Prof. G. DasDepartment of Chemical Engineering

The Drift Flux ModelLecture – 13

Indian Institute of Technology, Kharagpur

• Simplicity

• Applicable to a wide range of two phase flow problems of practical interest - bubbly, slug and drop regimes of gas-liquid flow

- fluidised bed of fluid particle system.

• Rapid solution of unsteady flow problems of sedimentation and foam drainage

• Useful for the study of system dynamics and instabilities caused by low velocity wave propagation namely void propagation.

Advantages

• A starting point for extension of theory to complicated problems of fluid flow and heat transfer where two and three dimensional effects such as density and velocity variations across a channel are significant.

•Important for scaling of systems.

•Detailed analysis of the local behaviour of each phase can be carried out more easily if the mixture responses are known

General Theory

05/03/23

Application- Bubbly flow, slug flow, drop regimes of gas-liquid flow as well as to fluidized bed

Volumetric Flux

Drift flux 21 2 TPj u j

12 11 TPj u j 1 211 TPj j j

2 21TPj j j

2 21

2

1TP

j jj j

1 1 2 2 212 1TP

TP TP

j j jj j

TPQjA

1 1(1 )j u 2 2j u

21 2(1 )u j j

Relative velocity between the phases taken care of by the concept of drift flux

11 1 1 1 1u u b f p

t

u

22 2 2 2 2u u u b f pt

Kinematic Constitutive Relation

In two fluid model, the momentum balance equations for unit volume of the individual phases in three dimensional vector form is:

1 11 1 1 1u u pu b ft z z

2 22 2 2 2u u pu b ft z z

For one dimensional flow, eqns can be resolved in the direction of motion to give:

110

1Fdpg

dz

220 Fdpg

dz

1 1 12 11 wF f F F

Under steady state inertia dominant conditions the aforementioned equations become:

Where F1 and F2 are the equivalent f’s per unit volume of the whole flow field. Thus

2 2 2 12wF f F F

Since action and reaction are equal

F1 = F2= - F12

Therefore the equations become:12

101Fdpg

dz

1220 Fdpg

dz

12 122 10

1F Fg

or

12 1 21F g

On subtracting momentum eqn for phase 1 from that of phase 2, we get

2 (1 )nju u

This gives:

21 (1 )nj u

F12=F12 (α, j21 )

The values of u2j for a few representative cases are as follows:

For the viscous regime,

4/3* *1/3 1.5

22 6/ 72 * *

2

11 ( )10.8

( )

d d

jd d

r rfgur r f

Where

1/ 2 11TP

f

3 4/ 7 0.75* *0.55 1 0.08 1d dr r

* 12

1d d

gr r

Where rd is the radius of the dispersed phase

For Newton’s regime ( 34.65)dr

1/ 21.5

2 6/ 71

18.672.43 1 ( )1 17.67

dj

r gu ff

For distorted fluid particle regime

where Nμ the viscosity number is given as:

8/30.11 1 /N

11/ 2

1

N

g

1.75

1/ 42

2 1 221 2.25

2 1

1

2 1

1

jgu

For churn turbulent flow regime

1/ 4

1/ 41 22 2

1

2 1jgu

1/ 4

1 222 g

It may be noted that in the aforementioned expression for u2j, the proportionality constant is applicable for bubbly flows and 1.57 for droplet flows.

2

21

0.35jg Du

In the absence of infinite relative velocity,

21

21

0 00 1

j atj at

Graphical Technique for solution of Drift Flux Model

Cocurrent Upflow

Cocurrent Downflow

Countercurrent flow with Gas Flowing up and Liquid flowing downfor a constant gas and different liquid velocities

Countercurrent flow with Gas Flowing down and Liquid flowing up

Drift Flux Model for Solid as the dispersed phase

Corrections to the one dimensional model:

2 21j j j

j dAj

dA

It may be noted that

j j

Since

dA jdAj

dA dA

0jCj

0CWhere is the ratio of averae of product of flux and concentration to product of averages or

0

1

1 1

j dAACdA jdA

A A

2 210

j jC j

212 1 20

jQ Q QCA A

2 21

0 1 2

Q A jC Q Q

021

2C

m n

[1 ]w

0CEstimation of

For fully developed bubbly flow (Ishii)

0 0 ,gl l

GDC C

Assuming power law profiles for α and j

For flow in a round tube

Co= 1.2 – 0.2 /g l

For flow in a rectangular channel

Co = 1.35 – 0.35 /g l

For developing void profile (0< α<0.25)

-180 gC = ( 1.2 - 0.2 ) ( 1- e ) round tube l

-180 gC = ( 1.35 - 0.35 ) ( 1- e ) rectangular channel. l

For boiling bubbly flow in an internally heated annulus

3.12< >0.212Co= 1.2 - 0.2 [1 - e ]g

l

In downward two-phase flow for all flow regimes

0C =(- 0.0214<j*> + 0.772) + (0.0214<j*> + 0.228) for (-20) <j*> < 0g

l

0.00848[<j*>+20] 0.00848[<j*>+20]oC =(0.2e +1) - 02e for < j*> < (-20)g

l

Where <j*> = 2 j

ju

Void profile changes from concave to convex due to• Wall nucleation and delayed transverse

migration of bubbles towards centre• Subcooled boiling regime• Injecting gas into flowing liquid through

porous tube wall• Adiabatic flow at low void fraction when small

bubbles tend to accumulate near the walls• Droplet or particulate flow in turbulent regime

05/03/23

Evaluation of terminal velocity

Bubbly flow

05/03/23

Bubble formation at Orifice

• Spherical bubble of radius Rb attached to orifice of radius Ro

• Largest bubble at static equilibrium

• Radius of bubble-blowing through-small orifice at low rates

• More accurate

• Ceases to be valid –orifice diameter comparable to bubble radius

34 ( ) 23 b f g oR g R

133

2 ( )o

bf g

RRg

13

1.0( )

ob

f g

RRg

12

0.5( )o

f g

Rg

05/03/23

Influence of shear stress

• Shear stress determine bubble size in forced convection /mechanically agitated system

• Shear stress influence –• Size of bubbles form away from point of formation

Max bubble size which is stable in flow

• Mechanical power dissipated/unit mass

3 25 50.725( ) ( )

f

pdM

pm

05/03/23

Formation of bubble by Taylor instability

• Formed by detachment from blanket of gas or vapor over a porous or heated surface

• Formation not identical with “Taylor instability” of a fluid below a denser fluid but physics similar

12

( )bf g

Rg

05/03/23

Formation by evaporation or mass transfer

• By evaporation of surrounding liquid/ release of gases dissolved in liquid

• Bubble form-nucleation centre-impurities in fluids/pits, scratches, cavities on wall

• contact angle in degrees• Valid for quasi-static case-not for bubbles formed during

boiling

12

0.0208( )

ob

f g

RDg

05/03/23

INFLUENCE OF CONTAINING WALLS 

In finite vessel: ub<u∞

ub/u∞ =fn(d/D), D=Tube diameter In region 5 for large inviscid bubbles  d/D < 0.125 , ub/u∞ =1  0.125 < d/D < 0.6 , ub/u∞ =1.13 e-d/D

 

0.6 < d/D , ub/u∞ =0.496 (d/D)-1/2 [Bubbles behaves like slug flow bubbles in an inviscid fluid] 

05/03/23

INFLUENCE OF CONTAINING WALLS CONTINUED 

• In viscous fluids:ub/u∞ =[1+2.4(d/D)]-1

 For bubbles behaving as solid spheres

ub/u∞=[1+1.6(d/D)]-1

 

For fluid spheres & µg << µf

If d/D > 0.6 ,ub/u∞ =0.12 (d/D)-2

 At d/D = 0.6 , ub/u∞ = 1-(d/D)/0.9 ( used to estimate ub for d/D < 0.6)

05/03/23

Formation of bubble by Taylor instability

• Formed by detachment from blanket of gas or vapor over a porous or heated surface

• Formation not identical with “Taylor instability” of a fluid below a denser fluid but physics similar

12

( )bf g

Rg

05/03/23

Slug flow