2 SlugFlowModeling Dukler Hubbard 1975

Two Phase Flow Modeling

Prepared by: Tan Nguyen

Two Phase Flow Modeling – PE 571

Chapter 3: Slug Flow Modeling

Dukler and Hubbard – Horizontal Pipes

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Dukler and Hubbard Model (1975)Introduction

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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)

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LU: Unit length of the slug

LS, LF: Length of the slug and the

liquid film

vTB: translational velocity

vLLS and vGLS: velolities of liquid and

gas phase in the slug body.

vLTB and vGTB: liquid film and gas-

pocket velocity in the stratified region

vTB > vLLS > vGLS > vLTB > vGTB

Introduction


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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


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s and HLLS 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


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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


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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(vS - vLTBe)

Hence, the pressure drop due to the acceleration is given

Accelerational Pressure Drop


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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


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vS is the slug velocity representing

the mean velocity of the fluid in the

slug body

vTB is the translational velocity

which is the front velocity of the

slug.

Velocities of the slug


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vS is the slug velocity representing the mean velocity of the fluid in the slug body

vTB is the translational velocity which is the front velocity of the slug.



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The tractor moves at a velocity of vS, 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 vS. 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, vTB, is equal to the slug velocity, vS, plus

the volumetric-scooping rate divided by the cross-sectional area of the slug

(additional velocity gained by the pickup process).



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Assuming that the total volumetric flow of the mixture is constant through any cross

section of the pipe.

Note that the total mass rate, WL + WG, is not constant at any cross section of the

pipe because of the intermittent nature of the flow.

qL + qG = constant



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Choosing a coordinate system moving at the translationnal velocity, vTB, The

continuity equation implies that the rate of pickup equals to the rate of shedding:

Defining c as

Therefore: vTB = vS + cvS = (1 + c)vS = cOvS.



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The parameter c can be

proved that it is a unique

function of the Reynolds

number ReLS.

ReLS = LvMd/L.

c = 0.021ln(ReLS) + 0.022



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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 vTB – vS.

The liquid film will flow backward with a velocity of vTB – vLTB.

Note that the vF increases as the cross-sectional area of the film decreases.

Hydrodynamics of the Film


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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:



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Note that is the average hydrostatic pressure acting on a cross sectional area of

the liquid film.

Hence, the film profile is given



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Where the shear stress force is given

The equilibrium level, hE, occurs when

The critical level, hC, occurs when



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The slug unit period, TU, 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


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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


TF

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Definition of pickup rate and relationship between vTB and vS:

vTB = vS + cvS = (1 + c)vS = cOvS.

Combining these two equations and assuming equilibrium liquid film: HLTB = HLTBe.

Let

Slug Length


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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 correlation gives

Slug Length


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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


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The length of the mixing zone is based on a correlation for the “velocity head” vH as

follows

Length of Mixing Zone


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1. Specify input parameters: WL, WG, d, fluid properties, HLLS and S.

2. Calculate the slug velocity, vS:

3. Determine c: c = 0.021ln(ReLS) + 0.022

4. Assume a value for LS, calculate LF:

5. Integrate numerically Eq. below from z = 0 - L and find HLTB(z), vLTB(z), HLTBe,

and vLTBe

Calculation Procedure


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6. Calculate Ls from

7. Compare the assumed and calculated values of LS. If no convergence is

reached, update LS and repeat steps 4 through 7

8. Once the convergence is reached, calculate the following outputs:

LS, LF, LU, vS, vTB, vLTB(z), HLTB(z), and HLTBe

vLTBe: from the final results of the integration

- pA from:

ReS, fS, - PF, - pU, and –dp/dL



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2 SlugFlowModeling Dukler Hubbard 1975

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