Trapo SS14 IIc
8
p. II.3/1 Transport Phenomena - Gerhard Eder, Institute of Polymer Science, Linz University, Austria - SS2014 Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation For a big number of flow problems the energy dissipation is negligible as compared to the mechanical energy changing from one form into an other. In those cases the Bernoulli equations has the following simple form: II.3 : Flow with negligible energy dissipation The first, second and third expression on the left hand side are usually called: pressure head, static head and velocity head. 2 2 p v h g g r + + = constant along a streamline manometer tube for pressure measurement Pitot tube for velocity measurement
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Transcript of Trapo SS14 IIc
p. II.3/1
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
For a big number of flow problems the energy dissipation is negligible as compared to the mechanical energy changing from one form into an other. In those cases the Bernoulli equations has the following simple form:
II.3 : Flow with negligible energy dissipation
The first, second and third expression on the left hand side are usually called: pressure head, static head and velocity head.
2
2
p vh
g gr+ + = constant along a streamline
manometer tube for pressure measurement
Pitot tube for velocity measurement
p. II.3/2
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
II.3.a : Flow of a liquid through an orifice
The basis of the calculations are the mass balance and the Bernoulli equation with a cross section of the vessel being much larger than the area of the orifice. Thus the velocity in the vessel is negligibly small. Thus one has:
However, in practice it is more convenient to use the actual area of the orifice and to introduce a correction factor for both the coefficient of area contraction and the friction. i.e.:
1 1 1 0
2' ' ( )V exitA v A p pf
r= = -
with A1' being the smallest diameter of the jet stream as given in the next figure
for a few examples.
p. II.3/3
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
For sharp edged orifices (example (a) in the previous figure) on has C
f =1 and A
1'/A
1 = 0.62 for sufficiently
high velocities, leading to C = 0.62 .For rounded orifices (example (b) in the previous figure) one has C
f = 0.95-0.99 and A
1'/A
1 = 1 , leading
to C = 0.95-0.99 .If a diffusor is attached to the orifice with an opening angle smaller than 8° in order to avoid eddying, one gets A
1'/A
1 > 1 and C
f = 0.95-0.99.
11 1 0
1
'2( )V f
ACA p p C C
Af
r= - =with
The flow coefficient C is given in the right table for a few important geometries of orifices:
p. II.3/4
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
II.3.b : Flow over weirs
In this case flow over weirs as function of the height of the liquid above the upper edge of the weir is considered:
On the upstream side of the weir at height h, the static pressure is 0 0( )p g h hr+ -
with negligibly small velocity v. The pressure in the overflowing part is everywhere p
0 . For the velocity v
h of the streamline coming from height h one
has due to Bernoulli: 20 0 0 0
1( ) 2 ( )
2 h hp g h h p v v g h hr r+ - = + = -and thus
For the volumetric flow rate one has0 0 0
30 0
0 0 0
2( ) ( ) 2 2
3
h h h
V V hd h v h W dh W g h h dh W ghf f= = = - =ò ò òIn reality this result must be multiplied by a friction factor C = 0.62 . If the width W is much larger than h0, one gets the following practical formula:
300.59V W ghf =
p. II.3/5
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
A few other cases of interest are:
p. II.3/6
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
II.3.c : Flow metersFlow meters measure the flow rate of a fluid in a stream. One distinguishes between:A. Displacement meters, which divide the stream into known volume unitsB. Flow meters, which measure the mean velocity either dynamic (based on flow laws – examples: orifice plate, Venturi tube, rotameter, Pitot tube, ...), kinematic (mechanical system – example: blade wheel, water meter), or thermic (via temperature due to friction).
1 1 2 2
2 21 1 2 2
1 1
2 2
m v A v A
p v p v
f r r
r r
= =
+ = +
A. Venturi tube:
The shape is chosen to avoid eddy formation. The mass balance and Bernoulli equation give:
12 2
22 1 2
1
1 2 ( )m w
AA p p
Af m r
-é ùæ öæ öê úç ÷= - -ç ÷ê úç ÷è øê úè øë û
Thus one obtains:
where additionally a parameter μw has been introduced to include slight energy
dissipation. The term in square brackets is called flow coefficient of Venturi tube.
p. II.3/7
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
B. Orifice plate:
simple constructioncheaper than Venturi tube
In contrast to Venturi tube one has considerable energy dissipation due to eddy currents
where the parameter μw has again been introduced to include slight energy
dissipation. The term in square brackets is called flow coefficient of the orifice plate.
2 0A Am=
12 2
00 1 2
1
1 2 ( )m w
AA p p
A
mf m m r
-é ùæ öæ öê úç ÷= - -ç ÷ê úç ÷è øê úè øë û
With
one has:
p. II.3/8
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Chapter II.3 – Flow Phenomena – Flow with negligible energy dissipation
C. Rotameter:
Here a force balance is used:
gravitational forces = pressure forces
1 2( ) ( )float float liquid floatV g p p Ar r- = -
2 ( )( ( )) ) liquid float liquid float
m b floatfloat
gVA h A
A
r r rf a
-= -
One has:
α is constant for sufficient high Re dependent on the shape of the float.Between 20% and 80% of the full scale one has a linear relationship
( )b floatA h A h- µ
since (r...radius of the float, δ...(small) opening angle of the rotameter) : 2 2( ) ( ) 2 2b floatA h A r r r r r r hp p p p d- = + D - @ D =
using the approximation tan( )r
hd d D@ =
