Two Phase Flow Modeling Prepared by: Tan Nguyen Two Phase Flow Modeling – PE 571 Chapter 3:...

Two Phase Flow Modeling

Prepared by: Tan Nguyen

Two Phase Flow Modeling – PE 571

Chapter 3: Stratified Flow Modeling

For Horizontal and Slightly Inclined Pipelines

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The mechanistic model of the stratified flow was introduced by Taitel and Duckler

(1976). Assumptions for this model are:

1.Horizontal and slightly inclined pipelines (± 100)

2.Steady state

3.Zero end effects

4.The same pressure drop of gas and liquid phase

Taitel and Duckler Model (1976)

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The objective of the model is to determine the equilibrium liquid level in the pipeline,

hL, for a given set of flow conditions.

Taitel and Duckler Model (1976)Equilibrium Stratified Flow

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Momentum equation for gas phase:

Momentum equation for liquid phase

Combined momentum equation


-

-

1

1

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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


d

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The wall shear stresses for the liquid, the gas and the interface are:

In this model, it is assumed I =WG (smooth interface exists and vG >> vI).


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From equation (1) gives:

Defining the dimensionless variables:


2

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Equation (2) can be written in a dimensionless form:

X is called the Lockhart and Martinelli parameter

Y is an inclination angle parameter


= 0 3

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All the dimensionless variables are unique functions of


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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:

L = 993 kg/m3 G = 1.14 kg/m3

L = 0.68x10-3 kg/ms G = 1.9x10-5 kg/ms

Calculate the dimensionless liquid level and all the dimensionless parameters.


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For horizontal, Y = 0. From the graph,

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Calculating the dimensionless variables:

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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:

Taitel and Duckler Model (1976)Stratified to Non-stratified Transition (Transition A)

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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


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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:


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For an inclined pipe, the stratified to non-stratified transition can be determined in

the similar manner.

Or:

Where


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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


4

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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.

Taitel and Duckler Model (1976)Intermittent or Dispersed Bubble to Annular (Transition B)

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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 > 0.35, the flow pattern will be slug or dispersed-bubble flow.


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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.

Taitel and Duckler Model (1976)Stratified Smooth to Stratified Wavy (Transition C)

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For s = 0.01, K is defined as:

Taitel and Duckler Model (1976)Stratified Smooth to Stratified Wavy (Transition C)

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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.

Taitel and Duckler Model (1976)Intermittent to Dispersed-Bubble (Transition D)

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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:


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Assuming that R ~ W,

The transition to dispersed bubble flow will occur when FT > FB.

Nondimensional form:

where


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1. Determine the equilibrium liquid level and all the dimensionless parameters

2. Check the stratified to nonstratified transition boundary.

3. If the flow is stratified, check the stratified smooth to stratified wavy transition

4. If the flow is nonstratified, check the transition to annular flow

5. If the flow is not annular, check the intermittent to dispersed bubble transition

Taitel and Duckler Model (1976)Procedures for checking the flow pattern

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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:

L = 993 kg/m3 G = 1.14 kg/m3

L = 0.68x10-3 kg/ms G = 1.9x10-5 kg/ms

Calculate the dimensionless liquid level and all the dimensionless parameters.

Flow Pattern PredictionExample

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For horizontal, Y = 0. From the graph,


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Calculating the dimensionless variables:


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Check for stratified to non-stratified transition

The criterion is not satisfied; The flow is stable and stratified flow exists


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Check for stratified-smooth to stratified-wavy transition

The criterion is satisfied; The flow is stratified wavy.


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Two Phase Flow Modeling Prepared by: Tan Nguyen Two Phase Flow Modeling – PE 571 Chapter 3:...

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