1. Two Phase Flow Modeling Prepared by: Tan Nguyen Two Phase Flow Modeling – PE 571 Chapter 3: Slug Flow Modeling Dukler and Hubbard – Horizontal Pipes

2. Two Phase Flow Modeling Prepared by: Tan Nguyen Dukler and Hubbard Model (1975) Introduction

3. Two Phase Flow Modeling Prepared by: Tan Nguyen Slug flow occurs in horizontal, inclined, and vertical pipes. SF and elongated bubble flow belong to the intermittent pattern. SF Characterized by an alternating flow of gas pockets and liquid slugs. The large gas pockets are called Taylor bubbles. The slugs are liquid which contains small entrained gas bubbles Introduction Dukler and Hubbard Model (1975)

4. Two Phase Flow Modeling Prepared by: Tan Nguyen L U : Unit length of the slug L S, L F : Length of the slug and the liquid film v TB : translational velocity v LLS and v GLS : velolities of liquid and gas phase in the slug body. v LTB and v GTB : liquid film and gas- pocket velocity in the stratified region v TB > v LLS > v GLS > v LTB > v GTB Introduction Dukler and Hubbard Model (1975)

5. Two Phase Flow Modeling Prepared by: Tan Nguyen Liquid slugs bridge the entire pipe cross-sectional area. They move at relatively high velocity (close to the mixture velocity) and overruns the slow moving film ahead of it, picks it up and accelerates it to the slug velocity creating a turbulent mixing zone in the front of the slug. At the same time, the gas pocket pushes into the slug, causing the slug to shed liquid from its back creating the film region. For steady state flow, the rate of pickup is equal to the rate of shedding. Mechanism of Slug Flow Dukler and Hubbard Model (1975)

6. Two Phase Flow Modeling Prepared by: Tan Nguyen  s and H LLS are the slug frequency and the liquid holdup in the slug body. Assuming homogeneous no-slip model flow occurs in the slug body. Input and Output Parameters Dukler and Hubbard Model (1975)

7. Two Phase Flow Modeling Prepared by: Tan Nguyen The total pressure drop across a slug unit consists of two components: Accelerational pressure drop in the mixing zone: due to  v: slug and liquid film Frictional pressure drop in the slug body: due to shear with the wall Pressure drop in the stratified region behind the slug is neglected. Total pressure drop gradient in a unit slug Total Pressure Drop in a Slug Unit Dukler and Hubbard Model (1975)

8. Two Phase Flow Modeling Prepared by: Tan Nguyen The pickup rate x, (mass/time): is the rate of mass picked up by the slug body from the film zone. The force acting on the picked-up mass equals to the rate of change of momentum: F = x(v S - v LTBe ) Hence, the pressure drop due to the acceleration is given Accelerational Pressure Drop Dukler and Hubbard Model (1975)

9. Two Phase Flow Modeling Prepared by: Tan Nguyen This pressure drop is due to the shear between the moving slug body and the pipe wall. Note that the flow in the slug body is assumed to be homogeneous no-slip flow with a fully developed turbulent velocity profile. Frictional Pressure Drop Dukler and Hubbard Model (1975)

10. Two Phase Flow Modeling Prepared by: Tan Nguyen v S is the slug velocity representing the mean velocity of the fluid in the slug body v TB is the translational velocity which is the front velocity of the slug. Velocities of the slug Dukler and Hubbard Model (1975)

11. Two Phase Flow Modeling Prepared by: Tan Nguyen v S is the slug velocity representing the mean velocity of the fluid in the slug body v TB is the translational velocity which is the front velocity of the slug. Velocities of the slug Dukler and Hubbard Model (1975)

12. Two Phase Flow Modeling Prepared by: Tan Nguyen The tractor moves at a velocity of v S, scooping the sand ahead of it. The sand is accumulated in the front of the scoop. The front of the scooped sand moves faster than v S. The front velocity of the sand is equal to the tractor velocity plus the volumetric-scooping rate divided by the cross-sectional area of the scoop. In other words, the translational velocity, v TB, is equal to the slug velocity, v S, plus the volumetric-scooping rate divided by the cross-sectional area of the slug (additional velocity gained by the pickup process). Velocities of the slug Dukler and Hubbard Model (1975)

13. Two Phase Flow Modeling Prepared by: Tan Nguyen Assuming that the total volumetric flow of the mixture is constant through any cross section of the pipe. Note that the total mass rate, W L + W G, is not constant at any cross section of the pipe because of the intermittent nature of the flow. q L + q G = constant Velocities of the slug Dukler and Hubbard Model (1975)

14. Two Phase Flow Modeling Prepared by: Tan Nguyen Choosing a coordinate system moving at the translationnal velocity, v TB, The continuity equation implies that the rate of pickup equals to the rate of shedding: Defining c as Therefore: v TB = v S + cv S = (1 + c)v S = c O v S. Velocities of the slug Dukler and Hubbard Model (1975)

15. Two Phase Flow Modeling Prepared by: Tan Nguyen The parameter c can be proved that it is a unique function of the Reynolds number Re LS. Re LS =  L v M d/  L. c = 0.021ln(Re LS ) + 0.022 Velocities of the slug Dukler and Hubbard Model (1975)

16. Two Phase Flow Modeling Prepared by: Tan Nguyen If we choose the interface of the slug as the coordinate, then The liquid will flow backwards in the slug body at a velocity of v TB – v S. The liquid film will flow backward with a velocity of v TB – v LTB. Note that the v F increases as the cross-sectional area of the film decreases. Hydrodynamics of the Film Dukler and Hubbard Model (1975)

17. Two Phase Flow Modeling Prepared by: Tan Nguyen The following analysis is carried out with an open channel flow. Assuming the pressure drop in the stratified region is neglected. The velocity of the liquid film: Hydrodynamics of the Film Dukler and Hubbard Model (1975)

18. Two Phase Flow Modeling Prepared by: Tan Nguyen Note that is the average hydrostatic pressure acting on a cross sectional area of the liquid film. Hence, the film profile is given Hydrodynamics of the Film Dukler and Hubbard Model (1975)

19. Two Phase Flow Modeling Prepared by: Tan Nguyen Where the shear stress force is given The equilibrium level, h E, occurs when The critical level, h C, occurs when Hydrodynamics of the Film Dukler and Hubbard Model (1975)

20. Two Phase Flow Modeling Prepared by: Tan Nguyen The slug unit period, T U, is the time it takes for a slug unit to pass a given point in the pipe, is given by the inverse of the slug frequency,  S : Slug Length Dukler and Hubbard Model (1975)

21. Two Phase Flow Modeling Prepared by: Tan Nguyen There are two different ways to carry out the mass balance for a slug unit: Integration with space: “freezing” a slug unit at a given time and checking the liquid Integration with time: Integrating the amount of liquid passing through a cross sectional area of the pipe at a given point along the pipe. Slug Length Dukler and Hubbard Model (1975) TFTF

22. Two Phase Flow Modeling Prepared by: Tan Nguyen Definition of pickup rate and relationship between v TB and v S : v TB = v S + cv S = (1 + c)v S = c O v S. Combining these two equations and assuming equilibrium liquid film: H LTB = H LTBe. Let Slug Length Dukler and Hubbard Model (1975)

23. Two Phase Flow Modeling Prepared by: Tan Nguyen The mass balance equation by applying the integration with time give This is equation can be simplified by using the assumption: equilibrium liquid film. Combining with the correlationgives Slug Length Dukler and Hubbard Model (1975)

24. Two Phase Flow Modeling Prepared by: Tan Nguyen The gas pocket velocity can be obtained from a mass balance on the gas phase with using the translational velocity coordinate system between two planes: This eq. implies that the rate of pickup = the rate of shedding for gas phase. Hence Gas Pocket Velocity Dukler and Hubbard Model (1975)

25. Two Phase Flow Modeling Prepared by: Tan Nguyen The length of the mixing zone is based on a correlation for the “velocity head” v H as follows Length of Mixing Zone Dukler and Hubbard Model (1975)

26. Two Phase Flow Modeling Prepared by: Tan Nguyen 1. Specify input parameters: W L, W G, d, fluid properties, H LLS and  S. 2. Calculate the slug velocity, v S : 3. Determine c: c = 0.021ln(Re LS ) + 0.022 4. Assume a value for L S, calculate L F : 5. Integrate numerically Eq. below from z = 0 - L and find H LTB (z), v LTB (z), H LTBe, and v LTBe Calculation Procedure Dukler and Hubbard Model (1975)

27. Two Phase Flow Modeling Prepared by: Tan Nguyen 6. Calculate L s from 7. Compare the assumed and calculated values of L S. If no convergence is reached, update L S and repeat steps 4 through 7 8. Once the convergence is reached, calculate the following outputs: L S, L F, L U, v S, v TB, v LTB (z), H LTB (z), and H LTBe v LTBe : from the final results of the integration -  p A from: Re S, f S, -  P F, -  p U, and –dp/dL Calculation Procedure Dukler and Hubbard Model (1975)

28. Two Phase Flow Modeling Prepared by: Tan Nguyen Calculation Procedure Dukler and Hubbard Model (1975)

29. Two Phase Flow Modeling Prepared by: Tan Nguyen Calculation Procedure Dukler and Hubbard Model (1975)