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Manmatha k. Roul, laxman kumar sahoo / international Journal of Engineering Research and

Applications (IJERA) ISSN: 2248-9622 www.ijera.com

Vol. 2, Issue 6, November- December 2012, pp.1047-1054

1047 | P a g e

Fig. 1(a) Idealized course of boundary stream lines

and (b) pressure profile for a sudden expansion.

(a)

Distance

P

r e s s u r e

∆ P

e

CFD modeling of pressure drop caused by two-phase flow of

oil/water emulsions through sudden expansions

Manmatha K. Roul*, and Laxman Kumar Sahoo

** *

(Department of Mechanical Engineering, Bhadrak Institute of Engineering & Technology, ODISHA, INDIA,PIN-756113)**

(Department of Mechanical Engineering, Bhadrak Institute of Engineering & Technology, ODISHA, INDIA,PIN-756113)

ABSTRACT

Pressure drop through sudden

expansions are numerically investigated for two-phase flow of oil/water emulsions. Two-phase

computational fluid dynamics (CFD)

calculations, using Eulerian – Eulerian model are

employed to calculate the velocity profiles and

pressure drop across sudden expansion. Axialpressure drops have been determined by

extrapolating the computed axial pressure

profiles in the regions of fully developed pipe

flow upstream and downstream of the pipe

expansion, to the transitional cross section. The

oil concentration is varied over a wide range of 0-97.3 % by volume. From the pressure-loss and

velocity data, the loss coefficients are obtained.

The loss coefficients for the emulsions are found

to be independent of the concentration and type

of emulsions. The numerical results are

validated against experimental data from the

literature and are found to be in goodagreement.

Keywords- Two-phase flow, pressure drop, loss

coefficient, velocity head, concentration,

emulsion.

I.INTRODUCTIONIndustrial piping systems are often

charged with two-phase flows. In contrast to thewell-known axial pressure profiles in the

transitional region between the flow separation andreattachment for single-phase liquid flow, thepressure profiles and the shape of streamlines in

two-phase flow through sudden change in flow areaare still unknown. Due to inherent complexity of two-phase flows through such sections, from aphysical as well as numerical point of view,generally applicable computational fluid dynamics(CFD) codes are non-existent. Two-phase flow of oil/water emulsions find application in a number of

industries, such as petroleum, pharmaceutical,agriculture and food industries etc. In manyapplications, pumping of emulsions through pipesand pipe fittings is required. Since a detailed

physical description of the flow mechanism is still

not possible for two-phase flow, a considerableeffort is generally needed to calculate the pressure

drop along the flow path. Several papers have been

published on flow of two-phase gas/liquid andliquid/liquid mixtures through pipe fittings. Hwang& Pal

1studied experimentally the flow of oil/water

emulsions through sudden expansions andcontractions and found that the loss coefficient foremulsions is independent of the concentration and

type of emulsions. Wadle

2

carried out a theoreticaland experimental study on the pressure recovery inabrupt expansions. He proposed a formula for thepressure recovery based on the superficial

velocities of the two phases and verified itspredictive accuracy with measured experimentalsteam-water and air-water data.

Tapucu et al.3

observed that emulsions canbe treated as pseudo-homogeneous fluids with

suitably averaged properties as the disperseddroplets of emulsions are small and are welldispersed. Consequently, the pressure loss for

emulsion flow in expansion and contraction shouldbe determinable in the same way as for single-phase fluid flow. Acrivos and Schrader

4observed

that significant velocity slip occurs at both sides of

the enlargement for two phase flow mixtures. Attouet al.

5developed a semi-analytical model for two-

phase pressure drop in sudden enlargements, basedon the solution of one-dimensional conservationequations downstream of the enlargement. Theycompared the predictions of three models

(homogeneous flow; frozen flow; and bubbly flow)with experimental data, with the latter modelproviding the best agreement with data. Abdelall et

al.6

studied the pressure drop caused by abrupt flowarea expansion and contraction in small channels

D1D2

Fullydevelo ed

Transitional

region

Fullydevelo ed

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and developed an empirical correlation for two-

phase flow pressure drop through sudden areacontraction. They indicated a significant velocityslip at the vicinity of the change of flow area.Salcudean et al.7 studied the effect of various flow

obstructions on pressure drops in horizontal two-

phase flow of air-water mixtures and derivedpressure loss coefficients and two-phase

multipliers.

In the present study, an attempt has been

made to simulate the flow through suddenexpansion using two phase flow models in anEulerian scheme. Fig. 1 shows a cross section of

the test section. At this section there is a sudden,sharp edged expansion. Fig. 1(a) shows theschematic diagram of the boundary streamlines forthe flow through a sudden expansion, while fig.

1(b) depicts the graph of the static pressure along

the flow axis for a steady state flow of anincompressible fluid across an expansion.

II.Mathematical FormulationsThe two-fluid or Euler - Euler technique is

considered for the present formulation. Thedifferent phases are treated mathematically asinterpenetrating continua, with each computationalcell of the domain containing respective fractions

of the continuous and dispersed phases. We haveadopted the following assumptions in our study

which are very realistic for the present situation.1. The fluids in both phases are Newtonian,

viscous and incompressible.2. The physical properties remain constant.

3. No mass transfer between the two phases.4. The pressure is assumed to be common to

both the phases.

5. The realizable k- turbulent model is appliedto describe the behavior of each phase.

6. The surface tension forces are neglected,

therefore, the pressure of both phases areequal at any cross section.

7. The flow is assumed to be isothermal, so the

energy equations are not needed.

With all the above assumptions the governingequations for phase q can be written as (Drew8):

Continuity equation:

0q q q q qvt

(1)

The volume fractions are assumed to be continuousfunctions of space and time and their sum is equalto one.

1q p (2)

Momentum equation:

( ) ( )q q q q q q q q q q q q

v v v p g M t

(3)

q , is the qth

phase stress tensor

q = eff T

q q q qv v

(4)

,

eff

q q t q (5)

Where q M is the interfacial momentum transfer

term, which is given by:d VM L

q q q q M M M M (6)

Where the individual terms on the right hand sideof Eq. (6) are, respectively, the drag force, virtual

mass force and lift force. The drag force isexpressed as,

3

4

d

q p q D p q p q

p M C v v v vd

(7)

The drag coefficient C D depends on the particleReynolds number as given below (Wallis

9; Ishii

and Zuber10):

CD = 24(1+0.15Re0.687

)/Re, Re 1000= 0.44, Re > 1000

(8)

Relative Reynolds number for primary phase q andsecondary phase p is given by

Re =q q p p

q

v v d

(9)

The second term in Eq. (6) represents the virtual

mass force, which can be described by thefollowing expression (Drew

8):

q q p pVM VM

q p VM p q

d v d v M M C

dt dt

(10)

where VM C is the virtual mass coefficient, which

for a spherical particle is equal to 0.5.

The third term in Eq. (6) is the lift force, and is

given by (Drew and Lahey11)

L L

q p L p q p q q M M C v v v

(11)

where LC is the lift force coefficient, which for

shear flow around a spherical droplet is equal to 0.5

2.1 Turbulence modeling

Here we considered the realizable per-phase k

turbulence model.

Transport Equations for k (FLUENT 6.2

Manual12

):

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,. .

t qqq q q q q q q q q

k

k U k k t

,

, ,. .

q k q q q q pq pq p qp q

t p t q p q p q pq p pq q

p p q q

G K C k C k

K U U K U U

(12)

Transport Equations for :

,

2

1 2 1

,

, ,

1

. .t q

qq q q q q q q q q

q q

q q q q q pq pq p qp q

qq t q q

q t p t q

pq p q p pq p q q

q p p q q

U t

C S C C K C k C k k k

C K U U K U U k

(13)Where,

qU

is the phase-weighted velocity. Here,

0.5

1max 0.43, , , 2

5ij ij

k C S S S S

The terms pqC and qpC can be approximated as

2, 21

pq

pq qp

pq

C C

(14)

Where pq is defined as

,

,

t pq

pqF pq

(15)

Where, the Langrangian integral time scale (,t pq ),

is defined as

,

,2

1

t q

t pq

C

(16)

Where,,

,

pq t q

t q

v

L

(17)

Where ,t q is a characteristic time of the energetic

turbulent eddies and is defined as:

,

3

2

q

t q

q

k C

(18)

And2

1.8 1.35cosC (19)

Where, θ is the angle between the mean particlevelocity and the mean relative velocity. The

characteristic particle relaxation time connectedwith inertial effects acting on a dispersed phase p isdefined as

1

,

p

F pq p q pq V qK C

(20)

Where, 0.5V C

The eddy viscosity model is used to calculateaveraged fluctuating quantities. The Reynoldsstress tensor for continuous phase q is given as:

, ,

2

3

T q q q q t q q q t q q q

k U I U U

(21)

The turbulent viscosity,t q is written in terms of

the turbulent kinetic energy of phase q:2

,

q

t q q

q

k C

(22)

The production of turbulent kinetic energy,,k qG is

computed from

, ,:

T

k q t q q q qG v v v

(23)

Unlike standard and RNG k models, C is

not a constant here. It is computed from:

0

1

s

C kU

A A

(24)

Where * ij ij ij ijU S S (25)

And 2ij ij ijk k

ij ij ijk k

Where, ij is the mean rate of rotation tensor

viewed in a rotating reference frame with the

angular velocity k .The constants A0 and As are

given by

A0= 4.04, As= 6 cos

Where 11cos 6 ,

3W

3

ij jk kiS S S W

S ,

ij ijS S S ,1

2

j iij

i j

u uS

x x

The constants used in the model are the following:

C1 = 1.44; C2 = 1.9; k = 1.0; = 1.2.

2.2 Boundary conditions Velocity inlet boundary condition is

applied at the inlet. A no-slip and no-penetratingboundary condition is imposed on the wall of thepipe. At the outlet, the boundary condition is

assigned as outflow, which implies diffusion fluxfor the entire variables in exit direction are zero.Symmetry boundary condition is considered at the

axis, which implies normal gradients of all flowvariables are zero and radial velocity is zero at theaxis.

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-20 -10 0 10 20 3

-10

-8

-6

-4

-2

0

2

4 =

P (

k P a )

L/D

V = 3.3 m/s(Present Work)



V = 3.3 m/s(Hwang & Pal)



-20 -10 0 10 20 3-10

-8

-6

-4

-2

0

2

4

= 0.2144

P (

k P a )

3.6 m/s(Present Work)



3.6 m/s(Hwang & Pal)

5.0 m/s(Hwang & Pal)6.8 m/s(Hwang & Pal)

III. NUMERICAL SOLUTION

PROCEDUREThe objective of the present work is to

simulate the flow through sudden expansion in

pipes numerically by using two phase flow modelsin an Eulerian scheme. The flow field is assumed to

be axisymmetric and solved in two dimensions.The two-dimensional equations of mass,momentum, volume fraction and turbulentquantities along with the boundary conditions havebeen integrated over a control volume and the

subsequent equations have been discretized overthe control volume using a finite volume techniqueto yield algebraic equations which are solved in aniterative manner for each time step. The finitedifference algebraic equations for the conservationequations are solved using Fluent 6.2 double

precision solver with an implicit scheme for allvariables with a final time step of 0.001 for quick

convergence. The discretization form for all theconvective variables are taken to be first order upwinding initially for better convergence. Slowly astime progressed the discretization forms areswitched over to second order up winding and then

slowly towards the QUICK scheme for betteraccuracy. The Phase-Coupled SIMPLE algorithmis used for the pressure-velocity coupling. Thevelocities are solved coupled by the phases, but in a

segregated fashion. The block algebraic multigridscheme is used to solve a vector equation formedby the velocity components of all phases

simultaneously. Pressure and velocities are thencorrected so as to satisfy the continuity constraint.The realizable per-phase k-ε model has been usedas closure model for turbulent flow. Fine grids are

used near the wall as well as near the expansionsection to capture more details of velocity andvolume fraction changes.

IV. RESULTS AND DISCUSSIONSThe sudden expansion considered in this

work is made from two straight pipes having innerdiameters of 2.037cm and 4.124cm. Axial staticpressure profiles are computed both upstream anddownstream from the expansion plane. Byextrapolating these pressure profiles to theexpansion plane the pressure drop is calculated.

The pressure differentials are computed withrespect to the reference pressure at 25D1 upstreamposition. The oil used in the present computational

work is Bayol-35 (Esso Petroleum, Canada), whichis a refined white mineral oil with a density of 780kg/m

3and a viscosity of 0.00272 Pa-s at 25°C.

Density and viscosity of water are taken as 998.2

kg/m3

and 0.001003 Pa-s, respectively. The volumefraction of oil is taken as 0, 0.2144, 0.3886, 0.6035,0.6457, 0.6950, 0.8042, and 0.9728. The emulsions

are considered as oil-in-water (O/W) type (water istaken as the continuous phase and oil as dispersedphase) up to an oil concentration of 62 % by

volume and water-in-oil (W/O) type (oil is taken as

the continuous phase and water as dispersed phase)beyond 64 % by volume (Hwang and Pal

1).

The streamlines and velocity vectors for α = 0.2144and v = 6.8 m/s are depicted in figs. 4 and 5

respectively. The streamlines take a typical

diverging pattern and a zone of recirculating flowwith turbulent eddies near the wall of the larger

pipe are created in the corner. This is due to the factthat the fluid particles near the wall due to their lowkinetic energy can not overcome the adversepressure hill in the direction of flow and hence

follow up the reverse path under the favourablepressure gradient( since upstream pressure is lowerthan the downstream pressure as depicted in figs. 2

and 3).

Figs. 6 and 7 show plot of e

P versus velocity

head 22V for various differently concentrated

oil-in-water and water-in-oil emulsions,respectively. It can be seen that

eP versus

22V data exhibit a linear relationship.

2K is the

slope of e

P versus 22V plots. Thus, the loss

coefficient for expansion eK which is equal to

1 2K K is calculated for various differently

concentrated oil-in-water and water-in-oilemulsions.

Here, 4

1 1 21K D D

. The

eK values for

different emulsions are plotted as a function of oil

concentration in Fig. 8. Clearly, the expansion losscoefficient is found to be independent of oil

concentration and has an average value of 0.432.

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-20 -10 0 10 20 3-10

-8

-6

-4

-2

0

2

4 = 0.3886

P (

k P a )

3.2 m/s(PresentWork)






-20 -10 0 10 20-12

-10

-8

-6

-4

-2

0

2

= 0.6035

P (

k P a )

V = 4.0 m/s(Present Work




V = 4.8 m/s(Hwang & Pal)V = 6.0 m/s(Hwang & Pal)

-20 -10 0 10 20 30-10

-8

-6

-4

-2

0

2

4

6 = 0.3050

P (

k P a )

L/D

V = 4.4 m/s(Present Work)V = 5.8 m/s(Present Work)V = 6.8 m/s(Present Work)V = 4.4 m/s(Hwang & Pal)V = 5.8 m/s(Hwang & Pal)V = 6.8 m/s(Hwang & Pal)

-20 -10 0 10 20-12

-10

-8

-6

-4

-2

0

2

4

6 = 0.1958

P ( k P

a )

V = 3.8 m/s(Present Work)V = 5.8 m/s(Present Work)V = 6.7 m/s(Present Work)V = 3.8 m/s(Hwang & Pal)V = 5.8 m/s(Hwang & Pal)


-20 -10 0 10 20 3

-10

-8

-6

-4

-2

0

2

4

6 = 0.3543

P (

k P a )

L/D

V=4.6m/s(Present work)



V=4.6m/s(Hwang & Pal)



-20 -10 0 10 20-10

-8

-6

-4

-2

0

2 = 0.0272

P ( k

P a )

V = 5.0 m/s(Present WorkV = 6.1 m/s(Present WorkV = 6.4 m/s(Present WorkV = 5.0 m/s(Hwang & Pal)

V = 6.1 m/s(Hwang & Pal)V = 6.4 m/s(Hwang & Pal)

Fig. 2. Pressure profiles for oil-in-water emulsions

flowing through a sudden expansion

Fig. 3. Pressure profiles for water-in-oil

emulsions flowing through a sudden expansion

Fig. 4. Stream lines for = 0.2144 and v = 6.8 m/s

Fig. 5.Velocity vectors for

= 0.2144 and v = 6.8 m/s

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0

2

4

6

8

10

12

0 5 10 15 20 25V

2 /2 (m

2 /s

2)

∆ P

e / ρ (

J / k g )

K2 = -0.5283 = 0.2144

Present Work

● Hwang & Pal

0

2

4

6

8

10

12

0 5 10 15 20 25

= 0 K2 = -0.5183

∆ P

e / ρ

( J / k

g )

Present Work

● Hwang & Pal

V2 /2 (m

2 /s

2)

0

2

4

6

8

10

12

0 5 10 15 20 25

V2 /2 (m

2 /s

2)

∆ P

e / ρ

( J / k g )

K2 = -0.4787 = 0.3886

Present Work

● Hwang & Pal

Computed as well as experimental pressure profiles

for oil-in-water and water-in-oil emulsions atvarious fluid velocities are shown in figs. 2 and 3respectively. The matching between thecomputation and that of the experimental

observation for the pressure drop seems to be pretty

reasonable in all these cases. It can be observed thatthe frictional loss in the inlet section causes the

decline in pressure. As the fluid reaches thetransitional section, the fluid is decelerated in theenlarged pipe area and there occurs a sudden rise inpressure. The pressure change at the expansion

plane eP is obtained by extrapolating the

computed pressure profiles upstream anddownstream of the pipe expansion (in the region of fully developed pipe flow) to the expansion plane.

The computed eK values for emulsions are

compared with the values obtained from thefollowing equations:(i) Borda- Camot equation (Perry et al.14):

2

1eK (26)

(ii) Equation of Wadle2:

2 1e

K (27)

Where is the ratio of the cross sectional area of

small pipe to that of large pipe. The value for

the expansion in the present work is 0.244. So the

value of eK obtained from Eqs. 26 and 27 are

0.5715 and 0.3689 respectively. The experimental

value of eK obtained by Hwang and Pal (1997) is

0.47. As shown in Fig. 8, the computed eK values

for all emulsions lie in between the two valuesobtained from Eqs. (26) and (27).

0

2

4

6

8

10

12

V2 /2 (m

2 /s

2)

= 0.6035

K2 = -0.468

∆ P

e / ρ

( J / k g )

Present Work

● Hwang & Pal

Fig. 6. ∆Pe/ρ versus V2 /2 for oil-in-water


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0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

K e

% Oil Concentration

O/W, W/O (Compn)

Perry et al. (1984)

Wadle (1989)

Average Ke(Compn)

V. CONCLUSIONSThe flow through sudden expansion has

been numerically investigated with oil-water

emulsions by using two-phase flow model in anEulerian scheme in this study. The majorobservations made relating to the pressure drop in

the process of flow through sudden expansion canbe summarized as follows:

1. The expansion loss coefficient is found tobe independent of the velocity and henceReynolds number.

2. The loss coefficient is not significantlyinfluenced by the type and concentration of

oil-water emulsions flowing through suddenexpansion.

3. Effect of viscosity is negligible on the

pressure drop through sudden expansion.4. The computed expansion loss coefficient is

found to lie in between the two valuesobtained from Borda- Carnot equation(Perry et al.

14) and equation of Wadle

2. It is

in relatively close agreement with thepredictions of Wadle2.

5. The pressure drop increases with higherinlet velocity and hence with higher massflow rate.

6. The satisfactory agreement between thenumerical and experimental results indicatesthat the model may be used as a simple,efficient tool for engineering analysis of two-phase flow through sudden flow area

expansions and contractions.

0

2

4

6

8

10

12

14

0 5 10 15 20 2 5

V2 /2 m

2 /s

2

= 0.3050 K2 = -0.5319

∆ P

e / ρ

( J / k g

)

Present Work

● Hwang & Pal

0

2

4

6

8

10

12

0 5 10 15 20 25

= 0.0272

2

2 2

Present Work

● Hwang & Pal

K2 = -0.4760

∆ P

e / ρ

( J / k g )

0

2

4

6

8

10

12

0 5 10 15 20 25

V2 /2 m

2 /s

2

Present Work

● Hwang & Pal

K2 = -0.527 = 0.3543

∆ P

e / ρ

( J / k g )

0

2

4

6

8

10

12

14

0 5 10 15 20 25

V2 /2 m

2 /s

2

= 0.1958K2 = -0.5393

Present Work ● Hwan & Pal

∆ P

e / ρ

( J / k g )

Fig.7. ∆Pe/ρ versus V2 /2 data for water-in-oil


Fig.8. Expansion loss coefficient as afunction of oil concentration

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REFERENCES1. C. Y. J. Hwang, and R. Pal, “Flow of Two

Phase Oil/Water Mixtures through SuddenExpansions and Contractions”, Chemical

Engineering Journal, 68, pp. 157-163.1997.

2. M. Wadle, “A new formula for the pressure recovery in an abrupt diffuser”,

Int. J. Multiphase Flow, 15(2), pp. 241-256, 1989.

3. A. Tapucu, A. Teyssedou, N. Troche, and

M. Merilo, “Pressure losses caused by

area changes in a single channel flowunder two- phase flow conditions”, Int. J.

Multiphase Flow 15(1), pp. 51-64, 1989.

4. A. Acrivos, and M. L. Schrader, “Steady

flow in a sudden expansion at highReynolds numbers”, Phys. Fluids 25(6),

pp. 923-930, 1982.

5. A. Attou, M. Giot, and J. M. Seynhaeven,“Modeling of steady-state two-phase

bubbly flow through a suddenenlargement”, Int. J. Heat Mass Transfer

40, pp. 3375 – 3385, 1997.6. F. F. Abdelall, G. Hann, S. M. Ghiaasiaan,

S. I. Abdel-Khalik, S. S. Jeter, M. Yoda,and D. L. Sadowski, “Pressure drop

caused by abrupt flow area changes insmall channels”, Exp. Thermal and Fluid

Sc. 29, pp. 425-434, 2005.7. M. Salcudean, D. C. Groeneveld, and L.

Leung, “Effect of flow obstruction

geometry on pressure drops in horizontalair-water flow”, Int. J. Multiphase Flow 9(1), pp. 73-85, 1983.

8. D. A. Drew, “Mathematical Modeling of

Two-Phase Flows”, Ann. Rev. Fluid Mech.

15, pp. 261-291, 1983.

9. G. B. Wallis, One-dimensional two-phase flow, McGraw Hill, New York. 1969.

10. M. Ishii, and N. Zuber, “Drag coefficientand relative velocity in bubbly, droplet or particulate flows”, AIChE J. 25, pp. 843 –

855, 1979.11. D. Drew, and R.T. Jr. Lahey, “Application

of general constitutive principles to thederivation of multidimensional two-phaseflow equations”, Int. J. Multiphase Flow 7, pp. 243 – 264, 1979.

12. FLUENT User’s Guide, Version 6.2.16,

Fluent Inc., Lebanon, New Hampshire03766, 2005.

13. W. L. McCabe, J. C. Smith, and P.Harriott, Unit Operations of Chemical

Engineering, McGraw-Hill, New York,

1993.14. R. H. Perry, D. W. Green, and J. O.

Maloney, Perry’s Chemical Engineers’

Handbook , McGraw-Hill, New York,1984.

15. S. V. Patankar, Numerical Heat Transfer

and Fluid Flow, McGraw-Hill,Hemisphere, Washington, DC, 1980.

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