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

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

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.

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

- Momentum equation for gas phase
- Momentum equation for liquid phase
- Combined momentum equation

1

1

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

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).

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

- From equation (1) gives
- Defining the dimensionless variables

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

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

- All the dimensionless variables are unique

functions of

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

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.

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

For horizontal, Y 0. From the graph,

Taitel and Duckler Model (1976)

Equilibrium Stratified Flow

Calculating the dimensionless variables

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

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

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

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

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

Taitel and Duckler Model (1976)

Stratified to Non-stratified Transition

(Transition A)

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.

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.

Taitel and Duckler Model (1976)

Intermittent or Dispersed Bubble to Annular

(Transition B)

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.

Taitel and Duckler Model (1976)

Stratified Smooth to Stratified Wavy (Transition

C)

- For s 0.01, K is defined as

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.

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

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

Taitel and Duckler Model (1976)

Intermittent to Dispersed-Bubble (Transition D)

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

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.

Flow Pattern Prediction

Example

Flow Pattern Prediction

Example

For horizontal, Y 0. From the graph,

Flow Pattern Prediction

Example

Calculating the dimensionless variables

Flow Pattern Prediction

Example

Check for stratified to non-stratified

transition The criterion is not satisfied

The flow is stable and stratified flow exists

Flow Pattern Prediction

Example

Check for stratified-smooth to stratified-wavy

transition The criterion is satisfied The

flow is stratified wavy.