Two Phase Flow Modeling – PE 571 Chapter 3: Stratified 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 t 1

5. Taitel and Duckler Model (1976) Equilibrium Stratified Flow d 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 >> vI).

7. Taitel and Duckler Model (1976) Equilibrium Stratified Flow 2 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 = 0 3

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/m3rG = 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) 4 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 > 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 > 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/m3rG = 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.