Research Papers: Fundamental Issues and Canonical Flows

Energy Dissipation Effect in the One-Dimensional Limit of the Energy Equation in Turbulent Compressible Flow

[+] Author and Article Information
Tor Ytrehus

Department of Energy and Process Engineering,
Norwegian University of Science
and Technology,
7491 Trondheim, Norway e-mail: tor.ytrehus@ntnu.no

Jan Fredrik Helgaker

Polytec Research Institute,
5527 Haugesund, Norway;
Department of Energy and Process Engineering,
Norwegian University of Science
and Technology,
7491 Trondheim, Norway
e-mail: jan.fredrik.helgaker@polytec.no

1Corresponding author.

Manuscript received July 6, 2012; final manuscript received January 23, 2013; published online April 8, 2013. Assoc. Editor: Ye Zhou.

J. Fluids Eng 135(6), 061201 (Apr 08, 2013) (8 pages) Paper No: FE-12-1318; doi: 10.1115/1.4023656 History: Received July 06, 2012; Revised January 23, 2013

The transportation of natural gas through high pressure transmission pipelines has been modeled by numerically solving the conservation equations for mass, momentum, and energy for one-dimensional compressible viscous heat conducting flow. Since the one-dimensional version is a result of averages over the pipe cross-section and the flow is normally turbulent, the order of averaging in space and time is an issue; in particular, for the dissipation term. The Reynolds decomposition and time averaging should be performed first, followed by the contraction to the one-dimensional version by the cross-sectional averaging. The result is a correction factor, which is close to unity, on the usual expression of the dissipation term in the energy equation. This factor will, to some extent, affect the temperature distribution along the pipeline. For low Reynolds numbers (Re104) it reduces the dissipation by as much as 7%, irrespective of roughness. For high Reynolds numbers (Re ≥ 107) and roughness in the high range of the micron decade, the dissipation is increased by 10%. If the pipeline is also thermally isolated such that the flow can be considered adiabatic, the effect of turbulent dissipation gains further importance.

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Fig. 1

Velocity profile of the inner wall in the inner variables u+ and y+. The figure taken from White [14] (p. 420).

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Fig. 2

Coordinate system used in integrating Eq. (17) over the pipe cross-section

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Fig. 3

Correction factor F as a function of the Reynolds number, surface roughness, and pipe diameter. Results are given for values of ε/D * 106 equal to 20, 10, 5, 3, and 1. Values for F have been computed from Eq. (27).

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Fig. 4

Local errors for p, m·, and T as a function of the grid points N

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Fig. 5

Outlet boundary condition for the mass flow for the smooth pipe

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Fig. 6

Results for the smooth pipe: (top) outlet pressure, and (bottom) inlet mass flow

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Fig. 7

Outlet temperature for a smooth pipe with and without the correction factor at Re = 3 × 107 and ε/D = 10−6. The difference in the outlet temperature is approximately 0.1 °C.

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Fig. 8

Boundary conditions: (left) inlet mass flow (f1(t)), and (right) inlet temperature (f2(t))

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Fig. 9

The Reynolds number and turbulent dissipation factor at the pipe outlet

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Fig. 10

Outlet temperature as a function of time with and without correcting the dissipation term

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Fig. 11

Temperature profile along the pipeline at t = 20 h with and without correcting the dissipation term

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Fig. 12

The difference between temperatures for the corrected and noncorrected dissipation term for the heat conducting (U = 2.5) and adiabatic (U = 0) cases

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Fig. 13

(left) Modeled inlet pressure, and (right) modeled outlet mass flow




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