HEAT TRANSFER FROM IMPINGING FLAME JETS

TR diss A

1559 Theo van der Meer

, HEAT TRANSFER FROM IMPINGING V FLAME JETS

HEAT TRANSFER FROM IMPINGING FLAME JETS

PROEFSCHRIFT

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft, op gezag van de Rector Magnificus,

prof.dr. J .M. Dirken, in het openbaar te verdedigen ten overstaan van een

commissie door het College van Dekanen daartoe aangewezen, op

10 september 1987 te 14.00 uur door

Theodorus Hendrikus van der Meer

geboren te Zoetermeer natuurkundig ingenieur

TR diss 1559

Dit proefschrift is goedgekeurd door de promotor prof.ir. C.J. Hoogendoorn

aan mijn ouders aan Funny

CONTENTS

1 . INTRODUCTION 1 .1 Background 11 1.2 Aims of this study 12 1.3 Outline of the investigation 13

2. LITERATURE SURVEY 2.1 Hydrodynamics 17 2.1.1 Turbulent free jets 18 2.1.2 The stagnation flow region 20 2.1.2.1 A bluff body in a uniform cross flow 21 2.1.2.2 The stagnation flow region of an impinging

jet 25 2.1.3 The wall jet region 28 2.2 Heat transfer of impinging flows 29 2.2.1 Stagnation point heat transfer 29

Influence of the turbulent length scale on stagnation stagnation point heat transfer 35

2.2.2 Heat transfer from cold impinging jets 36 2.2.2.1 The laminar impinging jet 37 2.2.2.2 The turbulent impinging jet 40 2.2.3 Heat transfer from flame jets 50

3. THEORY 3.1 The governing equations 53 3.2 Turbulence models 55 3.2.1 The k-£ model of turbulence 56 3.2.2 A low Reynolds number model 58 3.2.3 Drawbacks of the k-e model 60 3.2.4 The anisotropic model 62 3.3 The energy equation 67

4. THE NUMERICAL METHOD 4.1 The general finite difference equations 69

4.2 The hydrodynamic solver 73 4.3 The grid 74 4.4 The boundary conditions 76 4.5 Determination of the heat transfer coefficient 79

THE EXPERIMENTAL METHOD 5.1 Heat transfer from the isothermal jet 81 5.1.1 Experimental set-up 81 5.1.2 Temperature measurements with liquid crystals 83 5.2 Heat transfer from the flame jet 85 5.2.1 The experimental set-up 85 5.2.2 The Gardon heat flux transducer 88 5.3 The laser Doppler anemometer 91 5.3.1 The optical configuration 91 5.3.2 The electronic equipment 93 5.3.3 The seeding of the flow 9 5

RESULTS OF THE EXPERIMENTS 6.1 Introduction 97 6.2 Flow structure 97 6.2.1 Velocity and turbulence on the axis of the

free jet 97 6.2.2 The radial velocity gradient in the vicinity

of the stagnation point 1 07 6.2.3 The radial velocity profiles 110 6.2.3.1 The isothermal jet 110

H/d = 2 111 H/d = 6 113 The boundary layer thickness 116

6.2.3.2 The flame jet 117 H/d = 2 119 H/d = 6 119

6.2.4 Axial temperature distribution 122 6.3 Heat transfer 123 6.3.1 Stagnation point heat transfer 123

6.3.1.1 Isothermal jet 124 6.3.1.2 The flame jet 128

Radiation heat transfer 128 Convective heat transfer 130

6.3.2 Radial heat transfer distributions 137 6.3.2.1 The impinging isothermal jets 137 6.3.2.2 The impinging flame jets 141

Temperature distributions 141 The heat flux distributions 143 The Nusselt number distributions 145

7. RESULTS OF NUMERICAL SIMULATIONS 7.1 The laminar impinging jet 149 7.1.1 Comparison with literature data 154 7.2 The turbulent impinging jet 157 7.2.1 Comparison of results on different grids 159 7.2.2 Comparison of numerical with experimental

results 162 7.2.2.1 H/d = 6 162 7.2.2.2 H/d = 2 165 7.2.2.3 The stagnation point heat transfer 171

8. DISCUSSION AND CONCLUSIONS 8.1 The flow structure 173 8.2 Heat transfer 174 8.3 The simulated laminar impinging jet 175 8.4 The simulated turbulent impinging jet 176 8.5 Main conclusions 176

LIST OF PRINCIPLE SYMBOLS 179 LIST OF REFERENCES 183 SUMMARY 191 SAMENVATTING 1 93 CURRICULUM VITAE 195 NAWOORD 197

1. INTRODUCTION

1.1. Background

Heating, cooling and drying processes are often used in industry. In most applications high heat transfer rates leading to short processing times are required. The high heat transfer rates are especially needed in circumstances where the energy consumption of the process is relatively high. Obtaining short processing times is often needed for reasons of product quality. A very well-known technique for heating or cooling purposes is the application of impinging jets. The high heat transfer due to turbulent forced convection by impinging one or more jets of hot air or one or more flames on the object to be heated makes a relatively short exposure time possible. In the metallurgical industry this technique is called rapid heating. In cooling and drying a similar situation occurs; one or more jets of cold (dry) air impinge to cool (dry) a product.

Rapid heating of products in furnaces is a common process in, for instance, the glass and steel industry. To obtain a uniform heat transfer rate to the object in most cases radiative heat transfer is preferred over convective heat transfer. Radiative heat transfer can be achieved by firing gas, coal or oil in a radiation furnace. The walls of the furnace are heated and in its turn the object is heated by radiation heat transfer from these walls. Also often an electrically heated wall is used. In this way the control of the radiation temperature over the hot surface can easily be obtained. When a short exposure time of the object to the high temperatures is needed, it can be advantageous to enlarge heat transfer by impinging gas flames directly on this object. For this purpose high velocity burners are used, the major heat transfer is by convection. There are several other advantages of using these so-called impinging burners in rapid heating furnaces: - The furnace walls are less heated than in conventional radia-

11

tion furnaces, giving lower wall heat losses. Starting up and cooling down periods are much shorter which also result in an energy saving.

- Energy can be saved by switching on the burners only when heat is demanded.

- Compared to heating electrically the primary fuel demand is smaller.

- It is possible to heat locally. The energy savings compared to a conventional radiation furnace can be more than 50%. A major disadvantage of rapid heating furnaces can be nonuniformity of the heat flux distribution. With convective heat transfer it is much more difficult to obtain uniform heating of an object than with radiation heat transfer. It is possible that hot spots are created and overheating at such spots (for instance, at a stagnation point) can occur. For this reason it is important to know the heat flux distribution of a flame jet impinging on an object.

1.2. Aims of this study

The main purpose of the investigation presented in this thesis was to study the nonuniformity of the heat flux and to find out the influence of turbulence on the heat transfer for a single

Ufflk/ 1 I

''•Ml

or Fig. 1.1. The impinging jet.

12

flame jet impinging on a flat plate. The flow configuration is given in figure 1.1. The flame jets were produced by modified commercial rapid heating tunnel burners. The highest heat transfer rates applying impinging jets can be achieved for distances between the nozzle exit and the plate of 2 to 12 nozzle diameters. In this region for a turbulent jet the shape of the velocity profile and the turbulence intensity profile change with the distance from the nozzle. The jet shape also depends on the shape of the nozzle from which it originates. For these reasons one simple expression for the heat transfer to a plate on which the flame jet impinges cannot be given from literature. In this study both heat transfer and flow structure of impinging flame jets and of isothermal air jets from the same burners are thoroughly examined.

1.3. Outline of the investigation

From literature much data on stagnation flows and impingement heat transfer are available. In chapter 2 this literature is discussed. Since the flow around bodies of revolution has its analogies with the impinging jet on a flat plate these flows are discussed in the first place. Here an important parameter is defined: the gradient of the radial velocity near the stagnation point just outside of the boundary layer: aR = (3v/3r)r_,.0 A similar parameter can be defined in impinging jet flow: the gradient of the maximum radial velocity near the stagnation point: B = (3v x/3r) r . These parameters appear to depend on the shape of the body of revolution and the shape of the impinging velocity profile, respectively. The influence of the shape of the body or the shape of the impinging velocity profile on the heat transfer at the stagnation point can be expressed using the radial velocity gradients aR or g.

The governing equations for the flow and heat transfer are presented in chapter 3. These are the continuity equation, the Navier-Stokes equations, the energy equation and equations forming a model to calculate the turbulent viscosity of the

13

flow (the k-e model). Since the turbulence in a stagnation flow will be anisotropic due to the deceleration in axial direction and acceleration in radial direction, the commonly used k-e model has been extended with a third parameter, which takes the anisotropy into account.

In chapter 4 the numerical technique used, the finite volume method, is given. Together with the appropriate boundary conditions we have all ingredients to be able to solve the governing equations from chapter 3 numerically. The results from these numerical calculations are discussed in chapter 7. At first some experimental methods and set-ups for determining heat transfer and flow characteristics are given in chapter 5. The heat transfer measurements for the isothermal impinging jets are performed with a liquid crystal technique. Also a Gardon heat flux transducer is used for the determination of the heat transfer distributions of the impinging flame jets. Temperatures in the flame jets are measured with thin wire Pt-Rh thermocouples. At last in chapter 5 the laser Doppler anemometer for velocity and turbulence intensity measurements is discussed.

The next two chapters deal with the actual results from our study. The experimental results in chapter 6 and the numerical results in chapter 7. At first in chapter 6 results of the flow structures of the impinging jets are given. Important characteristics of the jets are: - the axial velocity decay and the axial turbulence development as a function of the distance from the burner. Comparisons between isothermal jets and flame jets can give insight into the effects of combustion on the turbulence.

- the radial velocity gradient near the stagnation point (6). The impact velocity profile will have its influence on this parameter. With (3 a first estimation of the heat transfer at the stagnation point can be made.

- the radial velocity profiles close to the plate. In the same chapter 6 the heat transfer results are discussed.

14

Stagnation point heat transfer coefficients, determined with the radial velocity gradient Q, are compared with stagnation point heat transfer coefficient, calculated from heat flux measurements. The influence of turbulence is examined. Heat transfer results from flame jets and from isothermal jets are compared and described as much as possible in a similar way. At last the radial heat transfer distributions of the impinging isothermal jets and impinging flame jets are discussed in this chapter.

Chapter 7 contains results on numerical calculations. Laminar impinging jets with three different impact velocity profiles (flat, parabolic and Gaussian) are simulated. With these calculations the influence of the impact velocity profile on the heat transfer distribution on the plate can be determined. Besides this the computer code can be validated by comparing the results with results from other investigators. Results of simulations of turbulent impinging jets are also given in this chapter. Calculations have been performed with the standard k-e model of turbulence with modifications for low Reynolds numbers and with a k-e model including a parameter for the anisotropy of the turbulence. The computed flow fields and heat transfer are compared with the measurements for validity for the two H/d values: H/d = 2 and H/d = 6.

Finally, in chapter 8 the conclusions from this study and their consequences for the practical use of impinging flame jets are discussed.

15

2. LITERATURE SURVEY

2.1. Hydrodynamics

Extensive studies on the hydrodynamics of stagnation flows have been done in the past. They will be reviewed in this chapter. Before entering into details a brief description will be given of the flow pattern of an impinging round jet on a flat plate. This flow can be divided into three regions (see figure 2.1): the free jet region, the stagnation flow region, and the wall jet region.

stagnation zone wall let

U *\ ►

Fig. 2.1. Flow regions of a jet impinging on a flat plate.

In the free jet region the flat plate has no perceptible influence on the flow. According to Schrader (1961) this region extends to a distance of 1.2 times the nozzle diameter (1.2d) from the surface. '-■

In the stagnation flow region the axial flow strongly decelerates and the radial flow accelerates giving rise to an increased pressure in this region. The characteristics''of the

17

stagnation flow region depend strongly on the dimensionless nozzle to plate distance (H/d). It extends from 1.2d in axial distance from the plate to about 1.1d in radial direction for small nozzle to plate distances (H/d < 12).

In the wall jet region the fluid spreads out radially over the surface in a decelerating flow.

In the following paragraphs the three regions will be discussed in more detail.

2.1.1. Turbulent free jets

The free circular turbulent jet has been studied thoroughly in the past. In this paragraph only a brief description will be given of the results of these investigations. More detailed information can be found in the handbooks of Rajaratnam (1976) and Abramovich (1963).

The free circular turbulent jet can be divided into three zones. Referring to figure 2.2 we have:

developed zone

developing zone

potential core zone

F i g . 2 . 2 . The f r e e j e t .

18

1 . The potential core zone immediately downstream of the nozzle. In this zone the potential core is the flow region where the velocity remains constant and equal to the velocity at the nozzle exit. Turbulence is being generated by the large shear stresses at the jet boundary and diffuses towards the axis. The length of the potential core depends on the initial velocity profile and on the turbulence intensity in the nozzle exit. According to Gauntner, Livinggood and Hrycak (1970) the potential core length varies from 4.7d to 7.7d.

2. The developing zone in which the axial velocity starts decaying. The velocity profile develops into a profile which is independent of the nozzle geometry.

3. The zone of fully developed flow, where the velocity profile has reached its final shape. Tolmien (1948) and Gortler (1942) calculated a radial velocity profile from boundary layer type equations with the use of Prandtl's mixing length theory. Reichardt (1942) performed measurements and found that a Gaussian velocity distribution fitted his results best.

It is shown by Rajaratnam (1976) that in the fully developed jet flow the jet broadens linearly and the velocity at the axis decays linearly. This has been justified by experimental results. For the axial velocity decay Hinze and v.d. Hegge-Zijnen (1949) and Schrader (1961) give the correlations:

u 6.39 Hinze and Zijnen: — = — (x/d i 8) (2.1) uQ x/d + 0.6

u 8.0 Schrader: — = — (x/d i 8) (2.2) uQ x/d + 3.3

If the jet has a different density than the surrounding fluid, a correction is required. Based on the conservation of momentum flux Thring and Newby (1953) find an equivalent nozzle-

19

diameter, :de, for non-constant density jets. Due to the high rate of entrainment the density within the jet will approach the density of the surroundings (Ps) within a short distance from the nozzle. The momentum flux is:

ird2 ird* G = — P o V = — f " Psuo' <2-3>

which leads to

d_ = d ( 2) (2.4) Ps

The relationships for isodensity jets can be used for non-isodensity jets using this equivalent diameter. Chen and Rodi (1978) come to the same equivalent diameter by dimensional considerations. Due to density differences also a buoyancy effect can occur. Chen and Rodi also give the limits within which a hot round jet will be non-buoyant, being:

Fr-2 (-°-)_4 - < 0.5 (2.5) Ps d

Here Fr is a densimetric Froude number

D u 3

Fr = 1°^°- (2.6) g(Ps - P0)d

This densimetric Froude number in our experiments was high enough to obey criterion (2.5). .

2.1.2. The stagnation flow region

In the stagnation flow region the axial flow strongly decelerates and the radial flow accelerates giving rise to an increased pressure. The characteristics of this region depend strongly on the dimensionless nozzle to plate distance H/d. The limits of the stagnation flow region too are determined by H/d. According to Schrader (1961) for nozzle to plate distances H/d < 10 the stagnation flow region extends to 1.2d from the

20

impinged plate in axial direction and up to about 1.1d from the stagnation point in radial direction. Before entering into details of stagnation of an impinging jet, the simpler flow around a bluff body will be discussed.

2.1.2.1. A bluff body in a uniform cross flow

The first solutions of the boundary layer equations for a two-dimensional shear layer along a cylindrical body, which is perpendicular to a uniform cross flow, were given by Blasius (1908), Hiemenz (1911) and Howarth (1935) (see Schlichting, 1968). They supposed the flow outside of the boundary layer to be a potential flow. The velocity along , the body can be expressed as:

V(x) = v.] z + V3Z + V5Z + . . . (2.7)

Here z is the coordinate along the surface of the body. The velocity profile in the shear layer was calculated as a similar polynomial in the coordinate perpendicular to the surface.

In the vicinity of the stagnation point the velocity decay due to stagnation; and the acceleration of1 the fluid flow along the surface just outside of the boundary-layer are given for axisymmetric flow by:

U = - 2 aRy and V = aRz (2.8)

and for plane flow by:j • ■ ■

U = - axy and V = axz' ' (2.9)

, Homann (1936) solved the boundary layer equations for the case of axisymmetric flow with assumption (2.8).. The constants aR and ax in equations* 2.8 and 2.9;' depend on-'the shape and size of the body of impingement and on the. uniform flow velocity.' For three different bluntr bodies it is found from potential flow solutions (see Kays; 1966 and Kottke, Blenke and Schmidt, 1977): ~ .

21

c i r c u l a r

sphere

cyl inder

d i s c aR

aR

a x

= 4

= 3

= 4

U^/dTr

U„/d

Uoo/d

(2.10)

(2.11)

(2.12)

where d is the diameter of the body involved and U^ the uniform flow velocity. More accurate experimentally determined values of aR and ax are given by:

Kottke, Blenke and Schmidt (1977) for a disc:

aR = Ujö (2.13)

Newman, Sparrow and Eckert (1972) for a sphere:

aR = 2.66 U^/d (2.14)

and Hiemenz (1911) for a cylinder:

ax = 3.63 Ujd (2.15)

Compared to the infinitely extended laminar flow around a body, the turbulent flow is far more complex. Let us consider the influence of turbulence.

Due to the experimentally found strong sensitivity of stagnation point heat transfer of cylinders and spheres to small changes in the intensity of free stream turbulence (see Kestin and Maeder, 1957; Kestin, Maeder and Sogin, 1961; Kestin, Maeder and Wang, 1961), Sutera, Maeder and Kestin (1963) and Sutera (1965) did a theoretical investigation into the vorticity amplification in stagnation point flow. In a basically two-dimensional flow vorticity was distributed periodically over the third dimension. The normal velocity far from the stagnation surface had a periodic waviness along the direction normal to the plane of the basic flow (see figure 2.3). They showed that such vorticity, having a sufficiently large scale, can enter the boundary layer and significantly alter the heat transfer at the wall. Vorticity with a scale larger than the neutral wave length Xm±n = 21I/(aT}/v)i or X •

1 K 'nun

22

Fig. 2.3. The distorted stagnation flow studied by Sutera, Maeder and Kestin (1963).

2Tr/(ax/v)5 will be amplified. The distortion of the velocity field seemed to be small. Figures 2.4 and 2.5 show the distortions of velocity and temperature of the mean flow along the surface compared to the undisturbed case for Pr = 0.74. The shear stress increased by 4.85%, the temperature gradient by 26%. Experimental verification of this theory is presented by Sadeh, Sutera and Maeder (1970). They conclude that turbulence, and hence vorticity, is being amplified by the deceleration of the stagnation flow and by stretching of fluid elements. The amount of amplification seems to depend on the direction of the vorticity as was predicted by the theory. For natural turbulence on a stagnation streamline the turbulence intensity they found is given in figure 2.6. The dependence of amplification on scale was also found to be in accordance with the predictions of the vorticity amplification theory.

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0.8

0.6

0.4

0.2

1—r

Fig. 2.4. The distorted stagnation velocity after Sutera, Maeder and Kestin (1963).

J L

v '

Fig. 2.5. The distorted temperature field on a stagnation streamline after Sutera, Maeder and Kestin (1963).

24

V7~

2.0

f.5

r.o 0.5

o 0 0.1 0.2 0.3

y

Fig. 2.6. The turbulence intensity on a stagnation streamline for natural turbulence (after Sadeh, Sutera and Maeder, 1970).

2.1.2.2. The stagnation flow region of an impinging jet

In the case of a bluff body in a uniform cross stream we have seen that the average flow field in the stagnation region is dependent of the shape of the body. So, it can be expected that in the case of an impinging jet on a flat plate, the average flow field in the stagnation region depends on the oncoming velocity profile. For a free turbulent jet the velocity profile changes from a flat profile into a fully developed one with a Gaussian shape. Thus the character of the centreline velocity decay in the stagnation region also changes. Similar to the definitions of a R and ax (equations 2.8 and 2.9) for a bluff body in a uniform cross flow, this same parameter ■ can bé defined for an impinging jet:

axisymmetric flow u = - 2 aRy and v m a x = aRr (2.16)

plane flow u = - axy and v m a x = axz (2.17)

Here v is the maximum velocity along the plate. The analogy

T 1 i r

25

between a bluff body in a cross flow and an impinging jet only exists in the direct vicinity of the stagnation point. Where for a bluff body in a cross stream the value of aR is determined by the shape of the body, this value for an impinging jet is determined by the shape of the oncoming velocity profile.

For an inviscid uniform impinging jet Strand (1964) calculated for the velocity along the deflecting surface (H/d = 1 ):

V = 0.9032 ^ ° - + . . . . (2.18) d

Scholtz and Trass (1970) obtained a similar expression for an inviscid parabolic impinging jet. They found for H/d = 0.25:

V = 2.322 ° — + . . . . (2.19) d

From experiments it appears that the value of aR for a disc with diameter d in a uniform cross flow is the same as for a uniform jet with diameter d impinging on a flat plate. Schrader (1961) and Dosdogru (1974) found for aR in the case of small nozzle to plate distances of a uniform turbulent impinging jet (1 1 H/d S 10):

H un Schrader aR = (1.04 - 0.034 -) -°- (2.20) d d

Dosdogru aR = (1.02 - 0.024 -) -°- (2.21) d d

Giralt, Chia and Trass (1977) correlate their radial velocity gradient in the stagnation zone with the impact velocity and the jet half radius at the beginning of the impingement region, where the axial velocity in the impinging jet becomes 98% of the axial velocity in the undisturbed jet. The impact velocity from measurements by Giralt (1976) being:

26

H H u„ = u^ (1.004 - 0.003 -) - S 5.5 (2.22) c ° d d H H u„ = u„ (1.35 - 0.066 -) 5.5 < - S 10.0 (2.23) "- ° d d

7.37 H u_ = u„ - > 10.0 (2.24) c ° 0.67 + H/d d

The length scale at the beginning of the impingement region is characterized by them as:

rni H H -iz- = 0.493 + 0.006 - 1.2 s - S 6.8 (2.25) d d ■ d

r, 1 H H -i* = 0.069 (1 + -) - > 6/8 (2.26) d d d

For the value of aR can then be found:

aR = U1 . ° - (2.27) 15

u-| is a function of H/d expressing the influence of the shape of the velocity profile. For H/d = 1 .2, where uc = u0 and r j. = id, they find:

aD = 0.916 2 (2.28) d *R

For H/d > 10, however,

aR 1.852 (2.29) di 2

Like in equations 2.18 and 2.19 one can see the strong influence of the shape of the oncoming velocity profile on the flow characteristics in the vicinity of the stagnation point.

The role of the turbulence in the flow field has been visualized by Yokobori, Kasagi and Hirata (1977). They showed that when the plate was positioned in the developing region of

27

the jet (4 < H/d < 10) large scale eddies existed in the stagnation zone. The eddies seemed to be much larger than the thickness of the laminar boundary layer, and appeared to be generated by the large *shea_r at the jet boundary'upstream. For H/d < 4 the stagnation zone looked laminar-like and for H/d > 12 the eddies seemed to be distorted and accompanied by small scale turbulence.

2.1.3. The wall jet region

Where the velocity essentially is parallel to the plate the wall;,jet region starts. Schrader (1961) gdves a correlation for the radius r at which the velocity along the wall starts decaying. This he defines as the beginning of the wall jet

. v ■ . 'i

region. The correlation fór r is: y

-2 = 1.09 (-)0-034 ' ' (2.30) d H

For the maximum velocity in the wall jet he finds:

U, : — — — : L- + K,(H/d - 1 .2)(— - 1 ) J0.. 1 +0.1.8. (H/d.- 1 .2);l ..2 , -.rgl

r -1.17 ( — ) - ' • " ' (2.31) rg

with K1 = 1.10 and K2 = 0.27 for H/d S 4.7 and K1 = 1.45 and K2 = 0.09 for H/d > 4.7. For the wall jet velocity profile Schrader found that already at r/d 6 2 the profile was similar to the profile calculated by Glauert (1956) for a fully developed wall jet (see figure 2.7). The validity of Glauert's calculations is shown by Bakke (1957) and Poreh, Tsuei and Cermak (1967) who did measurements for larger distances from the stagnation point (r/d > 10). Because our experiments 'are "restricted to small values of H/d (H/d i 12) and" r'/d (r/d s 5) no' details of their measurements are given.

28

1.2

0.8

0.4

O O 0.4 0.8 1.2 1.6 2.0

y/y*

Fig. 2.7. Wall jet velocity profiles measured by Schrader (nozzle diameter of 50 mm, H/d =• 2) and the profile calculated by Glauert (a).

2.2. Heat transfer of impinging flows

The heat transfer characteristics of impinging flows will be discussed in the next three paragraphs. Of course the heat transfer is determined by the hydrodynamics treated in the previous paragraph. Firstly, results from literature, of heat transfer at a stagnation point will be discussed, mainly for cylinders in a uniform cross flow. The next paragraph concerns local heat transfer distributions of impinging round jets with almost constant fluid properties. In the last paragraph impinging flame jets will be discussed where, due to the large temperature differences, the fluid properties (like dynamic viscosity, specific density and thermal conductivity) vary strongly.

2.2.1. Stagnation point heat transfer

Heat transfer at a stagnation point of a body of revolution has

I

y-* /

i

i i i

vo*L,

i i i

i o V

0 A

a

l

1 1 r=S0 mm r=60 r=80 r=100 r = 150

a

£? o

I

0 V

I

■ I

o

I

-

-

29

been studied extensively in the past. Knowledge of the heat transfer at this point is of importance because here the heat transfer will be at a maximum. From literature we know the solutions in expansion series from Pohlhausen (1921), Eckert (1942) and Merk (1958). Sibulkin (1952) solved the boundary layer equations for laminar heat transfer to a body of revolution near the forward stagnation point. His solution can be regarded as the basis of all other experimental and theoretical results. For the Nusselt number in the stagnation point of a body of revolution he found:

Nu = 0.763 d (-)5 Pr0'4 (2.32) v

In this equation 3 is equal to the velocity gradient just outside the boundary layer:

3r = <—>y = 6,r+o <2-33>

For a two-dimensional stagnation point flow Kays (1966) gives a similar equation, which comes to:

Nu = 0.57 d (-)5 Pr0*4 - - (2.34) v

For a sphere, cylinder and disc the values of 3 are known (see paragraph 2.1.2), leading to the corresponding stagnation point heat transfer results:

2.66 u n c r\ A

sphere 6 = aR = Nu = 1.2 4 Re^^Pr"*4 (2.35)

cylinder 3 = a„ = — Nu = 1.09 Re0*5Pr0-4 (2.36)

disc 3 = aR = — Nu = 0.763 Re°-5Pr0-4 (2.37)

For a uniform jet impinging on a flat plate equation 2.37 can

30

be applied. When an impinging jet has a nonuniform velocity profile (e.g. parabolic or Gaussian) the value of 8 will in general be higher leading to a higher heat transfer rate at the stagnation point. This in analogy with the heat transfer to the stagnation point of a cylinder or a sphere.

Much experimental and theoretical work has been done on the heat transfer of a cylinder in a turbulent cross stream in the past. From these studies much can be understood from the influence of turbulence on stagnation point heat transfer. In this paragraph a review of these studies will be given.

Kestin, Maeder and Sogin (1961) and Kestin, Maeder and Wang (1961) showed that the influence of free stream turbulence on the heat transfer rate on cylinders in cross flow was important. From experiments on heat transfer to a plate at zero incidence it was concluded that only in the presence of a pressure gradient the free stream turbulence had large effects on heat transfer coefficients. The biggest enhancement in heat transfer occurred at relatively low turbulence levels. Kestin, Maeder and Wang (1961) found that the local Nusselt number increased by amounts of 25%-50% when the turbulence intensity increased from 0.5% to 2%.

Sutera, Maeder and Kestin (1963) and Sutera (1965) presented a mathematical model for a steady plane stagnation point flow. They showed that probably the dominant mechanism of heat transfer enhancement by turbulence is vorticity amplification by stretching (see paragraph 2.1.2). Computations done by them showed that a certain amount of vorticity in the oncoming flow caused an increase of the wall shear stress of 4.85%, while the heat transfer was - augmented by 26% (at Pr = 0.74) .

Smith and Kuethe (1966) performed experiments in low-turbulence wind tunnels. They found that the influence of free stream turbulence increased with increasing Reynolds number. At high Reynolds numbers (Re > 105) a phenomenological theory for stagnation point heat transfer on a cylinder agreed with their

31

experimental, results,. The. theoretical- curve they found for the •heat transfer at the stagnation point on a cylinder is:

Nu - — = 1 + 0.0277,Tu/Re (2.38) /Re

The assumption was made that the eddy viscosity is proportional to the free stream turbulence and to the distance from the wall;. From their theory; Smith and Kuethe then concluded that Tu/Re; iwoyld be the single correlation parameter .to describe stagnation point heat transfer. From, their experiments at Re-number.s lower than 105 they found that Tu/Re was not the only parameter, instead there.- was another dependency on the Re­number : -'! . - . ! ■ • ■

Nu " " '" c " - — = 1+ 0.0277 Tu/Re (1 - exp (- 2.9 10~bRe)) (2.39) /Re -j. . ,. - , . . . , . . 1

■ -i Many ..-investigato.rs , later used the parameter Tu/Re in their correlations.. Some, of them used, the theoretical calculations ,of Frossling (1940). as a -basis for their turbulent heat transfer correlation. Frossling gives for the; laminar heat transfer ,at the stagnation point on a cylinder:

Nu . —-. = 0-34.45,. (2.40) /Re

which 'is the; same..as; .equation 2.36 with Pr =. 0.7 for air. J<estin- and Wood (1971) presented their experimental

results-using the Smith-Kuethe parameter Tu/Re and the result from-..Frossling. Thus they found- for the turbulent- heat transfer at ; the stagnation' point of a..cylinder: (7.5 lO1* < Re, ' < 1 . 2 5 1 0 5 )

Nu /Re

Tu/Re Tu/Re 0.945 +. 3.48 — — — - 3.99 ( ,-);* (2.41 ) 100 100

:. Sikmanovic, Oka and-. Koncar-Dj'urdj evic (1974) found, thati.at a relatively :low Re-number (Re ,= 19;000) the augmentation -of ;;the

32

heat transfer was absent for Tu < 2%. They found:

Nu Tie 0.945 + 1 .94

Tu/Re Tu/Re 2.41 (——-)• 100 100 (2.42)

This correlation comes very close to equation 2.39 for Re = 19,000.

Lowery and Vachon (1975) on the contrary did not find the dependency on the Reynolds number as in equation 2.39. This is not surprising because their Reynolds numbers vary from 1.10s to 3.10s, where with the exponent in equation 2.39 the effect hardly counts. Lowery and Vachon found that a turbulence intensity of 14% gave a maximum increase of the laminar heat transfer of 60%. Raising the turbulence intensity more did not seem' to increase the local heat transfer' anymore, however, it needed more data to justify this statement. They found the correlation:

Nu

/ R e

T u / R e Tu /Re = 1 .01 + 2 . 6 2 4 — 3 . 0 7 ( . . ) :

100 100 (2.43)

Fig. 2.8. Stagnation point heat transfer from cylinders.

33

The results expressed in equations 2.38, 2.41, 2.42 and 2.43 are gathered in figure 2.8.

Apart from the experimental results discussed so far, some investigators studied theoretically the stagnation point heat transfer on a cylinder in a crossflow. Already mentioned is the phenomenological theory of Smith and Kuethe (1966).

Galloway (1973) formulated a roll cell model, which has been simplified into an eddy viscosity. He used the findings of Sadeh et al. (1970) who showed the formation of roll cells in a two-dimensional stagnation flow. Galloway found a strong amplification for high Prandtl number flows.

Traci and Wilcox (1975) used the Saffman turbulence model in their partly analytic, partly numerical solution of stagnation point heat transfer. They considered three regions: the free stream flow, the still inviscid body distorted flow and the viscous wall region flow. Solutions of the three regions were matched to each other. In agreement with Sadeh et al. (1970) they find amplification of turbulent energy in the stagnation region, while their heat transfer calculations do agree with the known experimental results.

Miyazaki and Sparrow (1977) constructed a model for the eddy viscosity on the basis of measured turbulent velocity fluctuations. It contained a single unknown parameter which was determined from experimental heat transfer results. Their numerical calculations showed that the Nusselt number increased with the free stream turbulence but to a lesser extent as the turbulence intensity increases. The effect of turbulence on the friction factor was much less than on the heat transfer, which also was shown by previous investigators.

Gorla and Nemeth (1982) constructed a mathematical model in which the momentum eddy diffusivity depended on the free stream turbulence and the length scale. Available experimental data were used to find the eddy viscosity as a function of Tu/Re.

A more detailed numerical study on heat transfer

34

enhancement around the stagnation point of a cylinder was performed by Hijikata, Yoshida and Mori (1982). They added an extra equation to the k-e model of turbulence to take into account the production of turbulent energy due to anisotropy between the longitudinal and lateral Reynolds stress components in the free stream (see paragraph 3.2.4). A reasonable agreement with reported experimental data was found.

Influence of the turbulent length scale on stagnation point heat transfer

Next to the influence of the turbulence intensity on the heat transfer investigations were done to the role of the turbulent length scale. For the definition of a characteristic scale most investigators use the turbulent macroscale or integral scale being the scale of the energy containing eddies.The macroscale of turbulence can be found by integrating the area under the space correlation function:

Lv = ƒ R(x) dx (2.44) x o

with

u(X)2

Van der Hegge-Zijnen (1958) found a rapidly increasing heat transfer with increasing macroscale. He suggested the existence of an optimum value of the ratio between scale and cylinder-diameter which corresponds to a condition of resonance. Then, some frequency of turbulence coincides with the frequency of the eddies shed by the cylinder. The work done by Sutera, Maeder and Kestin (1963) and by Sutera (1965) has already been mentioned before. From their mathematical model it follows that wavelengths shorter than a so-called neutral wavelength:

35

Xmin =-2ir/(av) (2.46) cannot satisfy the governing equations. The physical signifi­cance of this is that vorticity with a scale smaller than the neutral scale is dissipated more rapidly due to viscous action than it is amplified by stretching. Recent experiments varied the length scales of the flow to measure its influence. Sikmanovic, Oka and Djurdjevic (1974) found that the Nusselt number slightly decreased with an increase of the turbulent macroscale in the region Lx/d = 0.05 to Lx/d = 0.182. In the region 0.015 < Lx/d < 0.095 Lowery and Vachon (1975) did not find a noticeable effect of the macroscale of turbulence. Neither did Katinas, Zhyugzhda, Zhukauskas and Shvegzhda (1976) in the region Lx/d = 0.16 to 0.36. Yardi and Sukhatme (1978) examined the effect of turbulent macroscale on the heat transfer . explicitly. They varied the macroscale over the wide range of Lx/d = 0.03.to Lx/d = 0.38. They found that the heat transfer coefficient at the front stagnation point increases by about 15% as Lx/d is reduced from 0.4 to 0.05. The effect seems to diminish as Tu/Re is increased. At the value of the parameter (Lx/d)/Re of about 10 the effect of the macroscale is at a maximum.

More recently Gorla and Nemeth (1982) presented a mathema­tical model to predict heat transfer from a cylinder in crossflow. They used an eddy viscosity model in which Tu/Re and (Lx/d)/Re are parameters. The dependence of the turbulent viscosity on Tu/Re was determined by fitting the results to the experimental data available. The measurements done by Yardi and Sukhatme (1978) were used by them to find the expression in the eddy viscosity for the length scale parameter.

2.2.2. Heat transfer from cold impinging jets

In the paragraphs 2.1.2 and 2.1.3 the flow structure in the stagnation flow region and in the wall jet region of an impinging jet has been described. The heat transfer as related

36

to this flow structure will be treated in the following paragraph, where we restrict ourselves to flows of fluids with constant fluid properties. Firstly, results from literature on laminar impinging jets will be discussed, followed by results on turbulent impinging jets.

2.2.2.1. The laminar impinging jet

Most of the results reported in literature on heat and mass transfer from laminar impinging jets are from theoretical studies, although some experimental works are also available. From these studies influences of some parameters on the transfer of mass and heat could be determined without the existence of turbulence in the flow. The effect of the Reynolds number on the Sherwood or Nusselt number in the stagnation region (a), the influence of the velocity profile of the impinging jet (b), the influence of the separation distance H/d between nozzle and plate (c) and the dependency of the transfer coefficient on the radial distance along the plate (d) will be discussed. Because of the sparsity of results on axisymmetric jets, also results on two-dimensional (slot) jets are considered.

a. As can be seen in paragraph 2.2.1 the Nusselt number at the stagnation point of a body of revolution in a uniform flow depends on Re 5 This same dependency has been derived by Scholtz and Trass (1970)' for a parabolic impinging round jet and by Sparrow and Lee (1975) for a nonuniform impinging slot jet. In both studies a solution for the inviscid flow field was obtained. This solution was employed as a boundary condition for the viscous flow along the impingement surface. Another way of predicting thé flow field and heat transfer from a laminar impinging jet is by solving the full Navier-Stokes equations with the appropriate boundary conditions. This is done using' a finite difference representation of the equations by Saad (1975), also

37

published by Saad, Douglas and Mujumdar (1977) for an impinging round jet and by Van Heiningen (1982), also published by Van Heiningen, Mujumdar and Douglas (1976) for an impinging slot jet. A conclusion for the round jet study was that Nu ~ Re for a parabolic velocity profile in the range of 900 < Re < 1950. For a flat velocity profile in the same Re-range they do find the 0.5 power of Re, as the boundary layer theory predicts. For the slot jet Van Heiningen et al. (1976) find for a flat velocity profile again agreement with the boundary layer theory: Nu ~ Re • . For a parabolic velocity profile, however, they find Nu ~ Re 0 , 6, which differs from the similar axisymmetric case. Finally, two references give experimental results on mass transfer in the stagnation region. Scholtz and Trass (1970) confirm their theoretical results and find experimentally the Re • -dependency of the Sh-number in the stagnation region of a nonuniform impinging round jet. Sparrow and Wong (1975) experimentally confirm the results of Van Heiningen et al. (1976) for a slot jet with a parabolic velocity profile: Sh - Re0-6.

The influence of the velocity profile. In paragraph 2.1.2 from equations 2.18 and 2.19 we can see the influence of the shape of the impinging velocity profile on the radial velocity gradient near the stagnation point. According to the theory of Sibulkin (1952) this velocity gradient (6) determines the heat transfer coefficient in the stagnation point. In the case of a uniform flow over a body of revolution the stagnation point heat transfer strongly depends on the shape of the body (see paragraph 2.2.1). In the same way the shape of the velocity profile of a jet impinging on a flat plate will have its effect on the stagnation point heat transfer. This influence has been shown by several authors. From the boundary layer theory Scholtz and Trass (1970) find for H/d = 0.5:

0.8242 Sc0*361 + 0.1351 (-) 3 Sc0-386 -R

0.0980(-)" Sc 0" 4 0 8 + . .' (2.47) R

This relation holds for a parabolic velocity profile. With the inviscid solution of a uniform impinging jet from Strand (1964) they calculate (H/d = 1.0):

0.3634 Sc0-361 + 0.03441 (-)2 Sc 0 - 3 8 6 -R

0.002531 (-)" Sc0'408 + . . (2.48) R

These two equations hold for 1 < Sc < 10. The numerical computations done by Saad et al. (1977) also show the importance of the velocity profile. Not only in the stagnation region, but also in the wall jet region the heat transfer from a parabolic impinging jet is higher than that from a uniform impinging jet, according to their calcula­tions .

c. The influence of the separation distance between nozzle and plate has been studied by" Saad, Douglas and Mujumdar (1977). They found from their numerical calculations on axisymmetric impinging jets with a parabolic velocity profile in the range 1 .5 < H/d < 12 a decrease in stagnation point heat transfer of 15% with increasing H/d at Re = 450. At Re = 950 they found no perceptible decrease of the Nusselt number in this range. At the same separation distances Sparrow and Wong (1975) measured mass transfer from a laminar impinging slot jet with the naphthalene sublimation technique. They found no influence of the separation distance on the heat transfer for H/d < 5 (277 < Re < 1700). At higher values of H/d they find turbulence effects.

d. The radial variation of the transfer rate is given by Scholtz and Trass (1963) from their theory by:

Sh /Re

Sh /Re

39

Sh = 0.4264 Re 3 / 4(-)" 5 / 4 g(Sc) (2.49) d

with

g(Sc) = 0.3733 Sc1/3 (2.50) ' for high Sc-numbers (Sc > 10). They find agreement of this correlation with experiments obtained with a liquid jet at a high Schmidt number (1000-4000). Later the same authors find agreement also for a Schmidt number of 2.45 (at r/d > 1.5) (Scholtz and Trass, 1970). Also Kapur and Macleod (1974) found agreement between their measurements and equation 2.49. They determined local mass transfer coefficients by holographic interferometry. Scholtz and Trass used for their solution of the boundary layer equation for the mass concentration the analysis of Glauert (1956). He obtained a solution of the boundary layer equations for the motion of axisymmetric wall jets on the basis of self-preservation of the form of the velocity profile. The theory of Scholtz and Trass, therefore, cannot predict the difference in wall jet heat transfer originating from a parabolic or a uniform impinging jet as observed by Saad et al. It gives a higher power of Re in the wall jet region than is found in the stagnation point region. However, the fully developed region may not yet be reached in the calculations by Saad et al. Their results for the wall jet region do not seem to agree with the theory of Scholtz and Trass (1963) and the experiments by them and Kapur and Macleod.

2.2.2.2. The turbulent impinging jet

In most practical applications of heat or mass transfer from impinging jets the flow will be turbulent. Exact solutions of the problem are then no longer possible. Because of the practical importance of this flow many investigators performed

40

experiments and tried to correlate the heat or mass transfer rate to the flow parameters. Also numerical studies with the help of turbulence models were performed. In this paragraph only results from axisymmetric impinging jets will be discussed.

There are several ways in which the heat transfer can be correlated to the flow parameters. One approach is correlating the Nusselt number (ad/X) to the relevant parameters by the Reynolds number in the nozzle Re = uQ d/v, the turbulence in the nozzle exit (Tu = /u 0' 3/u 0), the separation distance between nozzle and plate (H/d), the radial distance from the stagnation point (r/d) and the fluid properties. Thus a correlation would have the form:

H r Nu = f(Re, Tu, -, -, Pr) (2.51) d d

Correlations in'this form have been used in the past. The development of jet velocity, turbulence and jet velocity profile with x/d is accounted for by a single parameter H/d in this equation. The disadvantage of this method lies in the fact that results on heat transfer from impinging jets with jets from different orifices do not agree. Especially the turbulence level at the jet origin and the initial velocity profile influence the jet development and subsequently the transfer rates.

Because of the complexity of a result in the form of equation 2.51 the heat transfer at the stagnation point is often separated from the radial dependency.

Another way to describe the stagnation point heat transfer is to use local (impact) parameters of the flow. Parameters which describe the free jet at the plane of impact when the plate is not inserted. In this way a correlation can be found of the form:

Nu>5 = f(Re5,Pr,Tuc Y) (2.52)

41

Here the Reynolds number is based on the impact velocity; the turbulence grade Tuc is based on the impact turbulence intensity; y is a parameter" which'is a function of the shape of the impact velocity profile. Now all parameters in equation 2.52 are a function of x/d.

A review will be given of the most important contributions to literature on heat transfer from axisymmetric turbulent impinging jets.

Smirnov, Verevochkin and Brdlick (1961) correlated their heat transfer measurements together with results from Perry (1954) and Schmidt, Schuring and Sellschopp (1930) into one equation for the stagnation point heat transfer:

Nu = 0.034 d0-9 Re 1 / 3 p r0- 4 3 exp (-0.037 -) (2.53)

d The range of variables where this formula holds is: 0.5 < H/d < 10, 1600 < Re < 50,000 and 0.7 < Pr < 10. The dependence on the non-dimensional nozzle diameter d (in mm) (which varied from 2.5 mm to 16.5 mm) in this correlation is rather surprising and is not confirmed by later experimentalists,

Huang (1963) used the impact velocity measured by a pressure probe on the spot of impingement to correlate the heat transfer rate. He finds for the stagnation point (1 < H/d < 10 103 < Re < 10"):

Nu = 0.0233 Re c0- 8 7 p r

0- 3 3 (2.54)

Surprisingly he did not find any other dependency on H/d than that of the impact velocity alone.

It is difficult to verify these and former results because little is known of the characteristics of the jets that were used.

The first extensive experimentalists who studied the influence of turbulence on the heat transfer were Gardon and Gobonpue (1962) and Gardon and Akfirat (1965). They showed that in contrast to a laminar impinging jet the stagnation point

42

Fig. 2.9. Radial heat transfer distribution for a round impinging jet on a flat plate at H/d = 2 (from Gardon and Akfirat, 1965).

r/d

Fig. 2.10. Heat transfer for a round impinging jet on a flat plate for Re = 28,000 (from Gardon and Cobonpue, 1962).

43

heat transfer from a turbulent impinging jet increases when H/d increases from 0 to 5. This is due to an increasing turbulence level on the axis of a jet in this range where the velocity remains constant. Several peaks were found in the local radial heat transfer distributions as can be seen in figures 2.9 and 2.10. At small separation distances (H/d < 4) the maximum heat transfer rate was situated at r/d - 0.5. This can be explained by the existence of a minimum of the boundary layer thickness at this place as was predicted by Kezios (1956). At a higher radial distance (r/d = 1.9) an outer peak was distinguished which at low Reynolds numbers separated into two outer peaks at r/d = 1.4 and' at r/d . = 2.5. Two reasons for the possible existence of an outer peak were mentioned: 1) Penetration of turbulence into the boundary layer coming

from the mixing layer of the jet. 2) Transition from a-laminar to.a turbulent boundary layer. At higher values of H/d the inner as well as the outer peaks disappeared due to the higher turbulence level of the impinging jet for higher H/d. Experiments with turbulence promoters in the nozzle exit showed that turbulence indeed had an enormous influence:- at H/d =. 2 -the stagnation point heat transfer was augmented and the outer peaks disappeared. The results from this study were confirmed by Schliinder and Gnielinski (1967). Measurements of the turbulence intensity very close to the impingement surface (0.15 mm) showed a qualitative agreement between this turbulence intensity and the heat transfer coefficients. From this could, be', concluded that the outer peak in the radial heat transfer distributions at r/d ^ 1.9 is due to turbulent eddies penetrating the- boundary layer.

From mass transfer measurements Jeschar and Potke (1970) concluded that, for .1 ... s H/d. S 20 and for 5 s r/d é 40 the Nusselt number can be correlated with (r/d + 1)~ 1* 1 p r

0- 4 2

f(Re). For the stagnation point mass transfer in the ranges 10 < H/d S 20, 8,000 < Re < '30,000, they found:

44

Sh = 1.2 Re°-7(-)"1-1 Sc0'42 (2.55) d

It was possible to find a correlation without Tu as a parameter, because in this range of H/d the turbulence leyel does not vary significantly anymore.

Nakatogawa, Nishiwaki, Hirata and Torii (1970) made an attempt to correlate the heat transfer rate to local flow parameters. Their starting point is the correlation for heat transfer in a plane laminar stagnation point flow (see equation 2.34), however, they consider an axisymmetric flow. For the velocity gradient near the stagnation point, the axial velocity decay and the jet half width diameter, they use empirical relations. In spite of some poor assumptions, the experimental heat transfer results for small separation distances (H/d < 5) agreed quite well with their predictions. For higher distances H/d the experimental results were 1.25 to 1.5 times larger than the predicted values probably due to turbulence effects which were at H/d = 8 at the highest level. The dependency on the shape of the velocity profile was not accounted for by them. For the wall jet region theoretical solutions, obtained by assuming a velocity profile according to the 1/7th power law, agreed well with the experimental values.

The quantitative influence of turbulence has been studied by Donaldson, Snedeker and Margolis (1971a). They applied a correction factor to the laminar stagnation point heat transfer which is a function of the free stream turbulence level. For the theoretical description of the laminar heat transfer they, used a correlation from Lees (1956):

C_ dv j. -E- r {pu(-) r. n) 2 (2.56) , 2(PrT* ^^dr' r=° J

This relation is similar to Sibulkin's equation for the stagnation point heat transfer of a body of revolution (equation 2.32). For the radial velocity gradient (dv/dr)r=0 experimental values were evaluated from data given by

45

Donaldson, Snedeker and Margolis (1971b) who assumed:

dv , 1 32P i <^>r=o = t" > r / o > a <2'57>

Because extensive measurements were done on the flow structure of the free jet, the ratio of the theoretical laminar heat transfer could be determined as a function of the average relative turbulent intensity in the free jet, defined by

k = -(ü71" + 2 V71")^ (2.58)

In the range of 0.10 < k/u < 0.25 the ratio of turbulent to laminar heat transfer varied from 1.4 to 2.2. Although very much scatter was found they did not find any discernable effect of the Reynolds number on this ratio.

For the average heat transfer coefficients Subba Raju (1972) derived relations which fitted the experimental results of different authors. In the range of parameters 1 < H/d < 10, 2.10" < Re < 4.10s, 0.7 < Pr < 8.0 and 1 < D/d < 60 he found:

Nu\(-)0-5 = 1.54 Re0'5 Pr1/3 - s 8 (2.59) d d

Nu Pr"1/3(-)3 = 35.0 Re0*5 + 0.28 Re°-8( 8) - è 8 d d d

(2.60) This result gives an indication that for D/d s 8 the boundary layer along the impingement surface is laminar (Nu - Re 0* 5), while it is turbulent (Nu = Re0-8) for D/d a 8.

Kataoka and Mizushina (1974) investigated the local enhancement of the heat transfer rate by free stream turbulence. A minimum in heat transfer is found by them in the stagnation point for H/d < 0.5 and a maximum at r/d = 0.6. Here the large eddies coming from the mixing region penetrate the boundary layer. The local skin friction showed a secondary peak at r/d = 2.2. In contrast to this, the local Nusselt number had

46

a secondary peak at r/d = 4 (for 6 < H/d < 8.5). It should be noted that their measurements were performed at high Prandtl numbers (2420 to 3300).

The necessity of using local parameters of the impinging flow to correlate heat transfer was observed by Chia, Giralt and Trass (1977). They adopted the already mentioned boundary layer solutions obtained by Scholtz and Trass (1970) to the velocity and length scales proposed by Giralt, Chia and Trass (1977), discussed in paragraph 2.1.2. These scales are the collision velocity at the stagnation point Uc and the jet half width radius at the beginning of the impingement region. The result of this approach is a mass transfer rate for the stagnation region without the influence of turbulence:

Sh, 1 V-, r ( n-5>iam = "2 V1a{r,(Sc) + —Jd2'(Sc) ( ) 3 + . . . } Reiu.b lam I o v^ z r ^

(2.61) The functions c0'(Sc), d2'(Sc) etc. are tabulated by Scholtz (1965). The coefficients V-j , V3 etc. are tabulated by Giralt et al. (1977) for different nozzle to plate distances. The coefficients V , V, etc. take into account the varying impinging velocity profile.

The effect of turbulence is taken into account by:

Sh_- Sh, 7itT= (1 + Y i » <7ie->lam (2'62)

For Yf a form also used by Lowery and Vachon (1975) and Galloway (1973) for heat transfer to cylinders in a cross flow is assumed:

y± = a Sc1/6 (TU;L Re^ - b) (2.63)

Experimental results showed that beyond - H/d = 11.0 the variation of Sh^/ZRe^ with r/r-j. is universal, although the turbulence free mass transfer (Sh^//Re^ )-Lam is universal beyond H/d = 8.0. This is attributed to a still increasing effect of

47

turbulence between H/d = 8.0 and H/d = 11.0, while the velocity profile does not change shape beyond H/d a 8.0. For the enhancement factor Chia et al. (1977) found:

y± = 0 Tuj/Re < 4.0

Yi = 0.0156 Sc1/,6 (Tui/Re - 4) 4<Tui/Re < 34.0 (2.64)

"yi = 0.468 Sc 1/ 6 Tu±/Re > 34.0

These results are in qualitative agreement with results obtained .for heat transfer from cylinders in a uniform cross flow discussed in .paragraph 2.2.1. It should be mentioned that the results found by Chia et al. (1977) are based on measurements at a single Re-number (Re = 34,000) at Sc = 2 . 4 5 . However, the resultant mass transfer equations have been used to predict literature data and found to be consistent over a wide range of flow conditions.

' 'A similar approach of using the stagnation point heat transfer results of a, cylinder in a cross flow has been undertaken by Den Ouden and Hoogendoorn (1974).' They influenced the turbulence level at the nozzle exit by placing grids in the nozzle,.. It was found that for small separation. distances (H/d < 4) the experiments could be correlated with equation

Nu' ' Tu /Re Tu /Re - — = 0.497 + 3.48 3.99 ( ) 2 (2.65) /Re 100 100

Almost the same equation was found by Kestin and Wood (1971) for cylinders (see equation 2.41). At higher separation distances (H/d > 4) apparently the influence of the changing velocity profile was the cause that equation 2.65 did not hold anymore.

"Special attention to the radial distribution of the heat transfer coefficient has been given by Vallis, Patrick and Wragg (1978). They used an electrolytic mass transfer technique .in a Reynolds number range of 3.880 < Re < 23,000. Supposing a Pr1 "-dependency.they found for the stagnation point: .

48

Nu = 1 .93 R e 0 - 5 8 P r 1 / 3 ( _ ) - ° - 7 4 10 < - < 20 ( 2 . 6 6 ) d d

This result differs very much from that of Jeschar and Pötke (1970) discussed earlier. For the fully developed wall jet region Vallis et al. (1978) found:

Nu = 0.078 Re 0 - 8 2 Pr 1 / 3 (-)"1-05 8 < r/d < 17 (2.67) d

or

Nu r = 0.11 R e r0 * 8 2 P r 1 / 3 (2.68)

with the distance from the stagnation streamline r as a characteristic length scale in Re and Nur.

A very detailed study of the influence of turbulence on heat transfer in the stagnation region of a two-dimensional, submerged, impinging jet has been done by Yokobori, Kasagi, Hirata and Nishiwaki (1978). They observed the stagnation flow field with the aid of a flow visualization technique. Results are already discussed in paragraph 2.1.2. The large vortex-like motions they observed in the region 4 < H/d < 10 enhance heat transfer considerably. By fixing a fine cylindrical rod at the nozzle exit, it seemed possible to create artificially a pair of large scale vortices on the impinging wall. Even when the wall was positioned in the potential core, the vortices were observed. The heat transfer in this region was enhanced by the artificial eddies to the same level as the maximum increase in heat transfer produced by the mixing induced large scale eddies. This study thus demonstrates that heat transfer predominantly is affected by large scale structures.

More recently Kataoka et al. (1987) also studied the mechanism of the enhancement of stagnation point heat transfer by large scale turbulent structures. They demonstrated the existence of vortex rings at x/d = 1 as already shown by Yule (1978) and Strange and Crighton (1983). These coherent large scale structures are produced due , to the instability of the

°4 9

laminar shear layer. At x/d = 2.2 two vortex rings pair into one before breaking up into large scale eddies at the end of the potential core region. Autocorrelation of centreline velocity fluctuations indicate for x/d S 4 periodicity. With the characteristic frequency fe and the centreline velocity u a Strouhal number is defined as St = fgd/u. This number equals about 0.6 for 1 < x/d < 2 and 0.3 for 2 < x/d < 4. For x/d > 4 the Strouhal number is defined with the frequency of large scale eddies, determined from the integral time scale. This resulted in a Strouhal number of about 2 at x/d = 6 decreasing to about 1 at x/d = 10. Kataoka et al. correlated the heat transfer enhancement with a surface renewal parameter being the product of a turbulent Reynolds number Re^ = / ug' 2 d/v and this Strouhal number. The value of u s

, a has been obtained from measurements 5 mm upstream of the stagnation point. In this way they also show that enhancement of stagnation point heat transfer is mainly due to turbulent surface renewal by large scale eddies.

2.2.3. Heat transfer from flame jets

Knowledge of stagnation point heat transfer (see paragraph 2.2.1) and of heat transfer from impinging jets (paragraph 2.2.2) can be used when heat transfer from flame jets is studied. A large number of investigators have used the theoretical solution of Sibulkin (1952) for the boundary layer equations for heat transfer at the stagnation point of a body of revolution as a starting point for the prediction of heat transfer from flames. This theory leads for the heat flux density at a stagnation point to:

q" = 0.763 (pfyfg)0-5 (hf - h ) Pr"0-6 (2.69)

where p f, yf and hf are the density, viscosity and enthalpy in the flame just outside of the temperature boundary layer in the stagnation point, h is the enthalpy of the gasses at the wall.

Fay and Ridell (1958) extended this theory by taking into

50

account the dissociation of air and recombination of radicals in the boundary layer along a cooled object. Their theory can be applied when the chemical reactions of the flame are still present in the boundary layer along the impinged surface. This resulted in:

q" = 0.763 (^W)0.1 ( P U|3) 0- 5 (hf - h„) Pr"0'6 . PfUf

{1 + (Le0-52 - 1) -ii£} (2.70) hf

where Le = D/a is the Lewis number (D being the diffusion coefficient) and h _ D is the dissociation enthalpy.

Buhr, Haupt and Kremer (1976) found that for methane-air flames without preheating the radical concentrations are low. The heat coming free with the recombination of radicals was then found to be negligible. A number of studies concentrated on high temperature flames for which recombination of radicals in the boundary layer of a cooled surface is an important factor. Among these are studies from Conolly and Davies (1972), and from Kilham and Purvis (1971 and 1978).

Beer and Chigier (1968) reported results from an experimental investigation of a flame impinging at an angle of 20° on the hearth of a furnace. Their results show that heat transfer can be increased by a factor of 3 using direct impingement. The contribution of convection to the total heat transfer amounted 70%.

Milson and Chigier (1973) performed studies on methane and methane-air flames impinging on a cold plate. Both flames had a cool central core of unreacted gas giving rise to lower heat fluxes near the stagnation point than at some distance from this point (for 10 < H/d < 16). The heat transfer coefficient in the wall jet region was higher than in the impingement region due to the cool central core.

Horsley, Purvis and Tarig (1982) used impinging natural-gas-air flames from several types of burners. For the

51

stagnation point heat transfer they found that the results showed differences depending upon the turbulence structure of the free flames from the different burners. Yet all turbulent flames considered gave stagnation point heat transfer in the order of 1.2 to 1.6 times higher than calculated from Silbulkin's theory. This is in agreement with the findings of Giralt et al. (1977) for impinging isothermal jets discussed earlier.

52

3. THEORY

3.1. The governing equations

The flow of the impinging jet can be described by the full Navier-Stokes equations (the equations of motion) and the con­tinuity equation. For a two-dimensional axisymmetric flow these equations in cylindrical coordinates read (see Bird, Stewart and Lightfoot, 1960): - continuity equation:

3p 1 9 3 — + (rpv) + — (pu) = 0 (3.1 ) 3t r 3r 3x

- equations of motion:

3u „ 3u .. 3u 1 3 ~ 3?^, 3p p — + pu — + pv — = - { - — (rTrx) + — ^ } - — (3.2) 3t 3x 3r r 3r r x 3x 3x

3v : 8v _ 3v 1 3 _ 3T_V TOO 3D p — + pu — +pv — = -{ (rt__) + —£*■ ^ } -3t 3x 3r r 3r r r 3x r .3r (3.3)

with u, v, T and p momentary values. The components of the stress tensor T Q and TQ do not appear in these equations because they are supposed to be zero due to the absence of a El-dependency in the problem. The non-zero stress-components are:

3v 2 ->- ■+ u{2 (V.v)} (3.4) 3r 3

v 2 ->--»■ , g{2 (V.v)} (3:5) r 3

3u 2 ■+ ->■ - g{2 (V.v)} (3.6) 3r 3

53

3u 3v u ( — - — ) (3.7) dr 3x

Assuming incompressible flow which means that V.v = 0, the equations of motion can be reduced to:

3u „ 3u . _ 3u 1 3 3u 9 9u p — +pu — pv — = (pr — ) + — (u — ) + S~ (3.8) 3t 3x 3r r 3r 3r 3x 3x u

3v _ 9v .. 3v 1 9 9v 9 3v p — + pu — + pv — = (pr — ) + — (u — ) + S~ (3.9) 9t 9x 3r r 9r 3r 3x 3x v

with

1 3 3v 3 3ü 3p S~ = - _ (ru — ) + — (w — ) - — (3.10) u r 3r 3x 3x 3x 9x

1 3 3v 3 3u 2uv 3p S~ = - — (ru — ) + — (U — ) - -r- - — (3.11) v r 3r 3r 3x 3r ra 3r

If the flow under study is turbulent a time averaging of the equations over a time larger than the biggest time scales of the turbulence is appropriate. For this reason at first the Reynolds decomposition of the variables will be executed: the momentary value of a variable is the sum of the averaged value and a fluctuating value

u = u + u'

v = v + v' (3.1 2)

p = p + p'

Averaging of the equations for a stationary flow results in: - the continuity equation:

1 3 3 - — (rpv) + — (pu) = 0 (3.13) r 3r 3x

54

the equations of motion:

9u 9u 1 3 9u 9 9u 9 pu — + pv — = (ur — ) + — (y — ) - { — pu 9x 3r r 9r 3r 9x 9x 9x

1 9 + pru'v') + Sn (3.14) r 9r

3v 9v 1 9 9v 9 9v 9 pu — + pv — = (ur — ) + — (u — ) - { — pu'v' 9x 9r r 9r 9r 9x 9x 9x

1 3 Vfi'3 + prv'2 - p -2 } + Sv (3.15) r 9r r v

with 1 9 9v 9 9u 9p Su = - — (ru — ) + — (u — ) - — (3.16)

u r 3r 3x 3x 9x 9x 1 9 9v 3 3u 2yv 9p S„ = - — (ru — ) + — (u — ) - — (3.17) r 9r 9r 9x 3r r 3r v

The terms between brackets in equations 3.14 and 3.15 are called the Reynolds stresses. These express momentum transfer by turbulent motion, and will be treated as turbulent diffusion. Equations for the Reynolds stresses can be derived from the Navier-Stokes equations but the resulting equations contain higher order correlation terms which in their turn are unknown. This is the closure problem of turbulence. In the next paragraph it will be shown that by using turbulence models estimates are found for the unknown Reynolds stresses.

3.2. Turbulence models

The analogy between the Reynolds stresses and the viscous stresses is the basis for the Boussinesq hypothesis stating (Hinze, 1975):

3uH 9u^ 2 -pui'ui' = u. ( i + — 1 ) - - pkó^ (3.18)

3 Z 9Xj 9Xi 3 :

55

Herein u^ is the turbulent viscosity and k is the kinetic energy of turbulent fluctuations:

k = i (u,a + v'2 + v6'2) (3.19) Assuming that the turbulent viscosity is a scalar implies

that nonisotropic effects of the turbulence cannot be taken into account. The k-e model of turbulence, discussed in the next paragraph, makes this assumption. This model is very widely used. From literature we learn that it also is applied to nonisotropic flows, however, that is not fully justified.

Applying the Boussinesq hypothesis to the time averaged Navier-Stokes equations (3.14 and 3.15) leads to:

3u 3u 1 3 9u 3 3u pv — + pu — = - — (ru f f — ) + — (u-ff — ) 9r 3x r 9r e r r 9r 9x e £ t 3x

9 2 - — (p + - pk) + S„ (3.20) 9x 3

3v 3v 1 3 3v 3 3v pv — + pu — = - — (rupff — ) + —- (P-ff — ) 3r 3x r 3r e r r 3r 9x e r r 3x

3 2 3r — (p + - pk) + Sv (3.21 )

with the source terms:

3 3u 1 9 3v Su = ^ < M e f f ^ » + Ï J7 (rUeff ^ > (3.22)

3 9u 1 3 3v v S = — (ueff —-) + - — (rp-rr —-) - 2 u ff — (3.23) v 3x e r r 3r r 3r e t r 3r e r r r2

and with

Ueff = U + Ut (3.24)

3.2.1. The k-e model of turbulence

The value of the turbulent viscosity will be defined by means

56

of a turbulence model. One of the most frequently used models is the k-e model. The variables k and e represent indirectly the characteristic velocity- and length scale of the turbulent fluctuations. Their definitions are: - kinetic energy of turbulent fluctuations:

k = i ui'ui' (3.25) dissipation of k:

3u-' 3u, ' v i i- (3.26) 3x- dx-

The macroscale of turbulence in terms of k and e can be defined as:

'-D L = Cn k3/2/e (3.27)

If one assumes that, in analogy with the laminar viscosity, the turbulent viscosity is proportional to a characteristic

j. velocity scale of the turbulent fluctuations (k2) and to a characteristic length scale of the turbulent fluctuations (L) one can define:

ut = Cypk2/e (3.28)

Transport equations for k and e can be deduced from the Navier-Stokes equations (see Tennekes and Lumley, 1972). How­ever, the resulting equations contain a number of unknown correlations between fluctuating quantities. Rodi (1980) shows how these unknown terms in the equations can be modelled resulting in the following 'convection-diffusion' equations for k and e:

3k 3k 3 Ut. 3k 1 3 u<- 3k pu — + pv — = — (-£ — ) + (r -£ — ) + Sk (3.29) 3x 3r 3x 0^ 3x r 3r o^ 3r

3e 3e 3 u. 3e 1 3 uf 3e pu — + pv — = — (—£ — ) + (r —£ — ) + SF (3.30) 3x 3r 3x ak 3x r 3r a£ 3r t-: with

57

Sk = Pk - pe (3.31 )

se = c1 l pk - C2P ^ < 3' 3 2>

, 3u 3v v , 3u 3v Pk = Mfc{2 ( — )' + < —)* + (-)' + (— + —)'} (3.33) *■ u 3x 3r r 3r 3x

The five constants appearing in equations 3.28 to 3.33 are partly determined from measurements on well-defined turbulent flows and from computer optimizations. The set of constants used in general is:

ok = 1, oe = 1.3, a = 0.09, C1 = 1.44, C2 = 1.92

3.2.2. A low Reynolds number model

For a wall, where turbulence will go to zero very near to it, one needs special provisions in the model. It is customary to use wall 'functions. Herewith the steep gradients of the velocity along the wall and the variations of k and e near the wall do not have to be calculated, but are supposed to agree with universal profiles (see Launder and Spalding, 1972). For the flow under consideration these wall functions cannot be used. Especially in the stagnation point region the flow near the wall cannot be compared to a fully developed boundary layer flow for which the wall functions are valid. So the numerical calculations have to be extended till very close to the wall in order to determine the variations of u, v, k, e and T near the wall properly.

Then another problem arises: the k-e model is only valid at high Reynolds numbers. The fluid flow very close to the wall will not be simulated correctly by the standard k-e model. Several investigators have defined models for low Reynolds number flow with which laminarization can be predicted. Source terms for k and e are added to the equations or the constants in the model are made functions of a turbulent Reynolds number defined by

58

Re,. = — (3.34) ve

Patel, Rodi and Scheuerer (1981) give an evaluation of turbulence models for near wall and low-Reynolds number flows. The model developed by Chien (Chien, 1980) is used in this study. It implies that the k-e model described in paragraph 3.2.1 is.changed in five respects: - In the diffusion terms for k and e in equations 3.29 and 3.30 next to the turbulent viscosity the laminar viscosity is added.

- The dissipation term C2pe /k in the e-equation (3.30) is multiplied by a function fe, where

f_ = 1 - 0.22 exp {-(Jit)*} (3.35) e 6

By doing this the dissipation term fits experimental data of decaying homogeneous grid turbulence at both low and high Reynolds numbers.

- Since the diffusion of k very near a wall is finite an energy dissipation near the wall is needed to balance this. An extra dissipation term is added to the k-equation (3.29) which becomes effective near the plate:

2 vk (3.36) (H - x)

where H-x is the distance from the plate. A similar extra "dissipation" term is added to the e-equation (3.30):

2 ve C4 v*(H - x) exp { } (3.37) (x - H)

where v* is the friction velocity:

V* = {" U ( ë ) x = H}^ (3-38)

59

- The damping effect due to the presence of the wall is further taken into account in the definition of v .:

ka vt = Cy — {1 - exp (- C3 v* (H - x)/v} (3.39)

The two additional constants C3 and C4 are found by Chien by means of computer optimization for channel flow. The set of constants used in this model is:

Cy = 0.0 9 ,Cj = 1 .35 C2 = 1 .8 ok = 1 o£ = 1 .3 C3 = 0.00115 C4 = 0.5.

3.2.3. Drawbacks of the k-e model

The k-e model of turbulence has been widely used. It has already been mentioned that for nonisotropic turbulent flows the use of the model is not justified. Turbulence in a stagnation region is nonisotropic, as can be seen from the study of Hunt (1973). Yet several researchers simulate im­pinging jet flows with k-e-alike models with varying success. A 'one-equation model is used by Wolfshtein (1969) (who obtained agreement with experimental results) and by Bower et al. (1977) (who found disagreement for small nozzle-to-plate distances). Agarwal and Bower (1982) used a low-Reynolds number form of the k-e model and showed that the k-e model leads to better results than the zero- and one-equation models. Chieng and Launder (1980), however, predicted with a low-Reynolds number form of the k-e model heat transfer rates that were about five times too large in the vicinity of a stagnation point. Amano and Jensen (1982) used the k-e model to predict the flow and heat transfer of an axisymmetric jet impinging on a flat plate. With three different models for the calculation of k and e. at the grid points nearest to the wall their results mutually differed largely. For the stagnation point heat transfer they also found an overprediction.

The equations for k and e in the k-e model of turbulence

60

consist of terms which are modelled from the exact equations for k and e. It will be shown here that the modelling of one of the terms cannot be correct. The exact equation for the kinetic energy of turbulent fluctuations k derived from the Navier-Stokes equations reads (Tennekes and Lumley, 1972):

1 , ( i u W = , u-jP' ^ u » -3 I 3 u i n

- 2v u'iSij) - u'iu'jSij - 2v s i j S i j (3.40) IV V VI

where S ^ is the mean rate of strain:

3UJ 3U^ S Ü = i ( - + — 1 ) (3.41) x3 3XJ dx±

and s-i is the fluctuating rate of strain:

, 3u'H au'. Si- = i ( 1 + 1) (3.42) 13 * 3XJ dx±

The rate of change of the kinetic energy of turbulent fluctuations (I) thus is due to pressure gradient work (II), transport by turbulent velocity fluctuations (III), transport by viscous stresses (IV) and two kinds of deformation work (V and VI). The deformation work - u'^u'^S^ also arises in the energy equation for the mean flow, but with an opposite sign. This term describes the exchange between kinetic energy of the mean flow and kinetic energy of the turbulent fluctuations. It can be noticed directly that a deceleration of the mean flow leads to an increase of kinetic energy of turbulent fluctuations while an acceleration of the mean flow causes a decrease of the kinetic energy of turbulent fluctuations.

The flow of an axisymmetric jet impinging on a flat plate, which is subject of study, has very large normal velocity gradients near the stagnation point. The strongly decelerating mean axial flow will give its energy to the kinetic energy of

61

turbulent fluctuations. The strong acceleration of the mean radial flow extracts energy from the kinetic energy of turbulent fluctuations'. In the k-e model of turbulence the exchange term - u' u' S -i between mean and fluctuating flow is modelled by means of the Boussinesg hypothesis as mentioned before:

Su.: 3u.i 2 - U ' J U V = vf ( i + — 1 ) - - k 6±, (3.43)

This hypothesis, however, is not valid when the turbulence is anisotropic as is the case in a stagnation region, or more in general in accelerating or decelerating flows. Equation 3.43 has proved to be a good approximation for the non-diagonal components (i * j). For the diagonal components it turns out to be incorrect as can be seen from the resulting modelled equation for the turbulence energy in which - u'^u'-iS^ is modelled by:

" uiujSij = 2 vtSij* < 3"44> All components on the right hand side of this equation are positive, while the sign of the diagonal components on the left hand side is determined by the sign of the mean rate of strain. In this model accelerating and decelerating flows both produce turbulent energy. This can be seen clearly in the production term Pj, expressed in equation 3.33. If for many types of flows the k-e model gives fairly good results, this is due to the fact that the diagonal components of the Reynolds stresses in these flows can be neglected when they are compared to the non-diagonal components. However, this cannot be done in stagnation flows.

3.2.4. The anisotropic model

The k-e model of turbulence as we have seen in the previous paragraph assumes that the turbulence can be characterized by

62

one velocity scale. This, however, cannot be valid for a stagnation point flow where the influence of the wall gives rise to various velocity scales. Models are developed which allow for these different velocity scales. These models use transport equations for the individual stresses u^'iu1 and are called Reynolds stress models. If for the normally impinging jet the assumption is being made that the turbulent fluctuations in radial and tangential direction are similar and only the fluctuations in the direction normal to the plate deviate, then a model can be developed with only one extra equation. This being an equation for the difference in the axial and radial velocity scales (u12 - v' a). Such a model has been developed by Hijikata, Yoshida and Mori (1982). They studied the flow around and the heat transfer to a cylinder in a uniform air flow. The same type of model has been developed for the flow under consideration and will be discussed in this paragraph.

As distinct from a full Reynolds stress model where all Reynolds stresses are calculated by their modelled equations, here only two equations, namely for the normal stresses u,a and v12, will be used. The component vg'3 is supposed to be equal to v'2. The non-diagonal components of the Reynolds stress tensor are approximated by the Boussinesq hypothesis (equation 3.18). So it is assumed that the plate causes an anisotropy between the turbulent fluctuations parallel to it and the turbulent fluctuations perpendicular on it.

The transport equations for u 1 2 and v'2 can be derived from the Navier-Stokes equations (see Hinze, 1980). For a stationary axisymmetric flow with vg = 0 they read:

3u' 2 3u'2 3u'3 1 3 2 3u'p' u + v = vVu'2 (rv'u12) 3x 3r 3x r 3r . p 3x

diffusion

63

9u 9u 2 (u12 — + u'v' — ) 9x 9r

production

p' 9u' 9u' 9u' + 2 2 ( ) (3.45) p 9x 9XJ 3X:

redistribution dissipation

9v'2 9v'2 9u'v'2 1 9 u + v = vV2v'2 - (rv'3) 9x 9r 9x r 3r

9r p 9r

production redistribution

9v' 9v' 2 9v' vfi'2 - 2v ( vfi' + -S—) 9xi dx± . r2 ö 99 r2

dissipation

The unknown correlations in the diffusion, redistribution and dissipation terms have to be modelled.

The diffusion terms are modelled similar to the diffusion of k. ■ .

The redistribution terms (or pressure strain) consist of two parts; one part (11 -j ) due to the interaction of the fluctuating velocities; the second part (Ili 2) d u e to the interaction between main strain and fluctuating velocities. This can' be shown by diminishing the pressure fluctuation p' via a Poisson equation (see Launder, 1976). Rotta (1972) proposed for the first part:

64

e 2 Hi 1 = " C1 a - ^ i ' 2 " T k ) (3.47) i j 1 ' = k 3

One can see that IIi 1 is proportional to the anisotropy of the turbulence. In isotropic turbulence u^'a equals 2/3 k. The term can either act as a source or a sink term and thus redistribute the energy among the components. Shir (1973) gave a correction on n^ -j for wall effects. Near a wall the fluctuating velocity normal to it is damped while the fluctuating velocity parallel to the wall is enhanced relative to the fluctuating velocity in free shear flows. This effect is included in the following correction for JIJ 1 :

e k3/2 ni l' = C1 a' - <un'2 " 3 V u i ' 6ni> T (3.48) l,I I,g k n n l ni e ( H_ x)

where n denotes the direction normal to the wall and the function k ' /(H-x) is introduced to reduce the effect of the wall with increasing distance from it.

The part in the redistribution term due to interaction between main strain and fluctuating velocities can be modelled by (see Noat et al., 1970, or Reynolds, 1970):

ni,2 = " C2,g <Pi - \ Pk> <3-49>

In this equation P^ is the production of normal stress u^'s and Pk is the production of k. From this equation it can be seen that IT 2 i s proportional to the anisotropy of the production of u^'2. In isotropic turbulence P. equals 2/3 Pk.

For the dissipation terms in equations 3.45 and 3.46 it is assumed that the turbulence is locally isotropic, so that the amounts of energy dissipated in each energy component are equal. This assumption can be made because the energy dissipation takes place at the level of the smaller turbulent eddies. So for the dissipation terms can be written:

2 Ei = - e (3.50)

65

where e is defined by equation 3.26. Now the new parameter characteristic for the anisotropy of

the turbulence is defined:

g = 'u~r7 - V^ (3.51 )

From this definition, the definition of k (k = 5 (u'J + v'2 + Vg' 2 ) and the assumption v12 = Vg ' 2 one finds for the diagonal components of the Reynolds stress tensor as a function of g and k:

2 2 u'2 = _ g + - k (3.52)

3 3 — 1 2 v' 2 = vfl'2 = - - g + - k (3.53) ü 3 3

From equations 3.45 to 3.53 one can deduce the resulting transport equation for g:

9g 3g 3 Uf 3g 1 3 vt 3g pu — + pv — = — (u + —-) — + — — r (u + —-) — 3x 3r 3x ok 3x r 3r ok 3r

+ (1 " C2,g> Pg " PCi,g J g

2 2 k^ - 3 pC, ' (- k + - g) (3.54) 1'9 3 3 (H-x)

In this equation the diffusion term of g is presented in the same way reading: same way as the diffusion of k. P is the production term

2 8 3u 2 4 v Pg = P <- - 9 - - k) — + P (- g - - k) -y 3 3 3x 3 3 r

3u 3v + 2 vt {( — ) 2 - ( — )2) (3.55) u 3r 3x

The thus defined turbulence parameter g can be used to model the production term in the k-equation (term V in equation

66

3.40)

Pk = - uiUj S±j

In the standard k-e model all Reynolds stress terms are modelled by the Boussinesq hypothesis and equation (3.33) results. In the present model only the non-diagonal terms of the Reynolds stress tensor are modelled by the Boussinesq hypothesis. Together with the continuity equation (equation 3.13) this results in:

3u 3u 3v Pk = p (v,a - u'2) — + ut < — + — ) 2 K 3x z 3r 3x

+ p (v12 - v f l' 2)- (3.56) r

The last term equals zero, so that

3u 3u 3v P = - pg + u ( + ) ' (3.57) K 3x c 3r 3x This completes the anisotropic model consisting of the transport equations for k, e and g. This model does not have the drawbacks mentioned in the previous paragraph. In our study this model has been combined with the described model of Chien for low Reynolds number flows. The constants used in this model are partly suggested by Chien (1980) and by Hijikata et al. (198 2) . They are:

C y = 0 . 0 9

C 3 = 0 . 0 0 1 1 5

C 1 f g ' = 0 . 1 6

C1 = 1 . 3 5

C 4 = 0 . 5

C 2 = 1 .8

C 1 , g = 1 - 8

°k = 1 °£ C2,g = ° - 6

= 1

3.3. The energy equation

Since heat transfer to the plate is our primary concern, the energy equation has to be solved. For a two-dimensional axi-symmetric flow the energy equation reads:

67

3h 3h 3h 1 3 A 3h 3 A 3h p — +pv — +pu — = (r ) + — ( ) (3.58) 3t 3r 3x r 3r C 3r 3x C 3x Jr ïr

with h the momentary value of the enthalpy. Like the flow equations the enthalpy equation is averaged over a time larger than the biggest time scales of the turbulent flow. The Reynolds decomposition applied to the enthalpy results in an averaged (h) and a fluctuating value (h1):

h = h + h' (3.59)

Averaging equation 3.58 for a stationary flow then leads to:

3h 3h 1 3 A 3h pv — + pu — = — — r (— — - pv'h' ) 3r 3x r 3r C 3r

3 A 3h + — ( Pu'h') (3.60) 3x Cn 3x

In analogy with the Boussinesq hypothesis for the Reynolds stresses the heat transfer by turbulent fluctuations pui'h' is supposed to be linear with the enthalpy gradient:

A 3h Ui. 3h - pui'h' = -£. ( ) = — £ _ ( ) (3.61) Cp 3xi °h,t 3xi

This equation defines a turbulent diffusivity (At/pC = afc) being related to the turbulent viscosity by a turbulent Prandtl number an {.. Equation 3.61 in fact means that they are the same turbulent eddies that transport momentum as well as heat. With this hypothesis the averaged enthalpy equation becomes:

3h 3h x 3 r u m- 3h pv — + pu — = I ( + — ^ ) — } 3r 3x r 3r o h f l o h f t 3r

3 y Ui- 3h + — { ( — — + —)—} (3.62)

3x °h,l °h,t 3x

68

4. THE NUMERICAL METHOD

In this chapter the numerical method will be discussed which has been used to compute the flow and heat transfer in laminar and turbulent impinging jets. The governing equations are described in the previous chapter. References to similar calculations known from literature can be found in paragraphs 2.2.2 and 2.2.3. Most studies used a finite difference method or the hereto related finite volume method. The last method is also applied in this study. For the merits and demerits of the three methods we will suffice with a reference to Shih (1984) who gives a discussion on this subject!

4.1. The general finite difference equations

Starting point for the numerical calculations is the general finite difference equation. All transport equations which have to be solved are written in the same form as a convection-diffusion equation:

3 1 3 3 3<t> — (pud)) + (prvd>) = — (r\ ff — ) + 3x r 3r 3x $ < e t t 3 X 1 3 34> ■-■■■

r ^ <r ^ « Sr"' + S » (4"1)

Table 4.1 gives the different values of <t>, T* e£f and S^ for all equations that are solved in case of turbulent flow.

The finite volume method or also called the control volume method we apply here, has been described in detail by Patankar (1980). The solution domain is divided into a certain number of adjoining control volumes. Each of these volumes is surrounding a grid point. The convection-diffusion equation (4.1) is integrated over each of these volumes. After applying the divergence theorem this leads to balances for fluxes across the surface areas of the control volumes. The most attractive feature of this method is the integral conservation of the quantity 4> over each control volume and thus over the complete

69

TABLE 4.1 DIFFUSIVITY AND SOURCE TERMS FOR THE EQUATIONS TO BE SOLVED

FOR THE TURBULENT IMPINGING JET

' 1

continuity

x-impuls

r-impuls

enthalpy

kinetic energy

dissipation

anisotropy

3u p k - - pg Vx

p g = p (- \ g

0.4

*

i

u

V

h

k

E

g

♦ ut

8 1

-<R e

r*,eff

0

ueff

ueff

u ut

°h,l °h,t ut

u + —-°k

u + —-°E

u + —-°k

S* 0

3p 3 3u 1 3 3v 3x 3x e l r 3x r 3r e t t 3r

3p 1 3 3v 3 3v - —- + - — ( r u e f f — ) + — (upff — ) - (2 u t + u) 3r r 3r e r r 3r 3x e t r 3x z

0

2 uk R °C (x-H)2

E EJ 2 UE -C4Re l t 1 ,£ k k 2,£-'E,2 k (x-H)3

<! - C2,g> pg " PC1,g £ 9 " 3 PC1,g £ <J k + 1 9>

3u 3v 3v v v „, (— + — ) J + 2 v*. {-( — ) - + {-)') 3r 3x t 3r r r

3u 2 4 v , 3u,, ,3v-:11 k) — + p (- g - - k) - + 2 u t { ( — ) ' - <T-> ) 3x 3 3 r fc 3r 3x

2 t/6) 3

V T 7

k3/2 (X-H)E

Fig. 4.1. A control volume around grid point P with the four adjacent control volumes.

solution domain. A control volume around grid point P is shown in figure

4.1. The neighbouring control volumes are surrounding the grid points N, S, W and E. The interfaces of the adjacent control volumes (n, s, e and w) are situated halfway the grid points.

The total fluxes (convection + diffusion) are defined by:

3<t> J x = P u * " r < D , e f f g ^

3<t> j r = pvt - r ^ e f f _

With these definitions the equation can be written as:

3 1 3 3x x r 3r r <•>

Integration of equation 4.4 over a control volume gives:

rp <Je " Jw> + 'rnJn " rsJs> = rp 'v S* d V <4-5> The quantities Je, Jw, J and J are the integrated fluxes over the control volume faces, e.g.:

(4.3)

general convection-diffusion

(4.4)

71

Je = SJy. dr (4.6) or, if Jx does not vary over this area:

Je = «Pu* " r<t,,eff |^)e A r ( 4- 7 )

The source term S^ in equation .4.5 is linearized as follows:

S(t> = Sc + sp*P ( 4- 8 )

so that its volume integral leads to:

IS^ dV = (Sc + Spd)p) ArAx (4.9) l

Now the discretization scheme determines the values of Je, Jw, Jn and Js. We have used the hybrid scheme which is a combination of the central differencing scheme and the upwind differencing scheme. For the diffusion term in all circumstances central differences are applied:

3d) Ar (r4>,eff g^>e A r = <r4>,eff>e J^~ (<t,E " V =

De (<t)E - 4>p) (4.10)

The convection term (pu<t>)„ Ar {= Fe((t>)e) is either determined by central differences or by upwind differences. The decisive criterion is defined by the cell-Peclet number being the ratio between convection and diffusion through the concerning face of the control volume. For instance, for the east area of the volume:

pu6x Je f<t>,eff

pe = ~ = (- >e (4-11 >

Depending on this number three different approximations of the flux through the east area are made:

(4.12)

Pe s - 2

-2 < Pe < 2

Pe S 2

J e = F e * E

J e = J?Fe (d)p + <t>E) ■

J e = F e * P

p *E " *P S ( « x ) e

72

Similar expressions can be derived for the other three faces of the control volume. Together these yield to the general finite difference equation:

ap<t>p = ag^E + aW*W + aS(*,S + aN<')N + b (4.13) where .

ap = aE + aw + ag + aN - SpAxAr - . (4.14)

b = ScAxAr

and the coefficients aE, aw, ag and aN are dependent on the finite difference scheme.

4.2. The hydrodynamic solver

The momentum equations are particular cases of the general differential equation 4.1. A difficulty, however, in solving these equations lies in the unknown pressure field which indirectly is given via the continuity equation. In this paragraph it will be shown how the variables u, v and p can be solved by an iterative method. For this method where the primitive variables are calculated directly without eliminating the pressure, it is recommended to use staggered grids for the velocity components. Grid points for the velocities are defined on the faces of the control volumes for the pressure and other scalar variables. In this way it is prevented that an unrealistic zig-zag pressure field can be interpreted like a uniform pressure field by the momentum equations (see Patanker, 1980). In figure 4.2 a two-dimensional non-linear grid pattern with staggered u and v locations is shown.

The solution of the velocity field is obtained by applying the so-called SIMPLE procedure (Semi-Implicit Method for Pressure Linked Equations). At first a pressure field is estimated. With this estimation the momentum equations are solved giving estimated velocities. The obtained velocity field will not satisfy continuity. The continuity equation, turned

73

7 p-cell

2 v-cell

3 u-cell

Fig. 4.2. Control volumes in a staggered grid.

into an equation for the pressure correction, leads to a pressure correction. Next the velocities are corrected in such a way that the momentum equations are satisfying. Then with the help of this velocity field the other variables (h, k, E and g) are solved. The corrected pressure is regarded as a new estimate for the next iteration step. For details of this procedure see Patankar and Spalding (1972).

Lately other algorithms have been derived in order to try to improve the rate of convergence. Among these are SIMPLER (see Patankar, 1980) and SIMPLEC (see Doornmaal and Raithby, 1984). Both algorithms, when we applied them, did not give significant improvements in our case.

4.3. The grid

The definition of the grid used to perform the calculations is very important. Areas with large gradients require more grid points than areas with small gradients. Since we are interested in heat transfer at the wall we need many grid points near the wall. If a low Reynolds number model is used, a very fine mesh near the wall is needed to predict properly the damping out of turbulence. For the laminar and turbulent calculations different grids were used.

74 •

- the laminar case

This grid contained 20 points in radial direction and 30 points in axial direction. Most of the grid lines were concentrated near the axis of symmetry and near the plate. The grid was defined by:

x(i) = 6D{A(nx " 1 ) 3 + B("x " 1 ) 3 + C("x " i)} (4.16) nx - 1 nx - 1 nx - 1

n_ - i , n_ - i , r(i) = 3D{E(-^ ) 3 + F (-± )} (4.17) nr - 1 nr - 1

with nx = 30, nr = 20, A = 0.94, B = 0.05, C = 0.01, E = 0.8 and F = 0.2 this resulted in a grid shown in figure 4.3.

*-x

Fig. 4.3. The grid pattern used for the laminar cal­culations .

- the turbulent case

Since the gradients of velocity and of the turbulent parameters near the plate are much bigger in turbulent than in laminar flow, much more grid lines are required near the wall in the turbulent case. At least a few grid points should lie within the viscous sublayer of the wall jet. The x-coordinate of the grid lines is defined by:

75

x(i) exp{(-i - nx)/ax) - 1 'exp{(n - 1)/ax} - 1

(4.18)

In radial direction within 0.5D from the axis; of symmetry an equidistant grid is used with 1/5th part of the total number of grid points in this direction. For the region 0.5D < r < 3D a similar formula as equation 4.18 is used. The result is an almost linear grid in the radial direction.

The total number of grid points was varied from 40 x 40 to 40 x 60. The largest occurring aspect ratio of the control volumes for the finest mesh was 16. Much more grid points are needed for this aspect ratio to be in the order of unity.

4.4. The boundary conditions

The problem to be solved is defined by its boundary conditions. The boundary of the domain is divided into five regions as given in figure 4.4.

-m-w i

L ;

JL ^rrftfffTK

i

I

E -*-x,u

Fig. 4.4. The five regions for the boundary conditions.

For each of these regions the conditions were:

I. The solid wall:

<t> = 0(<t>=u,v,k,e,g) h = 2

76

For the effective viscosity in a near wall control volume the laminar viscosity is used.

I I . The a x i s of symmetry:

- 3d>/3r = 0 (4) = u , k , e, g , h )

- v = 0

III. The burner outlet

- v = 0 - h = 1 - u = uin The velocity profile at the burner outlet could have different shapes: • a flat velocity profile in the turbulent as well as the laminar case: uin = umax

- a parabolic velocity profile in the laminar case:

uin = umax <1 " ( ^ ) 2 } (4-19>

• a Gaussian profile which stretches itself over the regions III and IV of the boundary (laminar case):

uin = umax exP< <->'> (4.20) in max ■■ 4b D

The value of b was chosen such that at r = -jD: u^n = \ umax

= kin Measurements of the turbulent fluctuations gave:

" k = kin

u' 2 =16 v " (4.21 ) It is supposed that the tangential fluctuations are of the same magnitude as the radial fluctuations: Vg' 3 = v12. The kinetic energy of turbulent fluctuations at the burner outlet then equals:

77

9 (4.22) 16

With a measured turbulence level of 7% (/u,2/u = 0.07) this leads to the boundary condition for k at the burner outlet:

kin = • °-49 Uin2 ( 4' 2 3 )

9 = 9in With equations 4.21 and 4.22 we find for the anisotropy parameter at the burner outlet:

9in = < u " " v'2>in = J kin <4-24>

" e = ein The dissipation of turbulent fluctuations at the outlet can be calculated from

ein = C u 3 / 4 k3/2/!m <4-25> and from an expression for the mixing length given by Launder and Spalding (1972):

lm = 0.0375 D (4.26)

IV. The region next to the burner outlet

In practice next to the burner outlet there will be a wall or a free boundary. For the laminar case here a wall has been chosen: - u = 0 In the turbulent case for reasons of convergence it has been found that this boundary should be treated as an inflow. In this way a boundary with a free inflow is prevented.

The velocity of the inflowing mass was chosen such that there was enough flow rate to provide the jet with sufficient entrainment air, while the velocity itself was small enough to leave the jet undisturbed:

78

r

- u = ~ uin <4-27)

The other boundary conditions in this region were: - v = 0 - h = 1 - g = 0 - k and e were defined as small while Meff = u

V. The boundary with the free outflow

- 34>/3r = 0 (ct> = u, k, e , g, h) - For the radial velocity the condition of continuity has been applied. For every new iteration the velocities at the boundary were calculated from the velocities at the next last grid line.

4.5. Determination of the heat transfer coefficient

Since the first grid point from the wall in the turbulent case lies well within the viscous sublayer of the wall jet (y+ < 5), the determination of the Nusselt number for the turbulent and the laminar jet is similar. The heat transfer to the plate equals:

<' = X <^>x = H <4-28>

By definition this is also equal to:

qw" = a(TH - T») (4.29)

with a = heat transfer coefficient, TH = temperature of the plate and T , = temperature far from the plate. In case of temperature independent fluid properties, this leads to:

aD D 3h Nu = — = ■—- (— ) V _ H (4.30) A (hH - h j 3x 'x = H

79

From the numerical results this Nusselt number has been calculated in two ways: 1 ) (9h/8x)x=H has been calculated from the enthalpy of the

first grid point from the wall and the wall enthalpy, assuming a linear profile.

2) (3h/3x)x=H has been calculated from the enthalpies at the two nearest wall grid points and the enthalpy at the wall assuming a quadratic profile.

If the applied grid is fine enough, the two ways should give the same answer.

80

5. THE EXPERIMENTAL METHODS

5.1. Heat transfer from the isothermal jet

5.1.1. Experimental set-up

For the determination of the heat transfer coefficients of an isothermal jet impinging on a slightly heated plate an experimental set-up using a liquid crystal technique has been built as shown in figure 5.1.

©

-£-

©

°l MD

I®, 1 £-©

:© ©

1. thermostat bath 2. waterreservoir 3. copper plate 4. glass plate 5. liquid crystals 6. Isolation 7. burner 8. rotameter 9. calming vessel

10. pressure air line

® Fig. 5.1. Experimental set-up for the isothermal jet

heat transfer measurements.

Two thermostat baths provided a water flow of a constant temperature. This was forced to flow through a channel between a copper plate and a glass plate. These two plates were separated by a small distance of 2.9 mm. The water flowed along the copper plate into a reservoir behind this plate and from that reservoir back into the first of the two thermostat baths. In this way one side of the glass plate was kept to a nearly constant temperature, provided the flow in the channel was homogeneous and the heat transfer coefficient to the glass plate very high. The other side of the glass plate was covered with a thin layer of liquid crystals to measure its surface temperature (see paragraph 5.1.2).

81

A jet of air issued by a burner impinged on the plate with the liquid crystals on it. The air originated from compressed air and was reduced in pressure by a reducing valve. A constant flow was achieved by leading the flow through a large vessel. To determine the flow rate the air passed through calibrated flow meters. With this flow rate the Reynolds number (Reg) at the exit of the burner could be determined. Finally the temperature of the air coming from the burner was measured by a Cu-constantan thermocouple.

In a steady state the heat flux from the water to the glass plate equals the heat flux through the glass plate and also equals the heat flux from the glass plate to the air jet, or:

*w" = awp <Tw - Ta> = V d g (Ta " Tl > = a (T1 " Tj > + q"rad (5.1 )

where a,,„ = heat t r ans fe r coe f f i c i en t from the water to the g l a s s wp J

plate. The Reynolds number of the channel flow varied from 570 to 870. a was estimated to be 1000 W/m2K. T = the temperature of the water in the channel which was measured by thermocouples at five locations and varied from 50°C to 80°C. Ta = the temperature of the side of the glass plate in contact with the water. X = heat conductivity of the glass (1.0 W/mk). da = thickness of the glass plate (9.9 mm). T-, = temperature measured by the liquid crystals. T. = temperature of the air jet. q"racj = heat flux density due to radiation from the glass plate to the environment.

The thermal resistance of the thin layer of liquid crystals (50 urn) is neglected since it is much smaller than the uncertainty in 1/aw_. The heat flux by radiation equals:

q"rad = ea ( V - Tj*) = a r a d {T± - T j ) (5.2) With an emission coefficient of e = 0.9, a temperature of the

82

liquid crystal surface of 43.25°C and a jet temperature of 20°C, this gives

arad = 5* 8 w/ m 2 K ( 5' 3 )

Eliminating Ta from equation 5.1 gives for the heat transfer coefficient from the plate to the impinging air jet:

1 Tw - Tl 1/V + V Ag Tl " TJ «rad (5-4)

This leads to: T — T

a = 95.7 _w 1 - 5.8 W/m2K (5.5) Tl - Tj

With this equation the heat transfer coefficient can be determined with an accuracy of about 5%.

5.1.2. Temperature measurements with liquid crystals

In the transition from the liquid phase to the crystalline phase (the so-called mesophase) cholesteric liquid crystals have some peculiar optical properties. In the mesophase the molecules have a specific order in which the forces between the molecules are very weak. The molecular structure in this temperature range is instable and can be influenced very easy, changing the optical properties of the material. One of the most striking optical effects is the spectral reflectivity of light as a function of temperature. In the liquid phase as well as in the crystalline phase the material transmits all wave­lengths of the visible light while in the mesophase, when the temperature rises, its colour changes from red to yellow, green and violet. This phenomenon can be used to carry out temperature measurements in a non-destructive way (see for instance Fergason, 19 68 and den Ouden and Hoogendoorn, 1974).

Depending on the composition of a mixture of several cholesterics all desired temperature-colour characteristics can be obtained. One of the mixtures used in the experiments is

83

given in table 5.1.

TABLE 5.1 MIXTURE OF COMPONENTS USED IN THE EXPERIMENTS

parts in weight

10.4 3 33 24 35

component

butoxy benzoate chloride noanoate oleate oleyl carbonate

A calibration of this mixture gave a temperature-colour characteristic of table 5.2.

TABLE 5.2 TEMPERATURE-COLOUR CHARACTERISTIC OF THE MIXTURE

FROM TABLE 5.1

colour

red red-brown light brown yellow-brown yellow yellow-green light green blue green light blue blue dark blue violet

temperature in

42.8 42.95 43.1 43.2 43.25 43.3 43.4 43.65 43.85 44.15 44.6 44.8

°C

84

From table 5.2 it can be seen that very 'accurate temperature measurements are possible. Especially the colour yellow at 43.25°C could clearly be detected and gave an accurate measure of the temperature (± 0.1 °C).

The procedure of producing a layer of liquid crystals on the glass plate was as follows: The different components were brought to their liquid phase and mixed with a resin (neocryle B-723) in toluene. The obtained solution was then sprayed onto the glass plate and dried in a furnace for 24 hours. This resulted in a thin layer of about 50 urn.

By varying the water temperature at the other side of the plate, the colour yellow (representing 43.25°C) could be detected at different spots. With the help of a coordinate system engraved in the blackened copper plate behind the glass plate a radial distribution of the heat transfer coefficient could be measured.

5.2. Heat transfer from the flame jet

5.2.1. The experimental set-up

A small industrial rapid heating tunnel burner has been used to obtain a premixed flame jet. In its original version the burner had a tangential inlet of the gas-air mixture producing a swirl acting as a flame holder. In case of the isothermal jet from this burner the swirl still existed, however, the flame jet did not show any swirl. To gnt rid of this unwanted difference between isothermal jet and flame jet the tangential inlet connection was replaced by an axial connection. A small disk has been used as a flame holder.

Another alteration in the original burner design was made for a more practical reason. In order to be able to make a bigger version of the same burner, the shape of the inner burner wall was altered in such a way that it could easily be scaled. The final shape of the burner is shown in figure 5.2.

To determine the heat transfer coefficient from the flame

85

13.8mm

Alfi3

steel shield

flame holder

refractory material

inlet air/gas mixture

F i g . 5 . 2 . The b u r n e r .

jets to an isothermal plate the experimental set-up shown in figure 5.3 has been built.

Volume rates of air and natural gas fed to the burner are measured by rotameters before mixing these two flows in a mixing chamber. The air is supplied by a compressed air line, while the gas is available from bottled Groningen natural gas. The stoichiometric mixture of gas and air after equalizing in a vessel was led to the burner. A polished copper plate cooled by water from a thermostat bath was placed on the top of the burner. A Gardon heat flux transducer (Gardon, 1960) has been included in the plate. Surface temperatures of the plate are measured by eight thermocouples. The position of the burner. relative to the heat flux transducer in the plate can be varied by means of a 3-dimensional traversing mechanism in which the burner is placed.

Temperatures of the flame jets are measured using thin

86

ISOTHERMAL PLATE

X '

EQUALIZING CHAMBER

MIXING CHAMBER

«

natural gas

_£.

THERMOSTAT

thermometer

manometer

air supply

rotameter

"=j>i reducer

valve *

Fig. 5.3. Experimental set-up for flame jet heat transfer measurements.

87

10%RhPt-Pt and 30%RhPt-6%RhPt thermocouples with diameters varying from 180 uin to 50 urn. The junctions are butt-welded and to avoid contamination due to the high temperatures the wires are coated with a mixture of beryllium oxide and yttrium chloride as advised by Kent (1970). The thin wires are butt-welded to water-cooled supports of thicker wires of the same material. The radiation loss of the thermocouple junctions which becomes significant at temperatures over 1300°C is estimated by extrapolating the results of measurements with thermocouples with different junction diameters to a thermocouple with an infinitesimal diameter, and checked by a calibration.

5.2.2. The Gardon heat flux transducer

The heat flux transducer used in the experiments has first been introduced by Gardon (1960). The design of this transducer for our experiments is shown in figure 5.4.

cooling water

Isothermal plate

thermocouple

Isothermal surface

Fig. 5.4. The Gardon heat flux transducer.

88

A cylindrical copper body with cooling fins is cemented in the water-cooled isothermal plate. On the cylinder axis of this body a hole was drilled through. At one side this hole was closed by a thin foil of constantari in such a way that the surface of the copper body with foil was flushed with the isothermal plate. In the middle of the constantari f bil a thin copper wire is spot-welded. In this way a thermocouple is constructed by which the temperature difference between the centre of the foil and its edge, which is in contact with the copper body, can be measured. The surface temperature of the copper body, which is flat with the isothermal plate, is measured by a thermocouple and is kept equal to the plate temperature by an extra cooling.

When the constantan is imposed to a heat flux, a temperature gradient will exist over the radius of the foil. The measured temperature difference will then be a measure of the heat flux.

Gardon (1960) made a theoretical analysis of the performance of this heat flux transducer. For its sensitivity he found:

— = 2.29 103 — W/m2 mV (5.6) e R2

Measuring the emf. (e) of the thermocouple (in mV) from the transducer with a foil with radius R = 2 mm and a thickness S = 0.2 mm gives a heat flux density equal to:

q" = 11.43 10" e W/m2 (5.7)

Because of the uncertainty in its dimensions the Gardon heat flux transducer has to be calibrated. This has- been done in a black body cavity (Hohlraum). In a cylindrical opening at the bottom of this Hohlraum the transducer fitted exactly. The configuration factor was equal to 1 . The surface of the transducer has been painted black making the emission coefficient very close to 1. The heat flux density absorbed by

89

its surface is equal to: q" = O (Tg* - T0") + ac (Ts - T0) (5.8)

The temperature T of the furnace cavity was measured by a calibrated thermocouple. The temperature TQ of the heat flux transducer was controlled by the water cooling and measured by a thermocouple just beneath its surface.

The second term on the right hand side of equation 5.8 expresses the heat transfer to the surface by convection. According to McAdams (1954) the heat transfer coefficient for cooled square plates facing upward in air is:

ac = 0.12 (AT/L)0-25 (5.9)

with L = side of the square plate. With this equation ac in 5.8 can be approximated by stating L = 2R.

In figure 5.5 the results from the calibration are given. The sensitivity that is found appears to be linear for heat flux densities from 30 kw/m2 to 150 kW/m2 with the relation:

q" = (13.0 ± 0.4) 10" e W/m2 (5.10)

140-

120-

100-

80 -

60 -

40 -

20-

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 t[mV]

Fig. 5.5. Calibration curve of the heat flux transducer.

90

which comes reasonably near to Gardon's value (5.7). The highest temperature that could be reached with the furnace was 1100°C corresponding with a heat flux density of 150 kw/ma. Gardon found that these transducers had a linear relation between q" and e up to temperature differences of 185°C. For our transducer this corresponds with a heat flux density of 600 kW/m3. The result of the calibration expressed in equation 5.10 has been extrapolated up to this value.

5.3. The laser Doppler anemometer

Measurements of velocity and turbulence intensity have been performed with a laser Doppler anemometer (LDA). Details of this measurement technique can be found in textbooks (e.g. Durst, Melling and Whitelaw, 1976; Watrasiewicz and Rudd, 1976; Durrani and Greated, 1977). In this, paragraph the experimental set-up that actually has been used will be discussed without going into details on the technique itself. The experimental set-up consists of two parts: an optical and an electronic part.

5.3.1. The optical configuration

Figure 5.6 shows the optical configuration that has been used.

1 1/2-A plate 6 lens 2 beam splitter 7 diaphragm 3 glass rod 8 collector lens 4 Bragg cell 9 pinhole 5 wedge 10 photomultlpller

Fig. 5.6. The optics of the laser Doppler anemometer.

91

A laser beam passes a ^A-plate. With this retardation plate one is able to rotate the polarization plane into the direction suitable for th.e other optical components. Afterwards the beam is split by a beam splitter into two parallel beams of equal intensity. The beam separation is 50 mm. One of the beams passes, a Bragg cell operating at a frequency of 40 MHz. With this cell a positive or negative optical frequency shift of the laser light can be obtained, which makes it possible to detect the direction of the flow. Due to the acousto-optical effect several beams with shifted frequencies leave the Bragg cell. The cell can be adjusted in such a way that 85% of the diffracted light is in the first order beam having a frequency shift of 40 MHz relative to the original beam. Because the first order beam leaves the Bragg cell at a certain angle, a wedge is used to make it parallel to the original second beam and to restore the separation distance of 50 mm. The second beam is led through a glass rod to ensure that the shifted and unshifted beams have an equal beam path length. All beams other than the first order beam coming from the Bragg cell are blocked. A lens focuses the two remaining beams. The intersection of the two laser beams forms the so-called measuring volume. Scattered light from particles moving through the measuring volume is focused by a lens onto a pinhole. In order to reach an optimal signal-to-noise ratio only 70% of the width of the measuring volume (defined by the'1/e2-points of the Gaussian intensity distribution of the laser beams) is focused onto the pinhole. A photomultiplier is used to measure the intensity of sthe light coming through the pinhoie.

The Doppler frequency of the scattered light from the particles is equal to:

2 U sin' (6/2"}

where 6 isiri the angle between the two beams and U is the velocity component perpendicular to the optical axis.

92

Since the frequency (f) detected by the photomultiplier is the sum of the preshift frequency (fD) and the Doppler frequency (f^) the velocity component equals:

X u = — . (f - fn) (5.12)

2 sin (0/2) P For the measurements two different lasers were used: a 5

mW He-Ne laser and a 50 mW Ar-ion laser. The higher output of the Ar-ion laser more than compensates for the fact that the optics have been optimized for the wave length of the He-Ne laser. The characteristic properties of the set-up were:

Width of the measuring volume Length of the measuring volume Number of fringes in the measuring volume Number of fringes observed through pinhole Proportionality between velocity and frequency

He-Ne

188 um 1.13 mm 97 68

m/s 1 .925

MHz

Ar-ion

266 um 3.21 mm 90 60

m/s 2.94

MHz

5.3.2. The electronic equipment The electric signal from the photomultiplier has to be processed before results like averaged velocities and turbulence intensities can be calculated. Figure 5.7 gives an outline of the electronic equipment.

The photomultiplier signal passes an amplifier (20 dB) before it is downmixed to the desired frequency range. It means that the preshift frequency, which optically is equal to 40 MHz, is electronically brought down to a lower value (in steps variable from 0.01 to 20 MHz). Then the signal is analysed by a signal processing counter (TSI-1980A). In this counter the Doppler signal is band-pass filtered to remove high- and low-frequency noise and it is amplified. A threshold detector

93

power supply I photomultlplier

| pro-amplifier

mixer

| counter

HP • 1000 I micro-computer

graphic display

terminal

Fig. 5.7. The electronics of the laser Doppler anemo­meter.

senses the presence or absence of a Doppler signal by the amplitude of the incoming signal. A Schmitt Trigger converts the analogue signal into a digital pulse train. A timing circuitry, activated by the threshold detector, measures the time needed for a fixed number of pulses (= periods of the Doppler burst) to pass. This time is a measure for the velocity of the particle passing the measuring volume. The individual velocity measurements of the particles are read from the signal processing counter into a home-made microcomputer (see Cornelissen, 1980). Averaged velocities and turbulence intensities are calculated from 5,000 (N) measurements by:

individual

N ü = Z j = 1 , N

3 3 -j = i : (5.13)

N i,a = I j . i , u > 3 j=1 (5.14)

in which the weight factor w^ can be: - for arithmetic averaging: - for 1-D averaging (see Dimotakis, 1976): WJ

WJ = 1 ; = 1/|u,

94

Results of these calculations are routed to a terminal. From the measurements also a probability density function is made which is routed to a graphic display.

On basis of the observation of the probability density function a measurement can be accepted or rejected. The criterion for this decision is whether or not the probability density function looks like a Gaussian distribution, which can be expected. There are three possible ways of continuation: 1) If a measurement is accepted, the results are sent to a

HP1000 computer for storage and a new measurement can be initiated.

2) If a measurement is rejected, the same measurement will be repeated after eventual resetting : of the filters of the signal processing counter, the amplification factor or the preshift frequency.

3) If only a few of the 5,000 individual measurements appear to be far outside of the expected velocity distribution, a software filtering of the results can be applied, in order to exclude these apparently wrong individual measurements. Then the filtered results u, u'a and the probability den­sity function are again calculated from the filtered data and the results are stored by a HP1000 computer.

5.3.3. The seeding of the flow

The laser Doppler measurement technique requires light scattering particles that are small enough to follow the flow, and big enough to scatter sufficient light for detection. At high temperatures in flames one is limited in the choice of the particles used to seed the flow. For the measurements on the flame jets MgO is used.

According to Hjemfelt and Mockros (1966) MgO-particles with a diameter smaller than 2.6 um follow the flow within 1% if the frequency of the turbulent fluctuations does not exceed 1 kHz. Particles with diameters smaller than 0.8 um follow frequencies below 10 kHz. In the course of the measurements

95

some rude estimates were made of the sizes of the MgO-particles in the flow; Particles were made to collide on a small glass plate which was viewed under a microscope. The collided particles showed sizes of about 1 urn.

MgO has some disadvantages in its use as seeding material. It is hygroscopic as well as electrostatic. For this reason the powder- is firstly dried by keeping it for two or three days at a tempera'ture of 120°C. Then it is thoroughly mixed with Aerosol 972 and Al^On in percentages of weight: 94% MgO, 3% A"*"2°3' ^% Aerosol. The powder is then put into a vessel with a conical bottom (see figure 5.8). The mixture of gas and air flows radially into this vessel. In this way a swirling flow is created taking along particles from the bottom. .Only small particles reach the outlet in the centre of the cover of the vessel. In a second vessel big particles that possibly still are carried along, are intercepted. This particle generator •performed well for different flow rates by adjusting a valve in a bypass of the two vessels.

As' it was not necessary to measure with MgO-particles for the isothermal jet measurements, another particle generator was used. In an atomizer the fluid DIOP (di-iso-octyl-phtalate) was atomized. The signal-to-noise ratio achieved with these parti­cles was much better than with MgO particles. For a description of the atomizer see De Geus (1983).

Fig. 5.8.' The particle generator.

96

6. RESULTS OF THE EXPERIMENTS

6.1. Introduction

Experimental results of the flow structure and heat transfer are given and discussed in this chapter. A small and a large burner have been used from which both isothermal and flame jets are measured at different Reynolds numbers. Comparisons will be made between measurements of the flow structure and heat transfer of the small and the large burner. For both cases isothermal and flame jets are compared. To characterize the four different jets, the following symbols are introduced which will be used throughout this chapter:

si: small burner, isothermal jet sf: small burner, flame jet li: large burner, isothermal jet If: large burner, flame jet.

In the next paragraph the axial flow and the radial flow profiles, the stagnation point heat transfer, the radial distribution of heat transfer and finally the correlation between heat transfer and flow structure will be discussed.

6.2. Flow structure 6.2.1. Velocity and turbulence on the axis of the free jet

An important characterization of a turbulent jet is its axial velocity decay. The shape oc the burner determines the velocity profile and the turbulence profile at the exit of the burner. These burner exit conditions of velocity and turbulence are responsible for the way the jet spreads and decays in the initial part of 8 or 10 burner diameters. The jets issuing from the burners used in these experiments evidently differ from the jets from smooth nozzles as discussed in paragraph 2.1.1. As is mentioned earlier a jet with a high initial turbulence spreads wider and decays faster than a jet with a low initial turbulence.

97

1.0-

u/u0

O.S

0.0

i i » ! | 1

"

~

. . . . 1

i i i f 1 1 1 I -

O

O

& O

>fw A 7

*~t~

. , , i ■ • . ■

l , i i

Re=3S80 Re=6130 Re=8S60 Re =10600 Re =14400

I I I .

-

-

-

10 15 20 x/d

F i g . 6 . 1 . Axia l v e l o c i t y of i s o t h e r m a l j e t s from t h e smal l b u r n e r .

30

Tu(%)

20

10

-i i 1 1 1 1 i 1 1 r—i

o Re=3880 o Re=6130 a Re=8S60 o Re=10600 ° v Re=14400 9 v v o

» T $ 8 6 £ 2 2 2

T

v v v

I I L _

10 IS 20 x/d

Fig. 6.2. Axial turbulence levels of isothermal jets from the small burner.

98

The results on the axial velocity decay and the axial turbulence development for the isothermal jet from the small burner are given in figures 6.1 and 6.2. For the range of Reynolds numbers studied the axial velocity decay is slightly Re-dependent. At x/d > 10 the jets with the highest Re-numbers (10,600 and 14,400) decay faster than the jets with the lower Re-numbers (6,130 and 8,560). Precisely at that region (x/d > 10) turbulence is higher at high Re-numbers. From these measurements a potential core length is hard to determine. A 95% criterion is used here. This defines the potential core length as the distance from the burner exit to the place on the axis where the velocity has decreased to 95% of its initial value at the burner exit. This potential core length is about 4d for the isothermal jets from the small burner. The initial turbulence level in the burner exit varies from 6% to 8% without being correlated to Re (see figure 6.2). A turbulence level of about 20% is reached after 10d. Here the jets appear to be fully developed. Differences in turbulence intensity are found for the various Re-numbers at x/d > 8. Higher turbulence is generated at higher Re-numbers.

u/u0

1.0

0.8

0.6

0.4

0.2

0.0 0 2 4 6 8 10 12

x/d Fig . 6 . 3 . Axial ve loc i ty and turbulence development

of isothermal j e t s from the large burner .

Tu(%)

40

20

99

For the large burner the axial velocity and turbulence of two isothermal jets together are shown in figure 6.3. There seems to be a strong Reynolds dependency. The initial turbulence levels are about equal, however, the jets at the two Reynolds numbers are quite different. After the potential core length of 3.3d (Re = 4,240) and 3.9d (Re = 9,500) the jet with Re = 4,240 decays much faster than the jet with Re = 9,500. This may be due to instabilities that can arise at this low Re­number. The turbulence level of the low Re-number jet also rises to a very large value, while the high Re-number jet comes much closer to the expected values.

1.0 u/u0

0.8

0.6

0.4

0.2

2 4 6 8 10 12 14 16 18 20 x/d

Fig. 6.4. Comparison of axial velocities and turbu­lence levels of jets from the small and large burner.

In figure 6.4 the axial velocity and turbulence intensity of the jet from the small burner at Re = 10,600 and the jet from the large burner at Re = 9,500 are compared to each other. The jet from the large burner decays faster and shows at the same time a higher turbulence, level. Also the initial turbulence level of about 11% is a bit higher. The differences which are found here might be due to small differences in burner configurations (like wall roughness and position of the blunt body flame stabilizer).

100

- // Re=9500 - si Re = 10600

Tu(%)

30

20

10

_l I I L

The characteristics of the flame jets from the two burners are given in figures 6.5 up to 6.7. Figure 6.5 shows that the velocity decay on the axis of flame jets is much faster than the decay of the corresponding isothermal jets (see also figure 6.1). This is due to the density difference between the jet and its surroundings. This effect of density can be taken into account by introducing the effective diameter (see paragraph 2.1.1). For this reason a second horizontal axis is given in figure 6.8 with the results from the flame jets. This axis gives the axial distance non-dimensioned with an effective diameter of 0.37d. This effective diameter is based on the fact that the initial flame temperature is equal to the adiabatic flame temperature of a stoichiometric mixture. For the gas used in these experiments this temperature is 2133 K.

i i i i | i i i i | i i i i | i i i i

v Re=1047

Fig. 6.5. Axial velocity decay of flame jets from the small burner.

From figure 6.5 can be concluded that for a flame jet from the small burner there does not seem to be a Reynolds dependency of the axial velocity decay (as was the case for the isothermal jet from this burner). It is remarkable that the decay starts immediately at the burner exit. A potential core does not seem to exist, however, this conclusion must not be

101

drawn too quickly. With the 0.95-criterion about 1.5d is found for the potential core length. This matches to about 3.5 effective diameters, which corresponds well with the potential core from the isothermal jet from this burner.

30

Tu(%)

20

10

'0 5 10 15 20 x/d

Fig. 6.6. Axial turbulence levels of flame jets from the small burner.

The development of the turbulence level on the axis is given in figure 6.6. A very quick rise of the turbulence level over a length of the first four jet diameters corresponds to the sharp velocity decay in this region (see figure 6.5). The initial turbulence of the flame jets from the small burner varies from 6% to 12%. However, this does not give rise to different jet characteristics further downstream. The turbulence levels in the fully developed jets are about 26%, which already is reached after 5 jet diameters.

In figure 6.7 the velocity and turbulence on the axis of the flame jets from the large burner are given. In contrast with the results from the isothermal jets the flame jets at three different Reynolds numbers give similar axial velocity and turbulence developments. The 0.95-criterion here gives a

I I r-

Re=1047 Re=1771 Re=2376 Re=2736

o H °

102

Re 1900 2700 4226

U/U0 o o a Tu(%)

Fig. 6.7. Axial velocity and turbulence development of flame jets from the large burner.

potential core of only 1d (2.7 d„ff). However, the turbulence "level at x/d = 1 has risen up to 20% already, so there seems to be nearly no potential core at all. The initial turbulence level of 16% is also very high. The flame jets from the large burner are highly turbulent and develop very quickly. At x/d = 4 a fully developed turbulence level of about 29% is reached.

1.0 u/u0

0.8

0.6

0.4

0.2

i i i i i i i i i

^ ^ V sf • * - ^ \ If

\ \ \ \ \ \ - s-y^^^-T* /■ / - . \

/ / V \ - / / v ^v

/ / -^^^

1 1 1 1 1 1 1 1 I

6 8 10 12 14 16 18

Tu(%) 30

20

- 10

20 x/d _i

5 TO 15 20 25 30 35 40 45 50 */de/f

Fig. 6.8. Comparison of the flame jets from the small and large burner.

103

The results on the flame jets from the large and small burner are compared to each other in figure 6.8. The lines drawn in this figure and in the next three figures correspond to the averaged values of the measurements at different Reynolds numbers for the specific cases. Figure 6.8 shows that the flame jets from the two burners behave similar to the isothermal jets (see figure 6.4): A flame jet from the large burner also decays faster and has a higher initial and axial turbulence level than a flame jet from the small burner.

The comparisons between the flame jets and the isothermal jets from the small burner and the large burner are shown in figures 6.9 and 6.10 respectively. It should be noticed that the effective diameter is being used in this comparison. Both figures show the same remarkable result: Using the effective diameter concept on the axial velocity decay as well as on the axial turbulence development, the axial velocity of a flame jet decays slower than that of an isothermal jet. On the other hand the axial turbulence level of a flame jet increases faster with axial distance than that of an isothermal jet. In the isothermal jets the axial turbulence level starts increasing at a place where the axial velocity starts decreasing. Both effects occur at the axial position where the turbulence

1.0 u/u0

0.8

0.6

0.4

0.2

2 4 6 8 10 12 14 16 18 20

Fig. 6.9. Comparison of the isothermal and flame jet from the small burner.

_ i i i i _

Tu(%)

30

20

10

104

// Re = 9500 -If

Tu(%) 30

~~~ - .... - 20

- 10

J i 2 4 6 a 10 12 14 16 18 20

Fig. 6.10. Comparison of the isothermal and flame jet from the large burner.

originating from the jet boundary has penetrated to the axis. In the flame jets, however, the turbulence level starts rising before the velocity and also the temperature (see figure 6.29) on the axis significantly decrease. It might be due to the process of combustion taking place inside the burner.

This comparison between isothermal and flame jets from the burners points at the existence of combustion . generated turbulence which has no significant effect on the axial velocity decay.

Figure 6.11 summarizes all results on the axial velocity decay. In this figure also a comparison is made with some results from literature (see paragraph 2.1.1) given by Hinze (equation 2.2) and Schrader (equation 2.3). All jets show the linear axial velocity decay which is characteristic for round jets:

uo x/deff + b

Only the constants c and b in this equation are not equal for all jets. These are calculated by linear regression and together with the correlation coefficient given in table 6.1.

105

i i i i i 1 r

J I I I L

TABLE 6 . 1

THE CONSTANTS c AND b FOR EQUATION 6 . 1

s i

s f

l i

I f

c

8 .1

7 . 4

6 .1

7 . 5

b

3 . 3

- 0 . 8

1 .7

1 .3

c o r r . c o e f f .

0.9994

0.9988

0.995

0.9991

r e g i o n

x /d > 8

x / d > 4

x /d > 5

x /d > 3

7

6 u0/u

5

4

3

2

1

-

-

I

D

• A

O

I

I I I

// Re = 9500 si If sf

i l i

I I I

^

Hinze = — «0

Schrader = — «0

I I I

I I I

6.39 x/d+0.6

8.0 x/d+3.3

I I I

-

-

~

-

-

-

12 16 20 24 28 32 36 40 44 x/deff

Fig. 6.11. Comparisons of the axial velocity decays.

106

6.2.2. The radial velocity gradient in the vicinity of the stagnation point

An important parameter in the impinging jet flow is the radial velocity gradient in the vicinity of the stagnation point:

3V (—)y=6,r=o <6-2)

As explained earlier in paragraph 2.2.1 this parameter influences the stagnation point heat transfer strongly. In this paragraph measurements of 3 will be presented and discussed.

It is assumed that the velocity just outside the boundary layer in the vicinity of the stagnation point can be written as V = 3r. If viscous effects are negligible it satisfies the theorem of Bernoulli, which gives:

p + i p32r2 = constant = p s t (6.3)

where p ^ is the static pressure in the stagnation point. From measurements of the static pressure p distribution on the plate the radial velocity gradient can be determined:

The tangent at r=o in the graph of (2{pst - p(r)}/p)5 as a function of r leads to the value of 3. For the density in equation 6.4 in the case of an impinging flame jet the value is used of the local density on the axis of the free flame jet at the same axial distance from the burner exit (or: the density at the impact temperature).

The figures 6.12 up to 6 .~1 5 contain the results from the measurements of the radial velocity gradient. The expected value of 3 at small burner to plate distances is 8 = u0/d- F o r

this reason the graphs present 3d/u_ as a function of H/d. It can be seen from these figures that for small H/d-values where the velocity profiles can be expected to be uniform, the measurements correspond with the expected value. For higher

107

M "o 1.0

ÊË. "o

Fig. 6.12. iRadial velocity gradient g for the isothermal jet from the small burner.

Fig. 6.14. Radial velocity gradient 8 for the flame jet from the small burner.

1.4

1.2

r.oF 0.8

0.6

0.4

0.2

0.0

1

-

r -^ —

i

■ ~ - ,

i

o Re 4240 -o Re 9S00

\ Giralt \ \ Schrader

\ \.

o \

i

6 8

H/d

10

Fig. 6.13. Radial velocity gradient 3 for the isothermal jet from the large burner.

1.4

1.2

1.0

0.8

0.6

0.4

0.2

0.0

a Re = 1900 ° Re=2900 * Re=4226

Giralt Schrader

_i_ _ j _

6 8 10

H/d Fig. 6.15. Radial velocity gradient 8 for the flame jet from the large burner.

108

H/d-values B rises till a maximum will be reached. This maximum lies at H/d = 4 for the impinging isothermal jets and at H/d = 3' for the impinging flame jets. The increase of 3 at small H/d-distances must be due for the major part to the changing impact velocity profile. The influence of turbulence on 3 in this region cannot be distinguished clearly. In the figures 6.12 and 6.14 the turbulence level on the axis of free jets from the small burner has been given. The turbulence level in the isothermal jets has hardly been increased (from 7% to 9%) when 3 reaches its maximum (see figure 6.12). The turbulence level in the free flame jet on the other hand increases in the first three diameters from 8% to 21%. From the figures 6.12 and 6.14 a correlation of 3 with the free stream turbulence level for the region up to H/d = 4 cannot be found. This agrees with the flow visualization studies done by Yokobori, Kasagi and Hirata (1977). They found that for H/d < 4 the stagnation zone looked laminar. The large scale turbulence generated in the mixing layer of the jet has not yet been penetrated into the stagnation region.

At small values of H/d the results in figures 6.12 up to 6.15 are compared with results from Schrader (1961) and Giralt (1976). Schrader performed direct measurements of (9V/3r)r and found a slightly decreasing 3 up to H/d = 8. Since he used a jet issuing from a low-turbulence-level-nozzle the flat velocity profile was maintained over a considerably axial distance, which is the reason that his measurements do not show an increasing 6 till H/d = 4. Giralt gives from static pressure measurements a value of 3 at H/d É 1.2 : 3 = 0.916 uQ/d. For almost all cases for H/d < 2 there is a good agreement with values of 3 for the burners. They are equal or a little bit higher than those measured by Giralt and Schrader.

From the figures it can be seen that the Reynolds dependency of 3 is small as it is the case with the other flow properties. Again the same exception has to be made for the isothermal jets from the large burner.

109

Concerning the wall jet region this agrees well with the measurements done by Poreh, Tsuei and Cermak (1967). They found a turbulence intensity relative to the maximum wall jet velocity (/ v' 37vmax).100% of about 30% to 40% all through the wall jet. At s/d = 4 the same values were found from the present measurements.

Examining more closely the profiles of the turbulence intensities at r/d = 1 and r/d = 2 maxima can be found at about 2 mm from the plate. The humps in the profiles at this distance from the plate are probably due to what remains of the turbulence generated in the mixing layer of the free jet before impingement.

An important parameter~which characterizes the radial flow along* the plate is the maximum radial velocity v m a x. This parameter together with the rms-value of the fluctuating component at the location of v i/T' max ) is plotted in figure 6.18 (for the Reynolds numbers at H/d = 2). This value

v' m a xa seemed to be almost equal to the maximum of the rms-

value of the fluctuating radial velocity component along the

wmax

6

5

4

3

2

I 1 1

/ ^^^

/ / -*"" 1 / y 1 (I ,'

1/ '■' -jjy ^" 77 A'

r • i i V

i i

T L *^

~*

1 ' 1

1 1 1

D vmax ">

r=f \ Re = 6400

0 "max "I

v "max-)

TV.

**- ^^ —"- ' " ""— • ^ ^ " " " — ■ « ^ ^

1 I 1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r/d

3.0

2.5

2.0

1.5

1.0

0.5,

0.0

Fig. 6.18. Maximum velocity along the plate and fluctuating- component (si, H/d = *2) .

its'

'112

plate: ( /" max* The maximum velocity in the boundary layer increases up tó a distance of r/d = 1 . This radial distance corresponds with the edge of the stagnation zone. It has also been found by Schrader (1961) and Hrycak (1981) that the maximum of vm = v for small H/d-values is reached at r/d = 1 . The

IlldX value of / v' - ' reaches its maximum value at 1.5 < r/d < 2.0, which is remarkable because the average velocity in the boundary layer has already decreased significantly at this radial distance. The explanation probably is that the large eddies generated in the mixing layer of the free jet, which are deflected due to impingement penetrate into the wall jet only outside of the stagnation region and gives rise to the increase in turbulence.

H/d = 6

Figures 6.19 and 6.20 contain the measurement results for H/d = 6. Comparing the profiles of the average velocity at r/d = 1, r/d = 2 and r/d = 4 we find that at r/d = 2 the flow is constricted to the wall at the most. The variations in the turbulence intensities existing at H/d = 2 are not present here. Apart from small humps in the turbulence profiles within

6 y(mm)

5

4

3

2

1

0 0.5 1.0 1.5

Fig. 6.19. Radial velocity and turbulence intensity profiles (si, H/d = 6, Re = 3300; for legend see figure 6.16).

v 113

6 y(mm)

5

4

3

2

1

2 4 6 0.5 1.0 1.5

v(m/s) Vw*(m/s)

Fig. 6.20. Radial velocity and turbulence intensity profiles (si, H/d = 6, Re = 6400; for legend see figure 6.16).

half a millimetre from the plate at r/d é 1.0 the profiles look rather flat and similar.

Examining this little hump in the turbulence intensity close to the plate in the stagnation region we find that it also is present in the measurements at H/d = 2. This could agree with the theory of vortex stretching in the stagnation zone of Sutera, Maeder and Kestin (1963) and the measurements done by Sadeh, Sutera and Maeder (1970) on the turbulent properties in the stagnation region of a circular cylinder. However, these investigators found a more pronounced maximum. On the other hand Gutmark and Wolfshtein (1978) did not find an increase of the turbulence intensity in this region.

As for the measurements at H/d = 2 the values of / v'2/vmax.100% for r/d = 4 vary between 30% and 40%.

Figure 6.21 shows a comparison of the results presented in the figures 6.19 and 6.20. The profiles at r/d = 0.5, 1 and 2, scaled to the average velocity at the burner outlet, from the wall jets at Reynolds numbers 3600 and 6400, are compared to each other. The differences of the dimensionless velocity

114

\ I t l

.

-

..

i » r o

ft. \

K ^ " / » .«oW i

y (mm)

r/d=O.S y

(mm)

J I I I L. O.1 0.2

He=3600 Re=6400

r/d 1

y (mm)

0.1 0.2

,-*,

r/d=2

0.1 0.2

vl2/u0

y (mm)

r/d=0.S

Re-3600 Re=8400

y (mm)

0.5 1.0 v/ua

e

y (mm)

4

2 -

\\ \\

\\

r/d=2

*^ i i i

0.5 1.0 v/u0

0.5 1.0 v/u„

Fig. 6.21. Comparison of radial velocity and turbu­lence intensity profiles for two Reynolds numbers.

115

vmax 6

5

4

3

2,

1

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 r/d

Fig. 6.22. Maximum velocity along the plate and its fluctuating component (si, H/d = 6).

profiles as well as the dimensionless turbulence intensity profiles are small. There seems to be no Reynolds dependency of the present results, as can be expected.

Figure 6.22 shows the maximum average velocity in the boundary layer with the rms-value of its fluctuating component as a function of r/d. The maximum of v m a x is reached at r/d = 1.5. The stagnation zone extends itself till this distance. Again we see that the turbulence intensity does not vary much over the entire range. This is due to the turbulence in the free jet which at x/d = 6 is much more uniform than at x/d = 2.

i However, the fact that the turbulence intensity does not decrease 'in the same way as the average velocity does after r/d = 1.5 points at an influence of the large eddies induced at the edge of the jet.

The boundary layer thickness

Another characteristic parameter of the impinging jet flow is the boundary layer thickness defined by the distance from the plate to where the radial velocity reaches its maximum. In

116

Re=6400

Re = 3600

3.0

2.5

2.0

1.5

1.0

0.5

nn

S/d

0.20

0.16

0.12

0.08

0.04

- o a

-

-

-

-_a-

" I — I 1 H/d=6 Re=3600 Re=6400

o / /

1 1 1

J

-

/ " / -

-

1

S/d

0.20

0.16

0.12

0.08

0.04

H/d=2 o Re=3600 □ Re=6400

r/d 3 4

r/d

Fig. 6.23. The boundary layer thickness (si).

figure 6.23 the boundary layer thickness is given. In the stagnation zone (0.25 < r/d < 1 for H/d = 2 and 0.25 < r/d < 1.5 for H/d = 6) the boundary layer thickness increases only little. Where the wall jet becomes highly turbulent due to the large eddies from the jet mixing layer, the boundary layer thickness grows rapidly. From the measurements it can be seen that the lowest Reynolds number gives the thickest boundary layer.

6.2.3.2. The flame jet

Measurements.1; of the radial velocity profiles have also been performed with the impinging flame jet from the small burner. These measurements are difficult and from the scattering of the data it is o'bvious that the accuracy of- these measurements is worse than that of the isothermal flow measurements. The temperature gradients induce refractive index gradients. This caused the laser beams to deflect so the position in space of the measuring volume could not accurately be determined. While in the isothermal case measurements were possible up to a distance of 0.2 mm from the plate, with the flame jet

117

5r y(mm)

4

3

2

1 -

r/d ' ° 0.25

o 0.5 -

-■

f\TVf\\ *0d7S

\v\ ' u0

q

N& „ i.s

T^Sv * 4-°

-l / J X\ 8 12 16 20 24

v(m/s)

S y(mm)

4

6 8 10

4v^(m/s)

Fig. 6.24. Radial velocity and turbulence intensity profiles (sf, H/d = 2, Re = 1050).

6 y(mm)

5

8 12 16 20 24 28

v(m/s)

3 -

2

1

2 4 6 8 10 12

v'z(m/s)

Fig. 6.25. Radial velocity and turbulence intensity profiles (sf, H/d = 2, Re = 1900; for legend see figure 6.24).

118

measurements could be made up to a distance of 0.75 mm from the plate. Besides deflections of the beams due to the temperature gradients, the temperature fluctuations caused the laser beams to jitter. Also the higher range of velocities, resulting into higher Doppler frequencies with a worse signal-to-noise ratio, is a reason for the higher inaccuracy.

H/d = 2

Measurements on the radial velocities and turbulence intensities for the short burner to plate distance are gathered in figure 6.24 and 6.25. As in the isothermal case here also different shapes of profiles can be distinguished. Comparing the profiles at r/d = 0.75, 1.0 and 1.5 a strong constriction can be noticed of the flow along the plate at r/d = 1.0. From this distance on the strong mixing causes a flattening of the velocity profile (see r/d = 1.5). The temperature decay causes the strong velocity decay for higher r/d-values (r/d = 2 and 4). The profiles of the turbulence intensity show the same tendencies as the velocity profiles: an increase of turbulence intensity in the stagnation region till r/d = 0.75, then a decrease of turbulence intensity due to the strongly decreasing velocity.

Figure 6.26 gives the maximum average velocity and the corresponding turbulence intensity as a function of r/d. In this graph it can be seen more clearly that the maximum of vmax lies between r/d = 0.75 and r/d = 1.0. The maximum turbulence intensity is only slightly lagging behind. The influence of the mixing induced eddies is not as clear as for the isothermal impinging jet.

H/d = 6

At the larger burner to plate distance (H/d = 6, figures 6.27 and 6.28) the flow develops as can be expected. Like in the isothermal case turbulence in the stagnation region is rather

119

uniform. Figure 6.29, in which again the maximum velocity is plotted, shows that the maximum v m a x lies close to r/d = 1.5. The turbulence intensity already starts decaying before this maximum is reached which is quite remarkable. However, any influence of the free jet mixing layer is not expected, because of the large H/deff value.

28

Fig. 6.26. Maximum velocity along the plate and its fluctuating component (sf, H/d = 2 ) .

5 y(mm)

4

3

2

1

0 _i i i u 2 4 6 8 10

v(m/s)

2 3 4

v'2 (m/s)

Fig. 6.27. Radial velocity and turbulence intensity profiles (sf, H/d = 6, Re = 1050).

120

5 y(mm)

4

S y(mm)

4

8

2 -

12 16 2 4 6 8

v(m/s) -<W*(m/s)

Fig. 6.28. Radial velocity and turbulence intensity profiles (sf, H/d = 6, Re = 1900; for legend see figure 6.27).

Fig. 6.29. Maximum velocity along,the plate and.its fluctuating component (sf, H/d = 6 ) .

121

6.2.4. Axial temperature distribution Next to the axial velocity decay the axial temperature distribution of a flame jet is of importance. The hot flame jet mixes with the environmental air at room temperature; so the temperature of the jet decays. Due to the temperature dependent density of the flue gasses the velocity decay of the flame jets is a consequence of this temperature decay. The temperature difference between flame jet and plate is also an important parameter because it is the driving force for the heat transfer to the plate.

Measurements of the temperature decay for the flame jets studied are presented" in figure 6.30. The adiabatic flame temperature of Groningen natural gas is 2133 K (see reference "Basisgegevens over Gronings Aardgas", 1968). I,t is assumed that all gas has been burned within the burner. Looking at the axial temperature distribution this assumption is reasonable, because the temperature at the burner outlet almost reaches the

2000 T(K)

1800

1600

1400

1200

1000

800

600

I i i i i i 1 1 ' i i I I 2 3 4 5 6 7 8 9 10

x/d

Fig. 6.30. Axial temperature distributions of the flame jets.

122

adiabatic flame temperature and the axial temperature decreases rapidly after the flame leaves the burner. The temperature at the burner outlet of three flame jets is only 50 K lower than the adiabatic flame temperature. This difference can be attri­buted to a slight deviation of the stoichiometric ratio or to the radiation losses of the thermocouples which were used. The radiation losses become larger when the convection to the thermocouple decreases. This might be the reason that the exit temperatures at the two lowest Reynolds numbers for jets from the large burner (Re = 1900 and Re = 2900) are measured lower than in the other three cases. The different heat losses from the small and large burner might also contribute to a lower exit temperature in case of the large burner. The difference in heat loss is caused by the less effective thermal isolation of the actual combustion chamber of the large burner. This has been confirmed by the higher temperatures of the outer surface of this burner.

6.3. Heat transfer 6.3.1. Stagnation point heat transfer

Heat transfer at the stagnation point has been measured for both burners for the isothermal jet as well as for the flame jet. When mentioning an isothermal impinging jet it is meant that the jet is isothermal with the surrounding air and that the temperature difference between the air and the plate is below 30°C. The experimental set-ups and measuring techniques are described in paragraphs 5.1 and 5.2.

From the measurements of the velocity gradient at the stagnation point {3 = '5vmax/3r)r_0) and the theory of Sibulkin (1952) the stagnation point heat transfer can also be calculated from:

Q

NuR = 0.763 d (_)°-5Pr0-4 (6.5)

(see paragraph 2.2.1). Both results on stagnation point heat

123

i—! 1 ■ I i i i i 1 1 1 1 — r

— o — from heat flux measurements ~~ ""•• <Re=3300 —••— from /3 measurements

J L _ I I -. I I I I I 1 1 1 1 2 3 4 5 6 7 8 9 10 11 12

H/d

:.Flg. 6.31. Stagnation point heat transfer (si).

transfer from heat flux measurements and from velocity gradient measurements are discussed in this paragraph.

6.3.1.1. Isothermal jet .

Measurements on the stagnation point heat transfer for the •isothermal impinging jet from the small burner are gathered in figure 6.31. Heat transfer increases with H/d until at H/d = 5 a maximum is reached. There are several jet characteristics which are. important, to ■ heat. transfer for those first five .diameters, such as: the axial velocity, the.' turbulence intensity on the axis and the shape of the velocity profile-. As can be seen: from figures 6:1'.-and 6.2 the axial velocity'at x/d .= •'5 has decreased 'to: 90% of its initial value' at x/d ■.=: 0 .(causing a decrease of the Nusselt '.number), while the turbulence intensity has raised from 7% to 12% (possibly causing an increase of the Nusselt number). Since the axial velocity already decreased by 10%. the' velocity 'profile has developed from an originally uniform profile to a more or less Gaussian shaped profile (causing an increase of the: Nusselt

70-Nu

60-

50 -

40-

30 r 20-

10-

124

number). It is not possible to separate these three effects on the heat transfer quantitatively.

The influence of the shape of the velocity profile and the absolute value of the axial velocity can also be found in the heat transfer results calculated with equation 6.5 with the measured 3-values. As already stated in the discussion on the results of the B-measurements a separate influence of turbulence on 3 has not been found. Concerning the measurements at H/d < 4 this corresponded to flow visualization experiments done by Yokobori et al. (1977). The same investigators Yokobori et al. (1978) showed for a two-dimensional jet that the influence of turbulence on the stagnation point heat transfer was absent for H/d < 4. This fully agrees with our measurements. The two ways of determining heat transfer give corresponding results and since there was no influence of turbulence on B in this region, this influence is also absent in the direct measurements of heat transfer.

When the plate is positioned outside of the potential core of the jet, the two ways of determining heat transfer start to deviate. This is the region where the large eddies generated in the mixing layer of the jet penetrate till the axis. The turbulence in this region has a different character. While in the potential core region the scale of turbulence is small compared to the burner diameter, at axial distances higher than the potential core length the scale of turbulence is of the order of the burner diameter. For a jet with this large scale turbulence we can understand that the Sibulkin formula does not predict heat transfer anymore.

Figure 6.32 gives the same picture of the stagnation point heat transfer measurements from the isothermal jet from the large burner. It can be seen here that for the jet with Re = 4240 the heat transfer enhancement due to large scale turbulence already starts at H/d = 2. This can be explained by the shorter potential core length for this particular jet (see figure 6.3).

125

Nu

80

70 -

60

SO

40

30

20

10

Re=9500

y Re=4240

from heat flux measurements from 0 measurements

_j i_ 1 2 3 4 5 6 7 8 9 10

H/d

Fig. 6.32. Stagnation point heat transfer (li)

Nu

100 -

50

20

from heat flux measurements from |3 measurements Sibulkin theory

2,000 5,000 10" Re

Fig. 6.33. Stagnation point heat transfer at H/d = 2 (si + li).

126

The results show that heat transfer at the stagnation point is laminar-like at small burner to plate distances. The measurements from small and large burner at H/d = 2 are gathered in figure 6.33.

From linear regression on results from heat flux measure­ments of the four cases it is found that:

Nu = 0.57 Re 0- 5 2 5 (6.6)

From linear regression on the four results from B-measurements it is found that:

NUg = 0.60 Re 0- 5 2 2 (6.7)

From figure 6.33 it is clear that the measurements at H/d = 2 are very close to the theoretical correlation for laminar heat transfer from a uniform jet to the stagnation point of a flat plate. Besides this it can be concluded that the results at H/d = 2 from large and small burner match to each other.

The results presented in figures 6.31 and 6.32 are in close agreement with the findings of Giralt, Chia and Trass (1977), already mentioned in paragraph 2.2.2. With a similar formula as equation 6.5 and measured stagnation pressure profiles they found a "turbulence free" transfer coefficient (see equation 2.6 1). They found enhancement of (in their case) mass transfer by comparing the experimental results with this "turbulence free" transfer coefficient.

Our results can also be interpreted in the same way: The Nusselt number found with equation 6.5 is the laminar contribu­tion. The difference of the Nusselt number from heat flux measurements and from equation -6.5 gives the enhancement due to turbulence. This can be expressed by a turbulence enhancement factor defined by

Nu = (1 + Y)NUg (6.8)

This turbulence enhancement factor is given in figure 6.34 as a function of the turbulent Reynolds number Tu/Re for the

127

measurements on I both burners for several H/d-values. Different symbols are used for values of y at H/d < 3, H/d = 4 and H/d a 6. It can be seen from the figure that for H/d 2 3 the value of y is about zero even for values up to Tu/Re = 11. The results from .Giralt et al. (1977) are also given in the figure. The present results are in the same order of magnitude.

0.6

0.4

0.2

0.0

•0.2

4 8 . 12 16 20 24 28 32 36 40

TuJÏÏë .

.' Fig. 6.3 4. The turbulence enhancement factor (li' + ' ■ ■ s i ) .' •'

6.3.1.2. The flame jet

With the heat flux transducer the total heat flux density for flame jets can be measured, being the summation of convection and radiation. Before a discussion of the heat . flux measurements can be given, an estimation has to be made of the radiation heat transfer from the hot burner to the cold plate.

Radiation heat transfer

It is supposed that the radiation of the flame itself is negligible because of the very low emissivity of a hot gas layer of small thickness. Then there is only left the radiative heat transfer from the hot inner wall of the burner to the heat flux transducer. ' ,

Radiation1 from a cylindrical hole of finite depth- with

. H/d « 3 o H/d =4

- ° H/d >6 Gira,t -

J I I I I 1 I I L

1 2 8

diffuse reflecting walls at constant temperature is analysed by Lin and Sparrow (1965). They found an effective hemispherical emissivity e n which gives the total amount of energy Q leaving the mouth of the cavity in ratio to that emitted from a black walled cavity:

Q eh = -..-■. =~rr (6-9) TTR2O(T. )

where R is the radius of the cavity, o is the Bolzman factor arid Tw and Te are the cavity wall temperature and the temperature of the environment.

Now the problem of radiation heat transfer from the burner to the heat flux transducer is reduced to the problem of radiation from a circular disk to a plane element dA2 / being the surface of the heat flux transducer, parallel to the disk (see figure 6.35). The fraction of energy leaving surface

m Fig. 6.35. The burner exit A facing the heat flu?

transducer dA-

element A-j that arrives at element dA2 is defined as the geometric configuration factor F1 .2- Si'egel and Howell (1981) give for the configuration of figure 6.35:

1+2 id - .1 +. V /(Z2 - 4s2)

dA- (6.10)

with C = H/r, x, = R/r and Z = 1 + C2 + S2. The heat flux density absorbed by the heat flux transducer

can thus be estimated by:

u rad c* h V ' 1 + V /(Z2 -' 4i;2) XT * " T*)

(6.11)

129

where ec = the emissivity of polished constantan and T = the heat flux transducer temperature.

Reasonable estimates of the emissivities are: ec = 0.84 eh = 0.7 (from Lin and Sparrow (1965) with e =0.27)

A1203

The most uncertain and also most determining factor is the inner burner wall temperature. Figure 6.36 gives Q'Vad calculated at three different burner wall temperatures: T w = 1900 K, 1800 K and 1700 K. Especially at small burner to plate distances (H/d < 2) the contribution of radiation heat transfer to the total heat transfer cannot be neglected.

100

f'rad 80 (kW/m2)

60

40

20

0 0 2 4 6 8 10

H/d

Fig. 6.36. Heat transfer by radiation.

Convective heat transfer

Measurements of the stagnation point heat flux distributions were performed at two Reynolds numbers for the small burner and at three Reynolds numbers for- the large burner. These measurements are shown in figures 6.37 and 6.38. The dashed lines give the convective heat transfer after a correction for radiation calculated by equation 6.11 with Tw = 1800 K. The uncertainty of this correction is rather big because of the unknown inner burner wall temperature. Very high convective heat fluxes are reached at small burner to plate distances with

I I I

" l '

- J ' IV, - \l

- % - % - % - %

1 1 1

1 1 1 1 1 1 1

Tw=1900K

1700K

i ^ ~ - i 1 — , — i 1

--

-

-

-

130

Q (kW/m2)

600-

500

400

300

200

100

O

~T~ I

o Re=2700 o Re=1771 - - corrected for radiation

J L_ 1 2 3 4 5 6 7 8 9 10

H/d

Fig. 6.37. Stagnation point heat flux densities (sf)

q" (kW/m2)

400

350

300

250

200

150

100

50

0

«e=4226 fte=2900 fle=7900

- corrected for radiation

a 9 H/d

10

Fig. 6.38. Stagnation point heat flux densities (If)

131

the small burner: 570 kW/m2 at H/d = 1.5. The radiation heat flux for this H/d will be about 30 kW/m2. These high fluxes give rise to another uncertainty. The heat flux transducer could only be calibrated" up to heat fluxes of 150 kW/m2. Since the response of the transducer was linear over a big range, the calibration has been extrapolated.

A rapid decrease of the heat flux with increasing H/d is caused by the axial developments of velocity as well as temperature. Differences in results from large and small burner are also due to the differences in axial velocity and temperature development. As can be seen from the results in figure 6.8 the jet from the large burner develops much faster than the jet from the small burner. Figure 6.30 showed that the

* temperature decay on the axis of the flame jet from the large burner starts immediately at the burner outlet while this decay on the axis of the flame jet from the small burner only starts at x/d = 2. These differences give rise to the much faster decay of q" with H/d for the large burner than for the small burner.

The results from the flame jet can only be compared to those of the isothermal jet when they are given in dimension-less numbers. In doing so the problem arises at which tempera­ture the fluid properties has to be taken. In this respect the method proposed by Eckert (1942) has been followed. He stated that, in order to be able to compare the results with isothermal measurements, the fluid properties should be taken at a temperature (T*) where the, enthalpy across the boundary layer has the average value. So the Nusselt number can be calculated from:

q" C_(T*)d Nu = 2 (6.12 <haxis - hplt>*<T*>

The enthalpy on the axis (naxis' i s calculated from the temperature on the axis in the free flame jet at x/d = H/d (the impact temperature). The enthalpy near the plate is calculated

132

Nu 120

u 100

80

60

40

20

-\ 1 1 r H I I T"

A ffe=4226l o Re=2900 \ from heat flux measurements ° Re=1900i • Re=4226] • Re=2900} from/i measurements • Re=1900]

J I I L_ _l l_ 1 2 3 6 8 9 10

H/d

F i g . 6 . 3 9 . S t a g n a t i o n p o i n t hea t t r a n s f e r ( s f )

Nu 70

60

50

40

30

20

10

o Re=270&] } from heat flux measurements

o Re=1771) • Re=2700] • Re=1771j from p measurements

_i i i i i i_ 1 2 3 4 5 6 7 8 9 10

H/d

F i g . 6 . 4 0 . S t a g n a t i o n p o i n t h e a t t r a n s f e r ( I f ) ,

133

from the plate temperature. A Nusselt number can also be calculated from the velocity

gradient at the stagnation point (see equation 6.5). For the flame jet in a similar way this Nu-number can be calculated by:

NuR = 0.763 d (—- )°-5Pr0.4 (6.13) P v(T*)

The results of the Nusselt numbers as a function of the burner to plate distance are given in the figures 6.39 and 6.40. These Nusselt numbers are related to the convective heat transfer results. So first the corrections for the radiation heat flux were made.

From the plots it can be seen that between H/d = 2 and H/d = 4 to 5 the Nusselt number rises. Similar to the isothermal measurements this can be attributed to the changing velocity profile in this region. However, looking at the axial velocity and turbulence profiles, it is even more difficult as it is in the isothermal case to separate the effects of the shape of the velocity profile, the decreasing axial velocity and increasing axial turbulence level. Looking back at figure 6.8 it can be seen that the velocity in the region considered here decreases from 0.9 uQ to 0.5 uQ, while the turbulence level increases from 15% for the small burner and 20% for the large burner to about 30%.

Comparing the results from the heat flux measurements with the results from the 3-measurements it can be seen from figure 6.40 that for H/d £ 3 the agreement is rather good. At H/d = 3 the results from both measurements start to deviate. For H/d > 3 at the highest Re-number the difference between Nu and Nuo is at its biggest. The measurements on the small burner show the same effects less clearly. At Re = 1771 the Nusselt numbers calculated in both ways do not deviate much for the whole H/d region considered. However on the whole, figures 6.39 and 6.40 show the same trends as the results for the isothermal jets in figures 6.31 and 6.32.

134

No

100

50

30

-

-

o If 1 a sf J

■ sf j

1 ^ ^ ^

1 , 1

T . 1 1 |

from heat fluxes

from {'> measurements

Sibulkin theory

"o»

I I I I 1 i

-

-

-

3.000 5.000 10.000 Re*

Fig. 6.41. Stagnation point heat transfer at H/d = 1 (If + sf).

To show that even for the flame jets with rather high initial turbulence levels the influence of this turbulence on the heat transfer at small burner to plate distances is not big, all measurements at H/d = 1 are gathered in figure 6.41 . They are given here as a function of the Re*-number, which is defined as

Re* uo d

v(T*) (6.14)

where u_ is the velocity measured at x/d = 1 and v(T*) is the viscosity at the before mentioned "averaged enthalpy temperature". The results presented in this way agree very well with the theoretical results of Sibulkin for a laminar stagna­tion point flow, as was the case for the isothermal measure­ments. Here also the results of both burners match each other.

The turbulence enhancement factor defined by equation 6.8 for the impinging flame jets is given in figure 6.42 as a function of Tu/Re*. Again the same distinction is made between

135

o. e );

0.4

0.2

0.0 ■0.2

4 8 12 16 20 24 28 32 36 40

TuVRë*

Fig. 6.42. The turbulence enhancement factor (If + sf).

the results for H/d i 3 and H/d a 6. Despite of the rather high values of Tu/Re* (up to Tu/Re* = 28) the value of y is about zero if H/d S 3. For H/d a 6 it is noticed that y * 0 and that the turbulence enhancement factor is in reasonable agreement with the findings of Giralt et al. (1977).

In paragraph 6.2.1. it was concluded that the flow structures of the isothermal jets from the small burner differed from the isothermal jets from the large burner (figure 6.4). A similar conclusion was drawn concerning the flame jets from small and large burners (figure 6.8). Finally, the comparisons between the isothermal jets and the flame jets from the same burner (figures 6.9 and 6.10) showed that the flow structure of the isothermal jets is completely different from the flow structure of the flame jets. The heat transfer results discussed in this paragraph, however, show that the stagnation point heat transfer of all these different impinging jets (flame jets and isothermal jets) can be described in a similar way with 3 and y. This shows the wide applicability of this description.

136

T I I I I I 1 1 I

• H/d < 3 o H/d =4 o H/d >6 a Giralt -

J I I I I I I L

6.3.2. Radial heat transfer distribution

The heat transfer distribution as a function of the radial position (r/d) on the plate has been measured for three flow situations: the isothermal jets from the large as well as the small burner (li and si) and the flame jets from the large burner (If). The results will be discussed in this paragraph.

6.3.2.1. The impinging isothermal jets

With the liquid crystal technique described in chapter 5 radial heat transfer distributions have been measured. Results of these measurements for the small burner at different H/d-distances for two Reynolds numbers are given in figures 6.43 and 6.44. The low Reynolds number jet (Re = 3300) gives for all H/d-distances a maximal heat transfer at the stagnation point. For H/d = 2 and H/d = 3 the Nusselt number is constant over the stagnation region (0 < r/d < 0.5) and monotonically decreasing in the wall jet region (r/d > 0.5). For H/d a 5 the heat transfer coefficient distribution expressed by Nu = f(r/d) is bell shaped. The heat transfer for the high Reynolds number jet (Re = 6300, figure 6.44) differs in one respect. For H/d = 2 and H/d = 3 secondary peaks can be noticed in the heat transfer distributions at r/d = 2.

-i 1 1 1 1

Re=3300 H/d * 2 a 3

S* o 5 ^ ^ f c ~ o 7

J i i i Ê : 1 2 3 4 5 r/d

F i g . 6 . 4 3 . R a d i a l h e a t t r a n s f e r d i s t r i b u t i o n ( s i ) .

30

20

10

137

70

0\ 1 1 1 1 1 1 1 2 3 4 5 r/d

Fig. 6.44. Radial heat transfer distribution (si).

These peaks can also be noticed in the results for Re = 4250 from the large burner (figure 6.45) and even more pronounced for the higher Reynolds number jet from this burner (Re = 9500, figure 6.46). The reason for the existence of these secondary peaks in the heat transfer distributions around r/d = 2 can be the maximum of the turbulence intensity at almost the same radial location in the wall jet (see figure 6.18). The results from the large burner show the existence of a second maximum. Especially the results from the high Reynolds number (Re = 9500) at H/d = 2 and H/d = 3 show two maxima in the radial Nusselt number distribution. The already mentioned maximum at r/d = 2 and a maximum at r/d = 0.5. This has also been found by other investigators (see e.g. Gardon and Akfirat in figure 6.47) and is explained by the presence of a minimum in the boundary layer thickness. It is not clear why this maximum at r/d = 0.5 has not been found for the small burner.

A comparison of the present results at H/d = 2 has been made with results from Gardon and Akfirat (1965) for the same nozzle to plate distance. They presented results in the Reynolds number range: 2800 < Re < 28,000. Figure 6.47 gives

138

Fig. 6.45. Radial heat transfer distribution (li)

Fig. 6.46. Radial heat transfer distribution (li).

139

100

80

60

40

20

O 6 4 2 0 2 4 6 r/d

Fig. 6.47. Comparison of radial heat transfer dis­tribution with results from Gardon and Akfirat (1965).

the comparison for the large as well as the small burner results. Compared to our measurements the heat transfer results of Gardon and Akfirat show higher values. This has also been found by Hrycak (1983) who stated that the high values of the results of Gardon and Akfirat are due to inaccuracies of the heat flux transducers used by them.

The results compared in figure 6.47 show another difference: Gardon and Akfirat found three humps or peaks in their radial heat transfer distribution, while we find only two. The first peak found at r/d = 0.5 corresponds with the peaks found by us for the impinging jets from the large burner. Instead of the two outer peaks found by Gardon and Akfirat for Re < 10,000 we found only one outer peak. The peak at r/d = 1.4 according to Gardon and Akfirat is due to a transition from a laminar to a turbulent boundary layer. This peak vanishes for Re > 10,000. Our measurements of the flow structure close to the plate show that the wall jet also for r/d < 1.4 is turbulent (see paragraph 6.2.3), so this transition does not

140

occur in our experiments. Due to the higher initial turbulence of the jets from our burners the wall jets are fully turbulent at lower Reynolds numbers. In this respect our heat transfer results agree with the results of Gardon and Akfirat for Re > 10,000, where only one outer peak remains at the position r/d = 1.9.

:'•}■ "

6.3.2.2. The impinging flame jets

For the large burner at the Reynolds numbers of 1900, 2900 and 4226 at different burner to plate distances the maximum temperatures near the wall, the temperature of the plate and heat flux densities to the plate as a function of the radial distance from the stagnation point are measured. From these measurements with an equation similar to equation 6.12 in paragraph 6.3.1.2, radial distributions of the Nusselt number are calculated and presented in this paragraph. The Nusselt number is calculated from

q".c„(T*)d Nu = - 2J (6.15) <hmax " hPlt> A<T*>

The fluid properties cn and A are determined at the temperature (T*) at which the enthalpy of the gasses across the boundary layer has the average value of (hma^ + h lt)/2.

Temperature distributions

With the thin wire PtRho6%-PtRho30% thermocouples temperature profiles close to the wall have been measured." From these measurements the maximum temperature close to the wall has been found. Some of the results are presented in figures 6.48 and 6.49. Figure 6.48 gives the maximum of the' near wall temperatures for Re = 2900 and 2 é H/d S 10. The temperature in the stagnation region drops very rapidly with H/d. This agrees well with the axial temperature development given in figure 6.30. In the wall jet region for r/d > 2, however, the maximum

141

f 5 0 0

T Cc)

1000

800

\

. ^

Re

- .^

= 2900

«t

H/d ■■< 2

3 ' 4 • e ' a * 10

S B

°l oB ofi.

öfl o

e 0

8 4

8 0 D 1 i

H/d =2

ffe =

" f900 2900

- 4226

8 S 6 ° 1 .

° 1 i

4 e o r/d r / d

F i g . 6.48 and 6 . 4 9 . Maximum boundary l a y e r t empera­t u r e ( I f ) .

H/d=2

a f900 o 2900 •■ 4226

r/d r/d

Fig. 6.50 and 6.51. Radial distribution of the heat flux density (If).

142

wall jet temperature is independent of H/d in the region 2 ë H/d S 4. For r/d > 4 the maximum wall jet temperature is independent of H/d in the region 2 s H/d s 6. Figure 6.49 gives an example of the maximum temperatures close to the plate for the three different Reynolds numbers at a fixed H/d value (H/d = 2) . The maximum temperatures are independent of the Reynolds number as was also the case with the axial temperature developments of the free jets (see figure 6.30). Several measurements have been carried out twice to show the reproducibility. Almost all reproduced measurements coincided within 1 or 2%. Only a few deviated about 5%.

The heat flux distributions

For flames it is valuable for practical reasons to consider the measured heat fluxes themselves. The heat flux results are presented in figure 6.50 and 6.51. At Re = 2900 for 2 < H/d < 10 the radial heat flux densities are given in figure 6.50. The figure shows that the heat flux density decreases very fast when r/d decreases, especially at low H/d values. The nonuniformity of the heat flux to the plate is enormous. Even at H/d = 6 the heat flux drops a factor of 2 within two burner diameters into the radial direction. It is clear that it is only possible to reach a uniform heat flux distribution (for instance, with an array of burners) at higher burner to plate distances. Heat flux distributions at H/d = 2 for different Re­numbers are given in figure 6.51. This figure shows once more the large peak in the heat flux density at low H/d. In all cases the plate temperature is in the range of 50-70°C, while the maximum boundary layer temperatures range from 250-1500°C. Comparing figures 6.48 and 6.50 it is obvious that the sharp decrease of the heat flux density with r/d is to a large extent caused by the sharp decreasing boundary layer temperature.

143

H/d=2 Re=

'm t900 o 2900 * 4226

90

Nu

60

30

H/d=3 fle=

a 1900 a 2900

r/d r/d

H/d=4

a 1900 o 2900 ». 4226

30

r/d

Fig. 6.52. Radial Nusselt number distributions (If)

144

The Nusselt number distributions

In figure 6.52 the radial Nusselt number distributions calculated by equation 6.15, are presented. At H/d = 2 the distributions in the stagnation region have a slight maximum at r/d = 0.5. This has also been found in the results of the isothermal impinging jet of the large burner at small H/d-distances. At higher H/d the Nusselt number in the stagnation region monotonically decreases with r/d.

At the burner to plate distances H/d = 2 and H/d = 3 a second maximum in the heat transfer distributions can be distinguished at a radial position between 1.5d and 2.0d. Again similar maxima have been found for the isothermal impinging jet.

It can be concluded that the heat transfer distributions of isothermal jets and flame jets compare very well, despite of the large differences in heat flux densities and temperatures.

It was tried to correlate the radial Nusselt number distributions in two ways: 1) For the stagnation region: Nu ~ Re0-5Pr0,4

2) For the wall jet region: Nu ~ Re0,8Pr0-4

Ad 1 ) It has already been shown for the stagnation point that the Nusselt number is proportional to Re for small values of H/d (see figure 6.41). Correlating the results from figure 6.52 with the square root of the Reynolds number, it can be stated how far this proportion reaches. The correlation is presented in figure 6.53. It shows that the correlation holds for H/d = 2 and H/d = 3 in the region 0 < r/d < 1 . At H/d = 4 and H/d = 6 the correlation is bad. The conclusion is that not only at the stagnation point, but also in the whole stagnation region the heat transfer depends on Re^*5.

Ad 2) For the wall jet region the results are correlated with Re 0 , 8 in figure 6.54. A power of 0.82 has been found by Vallis, Patrick and Wragg (1978) (see equation 2.67) in

145

1.40

1.00

\ 0 . 6 0

0.20

1.40

«89»

8 °§8 D

a O

H/d=2 i

* « 0

ft

B

i —r

Re = a 1900

o 2900

» 4226

fc o o

0 ° ° 6 , □ 0 ,

1 1

•» 1.00-

0.60

0.20

Re = a 1900 o 2900

» 4226

■s g

H/d=4

r/d

Re = a 1900 a> 2900

« 4226

H/d=3

Fig. 6.53. Correlation of Nusselt number distribu­tions with Re°-5Pr0-4.

*» 0 »

o

« S 2

n

H/d = e 1 —

o

o

» S o S • B ° D 0 »

1 1 —

fle = a »900

o 2900

* 4226

. • S ' " ° * a

the wall jet region of impinging isothermal jets. However, this has been found for higher r/d-values than the r/d-values considered in our study. Yet, figure 6.54 shows that at H/d = 2 this correlation with Re 0 , 8 is possible for r/d > 3. For such a low burner to plate distance the wall jet will be established earlier than in the case of higher H/d-values. At H/d = 3 and H/d = 4 the

146

o.oa

0.06-

0.04-

0.02

o » . m □ 7900

o 2900

" 4226

H/d = 2

*.%8 o g 8

9 fl

H/d=3

a 7900

a 2900

« 4226

-

0.06

0.04-

0.02

O

* 0

o « a

o □

* o o o o o Q

' * * * » ° o ® . 1 * 0

H/d =4

a C3

a 6

Re = 7900 2900 4226

2 i

r/d

a o

' 8 „ 13

H/d = 6

* i * ffl

fle= a 7900

o 2900

.* 4226

^ t

6 0 r/d

Fig. 6.54. Correlation of Nusselt number distribu­tions with Re°-8Pr0-4.

correlation evidently improves when r/d increases from 3 to a higher value. For the measurements at H/d = 6 no correlation with Re could be found in the r/d region considered in our study.

This completes the discussion on the radial heat transfer distributions.

147

7. RESULTS OF NUMERICAL SIMULATIONS

7.1. The laminar impinging jet

Numerical simulations were first performed on laminar impinging jets. This was done in order to validate the computer code with a known case. Also a study could be performed on the dependence of the heat transfer on the shape of the impinging laminar velocity profile. Constant fluid properties were assumed for these calculations. The dependency of the heat transfer on H/d was expected to be very small for laminar jets at Re-numbers higher than 600 in the region 1 < H/d < 12. This in fact has already been found by Saad (1975). For this reason the burner to plate distance was fixed at H/d = 6.

To show the importance of the shape of the velocity profile on the heat transfer computed results on laminar jets with a parabolic and a flat profile are compared in figure 7.1.

35

Nu

25

15

5

0.5 1.0 1.5 2.0 2.5 3.0 r/d

Fig. 7.1. Radial heat transfer distributions from laminar impinging jets.

I I I I I

^ \ \ Re=600 \ \ parabolic

N flat \ \

- \ . - « • " " " " ^ "N . ^

^S^ ^S \ ^ N. \ — \. ^ >v \

> v \ \ w \

— ^ ^ s ^ ^ *"V

^ ^ " » — ^ ^ _ ^ - ^

I I I I I

"

"

_

—

149

The Reynolds number defined on the averaged velocity at the outlet is for both jets the same. This means that the velocity on the jet axis of the parabolic jet is twice that of the flat jet. Figure 7.1 shows that the heat transfer at the stagnation point also differs by a factor of 2. However, not only at the stagnation point, but for r/d È 2.5 at any radial position heat transfer from the jet with the parabolic profile is higher than heat transfer from the jet with the flat profile. The reason for this is, that the flow from the parabolic jet is more concentrated along the plate than the flow from the flat jet. This can clearly be seen from the figures 7.2, 7.3 and 7.4 giving numerical results of radial velocity profiles at r/d = 0.486, the maximum radial velocity and the boundary layer thickness as a function of radial distance. A comparison of the radial velocity profiles along the wall at r/d = 0.486 in figure 7.2 learns that the velocity gradient at the wall and

v (m/s)

0.8

0.6

0.4

0.2

' 0 2 4 6 8 y(mm)

Fig. 7.2. Radial velocity profiles of laminar im­pinging jets (Re = 600).

the peak velocity are much higher for the parabolic jet. The maximum velocity of the wall jet from the impinging parabolic jet is higher than that from the impinging flame jet at every radial position considered (see figure 7.3). At the same time the boundary layer thickness of the wall jet from the impinging parabolic jet- (defined by the distance from the wall where the

T i i 1 1 1 1 r

r/d = 0.486 parabolic flat

150

vmax (m/s)

0.8

0.6

0.4

0.2

i 1 1 l 1

parabolic ' ~ ~ X flat

f

- /

7 / / i i i i i

-

-

0.5 1.0 1.5 2.0 2.5 r/d

Fig. 7.3. Maximum wall jet velocities from laminar impinging jets (Re = 600).

5 (mm)

2

1

j r s

jT «•«*

"" j/r s*

, _ ^ ^ y "^^V ^^ ** — > v J S ^ «"^

^^' parabolic y

,s' flat

i i i i i i

_ -

-

---

0.5 1.0 1.5 2.0 2.5 3.0 r/d

Fig. 7.4. Boundary layer thickness of laminar im­pinging jet- (Re = 600).

maximum velocity is present) is smallest in case of an impinging parabolic jet (see figure 7.4). For r/d -► 0 we see that the difference is a factor of 2. The boundary layer thicknes of the impinging flat jet shows a minimum at r/d = 0.45. This corresponds with the maximal heat transfer at r/d = 0.40 (see figure 7.1). The comparison of the flow structure of both impinging jets gives reasons for their differences in Jieat transfer.

151

To find the influence of the shape of the velocity profile the Reynolds number has to be defined on a different diameter and velocity. We will take now Rei = u m, v di/v, with di being

2 max 2 2 the half width jet diameter (see also paragraph 2.1.2). In this way also a Gaussian shaped velocity profile can be taken into respect. Stagnation point heat transfer results of several simulations are gathered in table 7.1 and figure 7.5.

TABLE 7.1 COMPARISON OF HEAT TRANSFER CALCULATED FROM 0 AND FROM

THE HEAT FLUXES

flat

parabolic

Gaussian

Rei = 2

umax-di V

583

849 993

1202 1414 1697 1980 2122 2263 2405

691 1037 1383

3

53.3

240 281 340 397 486 561 606 668 687

108 162 221

Y = 1 u max 3 di

2 1 .04

1 .70 1 .70 1 .70 1 .69 1 .72 1 .71 1 .72 1 .78 1 .72

1 .88 1 .88 -1 .92

6 i 0.661 di(-) 2 2 V

16.3

25.1 27.2 29.9 32.3 35.7 38.4 39.9 41 .9 42.5

23.8 29.2 34.1

Nui = 2

a di 2 A

16.6

24.0 26.0 28.9 32.9 35.3 39.6 40.4 41 .4 41 .8

21 .6 27.6 32.9

The third column in table 7.1 gives the values of g = (8v/3r)/r->-0 determined from the calculated flow field. Then in the fourth column the influence of the velocity profile on 3 can" be seen by the parameter y = umax/(diB). with the formula

152

Nu 0.5

SO

30

20

10 300

Nuos = 0.909 Re0°s5

0.5

0.5

flat profile parabolic Gaussian

_L _i_ 500 1000 2000 Re, 0.5

5000

Fig. 7.5. Stagnation point heat transfer for laminar impinging jets.

of Sibulkin (equation 2.32)

Nu = 0.763 L (-)2 Pr 0.4 (7.1 )

the stagnation point heat transfer can be predicted. The results are given in column 5. The Prandtl number was Pr = 0.71 and the characteristic length scale L = dj.. Finally, in column

2 6 are given the computed values of the Nusselt number calculated from the computed temperature gradients at the wall obtained with the full simulation model. Comparing columns 5 and 6 the conclusion is that the Sibulkin theory can be applied to laminar impinging jets. The shape of the velocity profile influences g in the same manner as it influences the tempera­ture gradient. From the results in table 7.1 can also be concluded that from the three velocity profiles considered the Gaussian shaped profile gives the highest stagnation point heat transfer. Averaged values of y together with equation 7.1 give the following relationships:

153

- flat velocity profile:

Y = 1.04 Nui = 0.675 Rej.0-5 (7.2) 2 2

- parabolic velocity profile:

Y = 1.72 Nui = 0.868 Rej.0"5 (7.3) 5 2

- Gaussian velocity profile:

y = 1.89 Nui = 0.909 Rej.0'5 (7.4) 3 2

The resulting stagnation point heat transfer found from 8 and from the temperature fields are compared once more in figure 7.5. The results from the Gaussian jet and the parabolic jet come very close due to the nearly similar profiles near the jet axis.

7.1.1. Comparison with literature data

The results of the present laminar calculations can be compared with the numerical results of Saad (1975) and the analytical results of Scholtz and Trass (1970). The comparison is made in the figures 7.6 and 7.7 between present results at H/d = 6, results from Saad at H/d = 8 and results of Scholtz and Trass at H/d = 1.0. In this region of nozzle to plate distances (1.0 < H/d < 8) Scholtz and Trass found only a slight influence of H/d on heat transfer, so this comparison can be made.

Figure 7.6 gives the results from our simulations at Re = 1 ,000 and the results from Saad at Re = 950 both for a parabolic velocity profile. The heat transfer results are scaled with /Re, since a /Re-dependency at the stagnation point has been found by Scholtz and Trass. Their results are also shown in figure 7.6. At the stagnation point our result agrees with the result of Scholtz and Trass, but deviates about 5% with Saad's result. The little hump in the heat transfer just next to the stagnation point has only been calculated by us. Next to this hump our calculations show a faster decrease of heat transfer at increasing r/d. In the wall jet region Saad's

154

1.4

1.0

0.8

0.6

0.4

0.2

0.5 1.0 1.5 2.0 2.5

r/d

Fig. 7.6. Radial heat transfer distribution for a laminar impinging jet with parabolic velo­city profile.

predictions are higher, however, up to r/d = 0.5 the difference is only about 5%.

The same conclusion can be drawn from the comparison of the results from the flat velocity profile in figure 7.7. Our results deviate from Saad's results less than 5% for r/d 5 1. The maximum in heat transfer found at r/d = 0.4 at Re = 600 is not found by Saad at Re= 950. He found more or less a constant heat transfer coefficient in the stagnation region. The results from Scholtz and Trass also point at a maximum outside the stagnation point. This also has been found experimentally by them (Scholtz and Trass, 1970).

The stagnation point heat transfer results can also be compared. Here Saad found Nu ~ Re 0 , 3 6 (950 < Re < 2,000) for a parabolic velocity profile. This deviates from what Scholtz and Trass found experimentally: Nu ~ Re ' . As can be seen from

i i I i Scholtz and Trass Saad (Re=950) present ca/c. [Re = 1000)

155

Nu -Me

0.6

0.S

0.4

0.3

0.2

0.1

0.5 1.0 1.5 2.0 2.5 r/d

Fig. 7.7. Radial heat transfer distribution for a laminar impinging jet. with a flat velocity-profile.

figure 7.5 our results come very close to a proportionality of Nu with Re0-5.

There are two possible reasons for the deviations of the results of simulations done by Saad. In the first place Saad used an upwind differencing scheme, while in our calculations a hybrid differencing scheme is used (see paragraph 4.1). In the stagnation region for the five gridlines nearest to the plate the cell Péclet numbers were smaller than 2, so in this region our code used the central differencing scheme. The use of upwind differencing in this region may lead to "false diffu­sion" (see Patankar, 1980), because here the flow changes direction. In the second place the computational grids were not the same. Saad used 21 x 24 gridlines, while we used 30 x 20 gridlines. His first gridline near the plate was at a distance of 0.03d. Within this distance we had four gridlines with the

I I

Scholtz and Trass Saad (Re=950)

— present calc. (Re=600)

156

line nearest to the plate at 0.0026d. This means that the temperature gradient at the wall could be determined with a much higher accuracy.

With the computational results discussed in this paragraph the computer code can be considered to be sufficiently validated.

7.2. The turbulent impinging j et

Next to the numerical calculations of the laminar impinging jets simulations have been performed for a turbulent impinging jet. Two different models of turbulence have been used: 1 ) The k-e model with modification for low-Reynolds number

flows as proposed by Chien (see paragraph 3.2.2). 2) The anisotropic k-e model as modified by us with the low-

Reynolds number model as a basis (see paragraph 2.2.4). All calculations have been performed at a Reynolds number of 6500 at H/d distances between 2 and 6. At H/d = 6 three different grids have been applied on calculations with the Chien model. At H/d = 6, 4 and 2 one of these grids has been used on simulations with the anisotropic model.

Table 7.2 gives a summary of the cases that will be discussed here. The number of gridlines that has been used is equal to 40 x 40 or 40 x 60. The parameter a giving the degree

TABLE 7.2 THE NUMERICAL CASES DISCUSSED

Case

C1 C2 C3 A1 A2

H/d

6 6 6 6 2

turbulence model

Chien Chien Chien anisotropic anisotropic

nx

40 40 40 40 40

nr

40 40 60 60 60

ax

9 6

7.04588 7.04588 7.04588

157

of nonuniformity of the grid in the axial direction via equation 4.18 varied between 6 and 9. The two grids of the cases C1 and C2 are given in figure 7.8.

Case C2 had much more gridlines close to the plate than case C1. The grid of C1 appeared to be too coarse to determine heat transfer to the plate accurately. Heat transfer to the wall has been calculated in two ways (with a linear or a quadratic enthalpy profile near the wall, see paragraph 4.5).

*-X

Fig. 7.8. The two grids of case C1 (a) and case C2 (b)

158

This gave results that differed about 40% for grid C1 . It is not clear which of the two methods gives the best results. Evidently the number of gridpoints for the determination of the temperature gradient at the wall was too small for this case. The grid of case C2 gave much better results in this respect: less than 1% difference in the two results of the Nusselt number calculations. For the cases C3, A1 and A2 this difference was always less than 5%.

7.2.1. Comparison of results on different grids

The numerical results of the four cases C1 , C2, C3 and A1 differ, which is shown in the following figures on the axial velocity decay (figure 7.9), the axial development óf kinetic energy of turbulent fluctuations in the stagnation region (figure 7.10), the maximum velocity, in the boundary layer and the kinetic energy of turbulent fluctuations at the same location (figure 7.11), and the radial Nusselt number distribu­tion (figure 7.12).

At first these four cases are compared in figures 7.9 and 7.10. The computational grids of C1 and C2 differ in this respect, that in the axial direction C1 has more gridlines in

12.0

10.0 u

(m/s) 8.0

6.0

4.0

2.0-

0.0

Fig. 7.9. The axial velocity of four cases.

159

3.0

5.0

Fig. 7.10. Kinetic energy of tur­bulent fluctuations on the axis in the stag­nation region.

Fig. 7,11. The maximum velocity in the boundary layer and the kinetic energy of turbulent fluctua­tions at the same location.

r/d

the free jet region and C2 more gridlines close to the wall. Because of the fact that the C1 grid is finer in the free jet region, the jet development should be simulated more accurately with this grid. Case C1 results into less decay of the axial velocity compared to case C2, accompanied by lower values of the kinetic energy of turbulent fluctuations. Case C3 with more

160

Fig. 7.12. Radial Nusselt number distributions.

0 1 2 3 r/d

gridlines in the radial direction gives even less velocity decay and still lower values of k. The anisotropic model finally results in the same axial velocity decay as case C3, but the kinetic energy of turbulent fluctuations is lower with a peak before stagnation closer to the plate.

Figure 7.11 shows a continuation of the already noted differences. Case C2 has the lowest maximum velocities in the boundary layer and the highest values of k. Cases A1 and C3 differ only slightly if it concerns the maximum velocity, but the values of k for case A1 are significantly lower.

The differences in the flow give rise to different heat transfer results, as can be seen from figure 7.12. The deviation of C1 from C2 and C3 in this figure is rather big. As already stated, C1 had too few gridlines near the plate to predict the temperature gradient accurately. The Nusselt number distributions of C2 and C3 are very close. The lower radial velocities of C2 are compensated by the higher turbulence level. The rather big difference between A1 and C3 must be entirely due to the lower level of turbulence calculated with the anisotropic model.

Nu 120-

100-

8 0 -

60-

40-

20 -

0 -

rC3

r*\ici

^ i \ V \^

i 1

161

From these comparisons it can be concluded that the calculated temperature gradient at the wall as well as the jet development before impingement are dependent on the applied grid. Case C1 certainly did not have enough gridlines close to the plate, while case C2 did not have enough gridlines in the region of jet development. A very fine grid will be necessary to reach a high accuracy in both regions. It is not proven that the grid of C3 and A1 is fine enough. Therefore, calculations with much finer grids will be necessary. For the simulations with H/d = 2 the jet development region is much smaller. It is supposed that the applied grid is fine enough for this situation.

7.2.2. Comparison of numerical with experimental results

In this paragraph the results of the simulations carried out with the anisotropic model for burner to plate distances H/d = 6 and H/d = 2 will be discussed and compared to results from measurements.

7.2.2.1. H/d = 6

The computational results presented in figures 7.9 and 7.10 are not in good agreement with the experimental results presented in paragraph 6.2.1. The axial velocity decay of the simulated jet is much less than the decay measured on. the jets from the burners, and consequently the turbulence on the axis of the simulated jet is too low. The simulated jet has a longer potential core length and looks much more like a jet from a straight pipe with a well shaped nozzle (see figure 6.11). The boundary conditions in the burner exit correspond with the measurements on jets from the small burner: a flat velocity profile and a turbulence level of 7%. This, however, does not give rise to the same development of the jet in the first four or five diameters from the burner exit, where the plate does not yet have its influence on the flow. At x/d = 4 the velocity

162

on the axis of the simulated jet is still within 1% of the velocity in the burner exit. At x/d = 5 this decay is only 4%. The measurements on jets from the small burner gave a velocity decay on the axis of 4% at x/d = 4, and 10% at x/d = 5 (see figure 6.1 ). Apparently, the calculated jet does not mix with the surroundings in the same degree as the experimental jet. This cannot be due to the coarseness of the grid. As we have seen before, a finer grid resulted in less decay of the axial velocity.

There can be two reasons for the disagreement between simulations and experiments. At first, the specification of the profiles of u, k and z at the burner exit as flat profiles is not sufficient to be able to predict the jet development in the first few diameters. In this region the spread of a jet also depends on the boundary layer of the flow in the nozzle. According to studies performed by Yule (1978) and Strange and Crighton (1983) the development of free jets is determined by the presence at x/d = 1 of large scale axisymmetrically coherent vortical structures due to the instability of the laminar boundary layer. Within the first few diameters from the nozzle two vortex rings firstly pair into one before breaking up into several large scale eddies at the end of the potential core region. If the boundary layer in the nozzle is turbulent due to roughness of the nozzle inner wall, the coherent structures do not exist. The present model of turbulence cannot predict this behaviour of the flow.

A second reason for the disagreement between simulations and experiments may be the height of the Reynolds number. The convection, diffusion, production and dissipation of kinetic energy of turbulent fluctuations can only be well predicted if the Reynolds number is high enough. The low Reynolds number modification of the k/e model takes into account the behaviour of the turbulence in the low Reynolds number region close to the wall. The modifications from Chien are only activated in this region. It is not suitable, however, to predict the low

163

Reynolds number region at the boundary of the jet, where turbulence is being generated. The Reynolds number of the simulated jet may not be high enough to be sure that the k-e model leads to good predictions in this region.

The maximum velocity in the boundary layer along the plate together with the rms-value of the fluctuations of this velocity are compared to the experimental results at Re = 6400 in figure 7.13. It is clear that due to the less developed simulated jet the averaged velocities along the plate are higher and the fluctuating velocities are lower than measured. This is consistent with earlier experiences.

r/d

Fig. 7.13. Comparison of the maximum velocity in the boundary layer and the rms-value of its fluctuating component with results from measurements (H/d = 6 ) .

The final result of the simulations, being the heat transfer distribution to the plate, is compared to the results from measurements in figure 7.14. Although the anisotropic model results in the lowest heat transfer (see figure 7.12),

164

Nu 120

80

40

O 0 1 2 3

r/d

Fig. 7.14. Comparison of experimental and numerical results of radial Nusselt number distribu­tion at H/d = 6 .

this is still too high compared to the measurements. A maximum in the heat transfer is calculated at about r/d = 0.4. This maximum is much less extreme than the maximum calculated with the Chien model (C2 and C3 in figure 7.12). A similar maximum has been found in our measurements (see figure 6.45 and 6.46) and in the measurements of Gardon and Akfirat (see figure 2.9) at r/d = 0.5, but in those cases at lower H/d-values. The measurements of Gardon and Cobonpue (see figure 2.10) show that at a Reynolds number of 28,000 this maximum is not present anymore at H/d = 6. Again, this comparison and especially the presence of the maximum in the simulated heat transfer distribution, points at a less developed jet in our simulations.

7.2.2.2. H/d = 2

Results of simulations for a burner to plate spacing of H/d = 2 have only been obtained with the anisotropic model. At this low

I i simulations Re = 6500

experiments Re = 6300

1

165

x/d x/d

r/d r/d

Fig. 7.15. Contour lines of the turbulent parameters for H/d = 2.

166

H/d-value the development of the free jet has less influence. For this reason the simulations at H/d = 2 should be more accurate.

To get a qualitative idea of the turbulent parameters of the simulation the contours (lines of constant values) of k, e, g and Ueff are given in figure 7.15. The highest turbulence levels are reached in the mixing layer of the jet. Besides this, also in the wall jet region just before r/d = 2 all turbulent parameters have a maximum value. The kinetic energy of turbulent fluctuations reaches a third maximum around r/d = 0.5 very close to the plate. This maximum in k leads to a maximum in the turbulent viscosity at the same place, which can hardly be seen from the figure.

Further results of the calculated flow are presented in the form of radial velocity profiles and profiles of the rms-value of the fluctuating component of the radial velocity within the first 6 mm from the plate of impingement at six re­locations (figure 7.16). At four r/d-locations also the measured profiles for Re = 6400, already discussed in paragraph 6.2.3.1, are given. At r/d = 0 a peak in the rms-value of the fluctuating component is visible. This peak is in agreement with the theory of Sutera (1965), the measurements of Sadeh, Sutera and Maeder (1970) (see figure 2.6) and the recent measurements of Van Fossen and Simoneau (1987) on the stagnation line of turbulent flow around a cylinder.

At the next four locations r/d = 0.25, 0.5, 1.0 and 2.0 the results from simulations and experiments can be compared. The profiles of the averaged velocity agree within a few percent. Values of the turbulent velocity are predicted too low. This again has been caused by the fact that not enough turbulence is being generated in the mixing layer of the jet.

The excellent agreement between experiments and simula­tions can once again be noticed from figure 7.17. The maximum velocity in the boundary layer is predicted within a few percent. The predicted turbulence levels are too low for r/d >

167

y (mm)

4-J

2-

r/d=0.00

i — r

ï — i — i — i — i — i

i — i — i — i — i

\ \ \ r/d =1.00

■ \ \ \

1 '"V' ( i — i — i — i

i — i — i — i — i — i — i — i

0.25

1 1 1 1 1 1

I 1 1 1 1 1 1 1

S y

(mm) 4-\

2-

w I

A ' o.so

V

' | - ^ f i — r "i o 2 4 v e

I — I — I — I — I — I — I — l O O.S 1.0 1.5

v'Um/s)

\ 3.00

\

y i i i i i i—i

O 2 4 V 6

I 1 1 1 1 1 1 1 O 0.5 1.0 1.5

v'Um/s) V i V ï /==• } simulations r = , experiments

Fig. 7.16. Radial velocity profiles and profiles of the rms-value of its fluctuating component at H/d = 2 from simulations and experiments.

168

Fig. 7.17. Comparison of the maximum velocity in the boundary layer and the rms-value of its fluctuating component with results from measurements.

0.5. The significant maximum of the turbulence level found in the experiments at r/d = 2 is not found here, at least not at the location of the maximum velocity. Fig. 7.15 shows that this maximum is present in the simulations. From this figure together with figure 7.16 it can be concluded that this maximum lies somewhat further from the plate and is lower than in the case of the experiments. In general, the experiments and simulations of the flow field of the impinging jet at H/d = 2 agree excellent. The results affirm, that the disagreement found for H/d = 6 is due to the erroneous predictions of the jet development.

The heat transfer results of the simulated jet at H/d = 2 fit the experimental results to a lesser extent, as can be seen from figure 7.18. The simulations give a maximum in the radial heat transfer distribution, which was not measured for the jet

169

I

Nu

120

80

40

O 0 1 2 3

r/d

Fig. 7.18. Comparison of experimental and numerical results of the radial Nusselt number.

with Re = 6300 from the small burner. Our measurements on jets from the large burner and measurements performed by Gardon and Akfirat also show a maximum heat transfer at r/d = 0.5. These maxima, however, are less pronounced. They were attributed to a minimum' in the boundary layer thickness. The present calculations do not show a minimum . in the boundary layer thickness, but the maximum coincides with a maximum value of the effective viscosity very close to the plate at this value of r/d. .

Compared to the measurements the calculated heat transfer in the wall jet region is too low. This is most probably due to the lower turbulence predicted by .the simulations. The deviations between results from simulations and from experi­ments are mainly differences in turbulence or caused by these differences in turbulence. In case of a burner to plate distance H/d = 2 this difference in turbulence does not lead to different averaged velocities. Thanks to the fact that the

i i

simulations Re = 6500

- - - - - experiments Re = 6300

Gardon and Akfirat Re=7000

1 L

170

averaged velocities to a large extent determine the heat transfer, and that these averaged velocities are predicted well, the prediction of the heat transfer is in reasonable agreement with the measurements.

7.2.2.3. The stagnation point heat transfer

Like for the laminar jet, the stagnation point heat transfer from the impinging turbulent jet calculated from the temperature gradient at the wall can be compared to the heat transfer calculated by means of the radial velocity gradient 6 (via equation 7.1). This comparison has been made in table 7.3 for the cases discussed before. Only case C1 has been dis­regarded because of the inaccurate result of its heat transfer prediction. The values of 8 for the calculation of Nug have been determined from (8v m a x/3r) r = 0.

TABLE 7.3 COMPARISON OF Nu AND Nug

Case

A2 Exp H/d = 2 Exp H/d = 6 C2 C3 A1

Nu

54.5 56.4 59.0 90.6 81 .3 80.4

Nug

54.1 55.6 51 .2 63.8 63.3 66.1

For an impinging jet with a flat velocity profile it can be expected that the value of 8 will be equal to uQ/d. The jet with a Reynolds number of 6500 would give 8 = uQ/d = 526 re­sulting in a Nusselt number of Nug = 53.6. This agrees with the calculations done for H/d = 2 (case A2) . The temperature gradient at the wall leads to Nu = 54.5 and the velocity

171

gradient 3 to Nuo =54.1. At this short burner to plate distance the impinging velocity profile is nearly flat and the enhancement of heat transfer by turbulence is negligible as was confirmed by the measurements. This has been predicted well by the numerical model.

At H/d = 6 the experiments resulted in Nun = 51.2. This experimental value can be explained as follows. The decrease of 3 due to the decayed axial velocity in the free jet at x/d = 6 is almost compensated by the increase of 3 due to the changed velocity profile. As concluded before, the velocity decay of the simulated jet is less than that of the experimental jet. The relatively high value of the simulated Nup must be due to the shape of the velocity profile, while the axial velocity has hardly decreased.

The enhancement due to turbulence at H/d = 6 in the experimental case is 15% and in the numerical case (A1 ) 22%. This is rather high for the simulations considering the predicted low level of turbulence in comparison to the experiments.

From this closer look at the stagnation point heat transfer it can be concluded that with 3 determined from the simulated flow field a first estimate of the heat transfer at the stagnation point can be made with Sibulkin's equation.

172

8. GENERAL DISCUSSION AND CONCLUSION

The results of our experiments and simulations have led to a number of conclusions on heat transfer from impinging jets and flame jets from rapid heating burners. In this chapter the main conclusions from our study will be summarized and discussed.

8.1. The flow structure

The isothermal jets from the tested rapid heating burners differ from the flow structure of jets issued by a long straight pipe or by a well defined nozzle. The jets from the two burners used differ also mutually. Compared to the well defined jets (from pipe or nozzle) the jets from the burners are much more turbulent. This is demonstrated by the high turbulence level in the nozzle exit and the axial velocity decay after a small potential core length. Of course there is a reason for this high turbulence level. It is aimed to have a good mixing of air and gas in the burner.

The flame jets from the burners show an even faster axial development than the isothermal jets. The axial velocity decays faster due to the axial temperature decay. However, if an effective diameter is introduced which compensates for the density difference between jet and environment, it has been noticed that the flame jet axial velocity decays more slowly, while its axial turbulence level increases faster, compared to the isothermal jet from the same burner. This can point at turbulence being generated by combustion, which does not effect the mean flow.

The radial velocity gradient at the stagnation point (6) determined from the static pressure distribution around the stagnation point is about equal to uQ/d for burner to plate distances up to 2d for the isothermal jet as well as for the flame jet. Despite of the high initial turbulence levels of the flame jets in particular, this agrees with well defined impinging jets. At H/d = 2 the large eddies generated in the

173

mixing layer of the isothermal jet give rise to a maximum in the turbulence level in the wall jet outside the stagnation region (1.5 < r/d < 2.0). 'Maybe due to the strong temperature decay in the wall jet this has not been found for the impinging flame jet.

8.2. Heat transfer

Despite of the differences in flow structure of the jets from the burners, the heat transfer from impinging isothermal jets and flame jets from the rapid heating tunnel burners as used in our study agree well with results from other investigators on heat transfer from impinging jets. The heat transfer from the impinging isothermal jet and the impinging flame jet at the stagnation point can be described by an equation from Sibulkin for laminar heat transfer at the stagnation point of a body of revolution:

Nu = 0.763 d (-)0-5 Pr0-4 v

with the radial velocity gradient 3 determined from static pressure measurements in the vicinity of the stagnation point. For small H/d-values (H/d s 3) this equation predicts heat transfer well, despite of the high levels of turbulence (Tu/Re* up to 28) that exist in the free jets at the same axial distances x/d s 3. For higher values of H/d the Sibulkin equation underestimates heat transfer and a turbulence enhance­ment factor (y) has been defined by us. For the jets considered by us this factor is between 1 and 1 .5 for the region 4 < Tu/Re* < 32 (6 é H/d S 12). The impinging flame jet results agree quantitatively with the results of the impinging iso­thermal jets if the fluid properties are defined at the temperature belonging to the averaged enthalpy over the boun­dary layer. Then the stagnation point heat transfer for iso­thermal and flame jets can be described with the same (3 and y.

The radial heat transfer distributions of the impinging

174

isothermal jets agree to a certain extent with results on impinging jets from well defined nozzles reported in the literature. The higher levels of turbulence of the jets from the burners cause less pronounced peaks or humps in the radial heat transfer distributions.

The radial heat transfer coefficient distributions of the impinging flame jets compare well with the distributions of the impinging isothermal jets. Of course, the distributions of heat flux densities to the plate differ very much. The reason for this is, that for a flame jet impinging on a watercooled plate the heat flux density distribution follows both the heat transfer coefficient and the wall jet temperature distribution. Since the temperatures in the wall jet decrease very fast with r/d, the heat flux density is very nonuniform with a maximum at the stagnation point and a sharp decrease with increasing r/d. For smaller temperature differences between the flame jet and the surface of the object to be heated this nonuniformity will be less. In a furnace with recirculating hot gasses the temperature decay in the wall jet, and herewith the heat flux decay, will also be less pronounced than in our study. However, especially at the beginning of a heating process one has to be aware of this nonuniform heat flux density distribution.

8.3. The simulated laminar impinging jet

Results of simulations of laminar impinging jets have given insight into the dependency of the stagnation point heat transfer on the impinging velocity profile. If Nu and Re are defined on dj_, being the half width diameter, and on the

2 maximum velocity on the axis, the jet with the Gaussian velocity profile results in the highest Nusselt number at the stagnation point, compared to jets with a parabolic and a flat profile.

An excellent agreement between heat transfer calculated with Sibulkin's theory and heat transfer determined from the temperature gradient at the wall is noted.

175

8.4. The simulated turbulent impinging jet

Simulations of the turbulent impinging jet at a Reynolds number of 6500 and a burner to plate distance H/d = 6 were not accurate enough in the prediction of the jet development. The turbulence level of 7% in the burner outlet in the simulations did not lead to the same amount of mixing with the surroundings as in the experiments. This is most probably due to the turbulence model used. This with a parameter for anisotropic turbulence modified k-e model is not capable to predict the large scale turbulent structures that originate in the mixing layer of a free jet. The averaged velocities are simulated too high; the fluctuating velocities are simulated too low. As a result the simulated heat transfer is too high.

The simulations of an impinging jet at H/d = 2 agree much better with experiments. The averaged velocities agree within a few percent, the simulated fluctuating velocities are again too low. The heat transfer results from experiments and simulations deviated little, because of the differences in turbulence.

Future studies should concentrate on a better modelling of the turbulence of a jet in its first five or six nozzle diameters from the burner outlet.,— so that the present simula­tions can be extended to impinging flame jets.

8.5. Main conclusions

The most important conclusions of our study are:

- Very simple static pressure measurements can give a first estimate of the stagnation point heat transfer of an impinging jet; also if this jet originates from a burner. Together with temperature measurements in the free flame jet the static pressure measurements can give a first estimate of the stagnation point heat transfer from an impinging flame jet.

- Heat transfer . from an impinging isothermal jet and from an impinging flame jet can both be described with the radial

176

velocity gradient 8 and the turbulence enhancement factor y. - Heat transfer at the stagnation point of an impinging isothermal or flame jet is'only enhanced by turbulence if the impinged surface is placed outside of the potential core of the free jet. This holds even for very turbulent flame jets with an initial turbulence level of 12%.

- The heat flux density distributions of the impinging flame jets are very nonuniform for the high temperature differences used in our study between the surface of the plate and the impinging flame jet.

- The numerical simulations of an isothermal jet impinging on a flat plate at a nozzle to plate distance of H/d = 2 done with the k-e model of turbulence modified with a parameter for anisotropic turbulence agree well with the experimental results.

177

LIST OF PRINCIPLE SYMBOLS

a R ax

thermal diffusivity (m3/s velocity gradient in stagnation point (1/s velocity gradient in stagnation point (1/s

C specific heat at constant pressure (J/kg.K d diameter of nozzle (m

diameter of body of revolution (m de equivalent diameter (m D diffusion coefficient (m3/s D diameter of the plate (in eq. 2.5 and 2.60) (m Fr Froude number (-g parameter for anisotropy of turbulence (m2/sa g acceleration of the gravity (in eq. 2.6) (m/s3 G momentum flux (kg.m/sa h averaged enthalpy (J/kg h' fluctuating enthalpy (J/kg h momentary enthalpy (J/kg H nozzle to plate distance (m J flux (momentum or heat) k kinetic energy of turbulent fluctuations (m2/sa k mass transfer coefficient (m/s Lx macroscale of turbulence ■ ; (m nr number of gridpoints in radial direction nx number of gridpoints in axial direction Nu Nusselt number ( p averaged pressure (Pa p' fluctuating pressure (Pa p momentary pressure (Pa Pr Prandtl number (v/a) (-qw" heat flux density to the plate (W/ma Re Reynolds number (u d/v) (-Re turbulent Reynolds number (k3/ve) (-s.^ fluctuating rate of strain (1/s Sc Schmidt number (v/D) (r

17

Sh Sherwood number (kcd/D) SJJ mean rate of strain Si general source term t time T temperature T* temperature at averaged enthalpy Tu turbulence grade (/ u'a/u) u averaged axial velocity u' fluctuating axial velocity u momentary axial velocity v averaged radial velocity v' fluctuating radial velocity v momentary radial velocity Vg' fluctuating tangential velocity V > averaged radial velocity just outside the

boundary layer x axial distance y distance from the plate z distance along the surface from the stagnation

point (m)

a heat transfer coefficient (W/m2.K) ax parameter for nonlinearity of the grid 3 radial velocity gradient at stagnation point (1/s) Y turbulence enhancement factor (-) r e£j effective diffusivity öjj Kronecker delta 6X distance between two gridpoints in axial

direction (m)-e dissipation of kinetic energy of turbulent

fluctuations (ma/s3) X thermal conductivity (W/mk) u dynamic viscosity (Pa.s) Ueff effective viscosity (u + ufc) (Pa.s) ut turbulent viscosity (Pa.s)

( - ) ( 1 / s )

( s ) (K) (K)

( - ) ( m / s ) ( m / s ) ( m / s ) ( m / s ) ( m / s ) ( m / s ) ( m / s )

( m / s ) (m) (m)

180

v kinematic viscosity (m2/s) p density (kg/m3) a n -i laminar Prandtl number (-) a^ ^ turbulent Prandtl number (-) T stress tensor <t> general parameter

subscripts: c referring to the free jet value at the position

of the plate max referring to a maximum o referring to the nozzle exit r referring to the radial direction S referring to the surroundings x referring to the axial direction 3 referring to the radial position on which the

velocity has reached half of its maximum °° referring to the undisturbed flow

181

REFERENCES Abramovich, G.N., The Theory of Turbulent Jets. MIT Press,

Massachusetts (1963). Agarwal, R.K. and Bower, W.W., Navier/Stokes computations of

turbulent compressible two-dimensional impinging jet field. AIAA J. , 20., 577 (1 982) .

Ali Khan, M.M., Kasagi, N., Hirata, M. and Nishiwaki, N., Proc. 7th Int. Heat Transfer Conf. , Munchen, FC63, 363-368 (1982) .

Amano, R.S. and Jensen, M.K., A numerical and experimental investigation of turbulent heat transport of an axisymmetric jet impinging on a flat plate. ASME-Report, 82-WA/HT-55 (1982).

Bakke, P., An experimental investigation of a wall jet. J. Fluid Mech., 2, 467-472 (1957).

Basisgegevens over Gronings Aardgas. N.V. Nederlandse Gasunie (1968).

Beer, J.N. and Chigier, N.A., Impinging jet flames. Combustion and Flame, 1_2, 575-586 (1968).

Bird, B.R., Stewart, W.E. and Lightfoot, E.N., Transport Phenomena, John Wiley & Sons, New York (1960).

Blasius, H., Grenzschichten in Flüssigkeiten mit kleinen Reibung, Z. Math. u. Phys., 5_6_, 1-37 (1908).

Bower, W.W., Kotansky, S.R. and Hoffman, G.H., Computations and measurements of two-dimensional turbulent jet impingement flow fields. Proc. Symp. Turbulent Shear Flows, Pennsylvania, J_, 3.1-3.8 (1977).

Buhr, E., Haupt, R. and Kremer, H., Konvektiver Warmeübergang bei Verbrennung in der Grenzschicht. Westdeutscher Verlag GmbH, Opladen (1976).

Chen, C.J. and Rodi, W., On decay of vertical buoyant jets in uniform environment. 6th Int. Heat and Mass Transfer Conf., Toronto, MC 1 7, 97-102 (1978).

Chia, C.J., Giralt, F. and Olev Trass, Mass transfer in axisymmetric turbulent impinging jets. Ind. Eng. Chem. Fundam., 1_6, 28-35 (1977).

Chien, K.Y., Predictions of channel and boundary-layer flows with a low-Reynolds-number two-equation model of turbulence, AIAA paper 80-134 (1980).

Chieng, C.C. and Launder, B.E., On the calculation of turbulent heat transport downstream of an abrupt pipe expansion. Num. Heat Transfer, 2, 189-207 (1980).

Conolly, R. and Davies, R.M., A study of convective heat transfer from flames. Int. J. Heat Mass Transf., _1_5' 2155-2172 (1972).

Cornelissen, M.C.M., The 2900 System, an Emulation of the HP21MX/E, Graduation Report, Technical University Delft (1980) .

Dimotakis, P.E., Single scattering particle laser doppler measurements of turbulence. Agard Conf., St. Louis, France (1976) .

Donaldson, C.duP., Snedeker, R.S. and Margolis, P., A study of free jet impingement. Part 1. Mean properties of free and

183

impinging jets. J. Fluid Mech., 4_5, part 2, 281-319 (1971a). Donaldson, C.duP., Snedeker, R.S. and Margolis, P., A study of

free jet impingement. Part 2. Free jet turbulent structure and impingement heat transfer. J. Fluid Mech., J_5_, part 3, 477-512 (1971b).

Doornmaal, 'J.P. van, and Raithly, G.D., Enhancement of the SIMPLE method for predicting incompressible fluid flows. Num. Heat Transfer, 1_, 147-163 (1984).

Dosdogru, G.A., Dissertation TH Darmstadt, Synopsis in Chem.-Ing.-Technik, 4±, 1972, 1340 (1974).

Durrani, T.S. and Greated, C.A., Laser Systems in Flow Measure­ments (1977).

Durst, F. , Melling, A. and Whitelaw, J.H., Principles and Practice of Laser-Doppler Anemometry. Academic Press, London (1976).

Dyban, E.P., Epick, E.Ya., Some heat transfer features in the air flow of intensified turbulence. 4th Int. Heat Transfer Conf., Paris, FC5.7 (1970).

Eckert, E., Die Berechnung des Warmeübergangs in der laminaren Grenzschicht. VDI-Forschungsheft, 416 (1942).

Fay, J.A. and Ridell, F.R., Theory of stagnation point heat transfer in dissociated air. J. Aero. Sci., 2S_, 73/86

: (1958). Fergason, J.L., Liquid crystals in non-destructive testing.

Appl.. Optics, 1_, 1729 (1968). Fossen, G.J. van and Simoneau, R.J., A study of the -relationship between free-stream turbulence and stagnation ^region heat transfer. J. Heat Transf., 109, 10-15 (1987).

Frössling, N. , Verdunstung, Warmeiibertragung und Geschwindig-keitsverteilung bei zweidimensionaler und rotations-symmetrischer laminarer Grenzschichtstromung. Lunds. Univ. Arsskr., N.F. Ard. 2, 35, nr. 4 (1940).

Galloway, T.R., Enhancement of stagnation flow heat and mass transfer through interactions of free stream turbulence. AIChE J., 1_9, 608-617 (1973).

Gardon,. R., A transducer for the measurement of heat flow rate. J. Heat Transf., 8j2, 396-398 (1960).

Gardon, R. and Akfirat, J.C., The role of turbulence in determining the heat-transfer characteristics of impinging jets. Int. J. Heat Mass Transf., §_, 1261-1271 (1965).

Gardon, R. and Cobonpue, J., Heat transfer between a flat plate and jets of air impinging on it. Int. Developm. in Heat Transf., ASME, 454-460 (1962).

Gauntner, J.W., Livinggood, J.N.B. and Hrycak, P., Survey of literature on flow characteristics of a single turbulent jet impinging on a flat plate. NASA Rep. TND-5652 (1970).

Geus, A.C. de, Laser Doppler Snelheids- en Turbulentie-metingen in • een hete Gasvlam die een vlakke Plaat loodrecht aanstroomt. Graduation Report, Technical University Delft (1983).

Giralt, F., Ph.D. Thesis, University Toronto (1976). Giralt, F., Chia, C.J. and Olev Trass, Characterization of the

impingement region in an axisymmetric turbulent jet. Ind. Eng. Chem. Fundam., J_6_r 21-28 (1977).

184

Glauert, M.B., The wall jet. J. Fluid Mech., ]_, 625-643 (1956). Goertler, H., Berechnung von Aufgaben der freien Turbulenz auf

Grund eines neuen Nahrungsansatzes. Z. Angew. Math. Mech., 22., 240-254 (1942).

Gorla, R.S.R. and Nemeth, N., Effects of free stream turbulence and integral length scale on heat transfer from a circular cylinder in cross flow. 7th Heat Transfer Conf., Munchen, FC28, 153-158 (1978).

Gutmark, W. , Wolfshtein, M. and Wygnanski, I., The plate turbulent impinging jet. J. Fluid Mech., J3j , 737-756 (1978).

Hegge Zijhen, B.G. van der, Measurements of turbulence in a plane jet of air by the diffusion method and by the hot-wire method. Appl. Scientific Res., A7_, 293-312 (1958).

Heiningen, A.R.P. van, Mujumdar, A.S. and Douglas, W.J.M., Numerical prediction of the flow field and impingement heat transfer caused by a laminar slot jet. J. Heat Transf., 98, 654-658 (1976).

Heiningen, A.R.P. van, Heat Transfer under an impinging Slot Jet. M. Eng. Thesis, Chem. Eng. Dept, McGill University (1982) .

Hiemenz, K. , Die Grenzschichten einem in den gleichformigen Flüssigkeitsstrom eingetauchten geraden Kreiszylinder. Diss. Göttingen, Dingl. Polytechn. J., 326, 321-324, 344-348, 357-362, 372-376, 391-393, 407-410 (1911).

Hijikata, K., Yoshida, H.Y. and Mori, Y., Theoretical and experimental study of turbulence effects on heat transfer around the stagnation point of a cylinder. 7th Int. Heat Transfer Conf., Munchen, FC30 (1982).

Hinze, J.O., Turbulence. McGraw-Hill, New York (1975). .Hinze, J.0. and Hegge Zijnen, B.G. van der, Transfer of heat

and matter in the turbulent mixing zone of an axially symmetrical jet. J. Appl. Sci. Res., A1_, 435-461 (1949).

Hjelmfelt, A.T. and Mockros, L.F., Motion of discrete particles in a turbulent fluid. J. appl. Sci. Res., 1_6, 149 (1966).

Homann,; F., Der Einfluss grossen Zahigkeit bei der Stromung urn den Zylinder und urn die Kugel.. ZAMM 16, 153-164, and Forschg. Ing. Wes., 1_> 1 _ 1° (1936).

Horsley, M.E., Purvis, M.R.I, and Tariq, A.S., Convective heat transfer from, laminar and turbulent premixed flames. 7th Int. Heat Transfer Conf-., Munchen, FC70 (1982).

Howarth, L., On the calculation of steady flow in the boundary layer near the surface of a cylinder in a stream. ARC RM 1632 (1935).

Hrycak, P., Heat transfer from a row of impinging jets to concave cylindrical surfaces. Int. J. Heat Mass Transf., 24, 407-419 (1981).

Huang, G.C., Investigations of heat transfer coefficients for air flow through round jets impinging normal to a heat transfer surface. J. Heat Transf., 8J5, 237-245 (1963).

Hunt, J.C.R., A theory of turbulent flows round two-dimensional bluff bodies. J. Fluid Mech., 6±, 625-706 (1973).

Jeschar,__ R. and PÖtke, W. , Modellversuche iiber . den Warmeubergang zwischen' einem isothermen Strahl und einer

,.-- ebenen Plat.te. VDI-Berichte nr. 146, 129-136 (1970).

185

Kapur, D.N. and Macleod, N. , The determination of local mass-transfer coefficients by holographic interferometry - I. Int. J. Heat Mass Transf., 1_7, 1151-1162 (1974).

Kataoka, K. and Mizushina, T., Local enhancement of the rate of heat transfer in an impinging round jet by free stream turbulence. 5th Int. Heat Transfer Conf., Tokyo, FC8.3 (1974).

Kataoka, K., Suguro, M., Degawa, H., Maruo, K. and Mihata,' I., The effect of surface renewal due to large-scale eddies on jet impingement heat transfer. Int. J. Heat Mass Transf., 20, 559-567 (1987).

Katinas, V.I., Zhyugzda, I.I., Zhukauskas, A.A., Shvegzhda, S.A., The effect of the turbulence of an approaching stream of viscous fluid on local heat transfer from a circular cylinder. Int. Chem. Eng., 16, 283-293 (1976).

Kays, W.M., Convective Heat and Mass Transfer. McGraw-Hill, New York (1966).

Kent, J.H., A noncatalictic coating for platinum-rhodium thermocouples. Combustion and Flame, J_4» 279-282 (1970).

Kestin, J. and Maeder, P.F., Influence of turbulence on the transfer of heat from cylinders. NASA TN 4018 (1957).

Kestin, J., Maeder, P.F. and Sogin, H.H., The influence of turbulence on the transfer of heat to cylinders near the stagnation point. Z. Angew. Math. Phys., J_2, 115.-131 (1961).

Kestin, J., Maeder, P.F. and Wang, H.E., Influence of turbulence on the transfer of heat from plates with and without pressure gradient. Int. J. Heat Mass Transf., 2» 133-154 (1961 ).

Kestin, J. and Wood, R.T., The influence of turbulence on mass transfer from cylinders. J. Heat Transf., Trans. ASME, ser. C. , 92, 321-327 (1971 ) .

Kezios, S.P. Heat Transfer in the Flow of a cylindrical Air Jet normal to an infinite Plane. Ph.D. Thesis, Illinois Inst. of Technology (1956).

Kilham, J.K. and Purvis, M.R.I., Heat transfer from hydro­carbon-oxygen flames. Combustion and Flame, J_6, 47-54 (1971 ) .

Kilham, J.K. and Purvis, M.R.I., Heat transfer from normally impinging flames. Combustion Science and Technology, 2.8, 81-90 (1978).

Kottke, V., Blenke, H. and Schmidt, K.G., Messung und Berechnung des ortlichen und mittleren Stoffubergangs an stumpf angestromten Kreisscheiben bei unterschiedlicher Turbulenz. Warme und Stof f übertragung, 1_0, 89-105 (1977).

Launder, B.E., Heat and mass transport. Chapter 6 in Turbulence, ed. P. Bradshaw. Topics in Appl. Phys., vol. 12, Springer Verlag, New York (1976).

Launder, B.E. and Spalding, D.B., Mathematical Models of Turbulence. Academic Press, London (1972).

Lees, L. , Jet Propulsion, 2&-> 4 (1956). Lenze, B., Bestimmung der Geschwindigkeits- und Konzentrations-

profile im Kern- und Übergangsbereiche von Freistrahlen. Chemie-Ing.-Technik MS 545/77 (1977).

Lin, S.H. and Sparrow, E.M., Radiant interchange among curved

186

specularly reflecting surfaces - Applications to cylindrical and conical cavities. J. Heat Transf., BJ_, 299-307 (1965).

Lowery, G.W. and Vachon, R.I., The effect of turbulence on heat transfer from heated cylinders. Int. J. Heat Mass Transf., 18, 1229-1242 (1975).

Masliyah, J.H. and Nguyen, T.T., Mass transfer due to an impinging slot jet. Int. J. Heat Mass Transf., 2^, 237-244 (1979).

McAdams, W.H., Heat Transmission. McGraw-Hill, New York (1954). Merk, H.J. , Rapid calculations for boundary-layer transfer

using wedge solutions and asymptotic expansions. J. Fluid Mech., 5, 460-480 (1958).

Milson A. and Chigier, N.A., Studies of methane and methane-air flames impinging on a cold plate. Combustion and Flame, 21, 295-305 (1973).

Nakatogawa, T., Nishiwaki, N. , Hirata, M. and Torii, K., Heat transfer of round turbulent jet impinging normally on a flat plate. 4th Int. Heat Transfer Conf., Paris, FC5.2 (1970).

Miyazaki, H. and Sparrow, E.M., Potential flow solution for crossflow impingement of a slot jet on a circular cylinder. J. Fluids Eng. Trans of ASME, 249-255 (1976).

Newman, L.B., Sparrow, E.M. and Eckert, E.R.G., J. Heat Transf., 9±, 7-16 (1972).

Noat, D. , Shavit, A. and Wolfshtein, M. , Interactions between components of the turbulent velocity correlation tensor. Israel J. Techn. , 8_, 259 (1970).

Ouden, C. den, and Hoogendoorn, C.J., Local convective-heat-transfer coefficients for jets impinging on a plate; experiments using a liquid crystal technique. 5th Int. Heat Transfer Conf., Tokyo, 5_, 293-297 (1974).

Pamadi, B.N. and Below, I.A., A note on the heat transfer characteristics of circular impinging jet. Int. J. Heat Mass Transf., 2^, 783-787 (1980).

Patankar, S.V., Numerical Heat Transfer and Fluid Flow. McGraw-Hill, New York (1980).

Patankar, S.V. and Spalding, D.B., A calculation procedure for heat, mass and momentum transfer in three dimensional parabolic flows. Int. J. Heat Mass Transf., 1_5, 1787-1806 (1972).

Patel, V.C., Rodi, W. and Scheuerer, G., Evaluation of turbulence models for near wall and low-Reynolds number flows. Proc. 3rd Symp. Turbulent Shear Flows, Davis, Cal., 1.1-1.8 (1981 ) .

Perry, K.P., Proc. Inst. Mech. Engin., London, 168, N. 30, 775 (1954).

Pohlhausen, K., Z. angew. Math. Mech., 1_, 252 (1.921). Poreh, M., Tsuei, Y.G., Cermak, J.E., Investigation of a

turbulent radial wall jet. J. Appl. Mech., 457-462 (1967). Rajaratnam, N., Turbulent Jets'. Elsevier Scient. Publ. Comp. ,

Amsterdam (1976). . Rao, V.V. and Olev Trass, Mass transfer from a flat surface to

an impinging turbulent jet. The Canad. J. Chem. Eng., 42, 95-99 (1964).

Reichardt, H., Gesetzmassigkeiten der freien Turbulenz. VDI

187

Forschungsheft 414 (1942). Reynolds, W.C., Computations of turbulent flows-state-of-the-

art. Stanford University, Dept. Mech. Eng., Report MD-27 (1970).

Rodi, W. , Turbulence models and their applications in hydraulics. Presented by the IAHR Section on Fundamentals of Division II: Experimental and Mathematical Fluid Dynamics (1980).

Rohsenow, W.M., Hartnett, J.P. and Ganic, E.N., Handbook of Heat Transfer Fundamentals^ McGraw-Hill, New York (1985).

Rotta, J.C., Turbulente Strömungen. B.G. Teubner, Stuttgart (1972) .

Saad, N.R., Simulation of Flow and Heat Transfer under a Laminar impinging round Jet. M. Eng. Thesis, Chem. Eng. Dept., McGill University (1975).

Saad, N.R., Douglas, W.J.M. and Mujumdar, A.S., Prediction of heat transfer under an axisymmetric laminar impinging jet. Ind. Eng. Chem. Fundam., 16, 148-154 (1977).

Sadeh, W.Z., Sutera, S.P. and Maeder, P.F., An investigation of vorticity amplification in stagnation flow. Z. angew. Math. Phys. , 21, 717-742 (1970).

Schlichting, H. , Boundary Layer Theory. McGraw-Hill, New York (1968).

Schlünder, E.U. and Gnielinski, V., Warme- und Stoffubertragung zwischen gut und aufprallender Dusenstrahl. Chemie-Ing.-Techn., 39_, 578-584 (1967).

Schlünder, E.U. , Krotsch, P. and Hennecke, Fr.W., Gesetzmassig-keiten der Warme- und Stoffubertragung bei der Prallstromung aus Rund- und Schlitzdusen. Chemie-Ing.-Techn. , A2_, 333-338 (1970).

Schmidt, E., Schuring, W. and Sellschopp, Techn. Mech. Thermo-Dynam., Berlin, 1, 53 (1930).

Scholtz, M.T., Dissertation, University of Toronto (1965). Scholtz, M.T. and Olev Trass, Mass transfer in the laminar

radial wall jet. AIChE J., £, 548-554 (1963). Scholtz, M.T. and Olev Trass, Mass transfer in a nonuniform

impinging jet. AIChE J., 1_6_, 82-96 (1970). Schrader, H., Trocknung feuchter Oberflachen mittels Warmluft-

strahlen. VDI Forschungsheft, 484, Ausgabe B, Band 27 (1961 ) .

Shih, T.M., Numerical Heat Transfer. Series in Computational Methods in Mechanics and Thermal Sciences, Hemisphere Publ. Corp., Washington (1984).

Shir, C.C., A preliminary numerical study of atmospheric. turbulent flow in the idealized planetary boundary layer. J. Atmos. Sci., 10 , 1327 (1973).

Sibulkin, M., Heat transfer near the forward stagnation point of a body of revolution. J. Aeron. Sci., J_9, 570-571 (1952).

Siegel, R. and Howell, J.R., Thermal Radiation Heat Transfer. McGraw-Hill, New York (1981).

Sikmanovic, S., Oka, S. and Koncar-Djurdjevic, S. , Influence of the structure of turbulent flow on heat transfer from a single cylinder in cross flow. 5th Int. Heat Transfer Conf., Tokyo, FC8.6, 320-324 (1974).

188

Smirnov, V.A., Verevochkin, G.E. and Brdlick, P.M., Heat transfer between a jet and a held plate normal to flow. Int. J. Heat Mass Transf., 2_, 1-7 (1961).

Smith, M.C. and Kuethe, A.M., Effects of turbulence on laminar skin friction and heat transfer. Phys. Fluids, 9_, 2337-2344 (1966).

Sparrow, E.M. and Lee, L. , Analysis of flow field and impingement heat/mass transfer due to a nonuniform slot jet. J. Heat Transf., 9T_, 191-197 (1975).

Sparrow, E.M. and Wong, T.C., Impingement transfer coefficients due to initially laminar slot jets. Int. J. Heat Mass Transf., J_8, 597-605 (1975).

Strand, T., AIAA Paper nr. 64-424 (1964). Strange, P.J.R. and Crighton, D.G., Spinning modes on

axisymmetric jets.' Part I. J. Fluid Mech., 1 34, 231-245 (1983).

Strumillo, C. and Grabowski, S., The effect of free stream turbulence on the momentum, heat and mass transfer during flow around a sphere. Warme- und Stoffübertragung, JJ_, 277-282 (1978).

Subba Raju, K., Heat transfer in an impinging turbulent jet. Ind. Chem. Eng., 1_4, 13-17 (1972).

Sutera, S.P., Vorticity amplification in stagnation point flow and its effect on heat transfer. J. Fluid Mech., 21 /3, 513-534 (1965).

Sutera, S.P., Maeder, P.F. and Kestin, J., On the sensitivity of heat transfer in the stagnation point boundary layer to free-stream vorticity. J. Fluid Mech., 16/4, 497-520 (1963).

Tennekes, H. and Lumley, J.L., A first Course in Turbulence. MIT Press, Cambridge, Mass. (1972).

Thring, M.W. and Newby, M.P., Combustion length of enclosed turbulent jet flames. 4th Symp. on Combustion, Cambridge, Mass., 789-796 (1953) .

Tolmien, W., Calculation of turbulent expansion processes. NASA TM 1085 (1948).

Traci, R.M. and Wilcox, D.C., Freestream turbulence effects on stagnation point heat transfer. AIAA J., 1_3, 890-896 (1975).

Vallis, E.A., Patrick, M.A. and Wragg, A.A., Radial distribution of convective heat transfer coefficient between an axisymmetric turbulent jet and a flat plate held normal to the flow. 6th Int. Heat Transfer Conf., Toronto, vol. 5, 297-303 (1978).

Watrasiewicz, B.M. and Rudd, M.J., Laser Doppler Measurements. Butterworths, London (1976).

Wolfshtein, M., Some solutions of the plane-turbulent impinging jet. Trans, of ASME, J. Basic Eng., 20_, 5 7 7 (1969).

Yardi, N.R. and Sukhatme, S.P., Effects of turbulence intensity and integral length scale of a turbulent free stream on forced convection heat transfer from a circular cylinder in cross flow. 6th Int. Heat Transfer Conf., Toronto, FC29, 347-352 (1978).

Yokobori, S., Kasagi, N., Hirata, M., Characteristic behaviour of turbulence in the stagnation region of a two-dimensional submerged jet impinging normally on a flat plate. 1st Int.

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Symp. Turbulent Shear Flows, University Park, Pennsylvania, 3.17-3.25.(1977).

Yokobori, S., Kasagi, N., Hirata, M. and Nishiwaki, N., Role of large-scale eddy structure on enhancement of heat transfer in stagnation region of two-dimensional, submerged, impinging jet. 6th Int. Heat Transfer Conf., Toronto, FC (8b)-22, 305-310 (1978).

Yule, A.J., Large scale structure in the mixing layer of a round jet. J. Fluid Mech., 8J3, 413-432 (1978).

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SUMMARY

Impinging flame jets are used in the glass and steel industry for rapid heating purposes. Compared to a conventional radia­tion furnace, in a rapid heating furnace, which applies imping­ing flames, much higher heat flux densities can be obtained. The dominant heat transfer mechanism in a rapid heating furnace is convection. This study concentrates on the influence of turbulence on heat transfer and on the nonuniformity of the heat flux distribution to an object in such a furnace. In fact, we study the heat transfer from a premixed flame jet impinging perpendicularly on a flat plate. For this reason the flow structure and heat transfer of impinging flame jets as well as impinging isothermal jets from two rapid heating burners have been measured.

The separation distance between burner and plate in this study varied from 1 to 1 2 burner diameters. The Reynolds numbers of the examined isothermal jets from the burners were 3,300 é Re s 10,000. The Reynolds numbers of the flame jets, defined at the adiabatic flame temperature, were 1,700 S Re S 4,250.

Measurements of the flow field of free and impinging jets were performed with a laser Doppler anemometer. Heat flux den­sity distributions of isothermal jets impinging on a slightly heated plate were measured with a liquid crystal technique. Heat flux density distributions of the flame jets impinging on a watercooled plate were measured with a Gardon heat flux transducer.

Furthermore, static pressure measurements in the stagnation region were performed, in order to find the value of the radial velocity gradient just outside the boundary layer in the vicinity of the stagnation point. With this parameter a first estimate could be made of the heat transfer coefficient at the stagnation point of the impinging jets. It is shown that heat transfer from both isothermal and flame jets can be described in the same way with this velocity gradient and a turbulence

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enhancement factor. The results from flame jets agreed quantitatively with the results from isothermal jets if the fluid properties in the heat transfer correlation were taken at a temperature belonging to the averaged enthalpy of the boundary layer along the plate.

Radial heat transfer distributions of the impinging flame jets were very nonuniform, mainly due to the large temperature difference between the flame jet and the surface of the water-cooled plate.

Using a numerical model impinging isothermal jets were simulated. The two-dimensional Navier-Stokes equations, the continuity equation and the energy equation have been solved by the finite volume method. With simulations of laminar impinging jets the effect of the impinging velocity profile on heat transfer has been examined. Three different profiles (flat, parabolic and Gaussian) were used. The Gaussian profile re­sulted in the highest heat transfer at the stagnation point.

For simulations of a turbulent impinging jet (Re = 6500) the turbulence of the flow was taken into account by a low Reynolds number k-e model modified by us with a parameter for anisotropic turbulence. At H/d = 6 the results showed that the development of the free jet was not predicted well. The value of the radial velocity gradient, however, again yielded to a good first estimate of the heat transfer at the stagnation point.

At H/d = 2 the results from our simulations agreed well with the experimental results.

The most important conclusion from our study is, that from isothermal measurements and from measurements of the radial velocity gradient near the stagnation point a first approxima­tion of the heat transfer from impinging premixed flame jets can be made.

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SAMENVATTING

In de glas- en staalindustrie wordt veelal gebruik gemaakt van vlammen die loodrecht invallen op een te verhitten oppervlak. Vergeleken met een conventionele stralingsoven kunnen in een oven die gebruik maakt van deze gerichte vlammen, veel hogere warmtestroomdichtheden bereikt worden. In een dergelijke oven is convectie het belangrijkste warmteoverdrachtsmechanisme. Deze studie betreft de invloed van turbulentie op de warmte­overdracht en de niet-uniformiteit van de warmtestroomdicht­heden naar een object in een dergelijke oven. In feite bestuderen wij de warmteoverdracht naar een vlakke plaat die loodrecht wordt aangestroomd door een yoorgemengde vlam. Hier­toe zijn de eigenschappen van de stroming en de warmteover­dracht van loodrecht treffende vlammen, maar ook van loodrecht treffende isotherme stralen komende uit twee branders, experi­menteel bestudeerd.

De afstand van de brander tot de plaat varieerde in deze studie van 1 to 12 branderdiameters. De Reynolds getallen van de onderzochte isotherme stralen uit de branders waren 3.300 s Re é 10.000. De Reynolds getallen van de vlammen gedefinieerd bij de adiabatische vlamtemperatuur waren 1.700 S Re s 4.250.

Metingen van het stromingsveld van de vrije en de lood­recht treffende stralen werden met een laser Doppler snelheids­meter uitgevoerd. Warmtestroomdichtheden van isotherme stralen naar een weinig verwarmde plaat werden gemeten met een techniek gebaseerd op het meten van temperatuur met vloeibare kristal­len. Warmtestroomdichtheden van vlammen naar een loodrecht aan-gestroomde watergekoelde vlakke plaat werden gemeten met een Gardon warmtestroommeter.

Bovendien werden statische drukmetingen in het stuwpunts-gebied uitgevoerd, teneinde de waarde van de radiële snelheids­gradiënt in de nabijheid van het stuwpunt te vinden. Met deze parameter kon een eerste schatting worden gemaakt van de warmteoverdracht in het stuwpunt van loodrecht treffende stra­len. Het is aangetoond dat de warmteoverdracht van zowel iso-

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therme stralen als van vlammen op dezelfde manier beschreven kan worden met deze snelheidsgradiënt en een parameter die ver­hoging van de warmteoverdracht door turbulentie aangeeft. Kwan­titatieve overeenstemming werd gevonden, indien de stofeigen­schappen in de warmteoverdrachtscorrelaties gedefinieerd werden bij de temperatuur behorende bij de gemiddelde enthalpie in de grenslaag langs de plaat.

De radiele verdeling van de warmteoverdracht was sterk niet-uniform, hetgeen voornamelijk werd veroorzaakt door het grote temperatuurverschil tussen de vlam en het oppervlak van de watergekoelde plaat.

Met een numeriek model zijn loodrecht treffende isotherme stralen gesimuleerd. De tweedimensionale Navier-Stokes verge­lijkingen, de continuïteitsvergelijking en de energievergelij­king zijn opgelost met behulp van een eindige volumemethode. Met simulaties van laminaire loodrecht treffende stralen is de invloed van de vorm van het snelheidsprofiel op de warmteover­dracht bestudeerd. Er werden drie verschillende profielen ge­bruikt (vlak, parabolisch en Gaussisch). Het Gaussische profiel resulteerde in de hoogste warmteoverdracht in het stuwpunt.

Bij de simulatie van een loodrecht treffende turbulente straal (Re = 6500) werd gebruik gemaakt van een laag Reynolds-getal k-e model, dat door ons is gemodificeerd met een para­meter voor anisotrope turbulentie. Voor H/d = 6 wezen de resul­taten uit dat de ontwikkeling van de vrije straal minder goed werd voorspeld. Wel bleek weer dat de waarde van de radiele snelheidsgradient in het stuwpunt een eerste goede schatting gaf van de warmteoverdracht in het,stuwpunt.

Voor H/d = 2 kwamen de resultaten van de berekeningen goed overeen met de experimentele resultaten.

De belangrijkste conclusie die uit de resultaten van deze studie^ getrokken kan worden is, dat met behulp van isotherme metingen en metingen van de radiele snelheidsgradiënt in het stuwpunt de warmteoverdracht van loodrecht treffende voorge-mengde vlammen in eerste benadering bepaald kan worden.

1'94

CURRICULUM VITAE

13 juni 1951 geboren te Zoeterraeer

1957 - 1963 lagere school te Zoetermeer

1963 - 1967 St. Petrus Mulo te Zoetermeer

16 juni 1967 eindexamen Mulo-A en Mulo-B

1967 - 1970 St. Maartenscollege te Voorburg

12 juni 1970 eindexamen HBS-B

1970 - 1976 Technische Hogeschool Delft, afdeling Tech­nische Natuurkunde :

1974 - 1976 4e- en 5e-jaars werk in de subgroep Warmte­transport onder begeleiding van prof.ir. C.J. Hoogendoorn

15 juni 1976 doctoraal examen natuurkundig ingenieur; het afstudeeronderwerp was "Warmteover­drachtscoëfficiënten voor zeer viskeuze vloeistoffen in een statische menger"

augustus 1976 - wetenschappelijk medewerker in de subgroep Warmtetransport van de vakgroep Transport­verschijnselen van de faculteit Technische Natuurkunde aan de Technische Universiteit Delft

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NAWOORD

Dit proefschrift is tot stand gekomen met de hulp van veel mensen. Allen die een bijdrage geleverd hebben, wil ik hierbij mijn dank betuigen. Ik zal in dit nawoord volstaan met het noemen van diegenen zonder wie het resultaat in deze vorm niet voor u zou liggen.

Allereerst ben ik dank verschuldigd aan mijn promotor prof.ir. C.J. Hoogendoorn voor zijn directe begeleiding. Hij heeft mij in moeilijke momenten bijgestaan en gestimuleerd. Hij heeft mij de ruimte gegeven dit proefschrift te schrijven.

Dr.inz. Cz.O. Popiel van de Poznan Universiteit in Polen heeft gedurende het eerste jaar van mijn verblijf in de groep Warmtetransport met mij op dit onderwerp samengewerkt. Hij heeft mede richting gegeven aan dit onderzoek. Zijn bijdrage is van onschatbare waarde.

Voorts is er een aantal studenten geweest die in het kader van hun 4e- of 5e-jaars werk een deelonderzoek hebben gedaan. Dit waren Pieter Broerse, Gerard Burger, Henk Buys, Hans Dekker, Floris van Drunen, Jan Willem van Dijk, Aart de Geus, Dolf van Hattem, Albert van der Heiden en Ad Voets.

Een speciaal woord van dank gaat uit naar twee personen die bij de uitvoering van dit proefschrift een belangrijke rol hebben gespeeld: Bram de Knegt en Riny Purmer. Bram heeft de figuren verzorgd die zijn opgenomen in het proefschrift. Hij stond altijd voor mij klaar. Riny heeft het typewerk verzorgd. Zij heeft mij gewezen op vele fouten in het gebruik van de Engelse taal. De opmaak van dit proefschrift is haar werk ge­weest. De accuratesse van haar werk was verbluffend.

Een woord van dank gaat voorts uit naar die mensen van de algemene dienst van de faculteit Technische Natuurkunde, die hebben bijgedragen tot de totstandkoming van de experimentele opstellingen. De contacten zijn altijd zeer goed geweest.

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