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Two Phase Flow Modeling

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Two Phase Flow Modeling PE 571 Chapter 3: Stratified Flow Modeling For Horizontal and Slightly Inclined Pipelines * * * * * * * Calculating the dimensionless ... – PowerPoint PPT presentation

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Title: Two Phase Flow Modeling


1
  • Two Phase Flow Modeling PE 571
  • Chapter 3 Stratified Flow Modeling
  • For Horizontal and Slightly Inclined Pipelines

2
Taitel and Duckler Model (1976)
  • The mechanistic model of the stratified flow was
    introduced by Taitel and Duckler (1976).
    Assumptions for this model are
  • Horizontal and slightly inclined pipelines (
    100)
  • Steady state
  • Zero end effects
  • The same pressure drop of gas and liquid phase

3
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • The objective of the model is to determine the
    equilibrium liquid level in the pipeline, hL, for
    a given set of flow conditions.

4
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • Momentum equation for gas phase
  • Momentum equation for liquid phase
  • Combined momentum equation

1
1
5
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • The respective hydraulic diameters of the liquid
    and gas phases are given
  • The Fanning friction factor for each phase
  • Where CL CG 16 and m n 1 for laminar flow
    and CL CG 0.046 and m n 0.2 for turbulent
    flow

6
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • The wall shear stresses for the liquid, the gas
    and the interface are
  • In this model, it is assumed tI tWG (smooth
    interface exists and vG gtgt vI).

7
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • From equation (1) gives
  • Defining the dimensionless variables

8
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • Equation (2) can be written in a dimensionless
    form
  • X is called the Lockhart and Martinelli parameter
  • Y is an inclination angle parameter

9
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • All the dimensionless variables are unique
    functions of

10
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
11
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
  • Example a mixture of air-water flows in a
    5-cm-ID horizontal pipe. the flow rate of the
    water is qL 0.707 m3/hr and that of the air is
    qG 21.2 m3/hr. The physical properties of the
    fluids are given as
  • rL 993 kg/m3 rG 1.14 kg/m3
  • mL 0.68x10-3 kg/ms mG 1.9x10-5 kg/ms
  • Calculate the dimensionless liquid level and all
    the dimensionless parameters.

12
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
13
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
For horizontal, Y 0. From the graph,
14
Taitel and Duckler Model (1976)
Equilibrium Stratified Flow
Calculating the dimensionless variables
15
Taitel and Duckler Model (1976)
Stratified to Non-stratified Transition
(Transition A)
  • Kelvin Helmholtz analysis states that the gravity
    and surface tension forces tend to stabilize the
    flow but the relative motion of the two layers
    creates a suction pressure force over the wave,
    owing to the Bernoulli effect, which tends to
    destroy the stratified structure of the flow.
  • For a inviscid two-phase flow between
    two-parallel plates, following is Taitel and
    Duckler (1976) analysis

16
Taitel and Duckler Model (1976)
Stratified to Non-stratified Transition
(Transition A)
  • The stabilizing gravity force (per unit area)
    acting on the wave
  • Assuming a stationary wave, the suction force
    causing wave growth is given
  • Continuity relationship

17
Taitel and Duckler Model (1976)
Stratified to Non-stratified Transition
(Transition A)
  • The condition for wave growth, leading to
    instability of the stratified configuration, is
    when the suction force is greater than the
    gravity force
  • Where C1 depends on the wave size

18
Taitel and Duckler Model (1976)
Stratified to Non-stratified Transition
(Transition A)
  • For an inclined pipe, the stratified to
    non-stratified transition can be determined in
    the similar manner.
  • Or
  • Where

19
Taitel and Duckler Model (1976)
Stratified to Non-stratified Transition
(Transition A)
  • Approximately, c2 can be calculated as
  • Then, the final criterion for the transition A
    is
  • Equation (4) can be written in a dimensionless
    form
  • Where

20
Taitel and Duckler Model (1976)
Stratified to Non-stratified Transition
(Transition A)
21
Taitel and Duckler Model (1976)
Intermittent or Dispersed Bubble to Annular
(Transition B)
  • As the flow is under non-stratified flow and if
    the flow has low gas and high liquid flow rate,
    the liquid level in the pipe is high and the
    growing waves have sufficient liquid supply from
    the film. The wave eventually blocks the cross
    sectional area of the pipe. This blockage forms a
    stable liquid slug, and slug flow develops.
  • At low liquid and high gas flow rate, the liquid
    level in the pipe is low the wave at the
    interface do not have sufficient liquid supply
    from the film. Therefore, the waves are swept up
    and around the pipe by the high gas velocity.
    Under these conditions, a liquid film annulus is
    created rather than a slug.

22
Taitel and Duckler Model (1976)
Intermittent or Dispersed Bubble to Annular
(Transition B)
  • It is suggested that this transition depends
    uniquely on the liquid level in the pipe.
  • Thus, if the stratified flow configuration is not
    stable, 0.35, transition to annular flow
    occurs. If gt 0.35, the flow pattern will be
    slug or dispersed-bubble flow.

23
Taitel and Duckler Model (1976)
Intermittent or Dispersed Bubble to Annular
(Transition B)
24
Taitel and Duckler Model (1976)
Stratified Smooth to Stratified Wavy (Transition
C)
  • This transition occurs when when pressure and
    shear forces exerted by the gas phase overcome
    the viscous dissipation forces in the liquid
    phase.
  • Based on Jeffreys theory (1926), the initiation
    of the waves occurs when
  • In the dimensionless form, this criterion can be
    expressed as
  • Where s is a sheltering coefficient associated
    with pressure recovery downstream of the wave.

25
Taitel and Duckler Model (1976)
Stratified Smooth to Stratified Wavy (Transition
C)
  • For s 0.01, K is defined as

26
Taitel and Duckler Model (1976)
Intermittent to Dispersed-Bubble (Transition D)
  • This transition occurs at high liquid flow rates.
    The gas phase occurs in the form of a thin gas
    pocket located at the top of the pipe because of
    the buoyanc forces. For sufficiently high liquid
    velocities, the gas pocket is shattered into
    small dispersed bubbles that mix with the liquid
    phase.
  • This transition occurs when the turbulent
    fluctuations in the liquid phase are strong
    enough to overcome the net buoyancy forces, which
    tend to retain the gas as a pocket at the top of
    the pipe.

27
Taitel and Duckler Model (1976)
Intermittent to Dispersed-Bubble (Transition D)
  • The net buoyancy forces acting on the gas pocket
    (AG gas pocket cross sec. area)
  • The turbulence forces acting on the gas pocket
    (SI interface length)
  • Where v is the turbulent radial velocity
    fluctuating component of the liquid phase. This
    velocity is determined when the Reynolds stress
    is first approximated by
  • The wall shear stress

28
Taitel and Duckler Model (1976)
Intermittent to Dispersed-Bubble (Transition D)
  • Assuming that tR tW,
  • The transition to dispersed bubble flow will
    occur when FT gt FB.
  • Nondimensional form
  • where

29
Taitel and Duckler Model (1976)
Intermittent to Dispersed-Bubble (Transition D)
30
Taitel and Duckler Model (1976)
Procedures for checking the flow pattern
  • Determine the equilibrium liquid level and all
    the dimensionless parameters
  • Check the stratified to nonstratified transition
    boundary.
  • If the flow is stratified, check the stratified
    smooth to stratified wavy transition
  • If the flow is nonstratified, check the
    transition to annular flow
  • If the flow is not annular, check the
    intermittent to dispersed bubble transition

31
Flow Pattern Prediction
Example
  • Example a mixture of air-water flows in a
    5-cm-ID horizontal pipe. the flow rate of the
    water is qL 0.707 m3/hr and that of the air is
    qG 21.2 m3/hr. The physical properties of the
    fluids are given as
  • rL 993 kg/m3 rG 1.14 kg/m3
  • mL 0.68x10-3 kg/ms mG 1.9x10-5 kg/ms
  • Calculate the dimensionless liquid level and all
    the dimensionless parameters.

32
Flow Pattern Prediction
Example
33
Flow Pattern Prediction
Example
For horizontal, Y 0. From the graph,
34
Flow Pattern Prediction
Example
Calculating the dimensionless variables
35
Flow Pattern Prediction
Example
Check for stratified to non-stratified
transition The criterion is not satisfied
The flow is stable and stratified flow exists
36
Flow Pattern Prediction
Example
Check for stratified-smooth to stratified-wavy
transition The criterion is satisfied The
flow is stratified wavy.
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