Flow of fluids through mini channels

Manmatha K. Roul, Sukanta K. Dash / International Journal of Engineering Research and
Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 2, Issue 5, September- October 2012, pp.757-763
Flow of Single-Phase Water and Two-Phase Air-Water Mixtures
through Sudden Contractions in Mini Channels
*Manmatha K. Roula,*Sukanta K. Dash
*Department of Mechanical Engineering, Bhadrak Institute of Engg & Tech, Bhadrak(India) 756113
**Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur (India) 721302

Abstract
Pressure drop through sudden contraction pressure drop across sudden contractions. They
in small circular pipes have been numerically assumed the occurrence of the vena-contracta
investigated, using air and water as the working phenomenon and a dispersed droplet flow pattern
fluids at room temperature and near atmospheric downstream of the vena-contracta point. However,
pressure. Two-phase computational fluid Schmidt and Friedel [1], based on their experimental
dynamics (CFD) calculations, using Eulerian– results indicated that the vena-contracta phenomenon
Eulerian model with the air phase being did not occur in their system at all. Applications of
compressible, are employed to calculate the mini and micro channels in advanced and high
pressure drop across sudden contraction. The performance systems have been rapidly increasing in
pressure drop is determined by extrapolating the recent years. Some two-phase hydrodynamics and
computed pressure profiles upstream and heat transfer processes in mini and micro channels
downstream of the contraction. The larger and are different from larger channels [5-7], indicating
smaller tube diameters are 1.6 mm and 0.84 mm, that the available vast literature associated with flow
respectively. Computations have been performed and heat transfer in larger channels may not be
with single-phase water and air, and two-phase directly applicable to micro channels. Significant
mixtures in a range of Reynolds number velocity slip occurs at the vicinity of the flow
(considering all-liquid flow) from 1000 to 12000 disturbance in case of mini and micro channels
and flow quality from 1.9 103 to 1.6 102 . The (Abdelall et al. [7]). They observed that
homogeneous flow model over predicts the data
contraction loss coefficients are found to be
monotonically and significantly and with the slip
different for single-phase flow of air and water.
ratio expression of Zivi [8], on the other hand, their
The numerical results are validated against
experimental data and theory were in relatively good
experimental data from the literature and are
agreement. Pressure drops associated with single-
found to be in good agreement. Based on the
phase flow through abrupt flow area changes in
numerical results as well as experimental data, a
commonly-applied large systems have been
correlation is developed for two-phase flow
extensively studied in the past. But little has been
pressure drop caused by the flow area
reported with respect to two-phase flow through mini
contraction.
and micro channels. In the present work, an attempt
has been made to simulate the flow through sudden
Keywords: Two-phase flow; pressure drop; loss contraction in mini channels using two phase flow
coefficients; flow quality; two-phase multiplier. models in an Eulerian scheme. The flow field is
assumed to be axisymmetric and solved in two
1. Introduction dimensions. Before, we can rely on CFD models to
In the recent years, several papers have been study the two-phase pressure drop through such
published on flow of two-phase gas/liquid mixtures sections; we need to establish whether the model
through pipe fittings. Schmidt and Friedel [1] studied yields valid results. For the validation of results, we
experimentally two-phase pressure drop across have referred to the experimental studies conducted
sudden contractions using mixtures of air and liquids, by Abdelall et al. [7], who measured pressure drops
such as water, aqueous glycerol, calcium nitrate resulting from an abrupt flow area changes in
solution and Freon 12 for a wide range of conditions. horizontal pipes.
Salcudean et al. [2] studied the effect of various flow
obstructions on pressure drops in horizontal air-water 2. Mathematical formulation
flow and derived pressure loss coefficients and two- The Eulerian-Eulerian mixture model has
phase multipliers. Many of the published studies have been used as the mathematical basis for the two phase
assumed the occurrence of the vena-contracta simulation through sudden contraction. The volume
phenomenon in analogy with single-phase flow and fractions  q and  p for a control volume can
have assumed that dissipation occurs downstream of
the vena-contracta point [3-4]. Al’Ferov and therefore be equal to any value between 0 and 1,
Shul’zhenko [3], Attou and Bolle [4] have attempted depending on the space occupied by phase q and
to develop mechanistic models for two-phase

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phase p . The mixture model allows the phases to p
2
vq
Eq  hq   (9)
move at different velocities, using the concept of slip q 2
velocities.
and E p  hp for the incompressible phase, where hq
Continuity Equation: is the sensible enthalpy for phase q .
Continuity equation for the mixture is given by:
Relative (Slip) Velocity and the Drift Velocity:
 
 m   .  m vm   0 (1) The slip velocity is defined as the velocity of a
t secondary phase p relative to the velocity of the

where, vm is the mass averaged velocity: primary phase q :
  
v pq  v p  vq (10)
 
  q q vq   p  p v p The mass fraction for any phase p is defined as
vm  (2)
m pp
and  m is the mixture density: cp  (11)
m

The drift velocity and the relative velocity ( v pq ) are
m   q q   p  p (3)
connected by the following expression:
and  q is the volume fraction of phase q .   
vdr , p  v pq  c p vqp (12)
The basic assumption of the algebraic slip mixture
Momentum Equation: model is to prescribe an algebraic relation for the
The momentum equation for the mixture can be relative velocity so that a local equilibrium between
obtained by summing the individual momentum the phases should be reached over short spatial length
 p   p  m  
equations for all phases. It can be expressed as (Drew
[9]; Wallis[10]) 
scale. v pq  a (13)
f drag p

 mvm   . mvmvm   p  .  m vm  vm 
    T
where  p is the particle relaxation time and is given
t  
(4)
 
 
by:
  m g  F  .  q  q vdr , q vdr , q    p  p vdr , pvdr , p 
   
pdp2

where, m is the viscosity of the mixture: p  (14)
18 p
where d p is the diameter of the particles (or droplets
 m   q q   p  p (5) 
 or bubbles) of secondary phase p , a is the
vdr , p is the drift velocity for the secondary phase p :
   secondary-phase particle’s acceleration. The drag
vdr , p  v p  vm (6) function f drag is taken from Schiller and Naumann:
1  0.15Re0.687 Re  1000
Energy Equation: f drag   (15)
The energy equation for the mixture takes the 0.0183Re Re  1000
following form: 
and the acceleration a is of the form:

    v
 a  g   vm .  vm  m
t

 q q Eq   p  p E p   .  q vq  q Eq  p 

(7)
 t
(16)

In turbulent flows the relative velocity should contain
 
.  p v p   p E p  p   .  keff T   S E

a diffusion term due to the dispersion appearing in
the momentum equation for the dispersed phase.
where keff is the effective conductivity:

  p  m  d p2 a  m  (17)

keff   q  kq  kt    p  k p  kt   (8)

v pq 
18q f drag

 p D
q

and kt is the turbulent thermal conductivity. The
where ( m ) is the mixture turbulent
first term on the right hand side of Equation (7)
viscosity and (  D ) is a Prandtl dispersion coefficient.
represents energy transfer due to conduction. S E
It may be noted that, if the slip velocity is not solved,
includes any other volumetric heat sources. For the
the mixture model is reduced to a homogeneous
compressible phase,
multiphase model.

Volume Fraction Equation for the
Secondary Phase:

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From the continuity equation for the secondary phase
p , the volume fraction equation for the secondary
phase p can be obtained:

t
 p  p   . p  p vm   . p  p vdr , p  (18)
 

Turbulence modeling:
The k and  equations used in this model are as
follows (FLUENT 6.2 Manual [11]):

   
 mk      mvmk      t , m k   Gk , m  m (19)
t  k 
and

    
 m      mvm      t ,m    C1 Gk ,m  C2 m  (20)
t    k

where the mixture density and velocity  m and vm ,
are computed from

m   p  p   q q (21)
 
  p  p v p   q  q vq
vm  (22) Fig. 1: (a) Idealized course of boundary stream lines
 p  p   q q and (b) pressure profile for a sudden contraction.
The turbulent viscosity t , m is computed from
k2
t , m  m C (23)
 r Wall
and the production of turbulent kinetic energy, Gk , m
Mass flow Inlet

Wall
is computed from
0.8 mm

 

Outflow
0.42mm
  T 
Gk , m  t , m vm   vm  : vm (24)
The constants in the above equations are z
Axis of symmetry
C1  1.44; C2  1.92; C  0.09;  k  1.0;    1.3 0.4 m
3. Results and discussion Fig. 2: Computational domain for contraction section
3.1. Single phase flow pressure drop
Idealized course of boundary stream lines
and pressure profile for single phase flow of water
through sudden contraction is shown in Fig. 1. 10000
Computational domain with boundary conditions is 0
shown in Fig. 2. The flow field is assumed to be
axisymmetric and solved in two dimensions. The -10000 Reference
Pressure profiles for the single-phase flow of water -20000
Pressure
and air through sudden contraction is depicted in
Pc (Pa)

-30000
Figs. 3 and 4 respectively, for different mass flow Water
rates. It can be seen that the pressure profiles are -40000
m = 0.69 g/s
nearly linear up to 5 pipe diameters, both upstream -50000 m = 1.36 g/s
and downstream from the contraction plane. Because m = 2.31 g/s
there is a change in pipe cross-section and hence a -60000 m = 2.58 g/s
change in mean velocity, the slopes of the pressure m = 2.98 g/s
-70000
m = 3.69 g/s
profiles before and after contraction are different. The
-80000
gradients are greater in the smaller diameter pipe. It -25 -20 -15 -10 -5 0 5 10 15 20 25
can be observed from figs.3 and 4 that very close to Fig. 3: Pressure profiles for single-phase water
L/D
the contraction plane the static pressure in the inlet flow through sudden contraction
line decreases more rapidly than in fully developed
flow region. It attains the (locally) smallest value at a

759 | P a g e


4000 1.0
Air 0.9
Computation,---Kc=0.5
0 Theory, kd1=1.33, =2
m = 0.0234 g/s 0.8 Theory, kd1==1

Contraction Coefficient, Kc
-4000 m = 0.0245 g/s Ref
0.7 Experiment[7]
m = 0.0255 g/s pr
-8000 m = 0.0356 g/s
0.6
Pc (Pa)

m = 0.0365 g/s
-12000 m = 0.0373 g/s 0.5
m = 0.0524 g/s
m = 0.0574 g/s 0.4
-16000
m = 0.0625 g/s
m = 0.0681 g/s 0.3
-20000 m = 0.11 g/s 0.2 Laminar Turbulent
m = 0.116 g/s
-24000 m = 0.122 g/s 0.1
m = 0.142 g/s
-28000 0.0
0 1000 2000 3000 4000 5000 6000
-25 -20 -15 -10 -5 0 5 10 15 20 25
L/D Reynolds Number, Re1
Fig. 4: Pressure profiles for single-phase air Fig. 6: Contraction loss coefficient for
flow through sudden contraction single phase water flow
1.0
Computation
0.9
Theory, kd1==1
0.8 Theory, kd1=1.33, =2
Contraction Coefficient, Kc

Experiment[7]
0.7

0.6

0.5

0.4

0.3

0.2
Laminar Turbulent
0.1

0.0
0 2000 4000 6000 8000 10000 12000 14000
(a) Reynolds Number, Re1
Fig. 7: Contraction loss coefficient for
single phase air flow

0

Reference
-10 pressure
Pc (kPa)

-20

mL= 2.578 g/s
mG = 0.00493 g/s
-30 mG = 0.00624 g/s
mG = 0.00777 g/s
mG = 0.00931 g/s
(b) -40
mG = 0.0122 g/s
mG = 0.015 g/s
Fig. 5: (a) Velocity vectors & (b) Stream lines for single-
phase flow through sudden contraction ( mL  3.69 g/s )
 -20 -15 -10 -5 0 5 10 15 20
L/D
Fig. 8: Pressure profiles for flow of two-phase
air-water through sudden contraction

760 | P a g e

distance of about L/D = 0.5 after the contraction mass flow rates of either water or air. The velocity
section and depends only slightly on the mass flow vectors and stream lines for a particular mass flow
rates. The same is also demonstrated by the velocity rate of water and air are illustrated in Fig.9. It clearly
vectors and streamline contours in fig. 5, which demonstrates that unlike single phase flow, the two-
shows that the flow has minimum cross-sectional phase flow does not contract behind the edge of
area at about 0.5D downstream of pipe contraction, D transition and there is no change in the flow direction
being the diameter of the smaller pipe. Then, the i.e; a zone of recirculation is not observed. Thus
pressure gradually increases and, after reaching its vena-contracta is not a relevant term in two-phase
maximum, it merges into the curve of the pipe flow through sudden contractions. The computed
frictional pressure drop downstream of contraction. two-phase flow pressure drops caused by flow area
This local minimum value of pressure corresponds to contractions are displayed in Figs. 10 (a) and (b) as
the vena-contracta position. So, it can be concluded functions of Reynolds number ( Re L 0,1 ).
that vena-contracta is always obtained in the single-
phase flow of water and air through sudden
Where ReL 0,1   GD 1 L , (26)
contraction at a distance of 0.5D from the contraction
plane in the downstream direction. Simulated
velocity vectors are shown in fig. 5(a), which clearly Here G1 is the total mass flux
shows that eddy zones are formed in the separated corresponding to the smaller diameter pipe. The
flow region. The pressure change at the contraction computed as well as the experimental data are
plane  Pc  is obtained by extrapolating the compared with analytical model calculations
assuming homogeneous flow, and slip flow (Abdelall
computed pressure profiles upstream and downstream et al.[7]) with the aforementioned slip ratio of
of the pipe contraction (in the region of fully 1
developed flow) to the contraction plane. It is S    L G  3 . The homogeneous flow and the slip
observed that the pressure drop increases with the flow models are considered assuming a vena-
mass flow rates for single phase flow of both water contracta, and no vena-contracta (i.e., C c=l). It is
and air through sudden contraction. observed that calculations with the homogeneous
Assuming uniform velocity profiles at cross- flow assumption over predict the data monotonically
section 1 and 3, the contraction loss coefficient can be and significantly. The computational and the
found by applying one-dimensional momentum and experimental data ([7]) are found to be in relatively
mechanical energy conservation equations (Abdelall good agreement with the predictions of slip flow
et al.[7]; Kays [12]): model..
An attempt is made to correlate the two-phase pressure
1 2
Kc   u1    P2,3  P2,1    u1
2 
1
2
2
1    (25)
2 change data in terms of the Martinelli factor:

0.1 0.5
   1  x   G 
0.9

Where ( P2,3  P2,1 ) is obtained numerically. XM   L      , (27)
 G   x   L 
The contraction loss coefficients for single-phase
Fig. 11 shows the graph  L 0,c X M verses
water and air flows are depicted in Figs. 6 and 7,
respectively, where the experimental data [7] and X M ReL 0,1 considering both computational as well as
theoretical predictions are also shown. The data the experimental data for sudden contraction. The
points for water, although few and scattered, are resulting correlation for two-phase multiplier for
higher than that of the theoretical predictions and sudden contraction is given by:
in relatively close agreement with the  L 0,c
 160  X M Re L 0,1 
0.6877
experimental data. For turbulent flow in the smaller (28)
channel, contraction loss coefficient for single XM
phase water flow (Kc ) is found to be 0.5 (Fig. 6).
The contraction loss coefficient obtained for The comparison between the predicted
single phase air flow (depicted in Fig. 7), on the values of the two-phase pressure drops (calculated
other hand, agree with the aforementioned theoretical from Eq. 28) with the computed as well as
predictions as well as experimental data [7] well. experimental data [7] are depicted in Fig. 12. The
agreement is found to be quite good. It is notable that
3.2 Two-phase flow pressure drop the correlation can predict the two-phase pressure
Fig. 8 depicts the two-phase pressure drop drop quite well within 16% error. The correlation is
through sudden contraction for different mass flow valid in the range of 1000 < ReL0,1 < 12000.
rates of air keeping the mass flow rate of water
constant. Similarly for different mass flow rates of
water several numerical experiments are performed
and it is observed that the pressure drop through
sudden contractions increases with increasing the

761 | P a g e


100000 Computation
(Homogeneous flow model with vena-contracta)
90000
(Homogeneous flow model, no vena-contracta)
(Slip flow model with vena-contracta)
80000
(Slip flow model, no vena-contracta)
70000 Experiment[7]

Pc (Pa)
60000

50000

40000

30000

20000

10000

0
3512 3516 3520 3524 3528 3532 3536 3540 3544
(a) Reynolds Number, ReL0,1
(b)
Fig. 10(b): Two-phase flow pressure drop caused
by flow area contraction

0.6
Experiment[7]
0.5 Computation
Eq.(28)

0.4
L0,c/XM

0.3

0.2
(b)
Fig. 9: (a) Velocity vectors and (b) Stream lines for
 mL  2.319 g/s and mG  0.0203 g/s 
  0.1

0.0
0 20000 40000 60000 80000 100000 120000
80000
Computation XMReL0,1
(Homogeneous flow model with vena-contracta)
70000 (Homogeneous flow model, no vena-contracta) Fig. 11: Correlation for two-phase multiplier
(Slip flow model with vena-contracta) through sudden contraction
60000 (Slip flow model, no vena-contracta)
Experiment[7] 25
50000 Computation +16%
Pc (Pa)

Experiment[7]
40000 20
-16%
Predicted Pc(kPa)

30000

20000
15

10000
10
0
2576 2580 2584 2588 2592 2596 2600 2604
Reynolds Number, ReL0,1 5
(a)
Fig. 10(a): Two-phase flow pressure drop caused
by flow area contraction 0
0 5 10 15 20 25
Measured Pc(kPa)
Fig. 12: Comparison between predicted and
measured P c

762 | P a g e

4. Conclusions channels, Exp. Thermal Fluid science, Vol.
Pressure drops caused by abrupt flow area 26, (2002), pp. 389-407.
contraction in mini channels using water and air [7] F. F. Abdelall, G. Hann, S. M. Ghiaasiaan,
mixtures have been numerically investigated by using S. I. Abdel-Khalik, S. S. Jeter, M. Yoda, and
two-phase flow model in an Eulerian scheme. For D. L. Sadowski, Pressure drop caused by
single phase flow of water and air through sudden abrupt flow area changes in small channels,
contraction, the vena contracta is always established Exp. Thermal and Fluid Sc, Vol. 29, (2005),
at a distance of L/D = 0.5 after the contraction section pp. 425-434.
and its position depends only slightly on mass flow [8] S. M. Zivi, Estimation of steady state steam
rates. With turbulent flow in the smaller channel, void-fraction by means of principle of
approximately constant contraction loss coefficients are minimum entropy production, ASME Trans.
obtained for single phase flow of water and air. The Series C 86, (1964), pp. 237–252.
contraction loss coefficients for water are found to be [9] D. A. Drew, Mathematical Modeling of
slightly larger than the theoretical predictions and that for Two-Phase Flows, Ann. Rev. Fluid Mech,
air is in good agreement with the theoretical Vol. 15, (1983), pp. 261-291
predictions. The computed values of two-phase flow
pressure drops caused by sudden flow area contraction
[10] G. B. Wallis, One-dimensional two-phase
flow, McGraw Hill, New York, (1969).
are found to be significantly lower than the predictions
of the homogeneous flow model and are in relatively [11] FLUENT’s User Guide. Version 6.2.16,
close agreement with the predictions of slip flow Fluent Inc., Lebanon, New Hampshire
model. It is observed that there is significant velocity 03766, (2005).
slip in the vicinity of the flow area change. The data [12] W. M. Kays, Loss coefficients for abrupt
suggest that widely-applied methods for pressure drops in flow cross section with low
drops due to flow area changes, particularly for Reynolds number flow in single and
two-phase flow is not applicable to mini and micro multiple-tube systems, Trans. ASME 73
channels. It is also observed that unlike single phase (1950), pp. 1068-1074.
flow, the two-phase flow does not contract behind the
edge of transition and there is no change in the flow
direction i.e; a zone of recirculation is not observed.
Thus vena-contracta is not a relevant term in two-
phase flow through sudden contractions. Based on the
computational as well as the experimental data,
correlations have been developed (Eq.28) for two-
phase flow pressure drop caused by the flow area
contraction.

References
[1] J. Schmidt and L. Friedel, Two-phase
pressure drop across sudden contractions in
duct areas, Int. J. Multiphase Flow; Vol. 23,
(1997), pp. 283–299.
[2] M. Salcudean, D. C. Groeneveld, and L.
Leung, Effect of flow obstruction geometry
on pressure drops in horizontal air-water
flow, Int. J. Multiphase Flow, Vol. 9/1,
(1983), pp. 73-85.
[3] N. S. Al’Ferov, and Ye. N. Shul’Zhenko,
Pressure drops in two-phase flows through
local resistances, Fluid Mech.-Sov. Res,
Vol. 6, (1977), pp. 20–33.
[4] A. Attou and L. Bolle, Evaluation of the
two-phase pressure loss across singularities,
ASME FED, Vol. 210 (1995), pp. 121–127.
[5] S. M. Ghiaasiaan and S. I. Abdel-Khalik,
Two-phase flow in micro-channels, Adv.
Heat Transfer, Vol. 34, (2001), pp. 145-254
[6] S. G. Kandlikar, Fundamental issues related
to flow boiling in mini channels and micro

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Flow of fluids through mini channels

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