2 SlugFlowModeling Dukler Hubbard 1975
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Transcript of 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
Two Phase Flow Modeling
Prepared by: Tan Nguyen
Dukler and Hubbard Model (1975)Introduction
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)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
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)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
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)
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(vS - vLTBe)
Hence, the pressure drop due to the acceleration is given
Accelerational Pressure Drop
Dukler and Hubbard Model (1975)
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)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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).
Velocities of the slug
Dukler and Hubbard Model (1975)
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, WL + WG, is not constant at any cross section of the
pipe because of the intermittent nature of the flow.
qL + qG = constant
Velocities of the slug
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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.
Velocities of the slug
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Velocities of the slug
Dukler and Hubbard Model (1975)
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 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
Dukler and Hubbard Model (1975)
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)
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)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
Where the shear stress force is given
The equilibrium level, hE, occurs when
The critical level, hC, occurs when
Hydrodynamics of the Film
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
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)
TF
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
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 correlation gives
Slug Length
Dukler and Hubbard Model (1975)
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)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
The length of the mixing zone is based on a correlation for the “velocity head” vH as
follows
Length of Mixing Zone
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
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
Calculation Procedure
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
Calculation Procedure
Dukler and Hubbard Model (1975)
Two Phase Flow Modeling
Prepared by: Tan Nguyen
Calculation Procedure
Dukler and Hubbard Model (1975)