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TECHNICAL PAPERS

# A Finite-volume Approach for Simulation of Liquid-column Separation in Pipelines

[+] Author and Article Information
Mark A. Chaiko

PPL Corporation, Two North Ninth Street, GENPL5, Allentown, PA 18101machaiko@pplweb.com

J. Fluids Eng 128(6), 1324-1335 (Mar 26, 2006) (12 pages) doi:10.1115/1.2353271 History: Received May 20, 2005; Revised March 26, 2006

## Abstract

A finite-volume approach, based on the MUSCL-Hancock method, is presented and applied to liquid-column separation transients in pipelines. In the mathematical model, sudden closure of a valve on the downstream end of a pipeline initiates the hydraulic transient, while a head tank maintains constant upstream pressure. The two-phase fluid is treated as a homogeneous mixture, and changes in fluid pressure are assumed to occur at constant entropy. Effects of pipe elasticity on wave propagation speed are included in the model by coupling the circumferential stress-strain relation for the pipe wall to the local fluid pressure. In regions of the domain where the solution is smooth, second-order accuracy is achieved by means of data reconstruction based on sloping-difference formulas. Slope limiting prevents the development of spurious oscillations in the neighborhood of steep wave fronts. Data reconstruction leads to a piece-wise linear representation of the solution, which is discontinuous across cell boundaries. In order to advance the solution in time, a Riemann problem is solved on each cell junction to obtain mass and momentum flux contributions. A splitting technique, which separates flux terms from gravity and frictional effects, allows the compatibility equations for the Riemann problem to be expressed as total differentials of fluid velocity and an integral that depends only on the fluid pressure. Predictions for large-amplitude pressure pulses caused by liquid-column separation and rejoining are compared against experimental data available in the literature. Amplitude and timing of the predicted pressure response shows reasonably good agreement with experimental data even when as few as 20–40 computational cells are used to describe axial variations in fluid conditions along a pipeline of approximately $35m$ in length. An advantage of the current method is that it does not give rise to spurious oscillations when grid refinement is performed. The presence of nonphysical oscillations has been a drawback of a commonly used method based on discrete vapor cavities and characteristic treatment of wave propagation.

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## Figures

Figure 1

Simplified schematic of experimental apparatus used by Simpson and Wylie for study of liquid-column separation and rejoining (6)

Figure 2

Specified downstream fluid velocity V(t) in Eq. (8) derived from experimental data for Run 1 defined in Table 1

Figure 3

Finite-volume representation of computational domain

Figure 6

Comparison of finite-volume calculation of pressure at valve (z=L=36m) against experiment for initial fluid velocity of 1.125m∕s and reservoir pressure head of 21.74m (Run 2)

Figure 7

Comparison of finite-volume calculation of pressure at z=L∕4=9m against experiment for initial fluid velocity of 1.125m∕s and reservoir pressure head of 21.74m (Run 2)

Figure 8

Finite-volume calculation of void fraction at center of computational cell adjacent to downstream valve (Run 2 with 80 computational cells)

Figure 9

Comparison of DVCM calculation of pressure at valve (z=L=36m) against experiment for initial fluid velocity of 0.332m∕s and reservoir pressure head of 23.41m (Run 1)

Figure 10

Comparison of finite-volume calculation of pressure at z=L=37.2m against experiment for initial fluid velocity of 0.30m∕s and reservoir pressure head of 22m (Run 3)

Figure 11

Comparison of finite-volume calculation of pressure at z=L∕2=18.6m against experiment for initial fluid velocity of 0.30m∕s and reservoir pressure head of 22m (Run 3)

Figure 12

Comparison of finite-volume calculation of pressure against experiment for initial fluid velocity of 0.20m∕s and reservoir pressure head of 32m (Run 4)

Figure 4

Comparison of finite-volume calculation of pressure at valve (z=L=36m) against experiment for initial fluid velocity of 0.332m∕s and reservoir pressure head of 23.41m (Run 1)

Figure 5

Comparison of finite-volume calculation of pressure at z=L∕4=9m against experiment for initial fluid velocity of 0.332m∕s and reservoir pressure head of 23.41m (Run 1)

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