oss i n a round tube.

Because of the l ack of i nformati on th is agai n w l l be assumed. The cont r i

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wher e

z . i s the requi red represen ta t i ve l evel .m ^z. and z,, the respecti ve pressure point l evel s and

h Q = p (0 ) / pLg.

W th t he f o l l ow ng resul t s f o r the col umn

i

1

2

3

4

5

z.1

1 .01

2 . 5 1

A.54

6 . 5 7

8 . 6 0

z i + l

2 . 5 1

4 . 5 4

6 . 5 7

8 . 6 0

1 0 . 1 0

2 i + l " z i

1 .50

2 . 0 3

2 . 0 3

2 . 0 3

1 .50

m i d d l e

1 .76

3 . 5 3

5 . 5 6

7 . 5 9

9 . 3 5

z . (h = 12 .3 m)m i* o

1 .78

3 . 5 7

5 . 6 1

7 . 6 7

9 . 4 3

z . (h = 21 . 4 m)mi * o

1 . 7 7

3 . 5 4

5 . 5 8

7 . 6 1

9 . 3 7

u s e d

1 .77

3 . 5 5

5 . 6 0

7 . 6 4

9 . 4 0

The second probl em i s solved by cal culat i on of

The measured val ue i s w t hi n 2%of the requi red val ue and can be used w t hout

c or r e ct i o n.

A15 Friction number for bubble column loop with internal flow divider

Formul a (15. 3) must be adj usted for t he conf i gurat i ons w th i nternal

f l ow d iv i ders . As examp les we w l l cons i der a cen t ra l secto r p la te and a

concentr i c tube.

buti on at t he top w l l be neglected assum ng an open and rel at i vely l arge

disengagement tank. Equati on (15.3) f or t he i nternal cent ral baf f l e becomes

( w t h m = 1 )

K F = 4 f ( 2L /d R + 100) ( 2)

For a col umn w t h the same t otal cross- sect i onal area as t he exper i ment al

col umn thi s means ( L = 10. 34 m d = 0.25 m d = 0. 152 m) K = 4. 7.

For an open U-t ube col umn w t h m= 1 equat i on ( 15. 3) becomes

( 3)

: t otal area t hi s means (d = 0.177 m) K. v = 4.3.

Concentr i c tube.

Kubota et al . ( 1978) use for t he l 80°- beud

h " 1 0 dD/ d R W

whi ch gi ves l osses of the same order of magni t ude as the round t ube f ormul a.

The hydraul i c d i ameter of the annulus i s s i mply

( 5)

Again assum ng an open and rel at i vely l arge top sect i on equati on (15. 3)

beco mes

204 97/ A15

K = «( - =- + mF d D

+ 25 + 25) ( 6)

i n which t he contract i on loss i s cons i dered as an entry l oss (K as 0 . 5) .2 2

Expressed i n d and M = (m + l )/ m * " / i .

K . 4 f j ~ [ M2 - (M - 1) 2( M2 - 1 ) ) _ 1] + 500/ M + 50j ( 7)^ d R

9 7 205

A22 S ize and s tab i l i ty o f capt i ve s l ugs i n downf l ow

Abstract

Thi s appendi x descri bes t he formati on and stabi l i ty of capt i ve gas

s l ugs devel oped f r om an a i r i n ject i on po int i n to a water downf l ow. There i s

usual l y a wel l es tabl i shed s t abl e cavi ty l ength determ ned by the vo lumetr i c

gas f l ow rate and t he l i qui d vel oc i t y g i ven by the di raens i onl ess r e lat i onsh ip

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For a column w t h L ■ 10. 34 m d = 0.25 m and m = 0.2 or 1 t he values of K

are 4.7 and 10. 0 respecti vel y.

A18 Opti mal r i ser/ downcomer area rat i o f or bubbl e col umn l oop w t h

i nte r n al f l ow di v i d er

We w l l onl y cons ider t he in ternal concentr i c tube as di v i der .

Equati on (AL5.7) i s plot ted i n f i gure ( A18) and may be compared w th f i gure

( 1 8 . 1 ) . The aspect rat i o parameter i s now based on the total cross secti on and

i ts values are not comparable. The mni mum value i s f ound as expected f or

somewhat l ower val ues of m The losses are rel ati vely l arge in the r i ser

compared t o those of a bubbl e col umn l oop w t h exter nal downcomer and ci rcul ar

r i s e r . The f i gure shows thi s opti mal value t o be about 0.15, what corr espondsto a d iameter rat i o of 0 .36. The Bi l l i ngham pi l ot p l ant ( t abl e 4) coupl es an

area rat i o of 0 .64 w t h an aspect rat i o of 325 ( ! ) .

100 ( 1)

I t has been establ i shed that i n the 0. 15 m di ameter tube cert ai n combi nat i ons

of gas r ate and water vel oc i t y do not g i ve s tabl e cavi t i es .

Slug l ength L

The i nvesti gat i ons have been made i n the water t unnel descri bed i n

secti on 42. Ai r was suppli ed t hrough a 3 mm di ameter hol e i n the pipe wa l l . A

stabl e and wel l def i ned sl ug i s usual l y formed below the inj ecti on point . Gas

i n f l ow i nto the s l ug i s ba lanced by a cont i nual l oss f r om the lower sur f ace,

where most gas i s r emoved by di s rupt i on f r om the r i m of the vent i l ated cavi ty .

The highly tur bul ent wave f orms a closure whi ch i s dis t i nct enough to perm t

1 \ . V

T' \ " \. : = s r _ ^

X .—___

- r - ^ T - - " " " " " .

F i g u r e AZ2. 1; The length oi a capt i ve ai r s l ug i n downf l ou in a 0 .15 mdi ameter p ipe as a funct ion of f t

F lK- j ro A2Z.2: S t a bi l i t y ma r g i n s o f a c ap t i v e a i t s l u g i n do wn f l o *

206 97/A22 97/ A22 207

the cavi ty l ength t o be measured w t h an accuracy of about 10 percent or 1 cm

The peri pheral ski rt around the base of t he sl ug usual l y ext ends some two or

three cent i meters fur t her and i t i s t h is h i ghl y t urbu lent downstream shear

l ayer that i s r espons i b le for the gas d ispers i on. The d is t ance between the

nose of the cavi ty , wh ich i n s tabl e operat i on i s a l ways c l i ngi ng to the

i n ject i on poi nt , and the re l at i vel y f l at l ower sur f ace was determ ned for a

range of gas and l i qui d f l ow rates ( f i gure A22.1) . W t hi n a measuri ng accuracy

of about 10 percent a square root r e lat i onsh ip i s f ound (equat i on 1) .

7/22/2019 TR-diss-0001460(1)


0.74 n/ s and Q i

Sl ug s tabi l i t y

The experi ment showed that there are cl ear l i m ts to t he f l ow regi me

giv i ng stabl e cavit i es. A map of t he transi t i ons between the stabl e and

unstab le cond i t i ons i s gi ven in f i gure (A22. 2) . The poi nts shown are l i mt ed

to these determ ned i n the v ic i n i t y of the s tab i l i ty boundary . A t t empts to

operate i n the l ow l i qui d f l ow rate reg ion on the l ef t hand s i de wi l l l ead to

l arge gas bubbl es t hat break f ree and r i se agai nst the l i qui d downfl ow or are

entr ai ned downstr eam dependi ng on the gas f l ow rat e. At l ower gas rat es, i f

the gas r ate i s i ncreased whi l e mainta i n ing t he same super f i c i a l l i qui d

vel oc i t y , an i n i t i a l l y s tabl e cavi ty ( f i gure A22. 3) i s seen to broaden unt i l

the w deni ng eventual l y reaches back towards the i nj ecti on poi nt and f i nal l y

a l l ow ng a typ ica l Tay lor i ns tabi l i ty to devel op and to break away as a

separate bubbl e (f i gure A22. 4) . Since t he bubbl es f ound i n thi s way are

smal l er and get even smal l er by di spersi on at the wake, they are conveyed

downwar ds. Usi ng a hi gher gas rate ( above about 0.15 1/s) w t h an establ i shed

capt i ve s l ug c l i ngi ng to the or i f i ce, the cavi ty sur f ace i s seen to be

dist urbed by very l arge waves ( f i gure A22. 5) . Reduct i on of the water vel oc i t y

again l eads to an unstabl e cond i t i on, though th i s t i me i t i s character i sed by

a mass ive f ronta l escape of a i r f r om the s lug. The d is t urbed sur f ace i s i n

marked contr ast to t hat at l ow gas r ates . The sur f ace i ns tabi l i ty appears

rather l i ke an i nver ted water f a l l and i s cons is t ent wi th t he genera l

expectat i ons of a Rayl e i gh-Tay lor wave i ns tabi l i ty i n which a heavi er l i qui d

i s above a l i ghter one ( Tay l or , 1950) . Once ai r has escaped, t he rel eased

bubbl es r i se agai nst the l i qui d downfl ow. Thi s can occur even when the

super f i c i a l l i qu id veloc i ty i s much h igher than the vel oc i t y expected f or gas

s lugs i n s tagnant water i n a pi pe of t h is d iameter (C l i f t et a l . 1978).

The escape upst ream i s oft en not compl ete. As a resul t of break up by

i ns t abi l i t i e s or a t t he t a i l , the si ze of t hese detat ched bubbl es decreases

97/ A22 97/ A22

Thi s corr esponds w t h a squared actual vel oci ty 2.7 ti mes t he square of t he

l i qui d s up er f i c i a l v el oci t y . I t i s al s o noted that t h er e w l l be ano the r

tr ansi t i on in the stabi l i ty boundary above a sl ug l ength of 0.29 mwhere t he

l i ne for Q = 0.6 l /s touches the upper st abi l i ty boundary descri bed by

equati on ( 3) . More experi ments are cl earl y needed to establ i sh t he i nf l uence

of other var i abl es l i ke pi pe di ameter .

Concl uding

7/22/2019 TR-diss-0001460(1)


; A22. 6: P o s si b l e f o

Ther e ar e c l ea r l i m t s w th i n whi c h a s ys tem to i n j ec t g as i nto a

downf l ow ng l i qu id w l l work sat i s factor i l y . The length of the s tab le capt i ve

sl ugs i s i ndi cati ve of the general condi t i ons. Upper and lower bounds of s l ug

stabi l i ty have been def i ned f or an ins ta l l at i on w th a 0.15 m diameter p i pe.

A26 4" ventur i - i n jector no. 2 f or downf l ow

Constr uct i on deta i l s see f i gure (A26).

and so als o t hei r r i se vel oci t y. They woul d then be carr i ed back downwards,

oft en coal esci ng w t h the sl ug and general l y i nducing anot her upward surge.

Some hysteres i s was not i ced i n re-es tab l i sh ing the in t i ta l s tabl e s i tuat i on.

I n the l ower gas f l ow regi me, the stabi l i ty boundary may be

represented w th

2g( 2)

Thi s means that the energy of t he f ree fa l l i ng l i qu id sur f ace l ayer s t ar t i ng

f r om rest ( at the wal l ) w l l not i ncrease above that o f the bul k . The

si tuat i on at hi gher gas rates i s more complex but the stabi l i ty boundary I s

reasonabl y descri bed by

L + 2. 7 — = 0. 3 6

2g

( 3)

Thi s r esembles the Bernou l l i equat i on, a l though the l i qui d near the

base of t he bubble i s moving much f aster than the superf i ci al veloci t y and the

constant mer i ts some d iscuss i on. I t i s noted t hat i f the curvature of the s lug

surf ace i s t he same as that of the pi pe and the bubbl e extends j ust t o the

t'_:be centr e l i ne ( f i gure A22. 6) t he avai l abl e f l ow area i s reduced to 61%

A29 An a l t ernat i ve ventur i - type i n jector des i gn

Because of gome i nfor mati ve resul ts of a separat e experi ment i n a 6"

downfl ow water t unnel the resul ts ar e di scussed here bri ef l y ,

A str eami ned body (Gol dstei n, 1938), whi ch may be consi dered as an

i nverted vent uri , was constructed so t hat ai r coul d be i nj ected t hrough hol es

i n the "nose" of the body. Thi s was compared w th a "normal " ventur i i nj ector

(see tabl e A29) . L i qui d vel oci ty coul d be f r eel y chosen.

210 97A/ 29 97/ A29

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F i g u r e A29. 1: Co n

Compari son ult h C

The resul ts ( f i gure A29. 1) show that l i qui d vel oc i t y i s an impor t ant

parameter whi ch l argely determ nes i n whi ch mode t he inj ector operat es. Thi s

i s demonst rat ed w th t he di f f erent sl opes of the dependence of head l oss on

l ocal super f i c i a l gas vel oc i t y . Thi s means that the throat / p ipe cross -

sect i onal area rat i o i s an impor t ant des i gn parameter s i nce a smal l er rat i o

probabl y del ays the onset of the ai r pocket pr esence whi l e at the same ti me

si ngl e phase pressure drop incr eases.

TABLE A29

Ventur i conf i gurat i ons - 6" d i ameter p i pe

pi pe di ameter

thr oat di ameter

a r ea r a t i o

l ength

l ocat i on throat

l ocat i on hol es

di ameter hol es

number of hol es

venturi

150 mm

112 mm

. 56

260 mm

160 mm

128. 5 mm

3 mm

45

i nvert ed vent uri

150 mm

98 mm (outer di ameter )

. 57

470 mm

69 mm ( bel ow begi n convergi ng)

55 mm s ect i on)

1 mm

80

To exam ne the di spersi on process pict ures of the inverted venturi -

type- i n ject i on were t aken ( f i gure A29.2) which show the fo l l ow ng

p ) There ar e no ai r pockets cl i ngi ng to the 1 mm di ameter holes

D ) I n i t i a l bubbl e s ize depends on a i r i n ject i on rate

212 97/A29

c) I f air pockets are present these occupy t he l ow pressure, regi on of t he

d iverg ing sect i on (see al so f i gure A29. 3 where t he a i r i s put o f f ) . Wt h a

bett er st reami ned f orm thi s ai r pocket woul d have been smal l er or even

absent.

I t i s concl uded that the inver ted ventur i i s a promsi ng des ign s ince

i t coupl es a head loss equal to t he normal vent uri to a great er pract i cal

ut i l i ty . I nser t i on i n to the top of the tube I s s i mple and i n ject i on hei ght may

be adjust ed f reely and easi l y . The i nject or can be removed si mply for

mai ntenance.

97/ A32 213

vM = v L s (2a)

(2b)

For a si ngl e bubbl e at a l ocati on where

V L S= "V = VL <3 )

equati on ( 2b) becomes

7/22/2019 TR-diss-0001460(1)


A32 Notes on the Zuber and Findlay model

Cr i t i c i sm i s made by Wsman (1979). He not i ced t hat t he model

equat i on (32- 3)

( 1)

when pl ott ed i n the vr 1

- v p lane for a cer ta i n val ue of v greater t han

0 does not start f rom the v - axis as v approaches t o zero, but at the

l ocat i on def i ned by

; A32: Ref i nement of I

e r t . See f i gure 32 .4

and Findl ay model by Ws man (e<

f A32.

on A32-5) il lui ated with dati

V_. = vT + V ( 4)Gl Ls bw

Thi s means that for di f f erent val ues of v d i f f erent s t ar t i ng poi nts

prevai l so chat the model equati on (32-3) cannot be pl ott ed as one si ngl e

str ai ght l i ne. The success of the Zuber and Fi ndl ay model can then only be

used w th l ow, constant or a lmost constant l i qui d f l ow rat es .

W sman ( 1979) modif i ed t he model equat i on:

v„ . = vT + K v-, + v, ( 5)Gl Ls o Gs b<»

and repor ted t hat data f or l arge l i qui d f l ow rates ( 0.8 < v < 2. 5 m s ) f i t

to th is equat i on w th a s tandard devi at i on that i s about hal f that for t he

model equati on ( 1) . Fi nall y he suggested t hat an even more general model can

eas i l y be der i ved f rom the or i g ina l equat i on:

< 6 >

Tabl e ( A32) gi ves a summary of the corr el ati on based on the thr ee

model equati ons f or t he data used by W sman (1979) . No f i rm conc l us ions can be

drawn from these resul ts al though the f i ts are bet ter for the modi f i ed equa

t i ons . As an i l l us t rat i on the best f i t g iven in the tab le ( A32) ( equat i on (5) )

I s drawn together w t h the data agai n i n the V G1 " V M pl ane i n f i gu r e (A32) .

I t i s i nteresti ng to see that the data do indeed corr espond bett er t o a set of

s t r a ight l i nes . Thi s aspect mght t herefore a l so be i mpor t ant for bubbly f l ow

with l arge hydrostat i c ef f ects .

9 7 / A 3

TABLE A3;

Two phase

C ur v e f i t t i n g a c c or d i n g t o the moc

d a t a f r o m W s ma n ( 1 9 7 9 ) * .

v er t i c al u pf l o w

0. 8 < vT <2. 5 m sL s

0.4 <v G£ < 1.9 m s

number of d a t a p o i n t s : 10

i n a 0. 10 d i a me t e r

el eq

t u b e

j a t i o ns ( 1 ) , (5) a id (6) w th

97/A33 215

vG1 - <oVe> <o> = <a( vL + vr) >«x> » (1)

-j.v >/<a> + <Q

Af ter rearrangement f ol l ows:

( 2)

( 3)

7/22/2019 TR-diss-0001460(1)


e q ua t i o n

e q ua t i o n

e q ua t i o n

( 1 ) : C - 1 . 065 ,

( 5 ) : K = 1 . 1 8 8 ,

(6): 4 =1246C . = 0.928oL

V_ , 1 = 0. 390m/ s,Gdl

v, = 0. 368m/ s,

V - , , . = 0. 428 m sGdl

s( , -

s , =

B*

L e a vi n g out one v a l u e wh o s e d e v i a t i o n ( 6) be t we e n

( t h e s t a n d ar d d ev i a t i o n o;

to:

e q ua t i o n

e q ua t i o n

( 1 ) : C = 1. 080,

0( 5 ) : KQ = 1 . 1 8 4 ,

T h es e l a s t v a l u e s are the

( Re ma r k :

v el o ci t y

of 0. 5 m

a s l a r g e )

a n a b s o l u t e p o s s :

of 2 m s an e r r o r

s . For the Zuber

t he d ev i a t i o ns ) ,

v „ , . = 0. 363m/ s ,Gdl

v, = 0. 384m/ s,

same as g i v e n by t,

b l e e r r o r of 3s r =0

l e a ds

0. 080

0. 051

0. 038

meas ur ement

f or the

9 - 0. 057

s = 0. 035

i s ma n

0.12

i n <a> of 6% i n c r ea s i

a nd F i nd l a y mo de l

* Da t a f r o m S m s s a e r t ( 1 96 3 ) u s ed by W s ma n .

t h e s e

l n VG1jg to 2!

val ues

f i r s

means

X at

wou c

and fi t i s 3s, ,

t two equat i ons

for a gas

a gas vel ocit y

be about tw ce

A33 Notes on slip velocity based models

Mal nes ( 1966) expressed the gas vel oci ty w th two terms to separate

the rel ati ve vel oci ty:

Brown et al . ( 1969) fol l owed the same reasoni ng, but assumed fr om the start

that the rel ati ve vel oci ty i s constant:

<av >/<o.> = v (4)

whence equation (2) becomes

VG1 " " r + \ s / ( K " ' C > ) I5'

T hi s r e l a t i o n was p r e s e n t e d by Go me z p l a t a et a l . ( 1 9 7 2 ) a f t e r S t e p a n ek ( 1 9 7 0)

showed t hat t he o r i g i n al r es t r i c t i o n t o p ar a b ol i c - t y pe p r o f i l e s i mp os e d on i t

by Brown et al . was not n e c e s s a r y . Ma l n e s ( 1 9 6 6 ) had p r e v i o u s l y u s ed t he

a s s u mp t i o n of c o ns t a n t l o c a l r e l a t i v e v el o c i t y to c or r e l a t e hi s d a t a . B h ag a

a nd We be r r e a r r a nge d e qua t i o n (5) to

vG1 - vM/K + (1 - S i ) V r ( 6 )

and concl uded, by compari son w th equat i on ( 32- 3) , that

K = 1/ C0 ( 7)

and consequentl y, i n compari son w th equati ons ( 30- 17 and - 22) ,

1 - <a>/K = <ae>/<a> (8)

and hence t hat t he Ma l ne s / B r o w n mo de l was a s i mp l e va r i a t i o n of the Zuber and

F i n d l a y mo d e l . E q ua t i o n s ( 7) and ( 8) however are not c o r r e c t , b ec a us e C i s a

9 7/A33

funct i on of the re l at i ve vel oc i t y and Kn ot . F r om th e d ef i n i t i on of C

equati on ( 32-3) and equati ons ( 30- 12 and -13) one may show t hat

( 9)

wh i c h i s c l ea r l y n ot equ al t o l / K , equ at i on ( 3 ) . I t a l s o i mpl i e s t h at t he

paramet ers C and v„ , , do not separate t he ef f ects o f non uni f orm f l owo Gal

prof i l es fr om those of t he rel ati ve veloci t y as had been cl ai med by Zuber and

F i ndl ay ( 1965) among others . I n the genera l case i t i s i mposs i b le to do t h i s ;

97/ A33

C » C„ - 1 . (13)

F r om th e de f i n i t i on of t he sl i p ve l oc i t y i t f o l l ows th at

( U)

(15)

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i n the model of Mal nes ( 1966) and Brown et al . ( 1969) however the K- di st r i bu-

ti on parameter i s only af u nc t i on of t h e r adi a l di s t r i b ut i ons as aresul t o f

t he assumpti on of ac on stant l o ca l r e l a t i v e ve l oc i t y .

I n terms of the sl i p veloci t y equati on (5) becomes

(10)

Gomezpl ata et al . ( 1972) use equati on (10) i n combi nati on w t h equati on (33-3)

t o c or r e l a te da ta of o th er s , r es ul t i n g i n

K =1 - exp( - l l R&. 10 )/4 ( upward f l ow)

= 1 +exp( - 105 Re^ O )/2 ( downwar d, a < 0.28)

(11)

err or of about 10% I t should be noted that the way i n whi ch Gomezplat a et al .

( 1972) and Mal nes ( 1966) used the model v i ol ates the ori gi nal meani ng of Kas

expressed by equati on (32-23) . Al l i nhomogeneous eff ects are i ncorporat ed in

the K- va lue, whereas w th Mal nes t he ef f ect o f h indrance i s a l so added.

The Zuber and Fi ndl ay model i s i ncorporated i n tabl e 33 al so.

Rewri t i ng equat i on ( 32- 3) as

(12)

A36 Cal i brat i on of the two poi nt conduct i v i ty bubbl e probe

Vo i d f r ac t i on

I n f i gure ( A36. 1) t he cross -sect i onal average val ues of the voi d

f ract i on determ ned f r om the l ocal probe data (see sect i on 37) are compared

w th those deri ved fr om the vol umetr i c manometr i c method (i ndi cated by Ct). The

data f or t he downcomer ai r onl y mode (l ow top press ure) are l ower t han t he

others , i ndi cat i ng that other processes mght a lso be respons ib l e f or the

devi ati on. On average t he probe/averagi ng vol t meter combi nat i on val ues ar e

about 30% below those deri ved fr om the vol umetr i c manomet r i c met hod.

7 V nsar norma / 0 y Oh

- • *d3rC

°W

/ / y ^

I j f i u r e A 3 6 . 1 : C o m p a r i s o n b e t i

f l f i u r ^ A 3 6 . 2 : C a l i b r a t i o n of

I f o l l o ul n j

c a l c ul a t e d f r o m t he e l l i p t i c fi

2)8 97/ A36

Bubble vel oc i t y

The probe was i nsert ed i n a tube of 12 mm i nter nal di ameter at about

the same l ocat i on where two i nfrar ed l i ght beams centr al l y t raversed the tube.

These beams were 9 mm apart and at r i ght angl es t o each other. A passing

bubbl e i n ter rupts the recept i on of the l i ght by two photod iodes . The resu l t i ng

si gnal was cross- corr el ated i n the same way as the probe si gnal and the

resul t s are compared i n f i gure (A36. 2) . Bubbl e si ze was i n t he r ange 2- 3 mm

Both up and downfl ow could be used w t h bubbl e vel oci t i es bet ween 0 and about

2 m s. The method w t h t he crossed l i ght beams i s very accurat e, and was used

97/ A37 219

TABLE A37. 1

z ■ 8.74 m

normal Coppressureonl y r i s erai r i nj ec t i o n

l ow t oppressureonl y r i s er

V V W

Ls R GsR GR

m s mm s g/s0. 21 19 1.01

0. 28 29 1.65

0.19 17 0.24

0. 28 39 0.60

k o. s,a c o

m - 3% 10

0.62 4.6 86. 3 4.6 110.41 4.4 4.9

13. 6 4. 4 56

1.09 3.7 241.65 4.5 700.96 6.1 38

k v s-v c c

mm s mm s

0. 85 0. 56 263.9 0.58 170.89 0. 67 153.55 0. 71 26

0.74 0. 98 404.75 1. 01 350.76 1. 18 17

V V

w pr

m s mm s0.22 12

0.22 9

0. 50 49

0.57 59

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to cal i brate t he probe. As was expected t he probe system gi ves val ues that are

somewhat l ow- The devi ati on i ncreases wi t h vel oci t y, i n such way that t he

del ay of the bubbl e as it passes over the el ectr odes i s approxi matel y constant

at about 0. 3 ms ( f or a 2.5 mm bubbl e) .

A37 Fitting data. Riser void and bubble velocity profiles

See t abl es A37.1 and A37. 2. The f ormul ae used are:

2 ka* = (1 - k p ) 2 el l i pt i c voi d pr o f i l e3 2 \

v * = ( 1 - k p ) e l l i pt i c v el oc i t y pr o f i l e

1

< >= I * dp cr oss- sect i onal average

01 = «X*> di mensi onl ess average void f ract i onr

v= <v*> di mensi onl ess average bubbl e of gas vel oci t y

fa v

= <a*v*> d imens i onl ess av. f l ux or super f i c i a l gas vel oc i t y

di mensi onl ess one-di mensional gas veloci t y

p r of i l e c on t r i b ut i on t o s l i p v el oci t y

P = k k ; s = k + ka v a y

f a = 2(1 - ( 1 - k ar * 3) / ( 3 k a )

f v = 2( 1 - (1 - k / ' 5 ) / ^)

f av = * U " s / 2P ) ^ " s + P >% + s / 4 P +

2Ap l n 1 - s ( 2 p2)r — i n r — j —1 (1 - s + p) 2 + p 2 - s/( 2p2)

ai r i nj ec t i o n

l ow t oppressureonl y downcomerai r i nj ec t i o n

0. 31 53 0. 86

0. 34 69 1.16

0.17 20 0.30

0. 24 39 0.60

0. 27 53 0.86

2.4 6.8 590. 90 9.6 292.9 10.4 500.88 10.8 612.7 12. 0 68

1.02 3.7 232.25 4.1 480.99 5.9 262.45 6. 5 440.95 7.5 492.6 8.2 61

5.05 1. 20 120.81 1. 30 445.2 1.31 480.85 1. 39 574.7 1.39 58

0.80 1. 10 603.95 1. 16 600.72 1. 18 334.8 1.22 230. 83 1. 29 544.65 1. 30 54

0. 54 66

0.58 55

0.53 67

0. 62 57

0.49 60

TABLE A37. 2

2 = 2. 13 m

normal t oppressureonl y r i serai r i nj ec t i o n

l ow toppressureonl y r i s erai r i nj ec t i o n

l ow t oppressureonl ydowncomerai r i nj ec t i o n

VLs R VGsR WGR

m s mm s g/s

0.28 19 1.65

0.19 6 0.24

0. 28 13 0.60

0. 31 19 0.86

0. 34 25 1. 16

0.17 7 0.30

0. 24 13 0.60

0.27 19 0.86

k a s xa c o

m - 3% 10

0.94 1.3 692.8 1.4 55

0. 90 1.2 952.65 1.4 1020.85 2.4 493.4 2.6 450.91 2. 1* 433.0 2.3* 440.88 4.4 284.0 4.5 29

1.24 1.2 2341.6 1.3 670.90 1.5 263.75 1.6 220.23 2.0 39

29.3 2.0 44

k v s,V C 0

mm s mm s

0.88 1. 18 694. 45 1. 18 69

0.44 0.51 5310. 6 0.51 530.64 0.60 606.3 0. 60 590.63 1. 00* 586.35 1. 02* 52

0.67 0. 68 805.8 0.70 70

0.65 0. 80 625.6 0. 82 520.71 0. 66 365.3 0.68 260. 63 0. 66 237.6 0.65 37

V V

w pr

m s mm s

0.41 63

0. 38 10

0.36 15

0. 61* 31*

0.39 21

0.48 33

0.36 23

0.40 3

* obvi ousl y wrong value

A38 Singl e bubbl e ter m nal vel ocit y

I n l i t e ra tu re suf f i c i ent data are reported on s ingl e bubbl e term nal

vel oc i t y . Except fo r very c l ean l i qui ds term na l vel oc i t y i ncreases w th

bubbl e di ameter w t h a constant vel oci t y r ange for bubbl es bet ween about 3 to

8 mm

Expr essi ons are needed f or the computer al gor i thm The concept

i ntr oduced by Wessel i ngh ( 1984) on the fol l ow ng number s or di mensi onl ess

quant i t i es i s adopted here.

di ameter «

97/A38

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d* = d/d = Ga l / 3 w th d = ( nA r ) 1 / 3 ( 1)

v el o ci t yVK J V . = Re' d* wi t h v. = C

1 1^ )

1 7 3 ( 2)

surf ace tensi on ,a* = o/ or = d*2/ Eo w t h ar = (£±J£-)1/3

( 3)

P

where Ga i s the Gal i l eo number and Eb t he Eötvö's number.

The term nal veloci ty i s then descri bed by (gas bubbles i n l i qui d)

r egi on 1 u* = d* / 18 (s tokes r egi me) ( 4)

regi on 2 u* = 0. 23 d* (r i gi d sphere) (5)

1/ 6regi on 3 u* ■ 30* (wobbl i ng bubbl e, i ndependent of si ze) (6)

(spheri cal cap bubbl e) (7)

I n f act r egi on 2 bubbles do not behave as ri g id spheres ent i rel y.

Dependi ng on the amount of contam nat i ons i n the syst ems t he ter m nal vel oci t y

i n thi s reg ion w l l l ay between that fo r r i gi d spheres and the term nal vel oci -

„ , , , 0. 76- 0 . 52 0. 52, 1.28vbo= = 0. 136 g n p d ( 8)

u* = 0. 136d* ( 9)

For t he ai r / t ap water case the terms are

d = 0.047 10 mr

r . » 0. 021 m s

* = 3300I n the computer program the fol l ow ng approxi mate expressi on i s used

/ * -1 0 , * -1 0. 0 . 2 , , +1 0 . . 1 0- 0 . 2 - 0 . 5u* = (u* + u* ) + ( u* + u* ) ( 10)

which i s compared w th l i terature data i n f i gure (A38).

A42. 1 I n s i tu cal i b rat i on of the i nduct i ve flow meter

From l i tera tu re (M l l e r , 1983 ; Bern ier and Brenner , 1983) i t i s known

that i nducti ve f l ow meter s are t rue vol umetr i c f l ow meter s even when sol i ds or

gas are pr esent i n the l i qui d f l ow. For experi mental reasons the f l owmeter wasonl y cal i brated aft er the mai n seri es of measurements had been compl eted. Thi s

was done by di vert i ng the water f rom the ci rcul at i on l oop i nto a cali brated

vessel . Resul ts are shown i n f i gure (A42. 1). On cl ose i nspect i on of t he

i nsta l l a t i on i t was r eal i sed that the devi a t i on f rom l i near i ty was p roduced by

the bui l d up of a large ai r pocket before t he valve i n the hori zontal sect i on

222 97/A42 9 7 A / 4 2 2 2 3

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Figure A42.1 : In s itu calibration of the Inductive f lowraeter with and without air p r es t . i t . Broken UlM Is thetw o p h as e ca l ib r a t io n cu r v e .

downstream of t he f l owmeter , wh ich was therefore onl y par t i a l l y f i l l ed w th

l i qui d gi v ing er r ors i n es t i mates of the l i qui d f l ow rat e. Bu i l d -up of a i r was

ti me dependent but an equi l i br i um value was reached wel l w t hi n the ti me

needed for the Laser Doppl er and bubbl e probe measurements . The broken l i ne in

the f i gure g ives th i s equ i l i b r i um and was used for i n terpretat i on of the phase

t wo data.

A42.2 Processi ng of the bubbl e i nterr upted Laser Doppl er si gnal

I n vi ew of the low s i gnal presence as a resu l t o f the i n ter rupt i ng

bubbl es i t was necessary to del ete the "hol d" -s i gnal f rom the tota l output of

the t racker. Thi s i s done in a way si m l ar to t hat descri bed by Ohba and

Yuhara ( 1979) . W th t he hel p of a log i ca l output of t he t racker ("drop i n"

s i gnal ) onl y the val i d par t s o f t he s i gnal were of f ered to the contro l l ed

corr el ati on and volt age averager.

However under t he ext reme ci r cumst ances encounter ed t he tr acker

showed some syst emati c er r ors. (Van der Lans and Vel l i nga, 1982) :

a) Aft er a " hol d" the si gnal was decl ared val i d bef ore t he fr equency wasactual l y t r acked.

b) The t r acker tended to search f i r s t for l ower f requenc i es , l ead ing to a

l onger del ay before a f requency higher t han that corr espondi ng to t he

"hold"- value was actual l y t racked than when a drop i n fr equency occurr ed.

Thi s l ed to va l ues f or the l i qui d vel oc i ty , that were systemat i ca l l y t oo l ow,

O 05 ID

FlRnre A44: Sea rext.

i ncreas ingl y so w th i ncreas i ng voi d f ract i on. The cor rect i ons were 7 percentor l ess for voi d f ract i ons of 3 percent or l e ss .

A44 Averaging local voi d fraction

I n f i gure A44 t he f i tt ed curves are s hown t hat have been used for the

determ nat i on of the cross -sect i onal average voi d f ract i on by probe.

A45 Some bubble s i z e distributions and detection probability

Di f f e r enc es i n si z e d i s t r i b ut i ons as a f u nc t i on o f l i qui d f l ow r a te

are not promnent ( f i gure A45.1) . F or l ow v oi d f r ac t i ons ( 0 . 5 % th e d i s t r i b uti ons are consi st entl y smal l er and shi f ted t owards smal l er di ameter s

( f i g u r e A45.2) .

I t i s noted here that a l l the di s t r i but i ons presented up to now are

based on r aw, uncorr ected dat a. Burgess and Cal derbank ( 1975) point ed out t hat

probe data should be corrected for the hi gher probabi l i ty l arger and f aster

i

f \ i-

i . 1 0 V.VLl lmls

: isN

s

rlR 3h

3?

ü j ,

39

97/ A45 225bubbl es have of bei ng detect ed. Since i n our case bubbl e vel oci ty and si ze di d

not cor re l ate at a l l ( t he range of t hei r term nal ve loc i t i es was smal l

r e l a t i v e t o t h e tur b ul ent v el oci t y f l u ctu at i ons ) a c o r r ect i on f or t h e v el oci t y

seems unnecessary. The corr ecti on for s i ze i s cal culat ed by Burgess and

Cal derbank to be proport i onal to the (to t he probe) proj ected area for

spher i ca l bubbl es . I n pract i ce however th i s re l at i on i s f ar more compl i cated

due to t he form of t he bubbl es. (For i nstance l arger bubbl es are more

el l i psoidal and are accepted more easi l y by the di scri m nati ng l ogic; however

th i s i s not so for the l argest , spher i ca l cap bubbl es). As an i l l u st r a t i on t h e

proposed corr ecti on has been carr i ed out w th one of t he di st r i but i ons and

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Fifiure A45.1: The I n f l u en ce o : l i q u i d v e l o c i t y o n t h e bu bb l e s i z e di s t r i l

1A

A

■ ré \

1

v l

■

.O-Mmfi : rl H .o5

Ei«5

d h

39

a „

S

V^-»*—•£<•-—■*_

-

é&

Figure A45.2; Only Foe very low void fraccione (0.5%) [here Is

distribution.

' A

1

; . : : : „ ' "

ElrWcr^

-

1

,.'■'''

i f//

.. //

ƒr l ure Afr5.3: Compariso n between uncorrected and corrected bub ble size distribut ion. Correc ted by uelgh ln i

ha, i / d2.

FlRure A45.4: Cumulative bubble size distribution on probability plot. For symbols see f i gure 445.3.

compared w th the raw data (f i gure A.45. 3). The di st r i buti on seems t o becomel ess skew and more Gaussi an. A cumul ati ve probabi l i t y cur ve however s hows t hat

thi s i s not t he case and that t he di st r i buti ons are tending to l og- normal or

are perhaps based on a combi nati on of t wo nor mal ones ( f i gure A45. 4) .

A detai l ed di scussi on on the underl y i ng processes l i es far beyond the

scope of t hi s thes i s .

A4 8 Log ar i t h m c l i qui d v el oci t y d i s t r i b ut i ons

Turbu lent prof i l es are of t en presented i n l ogar i thm c form (Hi nze

1975):

( 1)

v " « v / v f ( 2)

and

y ■ vf y /v (3)

Report ed val ues of A and B dif f er w del y. Tennekes and Lum ey ( 1980)

menti on A = 2.5 but i ncreasi ng w t h Reynol ds number, and B = 5; whi l e Ooms( 1980) gi ves

v + = 2.44 I n y+ + 4.9 ( 4)

Thi s express ion f i t t ed our s i ngl e phase data wel l ( f i gure A48.1) . The

22697A/ 48 97/A55

227

F i g u r e A46. 1: L i q u i d v e l o c i t y d i s t r i b ut i o n of t u r b ul en t c o r e o f d o wn wa r d p i p e f l o w S i n g l e p h a se .

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F i g u r e A46.2: A s f i g u r e A48.1. Two phase, v - 0 .62 m s-

F i g u r e A48.3: A s f i g u r e A48.1. Two phase, v = 0.66 m/e.

F i g u r e A48. 4: As f i gure A48.L . Two phase, v Lg - 1 . 4 7 m s .

f o l l ow ng f i gur e s (A48. 2, 3 and 4) show that the prof i l es keep t hei r

l ogar i thm c f orm i n the case of two phases , but t end to rot ate around a

cert ai n val ue. The dat a are however too f ew to show systemati c t rends.

A5 5.1 Adaptation of the proposed steady state model for operation with

overflow

W t h an over f l ow i t i s the top pressure at the over f l ow l evel(z = L ) r ather than the bott om pressure which i s const ant . The i n i t i a l

hydrost at i c pr essure may be expr essed as:

P8( Z ) - P t + P L g( L 0 - z)

Fifiure A55: Circulation velocity with and without overf low.

Equati on ( 55-10) becomes

q(z) m VVa (L ) P U)o s

Hence i n tegrat i on of ( 55- 8) yi e lds

p ( z)p(z ) = Ps( z ) - o. ( Lo) pt I n —■

P t

whi ch may be si mpl y i ncorpor at ed i n the computer pr ogram Some measur ement s

were made to ver i f y t he adaptat i on. S ince the over f l ow l evel i n t he exper i menta l co lumn is h igher then the in i t i a l l i qui d height w thout operat i on w th

overf l ow (L = 11. 15 m L = 10. 5 m) , the two cases must be i denti cal at ao ' "

c er t a i n g as i n j ec t i on r a te . I n f i gur e ( A5 5. 1) t h i s i s s een t o oc c ur f o r an a l

i n j ec t i on r ate w th i n t h e us ua l ope r at i on r an ge . Thi s i mp l i e s t h at t h er e i s

l i t t l e d i f fer ence between both operat i on modes .

A55. 2 Computerprogram for steady state model STS

PRCJ RAM RLSTU( 3, 9b), STEADY STATE MODEL general i zedCOMMON AR, AD, RLE, OT, PTf X,CAl f AR, CALFAD, ALFART, ALFADO, VLSD, I TESTCOMMON Y, RL, CALFAY, PRY, ALFAKY, VSR, VGSR, VBI NDB, VSD, VGSDEXTERNAL ALFAR, ALFAD, PR, PD, VBI NFDI MENSI ON AUX( 41 )LUT=LOGLU( SYLU)I F(LUT.EQ. 0)LUT-1

C Def aul t parameter s of the t

DR= 224DD= 1RLE=10. 34

m ( di amet er r i s er )m (di ameter downconer)m (downconer l ength)

n(RM ( RM+ ) )) ) +480*RM ( RM+ )+S0)

WRI TE( LUT, 35) I TESTFORMATf "Whi ch out put?" / "0 - normal "/

h" l - / a l f aD+"/ "2 - f ( vLsD) " / "3 ~ constant ve lo c i t y 'V l l , "READ( LUT, *) I TESTWRI TE(L UT, 37)FORMATC' For output t o pr i nter: 6 _" )LUP=LUTREAD( LUT, *) LUP

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RL=11. 15X=2. 9Y-C.FW=0. 005TEMP- 20PT=10. 33Hl =1. 01H2-10. 1CV1= 6CV2=8.CV3=2.DWG= 1RATI 0=1PI =3. 141593WRI TE(LUT, 10)DR, DD, RLE,RL, TEMPFORMAT! "Gi ve ri ser di ameter( ", F5. 3,*' ) and downcomer( ", F5.3, ")

n", / f"downcomer l engt h f " , F 6. 2 , " ) . i ni t i al l i qui d hei ght ( " , F 6. 2, " ) "t i , / , "and t emperat ure (",F6 2 " ) " )READ( LUT,* ) DR, DD, RLE, RL, TFJ MPW R I T E < L O T » 1 1 > Y , XF OR MA TC 'G iv e r i s e r i n j e c t i o n h e i g h t ( " , F b . 2 , " ) "

t ï , / . " a n d d ow n c om e r i n j e c t i o n h e i g h t ( " , P 6 . 2 , " ) " )

READ( LUT, *) Y, XWRI TE!LUT, 12) I DOFORMAT! "I s downcomer ext ernal ( O) or i nternal ! 1)?", 11, "_" ]READ( LUT, Ml DOWRI TE ( LOT, 13) CV1, OJ2, CV 3FORMATC' Gi ve coef f i ci ents downcomerREAD( UJ T, *) CV1, CV2, CV3W R r T E ( L U T , 3 0 ) F TF O R M AT C 'G i ve t o p h e a d ( m H 20 ) D e f a u l t : " , P 6 . 2 )READ( LUT, *) PTCP= 02926*( 273+TEMP)AR=DR*DR*PI / 4AD=DD*DD*PI / 4I Ft : CO. B2. 1)AR=AR- AD

: - : ■ ■- : ■ . . . ' ,

C\ - ; =4*™ *( RM*RM*RLE/ DR+125+RLE/ DR/ SQRTtRM) )I F( I D0. EQ. l ) Cw=4*FW*( RLE/ DR*<SCOT(( RM+ ) / RM) +RM*RM ( l - S0j RT

m ( d i s per s i o n hei ght )m (downcomer a i r i nject i on hei ght )m ( r i s er i nj ec t i on hei ght )

( f r i c t i o n f a ct o r )oC ( l i qui d t emperat ure)m ( t op head)m ( l ower r i s er p r es s ur e poi n t )m (upper r i ser pr essur e po int )s ( i njecti on head l oss parameter)! ( i njecti on head l oss parameter)! ( i njecti on head l oss parameter)g/ s(defau t output parameter )

(r ati o WG/ WGD if constant )

i nj ec t o

I F( LUP.NE. 6)L UP=LUTI F( I TEST. EQ. 2)GOTO910

C I ni t i al vel oc i t yC

VL£D=1I F! I TEST. EQ. 3)GOTO810

CC Choi ce bet ween vari abl e r i ser or downcomer i nj ecti on.C

WRI TE(L UT, 110) 1W110 FORMAT( " WG ( 1 ) , WGD (0) or WG/ WGD (-1) const ant" , i 2, " ")

READ(LUT, *) I W S I F(I W 161,121, 111CC Ri ser i nject i on constant .C1 1 1 I R E T = 1

W R I T E ( L U T , 1 1 5 ) W G , W G D S , W G D E , D W G1 1 5 F O R MA T C 'G i ve W G, WGD s t a r t , WGD e n d ( i n g / s ) a n d s t e p s i z e " , 4 F 6 . 2 )

!• ! ,V II LOT, * )WG,WGDS,WGDE,DWGI F ( L U P . E Q . 6 ) C A L L LU RCji 1 B , 6 , 1 )W R I T E ( L U P , 1 1 7 ) A R ,A D , R L E , T E M P , P T , X ,Y , C W , C v l , C V 2 1 C V 3

" ■ C ' A R = A D = R L E = T E MP= FT = X ~ Y = C " ,ti"W = C V 1 - C V2 = C V 3 = " , / X , 2 F 8 . 5 , 6 F 7 . 2 , 3 F 5 . 2 / / ,h" 1

h " 1 0 " , / ,h" WG WGD al f aR al f aD vLsD dpv/L al f aRT pRY al f aRY"h" pO")

I ST=( WGDE- WGDS) / DWG+ 0001I P(I ST)118, 119,119

118 I ST=- I ST S DWG- - DHG119 J —1120 J = +1

WGD=W3DS+ *DWG. ! . 1ST) GOTO 140GOTO 275

CC Downccmer i n ject i on constant

WRI TE( LUT, 125) WGD, WGS, WGE, DWG125 FORMATC' Gi ve WGD, WG st art , WG end ( i n g/ s) and st epsi ze" , 4F6.2)

READ( LUT, * ) WGD, WGS, WGE, DWG

8 1 0 I R E T = 4W R I T E ( L U T , 8 1 1 ) W G S , W G D E , D W G

8 1 1 F O RM A TC 'G iv e WG s t a r t f u p p e r b o u n d a ry ) , W G D e n d a n d s t e p s i z e ( g / s ) "h 3 F 7 . 3 )

210

250

ENDIFIF(ITEST.EQ.2)AMET=ALR-ALD-DPVIF(ITEST.EQ.3)THENAIF=(VLSD-VPR)*(VLSLX>VPR)DIF=(VLSD-VLSDO)*(VLSD-VPR)

IF(IW*AIF.GT.O)THEN S IPRINT=1 5 ELSE $ IPRINT=0 $ ENDIFIF(AIF.LT.0.0R.BIF.GT.0)IW=(-1)*IWIF(AIF.GT.0.AND.BIF.GT.0)IFL=(-1)*IFLVLSDOVLSDENDIF

IF{ITEST.NE.3.0R.ITEST.EQ.3.AND.IPRINT.EQ.1)nWRITE(LUP,250}WG,WGD,AIJ^A] fVLSD,DPV.ALFART,PRY,ALFARY,PR(0)

FORMAT(X,2(F7.3,X),5F7.4,F7.3,F7.4,F7.2)

REAL. FUNCTI ON PR{Z)COMMON AR,ADRLE,OT,PT,X,CALFAR,CALFAD ALFART, ALFADOVLSDI TESTCOMMON y,RL,CALFAY, PRY, ALFARY, VSR, VGSR, VB NDB VSDVGSD

C Riser head

I F(Z. LT. Y)THENPR=PT+RL- Z-ALFART*PT*ALCG l + RL-Y) / PT)

n -ALFARY*PRY*ALCG {PT+RL- Z)/ (PT+RL- Y) )ELSE

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C350

400

GOTO (1 20,13 0,92 0,820,1 70) ,IRE T

HRriEtLUP,400)FORMATC DIVERGENT")VLSD=1

I F( I RET.EQ4)GOT0812

470

480

500510

I F(LUP.Eq. 6)WRI TE(6, ' ( "1"}' )I F(LUP.EQ6)CALL LURQ 0B 6, 1)WRITE(LUT,300)I 0PNFORMAT("Again? " , I 1 , " _" )READ LUT, *) I OPN S I F( I OPN450, 45O5

WRITECLtJr,460)IVZ

FORMATCPrint ax ial void di st ri but io n? ",I I," _"}READ(LUT,*)IVZ $ IF(IVZ)610,610,470

IF(LUP.EQ.6)CALL LURQ(1B,6,1)

WRITE(LUP,480)

FORMAT( " Posi ti on voi d head v sl i p vgsDO 500 1=0,28Z=RLE/28*I + 00001VOD=ALFAR(Z)DRUK PR(Z)- PT-RL+ZWRITE(LOP, 510) Z,VODDRUK,VSR, VGSR, VB NDBFORMAT(X, F6. 2, 5X,F6. 4, 5X,F6. 3, 3X,P6. 3, 3X,F6. 4, 3X,F6. 3)I F( LUP. EQLÜT)READLUT,*)DUMMYWRITE(LUP, 480)DO 600 1=0,28Z=RLE/ 28*IVOD=ALFAD Z)DRUK=PD Z)-PT-RL+ZWRITE(LUF, 510) Z,VODDRUK,VSDVGSDI F(LUP.EQ6)WRI TE(6,' ( "1") ' )

I F(LUP.EQ6)CALL LURQ 0B 6, 1)

WRITE(L0T, 620) I SFORMAT("Stop? " , I 1 , " ")READ LUT, *) I S $ I F( I S75, 5, 999END

PR=PT+RL-Z-ALFART*PT*ALCG 1+RL~Z)/ PT)ENDF

PR=PR+0.0004385*(VLSD*ADAR)**1. 75*(RLE-Z' ) / AR**. 625

REAL FUNCTI ON PD Z)COMMON AR,AD RLE,CWPT, X,CALFAR,CALFADALFART, ALFADOVLSD I TESTCOMMON Y, RL,CALFAY, PRY, ALFARY, VSR, VGSR, VB NDB VSDVGSD

C0 Downcomer head.C

XHULP=XI F(Z.LT. X) XHULP=ZP0=PR( 0)PD=PO-Z- ALFAD0*{PT +RL) *ALCG(1- XHULP/ ( PT+RL})PD=PD+ 0004385*VLSD**1.75*( Z+84. 63*SQRT(AD) / AD**. 625RETURNEND

REAL FUNCTI ON ALFAR(Z)COMMON AR,ADRLE, a-J , PT,X,CALFAR,CALFADALFART, ALFADOVLSDI TESTCOMMON Y, RL,CALFAY, PRY, ALFARY,VSR, VGSR, VB NDB VSDVGSD

C Ri ser voi d fracbii

VSR=3VLSR=VLSD*ADARPRZ=PR(Z)

I F( Z.LE.Y)THEN S VGSR=CALFAY/ PRZ

ELSE $ VGSR=CALFAR/ PRZ $ ENDFI F(VGSR.LT.. 0001)RETURNDB= 006*( PD X)/ PRZ)**( l / 3. ) S VB NDB=VB NF(DB)VSR=VB NDB+ 2*VLSR*( VLSR+VSR)**2/(VLSR+VSR-VGSR)**2+VGSR/2

CALL QATR(0, Y, . 0001, 40, ALFAR,YY, I ER, AUX)CALL QATR(Y- r. 0001, RLE, . 0001, 40, ALFAR,YR, I ER,AUX)CALL QATR(0, X, . 0001, 40, ALFAD YD I EDAUX)

9 7 / A 5 5

FiRure A64.1: Computed influence of inertia following a positive Bti

97/ A64 23:i nverse po l ynomal s . Neg lect i ng local voi d f r act i on e f f ects t he change f r om

the one d ist r i but i on to the other i s descr i bed by the t r ans l a t i on o f a f ront

th rough t he r i ser .

The funct i on f i s then

L2 2 gI \ f e

f ( t ) = k v + - ~ (a - a )dz (6 )o L 1 e o

e 0

(a star t , a end va lue) .o ' eThe t i me i s brought i n w th the tr ansf ormati on

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downcomer a i t In ject io n) based on a simple , ex p l ic i t s imulat ion of Idevelopment .

F1 S" r ' f i gure A64.1 f or

A64 Solution of the differential equation of motion with the help of an

approximate time function

Equat i on (64-1) may be rewri t t en i n a general f orm

4 ? + k v 2 = f ( t ) CD

W L e

( 2 )

( 3 )

f ( t ) R ( « R aD Pv ) ( 4 )

V ( 5 )

The funct i on f i s a compl i cated f unct i on of t i me si mulat ed wi t h an

i tera t i ve program i n chapter 6 . To test the i n f l uence of the i nert i a term

(vel oci ty t i me der i vat i ve) th i s f unct i on was approxi mated f or a si mple case.

Worki ng i n the air l i f t mode fol l ow ng a step change i n the i nput gas rat e the

axi al voidage di st ri but i on changes i n t i me to a new steady st ate one. These

di st ri but i ons are descri bed i n the steady st ate model and were f i t t ed by

( 7)

wher e

'c-Hv0

For the gas veloci ty a si mple approxi mati on was used

( 9)

The f unct i on f( t ) al l ows t he use of a Runge Kut ta numeri cal sol ut i on of

equat i on ( 1) . Result s are gi ven i n f i gures (A64. 1) and ( A64. 2) and these show

that the devi a t i on stays be l ow f our percent o f actua l vel oc i ty . Furt hermore i t

demonst rat es that t he l ocal voidage eff ects can not be negl ected.

The inert i a term i nfl uence on a step response i s of second order and

as a f i rst approxi mati on may be omt ted.

A65.1 Computerprogram DYD for the quasi stationary model with flow sheet

and description

I t should be noted t hat the program i s a product of cont i nuous

i mprovements and updates unt i l the l ast moment . The f l ow sheet i s gi ven i n

f i gure (A65. 1) and ref erred t o i n the comments i n the pr o gr a m l i s t i ngher eaft er. The comment s shoul d be suf f i ci ent f or a descr i pt i on.

As an i l l ust r a t i on in f i gure (A65 .2 ) t he computed ax i a l voi dage

d ist r i but i on is shown as i t deve lops dur i ng start up. I t demonst r a tes the

t r ansl a t i on of the f i rst gas mass f r ont th rough the r i ser .

PROGRAM RLDYD{3, 99) , DYNAMC SIMULATION WTH DOWNCOMER.DMENSION I PR(5) , PR(100), ALFR(100, 2) , ZR(100, 2) , VLSR(2), I VR(100)DMENSION PD50), ALFD50, 2) , ZD50, 2) , VLSD2) , WD50)DMENSION ZH(20, 2) , WH(20, 2) , IDCB(144), I SI ZE(2) , NAMF(3), IOBS(4)DMENSION NMF{3)COMMON WGDOWGDN I FSUB LUTLOGCAL I FBRK, RLANSEXTERNAL WGD

C DECLARATION OF VARIABLES

: ALFR: ALFD

voi d fracti on ri ser, , downcomer.

:

CP=8.579C t i me step control

TB=0TL=200DT=1IPI=1

C pressure point posi ti ons ( m81=1.01H2=10.1WRITE(LUT,10)AR,AD

10 FORMAT!"Ri ser/ downc. cross secti on", 2F8. 5, " _" )READLUT,*)AR,ADWRITE!LUT,11)RL,DL,HL

11 FGRMAT("Col umn l ength,downc. i nfecti on l eve , l ength of hor. part",

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: WGR: WGD

: VLSR: VLSD

gasinput ri ser ( kg / s ) ., , downcomer.■ gas mass i n a ri ser bl ock ( kg)■ i n a down, bl ock' i n a bl ock of the hori zontal part: pressure in ri ser (m: pressure i n downcomer: positi on of a ri ser block (m .■idemfor downccmer.■ i demfor hori zontal part; ri ser superf i ci al l i quid ve oci ty ( m s ) .■ i demdowncomer.: eff ecti ve sl i pve oci ty ri ser.■idemdowncomer.

Yi3F72)READLUT, *)RL, DL,HLWRITE(LUT,12)CWCP

12 FORMAT!"Fr i cti on number, gas constant. ", 2F7. 3, " _" )READLUT, *)CWCPWRITE(LL1T,13)H1 H2

13 FORMAT("posi ti on pressure points", 2F7.2, " _" )READLUT,*)H1, H2

CC Co u i tia val ues

C l i st devi ce control

C LÜT=LCGLU(SYLU)I F(LUT. EQ0)LUT=1WRITE(LUT,5)

5 FORMAT!"Output to pri nter?: 6 _")READLUT,*]LUPI F{LUT.HE.6)LUP=LüT

SWH=0MAXR=1MAXD=1MAXH=1NMAXD=1

NMAXH=1VLSR(1)=0VLSD1)=0VSD= 15WGDCK)DP=0I FSuB=0

iss secti on (m2)

constants

ri ser and downcome:AR=0398AD= 00785

! ri ser/ hori zontal part/ downcomer l ength (mRL=10.34HL=85DL=2. 9

' f ri cti on numberCW=4. 5

' bottomhead (m£0=11. 7

: gas constant (head c - densi ty)( 20 Cg)

di ng ini ti a val ues fromf i l e

NAMF(1)=2HSJNAMFC2)=2HSDNAMF( 3)=2HY1WRITE(LUT,200) NAMFFORMAT("Name fi l e to read(i f not: EX)? Defau t: ", 3A2, " _")READLUT,250)NAMFFORMAT(3A2)I F( NAMF. EQ2HEX) GOTO450I F( IANDNAMF, 177400B)+40B NE. 2HS ) GOTO50CALL OPEN l DCB I ROP. NAMF)CALL RLDRC( I DCB I OBS,I RRC,LUT) !data readVLSD1)= OBS(1)/ 1000.

VLSR(1)=ADAR*VL5D{1)MAXR= 0BS(2)MAXD= OBS(3)MAXH= OBS(4)CALL RLDRC( I DCB I OBS,I RRC,LUT)WGR=OBS(l ) / 1. 0Eb

C Downccmer gas in put st epC

DUMMY=WGD(0)

WRITE(LUT,550)TBTL,DT, I PI550 FORMAT("Start, running and stepti me, number of steps between ",

PD( 1) =P0- DL

- Sta r t o f i t e r at i on t i me l o op—

DO 6000 XT=1,ISTTYD=TYD+DT/2WD(l)=WGD(TYD)*DT/2DELTA=1IFLAGOJFLAG=0KFLAG=0

CC In i t i al val ues f or f ol l ow ng t i me s tep.

C Axi al voi d di st r i buti on i n downcomer.C2050 CO 2100 I =2, NMAXD2100 ALFD(I , 2)=WD(I )* CP/AD*2/( ( PD(I - 1)+PD(I ) )* ( ZD(I - 1,2)- SD( I , 2)) )

ALFDI NMAXD, 2) ={1- FTR) *ALFDI NMAXD, 2)CC Axi al pressure di st r i but i on i n downcomer.C

PD( NMAXD) =PO- ZD( NMAXD, 2)DO 2150 I =NMAXD, 2, - 1

2150 PD( I - l )=PD( I ) - {ZD( I - l , 2 ) - ZD( I f 2) ) * ( l - ALFDU, 2) )CC Pos i t i ons in hor i zonta l par t .C

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C PR( MAXR+1) =PR( MAXR)PD( MAXD+1}=PD( MAXD)DO 1000 I=MAXR,1,-1WR(I+1)=WR<I)

1000 AL FR (I+ 1 ,2 )= AL FR (I ,1 )DO 1050 I=MAXD,1,-1WD(I+1)=WD{I)

1050 AL FD(I+ 1 ,2 )= AL FD(I ,1 )MAXR=MAXR+1VLSR( 2) =VLSR( 1)VLSD( 2) =VLSD( 1)

CC spat i a l i t e r at i onC1200 IF((W D(1).EQ .0).AND .(MAX D.EQ.1) ) GOTO 4000CC C a l c u l a t i o n o f p o s i t i o n s m d o w nc o m er .C

IHD=1FTRK)AL FD(1 ,1 )= AL FD(2 ,2 )DO 2000 1=2, MAXD+1VGD- f VLSD(2)/ ( l - ALFD(I , 2))+VLSD(l )/ ( l - ALF, D( I - l , l ) ) ) / 2 - VSDI F( VGD. LT. O) GOTO 6025ZD{I , 2)=ZD( 1-1, 1)- VGD*DT/ 2I F( ZD( I , 2). GE. 0) GOTO 2000I HD= HD+1ZH( I HD, 2) =- ( VGD+VSD) / VGD*ZD( 1, 2 )WH( I HD, 2) =WD(I )

2000 CONTI NUENNAXD=HftXD*- lI F( I HD. EQ. l ) GOTO 2050

NMAXD=NMAXD-IHD+2FTR=ZD( NMAXD, 2) / ( ZD( NMAXD, 2) - ZD( NHAXD- 1( 2) )WH(2, 2) =FTR*WH(2, 2)ZD( NMAXD, 2) =0

SWH=0I HH=1C I F no gas f l ow to hor i zonta l par t then go to r i ser .

I F( I HD. EQ. l ) GOTO 4000DO 3000 I =2, MAXHZH( I HD+ - 1,2) =ZH( I , 1)+ZH( I HD, 2)

3000 WH( I HD+ - 1, 2)=WH( I , 1)NMAXH=MAXH+1HD- 1DO 3050 I =2, NMAXHI F(ZH( I , 2). LT. HL) GOTO 3050I HH= HH+1SWH=SWH+WH( I , 2)

3050 CONTI NUEC I F no gas f l ow to r i ser then qo to r i ser .

I F( I HH. EQ. l ) GOTO 4000NMAXH=NMAXH- I HH+2m I WMCH, 2)=( HL*- aH( Kft XH- l »2) ) / CZH( I »aXH, 2)- ZH( ! «OU{R- l , 2) ) *

■nWH(NMAXH, 2)ZH(NMAXH, 2) =HLSWH=SWH-WH (NMAXH, 2 )f f i ( 2) =«RU) +SWH

cC Po si t i ons in r i ser .c4000 DO 4050 1-2 MAXR

I F(PP(I ) . LT.. 0) GOTO 6025VS2= 0473+ CO4b51*( PO/ FR( I ) )* *( 2/3. )VS=SQRT(VS2)+ 04+ 02*( PO/ PR(I i - l }

4050 ZR( I , 2)=ZR( I - l f l ) +( VS+VI£R(2)/ ( l - ALFRa, 2)) +VS+VLSR( l )' n/ ( l - AL FR( I - l , l ) ) ) * DT/ 2 / 2

CC Ax ia l r i ser voi d di s t r i but i o nC

DO 4100 1=2, MAXR4100 ALFR( I , 2)=WR(I ) *CP/ AR*2/( (PR(I )+PR( I - 1)) *( ZR(I , 2)- ZR<1-1,2) ))CC Ax ia l r i ser pr essur e di s t r i but i on

DO 4150 1=2, MAXRPR( I )=PR( I - 1 ) - (Z R( I , 2) - ZR( I - 1 , 2) ) *U- ALFR( I , 2) )I F(ZR( I -1 , 2) . LT. H1.AND. ZR( I , 2) . GE.H1) P1=PB( I )+(PR( I ) - PR( I -1 ) )

n / (Z R( I , 2) - ZR( l - l , 2) ) * (Hl - 2 R{I , 2) )4150 I F(ZR( I -1 , 2) . LT. H2. AND. ZR( I , 2 ) . GE.H2) P2=PR( I )+(PR( I ) " PR( I "! ) )

C Shi f t cal cul ated t o i n i t i al va l ues .C5350 VLSD( 1) =VLSD( 2)

97/A65

0

H S T S S S

- A

1

um * ls l e j

i ^ ^ ^ ^ ^ ^ ^ ^

/ ^

■

M

BO /

7 / "7°v

ƒ / /

-«■ halghLlm

i

state_

-

Fissure A 6 5 . 2 : Coinputf

97/A66

%y^W^

Flf iur e A66 : Illus t

a) averaging each ft

atloci of the smooching process performed by

r successive samples (middle l ine)

7/22/2019 TR-diss-0001460(1)


Figure Abo. 3;

Ami ) Uc/.(i , j ) - a(i - l , j )( l - exp( £ _ At) )ARp(i ) Az/ c (3)

(Az)

( for nomencl ature see program d es c r i p t i o n ) . Wth an esti mate for D of

0 1 m Is ( J oshi and Shah, 1981) the effect was computed for a step i n the

ri ser air i nj ecti on (ai r l i f t mode) and shown to inf l uence on y the overshoot

(f i gure A65 .3 ) . I n vi ew of the f i gures gi ven in secti on 66 gas di spersi on i s

probabl y smal l er. Si nce the esti mated value is based on l i terature for bubble

col umns w th very smal l l i qui d rates thi s was expected. Gas di spersi on was not

i ncorporated i n the f i nal formof the programbecause of the uncert ainty about

the val ue of D and i ts smal l i nf l uence.

A66 Smoothing and differentiating program BEW

Despi te prefi l teri ng ( f i rst order, T = 1 25 s) consi derable noise was

present i n the si gnal f romthe di f ferenti al pressure c e l l . A programwas

developed to smooth fi rst the si gnal and then restore the ori ginal si gnal and

b) p o l yno m i al I l t t t r through fi ve successive samp es (l ower l i n e) .

;ti OD of samp ed data ( samp e ti me 200 ms) of pr<

vel oci ty ( fromthe f l ow meter si gnal ) by di f ferenti ati ng. The method used

fi rst averaged over four samp es ( reducing the data by a factor four, see

f i gure A66) and then fi tt i ng a second order polynomal w th the l east squares

method through 5 poi nts (see again the f i gu r e) . The pol ynomal cou d be used

for the di f ferenti ati on ( appendi x 1 3. 1 ) .

A77. 1 Characteri sti c val ues for a l oop reactor

B enke' s expressi ons (77-2 to -4) are based on the compari son w th

bubble col umns w thout i nternal s. The characteri sti c di ameter i s then the

col umn di ameter. Characteri sti c l ength and vel oci ty fol l ow from the

cir cu ati on ti me

(1)

To general i ze equati on (77- 3) for a column wth changing cross secti ons

(notabl y of the downcomer of our experi mental col umn) we use equati ons (1) and

(77- 4) (i refers to the secti ons w th di f ferent di ameters)

STTd /4)L■ I

L. 2(2)

2 V 2d L = 2 ^d L.

2 L .1 1

(3)

242 97/A77The charact eri st i c l ength is s i mply anal ogous to Bl enke' s expressi on

L - T L . ( 4)C *r* il

I n th i s way the di scharged l i qui d vo lume i s a l so wel l def i ned

VL = « ï ï d2c/ 4 ) L c ( 5)

Note t hat w th Bl enke' s f ormulae the character i s t i c l ength i s tw ce the hei ght

of the co lumn, which di f fer s f rom that used as the character i s t i c l ength in

the Pécl et-

number f or bubbl e col ums w thout i nter nals .

97/ A77 2'

Hydrostat i c character i s t i c val ue

We w l l express t he character i s t i c val ue as f o l l ows:

( 2)

i c

pressure. We w l l onl y d iscuss the s i t uat i on a i r / water where t he

character i s t i c pressure cor responds w th t he hei ght h above the i n let .

S tar t i ng po int s

7/22/2019 TR-diss-0001460(1)


A77. 2 The character i s t i c super f i c i a l gas vel oci ty

I n t r oduc t i on

Overal l desi gn parameters for bubbl e col umns, such as axi al di sper

s i on are of t en cor r e lated w th the onl y i ndependent var i abl e, t he gas f l ow

r a t e . General l y these parameter s are st rongly coupl ed to the l ocal void

f r a c t i o n

< « > - ^ ( UG

When hydrostat i c eff ects are i mport ant, overal l parameter s shoul d be descri bed

w th a r epr es enta t i v e va l u e of v„ wh i c h we w l l c al l t h e c ha r ac te r i s t i cGs

Gas veloc i t i es encountered i n var i ous l i terat ure sources i nc lude,the outl et val ue, normal l y based on barometr i c pr essure and actual t emperature v„ , or on standard condi t i ons;

Gsb '

t h e i n l e t v a l u e , v_ , ;Gsit h e l o ca l v al ue (H i l l s , 1 97 6) f or c or r e l a t i ng l o ca l v o i d f r act i ons ;

the value hal f way along the col umn ( Fuj i e et al . , 1980) ;

the ari t hmeti c mean val ue of i nl et and out l et ( Freedman and Davi dson, 1969) .

- I d ea l g as

v« h = constant ( 3)

- I n f l uence vo id f ract i on on pressure

h m h + LZ ( 4)

wher e h = p / pTg, normal l y h, , and E i s t he mean l i qui d vol ume f ract i on

between the top and a gi ven vert i cal coordi nate z.- The character i s t i c va lue i s t he ar i thmet i c mean j , a s i mple approxi mat i ng

express ion i s prefer r ed.

We can choose between the fo l l ow ng poss i b i l i t i es :

- Out l et val ue

h - h (5)c t

- Hal f way

h - h + k H (6)

fch/ + M h , + EH)1 (7 )

9 7 / A 7 7

equation (A77-5!

50 10

97/ A77 Z4 D

two possi bl e approxi mati ons wi l l be consi dered

e - i c«)and

These combi ne t o gi ve

h* ~ = x ( 5)c

= 1/(1 + W )( 6 a )

7/22/2019 TR-diss-0001460(1)


r f i c i a l gas vel oc i t y f rom the commonly used out l et val u i

/ • 1 2 0 % , - 6 0%

F i g u r e A77.2; Compari son of t he var i ous models f or the rel evant

d a t a a t t h e s a me c i r c u l a t i o n ve l o c i t y b ut d i f f e r en t t o p pr es s ur i

- Mean

h = _>>_c H

f dz

J h

i u p e r f l c l . 1 g . s v e lo clC y ,

( 8)

where £ i s t he mean l i qui d volume f ract i on of the rel evant part of the col umn

and H the correspondi ng mxt ure hei ght .

Even if an approxi mati on f or z based on a const ant gas vel oci t y i s

used, th i s l eads to express ions t hat are too complex . Theref ore t he f o l l ow ng

- 1/(1 + 6TH*) (6b )

- (1 + f eH*) / ( l + ~H*) O)

= l n(l + H*) / H* ( 8a)

= l n ( l +I ) / l H* ( 8 b)

where h* = h /h and H* = H/ h_ are pl ott ed i n fi gur e (A77. 1) . I t showsc c t t

that the l i near ( 6) and logar i thm c ( 8) express ions are comparab l e, and t hatthe cor rect i on compared t o the out l et val ue (5) i s a l ready near l y 10%f or a

col umn of onl y 2 mhi gh.

Compar i son

A very sensi t i ve compari son w t h t he exper i mental col umn can be made

between operat i on under barometr i c and l ow pressur es. Total l y di f f erent baro

metr i c gas f l ow rate val ues v Q l ead to the same c i rcu l at i on veloc i t i es

( and m x i ng c ha r acte r i s t i c s ) . Th e c ha r ac te r i s t i c v a l u es of t h e r i s er b as ed

super f i c i a l gas veloc i ty ca l cul ated w th equat i ons ( 5) - (8) are compared for

the t wo operat i on modes i n f i gure (A77. 2) . The top press ure was approxi matedwi th

M L ) = h + c „ (H " L ) O )

^ b 97 /A 77

F = 1 ~ — H A l n ( l + H * ) ( 1 0 )

b a s ed o n t h e a s s u mp t i o n of c o n s t a n t g a s v e l o c i t y .

T h e f i g u r e s h o ws Ch at t h e i n f l u e nc e of v o i d f r a c t i o n i s s l i g h t . T h e

l a r g e s t v oi d f r a c t i o n en c ou nt e r e d wa s 1 0%. Th e r e i s no s i g n i f i c a nt d i f f e r e n c e

b e t we e n t h e l i n ea r a nd l o g ar i t h m c c o r r e c t i o n s an d t h es e ar e c l e ar l y b e s t ,

C on c l u s i o n s

- A c h ar a c t e r i s t i c v a l u e f o r t h e g a s f l o w r a t e s h o ul d b e u s e d i n s t e a d o f t h e

9 8 L i s t o f r e f e r e n c e s

247

Abuaf. N. , O C.J ones, and G. A.Zi mmer, 1978 42Proc. US Nucl ear Reg. Gum Rev.Group Meeti ng on Two Phase Fl owI nstrumentati on, New York, l b.Opti cal pr obe for l ocal voi d f racti on and int erface vel ocit y measurements

Adler, I . , W- -D Deckwer,and K.Schü' gerl , 1982 2Q em& g.Sci . , 37,271 S 417Performance of tower l oop bi oreactors, I 5 I I

A l v a r e z - C u e n c a , M . , C . G . J . B a k e r , a n d M . A . B er g o u gn o u , 1 9 80 8 3C h e m . E n g . S c i . , 3 5 , 1 1 2 1Oxygen mass tr ansfer i n bubbl e col umns.

Alvarez- Cuenca, M , and M A.Nerenberg, 1981 81A. I . Ch. E . J l , 27, 66Oxygen mass t ransf er i n bubbl e col umns worki ng at l arge gas and

l i qui d f l ow ra tes .

7/22/2019 TR-diss-0001460(1)


b a r o me t r i c o r o ut l e t v a l u e wi t h t a l l c o l u mn s an d f o r c o mp a r i s o n o e t w ee n

c o l u mn s e ve n wi t h s ma l l c o l u mn s i f t h e y d i f f e r i n h e i g h t .

- A s c h ar a c t e r i s t i c v a l u e ma y be u s e d t h e l o g a r i t h m c me a n v al u e

\( 1 1 )

f o r a c o l umn ope n t o t he a t mo s phe r e . A l s o t he ha l f wa y v a l ue ( e qua t i o n ( 6 ) ) ma y

be used.

- T h e i n f l u e n c e o f v o i d f r a c t i o n s be l o w 1 0 % ma y be n eg l e c t e d .

- C or r e l a t i o ns f o r ma s s t r a n s f e r , d i s p er s i o n c o e f f i c i e n t s , me an v oi d f r a c t i o ns

a nd t h e l i k e , ba s ed on o ut l e t o r i n l e t v a l u es c a nn ot g e n er a l v a l i d i t y .

N o t e

T h e c h ar a c t e r i s t i c s up er f i c i a l g as v e l o c i t y , a s d ef i n ed i n eq u at i o n

( 1 1 ) , i s di r e c t l y r e l a t e d t o t h e p o we r d i s s i p a t i o n p e r v ol u me i n a b ub b l e

c o l u mn . F r o m R oe l s a nd He i j n en ( 1 9 8 0) :

P - QG£ I l n ( p . / p t ) ( 1 2 )M

3 3

( QG g a s f l o w r a t e ( m Is); V mo l a r v o l ume ( 0 . 0224 m / mo l ) ; R u n i v e r s a l g a s

c o ns t a n t ( 8 . 3 J / mo l K ) ; T t e mp e r a t u r e ( K ) ; p ± an d p p r es s ur e at i n j e c t i o n

a n d t o p l e v e l ( P a ) ) ; wi t h ( 1 1 ) a nd ( 1 2 )

p/v" M W ( 13)

Ar i s .R. , 1959 ?1Q emEng.Sci . , 9,266Notes on the di f fus i on- t ype mode l fo r l ongitudinal mxi ng in f l ow

Bai rd,M H. I . , and R.G. Rice, 1975 72, 74ChemEng. J l . , 9,17Axi al di spersi on in l arge unbaff l ed columns

Barnea,D., O Shohamand Y.Tai tel , 1982 41ChemEng. Sci. , 37, 741Flow patt ern tr ansi ti on for vert i cal downward two phase f l ew

Beek.W J . , 1965 53Proc. Symp. on Two Phase Fl ow Exeter J une 1965, F401Osc i l l a t ion i n a batch o f l i qui d susta ined by a gas f l ow

Bergl es ,A.E. . J .G .Col l i e r , J .H .De lhaye ,G.F . Hewt t , F .Mayi nger , 1981 16Hemsphere Publi shing Corporati on, Washi ngtonTwo-phase fl ow and heat tr ansfer i n the power and process indust ri es.

Berni er, R.H., and C.E. Brennen, 1983 A42I nt . J l .Mult i phase F low 9,251Use of the elect romagneti c f l owmeter i n a two-phase El ow

Bhaga, D. , and M E.Weber, 1972, 32,35,A33Can.J l . ChemEng. , 50,323 & 329Hol dup i n vert i cal two and three phase f l ow

Bl enke, H. , 1979 1,21, 52,74,77,A77Advances in Biochemcal Engi neeri ng,13,121Loop reactors

Bol ton,D.H. ,and J . C.Ousby, 1975 4Workshop on Lar ge Wastewat er Treatment Pl ants, ViennaI n t . As s . on Water Pol l uti on ResearchThe ICI deep shaft eff l uent t reatment process and i ts potenti al forl arge sewage works .

Eol ton,D.H., D. A. H nes, and J . P.Bouchard, 1976 431st Annual Purdue I ndustr i al Waste Confer ence, Lafayette (I ndi ana)The appl i cat i on of the I CI deep sha f t process to i ndust r i a l e f f l uents

Bott on, R., D. Cosserat, and J . C. Charpenti er, 1978 34ChemEng. J l . , 16,107I nfl uence of col umn di ameter and high t hroughputs on the operati on of abubbl e col umn

Brown,L. C. , and C.R.Bai l l od, 1982 83J l . Envir onmental Engineeri ng D vi si on, Proc.ASCE,103, 607Modeli ng and i nterpreti ng oxygen t ransfer data.

Brown,R.W, A. Gctnezplata,and J .D.Pr i ce, 1969, A33,35ChemEng. Sci. , 24, 1483A model to predict voi d-f racti on i n two phase f l owSee al so Gomezplat a et al . (1972) and Stepanek(1970)

Bruij n,J ., and H. Tui nzaad, 1958J l , AmWater Works Associ at i on,50,879The rel at i onshi p between depth of U-t ubes and the aerat i on process.

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Siebers . H. H. en R.G. Veldkamp, 1980 5H20,13, 425Simula t i e van het deep shaft proces

Sm t h , J . M , E . S t a mme r s , a nd L . P . B. M J a ns s en , 1981 15DUH 1981, Del f tFys ische Transpor t verschi j nsel en I

SÖderberg,D. J . , 1980 5,14,15,21,41,51,Ph. D . T hes i s , Un i v e r s i t y of Cambri dge 53,54, 63,64, 92T he s t a bi l i t y of c i r c u l a t i o n i n bubbl e columns

Spe dd i ng, P . L . , a nd J . J . J . C hen , 1978 98ChemEng.Sc i . , 33,403Ve r i f i c at i o n of a two- phase hol dup re l a t i on

S p e d d i n g , P . L . , a n d V an T h an h N g u y en , 1 9 8 0 4 1ChemEng.Sc i . , 35, 779Regi me maps f o r ai r wat er t wo phase f l ow.

Speece,R. C., D. Gal l agher, C.Kri ck. and R.Thomson, 1980 5Prog.Wat. Tech. , 12, 395Pi l o t per fo rmance of deep U- t ubes.

Spe ur , J . A. , 1983 A65Ma s t e r s t he s i s . De l f t Un i v er s i t y of TechnologyHydrcdynamca van een bel l enko lom met r e ci r c ul a t i e

Stepanek , J . , 1970, A33, 98Chem I ng. Sc i . , 25 , 751A mod'- l t o p r e di c t v oi d f r a c t i o n i n t wo phase f l ow

J . F l u i d Mech., 116,343Turbulence i n two-phase di spersed fl ows.

Towel l , G. D. , C.P. Str and, and G. H.Ackerman, 1965 33R. I .Chi . E- I .ChemE. -sympos i um M xing-Theory re l a ted t o P r a c t i s e ,London, paper 10.10, 90M x i ng and mass tr ansfer m l arge diameter bubbl e col umns

Towel l , G. D., and G. H. Ackerman, 1972 72Proc .5th Eur . 2nd. I nt . Symp.Chem React . Engg. , E l sev i e r , Amsterdam B3-1Ax i a l m x i n g of l i q u i d and gas i n l a rge bubbl e reacto rs

Val uk ina ,N .V . , B .K .Koz 'menko ,and O. N. Kashinsk i i , 1979 31J l . E ngng. Phy s . , 36, 695 . t r a nsl a t i o n UDC 532. 529.5Char a c t e r i s t i c s of a f l o w of moncdisperse gas- l i qui d mxtur e i n av er t i c al t ubefr om I nzhenerno-F i z i chesk i i Zhurna l , 36, 695

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99

UPPERCASE

H H e nr y ' s c o n s t a n t (m/ s ) ( 8 3- 2 )e

I HL i n j e c t i o n h ea d l o s s (m ( 7-5)

K d i s t r i b ut i o n c o e f f . ( A 3 3- 3 )

L l e n g t h (7-3)

L c h ar a c t e r i s t i c l e ng t h ( 7 7 - 2 )c

I e f f e c t i v e h ei g ht (7-3)

eL l i q ui d o v er f l o w h e i g h t ( A 5 5 . 1 )

o *L s l u g l e n g t h ( A 2 2- 1 )

s u bs c r i p t s

t h e l i q ui d p ha s e

a i x t u r e of b o t h p h a s es

( 7 - 3 )

( 3 0 - 9 )

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E ur . J . Ap pl .M c r o bi o l .B i o t echn ol . , 1 2,1 50I nves t igat i on of thes t ructure of two-phase f l owsreactors . VI . Turbul ence s tuctures .

S l o ka r ni k , M , 1982ChemI ng. Techn., 54, 939Verf ahrenstechni k der aeroben Abwasserr ei ni gung

Zuber,N., and J. A Find ay, 1965,J l . o f Heat Trans fer , Trans .ASME,87, 453Average vol umetri c concentr at i on int wo phase flo*f

Zu n, I . , 1980I n t . J l .Mu l t i p ha se F l o w6 ,5 83The t ransverse mgrat i on of bubbl e:bubbl y f l ow

i bubbl e column bi o-

30, 31,32,A32, 33,A33,35, 37, 38, 39,41, 43, 52, 95

influenced by walls in vertical

List of symbols

Dimensions are according to S.I .

UPPERCASE

A

Ae

At

B

C0

ClD

D

F

G

H

area

ef fecti ve area

total area

di stri buti on coef f .

di stri buti on coef f .di spersi on coeff i ci ent

constant

tr ansfer f uncti on

hei ght

( 15- 2)

( 64- 1)

( 92- 1)

( 32- 2)

(A33- 13)( 71- 1)

( 63- 1)

( 75- 1)

( 77- 8)

subscri pts

c. mass trans

downcomer

fr i cti onal

the gas phase

hydraul i c

( 8 6 - 1 )

( 7 - 3 )

( 7 - 4 )

( 1 2 - 1 )

( A 1 5 - 1 )

M ma s s, mo l e c ul a r we i g ht

M

P

Q

R

R

T

T

U

V

W

( kg/mol )

eff ecti ve mass

power

vol ume fl ow

pipe radius

uni versal gas constant

temperature

ti me

vol tage

vol ume

mass f l ow

LOWERCASE

( 99)

( 64- 1)

( 92- 1)

( 30- 1)

( 32- 7) ri ser

( A77. 2-12)

( 36-1)

( 64-2)

( 12-1)

( 7 - 3 )

s u bs c r i p t s

s p ec i f i c a r e a

c o n s t a n t

c o nc e nt r a t i o n

o x yg e n s a t u r a t i o n

c o nc e nt r a t i o n

c o n s t a n t

g a s c o n s t a n t ( R T / M)

( 8 1 - 1 ) ac c e l e r a t i o na l

( 5 2 - 1 ) c o nc e r n i n g v oi d f r a c t i o n

a v e r a g e

( 5 2 - 1 ) bu b bl e

b a r o me t r i c

( 7 1- 1 ) c h a r a ct e r i s t i c

( 8 5 - 5 )

( 5 3 - 6 )

c i r c u l a t i o n

( 7 - 1 )

( 3 7 - 1 )

( 7 7 - 2 )

( 2 8 - 1 )

( 5 2 - 1 )

( 5 2 - 1 )

( 5 2 - 1 )

258

LOWERCASE

di ameter ( 14- 1)

character i s t i c d i ameter (77-3)

i n i t i a l bubb le d iameter (38-9)

hydraul i c di ameter ( A15- 1)

i n jector nozz l e d i ameter ( 28)

tube (col umn) di ameter ( 14- 1)

vol ume/ sur f ace mean

bubbl e di amet er (34-4)

c en t r e l i n e

d r i f t(32-5)

( 30- 10)

99

LOWERCASE

subscr i pts

k cor r ect i on factor (86-1)

k exponent ( 32-6)

m c er t a i n gas s mas s ( 65- 1 )

m exponent ( 3 2- 5 ) meas ur ed (2 3- 1 )

m ar ea r at i o ( 15- 2 ) m x i ng ( 77- 1 3)

m oxygen c onc ent r a t i o n

r a t i o ( 8 3- 2 )

m t r acer c oncent r at i o n

r a t i on exponent

( 77- 11)

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t o t al di f f e r ent i a l

eq

f

f

f

f

g

h

h bh

0h

t

f r i c t i o n f a c t o r

f r equency

funct i on

di mensi onl ess mean (def

gr avi t a t i ona l ac ce l .

head

barometr i c head

bott om head

top head

(14-1)

. A37)

(7-3)

( 12- 1)

(77-8)

( A13.2- 8)

(12-1)

e f f ec t i v e

e l l i p t i c

end or f i nal val ue

equi valent

f r l c t i o na l

f i sh or i nver ted ventur i

f r i c t i o n

g r av i t a t i ona l

natur al number

i magi nar y number (63-1)

( 30- 12)

( 30- 18)

vol umetr i c f l ux

d r i f t f l u x

c ha r ac te r i s t i c v e l oc i t y ( 3 0- 2 3)

natur al number

mass t r ans fer coef f . (81-1)

pa r ameter i n el l i p t i c

curve [ - ] (37-1)

pa r ameter i n el l i pt i c

curve [ - ] (37-3)

(7 - 3 )

(A64- 6)

( 74- 1)

( 7 - D

( A29)

(48-4)

(7 - 1 )

i nl e t va l u e

i n j ec t i on o r i nj ecto r ( 2 3- 7 )

i nhomogeneous + hi nderi ng ( 35-19)

n n umber

p pressure

p head

p exponent

pr

r r adi a l c oo r di n ate

s l apl ace v ar i abl e

s s tandard dev i at i on

Sr standard devi ati on

of the devi ati ons (T32)

s hydrostat i c val ue (53-4)

s sl ug ( A22-1)

s saturat i on val ue (85-5)

sf s i ngl e phase ( F43. 1)

t (r esi dence) t i me tube or col umn ( 14- 1)

t c i r c ul a t i on t i me (7 6- 7 )

t m x ing t i me (77-13)

t t op ( a t l i qui d l eve l ) ( 1 2- 1 )

t t o ta l ( 9 2- 1 )

u volt age ( A13. 1-1)

v vel oci ty (7-4) concerni ng the vel oc i t y (37-3)

(31-2)

(7-1)

( 65- 9)

( 34- 1)

( 32- 7)

( 75- 1)

out l et val ue

s ta r t ( z e r o ) va l u e ,

ove r f l ow

pressure

probe

p r o f i l e

r e l a t i ve

s l i p

s up er f i c i a l

bot tom value

( A55.1)

(53-6)

(F42. 7)

( 35- 19)

(30-5)

(30-1)

(15-1)

,

260

v s i ngl e bubbl e

t e r m n al v el o ci t y ( 3 1- 1 )

v c i r c ul a t i o n v el o ci t y ( 5 2- 1 )

v character i s t i c vel oci ty ( 77- 4 )

v ( l ocal ) gas vel oci ty (30 -5 )

v_. one-di mensi onal gas

vel oci ty (30 -3 )

v Gd dr i f t v el o ci t y ( 3 0- 1 7 )v super f i c i a l gas vel oc i t y (30 -6 )

99GREEK

Ü

TT

P

P

per i od (63-1)

Pi

densi ty (7 -3 )

dlmensi onl ess radi al

coordi nate r /R ( 32- 7)

SPECI ALS

CO i n f i n i t e

subscr i pt

af t e r i nf . t i me, i n i n f . medi um

261

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v m xtu re vel oci ty (30 -9 )

v ( l o ca l ) r e l a t i v e v el o ci t y ( 3 0- 5 )

s l i p vel oc i t y

f r i c t i o n ve l o ci t y

(30-1)

(48-4)

venturi (26-1)

( 32- 6)

var var i ance

w

x qua l i ty (F44. 3 )

x i n jec t i on hei ght

( downcomer ) ( 19)y i nj e ct i on he i g ht ( r i s e r )

y mol f ract i on ( 83- 2)

y d i s t ance f r omwal l (31 -3 )

z axi a l o r hei gh t

coord i nate (7 - 1 )

subscr i pt

A di f f e rence

2^ summat i on

a voi d f r act i on (7 - 2 ) cone , the voi d f r ac t i on (23 -6 )

Ó dev i a t i on concerni ng the dev i a t i ons

E l i qui d hol d up ( 30- 1)

j j dynam c v i scos i t y

v k i nemat i c v i s cos i ty (31 -3 )

{'Ó

."V

<A

-1

32—

<><>=

i nt e gr a t i o n

par t i al di f f e r ent i al

(axi al ) mean

dev ia t i on f r om

t i me-aver age ( 48-1)

root mean square val ue ( 48-3)

* cross- sect i onal average (30-15)

est i mate

about equal to

proport i onal

l ess than

greater than

l ess or equal

greater of equal

corr esponds with

SPECI ALS

*+

dl mensi onl ess def i ned quanti tydl mensi onl ess def i ned quanti ty

dl mensi onl ess def i ned quanti ty

one- di mensi onal

vol ume/ surf ace mean

( 30-

(Sau ter ) (34

- 3)

- 4)

262

Dl mensi onless numbers and quanti t i es

B* Mass tr ansf er group ( 85-12)

c* = c/ c c (71-2)

c * = ( c - c ) / ( c M - c ) ( 7 3- 4 )

Fr Fr oude number

Fr

Fr

h*

H*

K FP*

- ' G. K8V= VL 8D / gL e= h/ h

= H/ h

Fri ct i on number- p/ p(0)

(35-21)

( 55- 15)

( A77.2- 5)

( A77.2- 6)

(7 -4 )(38-4)

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P°

Pé

Re

= p/ PgLe

Pécl et number

Reynol ds number

(56-1)

(71-3)

Re c = Vc d c / P ( 7 7 " 1 )

Re centr e l i ne value

ReM " W Ls* = s t l )

t * - t / t cv* = v/ v

+ . c

v = v/ v:t 2

( v a r ) * = var / t *x* = x/L+ "

y = v f y /Z * z/L

*a = a/a

c% ■ °D + P°

( C4)

(76 -3 )

(71-2)

(37-3)

( A48-2)(71-6)

( 19)

( A48-3)

(71-2)

(37-1)

(55-15)

TR-diss-0001460(1)

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