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

Two Phase Flow Modeling Prepared by: Tan Nguyen 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 (± 10 0 ) 2.Steady state 3.Zero end effects 4.The same pressure drop of gas and liquid phase Taitel and Duckler Model (1976)

Two Phase Flow Modeling Prepared by: Tan Nguyen The objective of the model is to determine the equilibrium liquid level in the pipeline, h L, for a given set of flow conditions. Taitel and Duckler Model (1976) Equilibrium Stratified Flow

Two Phase Flow Modeling Prepared by: Tan Nguyen Momentum equation for gas phase: Momentum equation for liquid phase Combined momentum equation Taitel and Duckler Model (1976) Equilibrium Stratified Flow - -  1 1

Two Phase Flow Modeling Prepared by: Tan Nguyen The respective hydraulic diameters of the liquid and gas phases are given The Fanning friction factor for each phase: Where C L = C G = 16 and m = n = 1 for laminar flow and C L = C G = and m = n = 0.2 for turbulent flow Taitel and Duckler Model (1976) Equilibrium Stratified Flow d

Two Phase Flow Modeling Prepared by: Tan Nguyen 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 v G >> v I ). Taitel and Duckler Model (1976) Equilibrium Stratified Flow

Two Phase Flow Modeling Prepared by: Tan Nguyen From equation (1) gives: Defining the dimensionless variables: Taitel and Duckler Model (1976) Equilibrium Stratified Flow 2

Two Phase Flow Modeling Prepared by: Tan Nguyen 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 = 0 3

Two Phase Flow Modeling Prepared by: Tan Nguyen All the dimensionless variables are unique functions of Taitel and Duckler Model (1976) Equilibrium Stratified Flow

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Equilibrium Stratified Flow

Two Phase Flow Modeling Prepared by: Tan Nguyen Example: a mixture of air-water flows in a 5-cm-ID horizontal pipe. the flow rate of the water is q L = m3/hr and that of the air is q G = 21.2 m 3 /hr. The physical properties of the fluids are given as:  L = 993 kg/m 3  G = 1.14 kg/m 3  L = 0.68x10 -3 kg/ms  G = 1.9x10 -5 kg/ms Calculate the dimensionless liquid level and all the dimensionless parameters. Taitel and Duckler Model (1976) Equilibrium Stratified Flow

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Equilibrium Stratified Flow

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Equilibrium Stratified Flow For horizontal, Y = 0. From the graph,

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Equilibrium Stratified Flow Calculating the dimensionless variables:

Two Phase Flow Modeling Prepared by: Tan Nguyen 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen The condition for wave growth, leading to instability of the stratified configuration, is when the suction force is greater than the gravity force: Where C 1 depends on the wave size: Taitel and Duckler Model (1976) Stratified to Non-stratified Transition (Transition A)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen Approximately, c 2 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) 4

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Stratified to Non-stratified Transition (Transition A)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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. Taitel and Duckler Model (1976) Intermittent or Dispersed Bubble to Annular (Transition B)

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Intermittent or Dispersed Bubble to Annular (Transition B)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen For s = 0.01, K is defined as: Taitel and Duckler Model (1976) Stratified Smooth to Stratified Wavy (Transition C)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen The net buoyancy forces acting on the gas pocket (A G : gas pocket cross sec. area): The turbulence forces acting on the gas pocket (S I : 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)

Two Phase Flow Modeling Prepared by: Tan Nguyen Assuming that  R ~  W, The transition to dispersed bubble flow will occur when F T > F B. Nondimensional form: where Taitel and Duckler Model (1976) Intermittent to Dispersed-Bubble (Transition D)

Two Phase Flow Modeling Prepared by: Tan Nguyen Taitel and Duckler Model (1976) Intermittent to Dispersed-Bubble (Transition D)

Two Phase Flow Modeling Prepared by: Tan Nguyen 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

Two Phase Flow Modeling Prepared by: Tan Nguyen Example: a mixture of air-water flows in a 5-cm-ID horizontal pipe. the flow rate of the water is q L = m3/hr and that of the air is q G = 21.2 m 3 /hr. The physical properties of the fluids are given as:  L = 993 kg/m 3  G = 1.14 kg/m 3  L = 0.68x10 -3 kg/ms  G = 1.9x10 -5 kg/ms Calculate the dimensionless liquid level and all the dimensionless parameters. Flow Pattern Prediction Example

Two Phase Flow Modeling Prepared by: Tan Nguyen Flow Pattern Prediction Example

Two Phase Flow Modeling Prepared by: Tan Nguyen For horizontal, Y = 0. From the graph, Flow Pattern Prediction Example

Two Phase Flow Modeling Prepared by: Tan Nguyen Calculating the dimensionless variables: Flow Pattern Prediction Example

Two Phase Flow Modeling Prepared by: Tan Nguyen Check for stratified to non-stratified transition The criterion is not satisfied; The flow is stable and stratified flow exists Flow Pattern Prediction Example

Two Phase Flow Modeling Prepared by: Tan Nguyen Check for stratified-smooth to stratified-wavy transition The criterion is satisfied; The flow is stratified wavy. Flow Pattern Prediction Example