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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