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Paulo M. Coelho and Carlos Pinho

Paulo M. Coelho
pmc@fe.up.pt University of Porto Rua Dr. Roberto Frias, s/n 4200-465 Porto, Portugal

Considerations About Equations for Steady State Flow in Natural Gas Pipelines
In this work a discussion on the particularities of the pressure drop equations being used in the design of natural gas pipelines will be carried out. Several versions arepresented according to the different flow regimes under consideration and through the presentation of these equations the basic physical support for each one is discussed as well as their feasibility. Keywords: natural gas flow , pressure drop, gas pipelines, Renouard

Carlos Pinho
Member, ABCM ctp@fe.up.pt CEFT-DEMEGI - Faculty of Engineering University of Porto Rua Dr. Roberto Frias, s/n 4200-465Porto, Portugal

1 The design of gas pipelines and networks is commonly presented through a series of numerical procedures and recommendations, and usually flow equations are recommended by the several authors according to common design and calculation practice, without a deep analysis of the basic physical reasoning that is behind each one of such equations. In this work adiscussion on the particularities of the pressure drop equations being used in the design of gas pipelines will be carried out and several versions presented. The development of the flow equation is commonly found in several books and publications in Fluid Mechanics or connected to industrial gas utilization technologies (Pritchard et al., 1978; Katz and Lee, 1990), consequently it is useless to presentonce again such derivation. The reader can consult the work of Mohitpour et al. (2000) where such analysis is presented for steady and unsteady state compressible fluid flow.

Tst = standard temperature, 288.15 K zavg = gas compressibility factor, dimensionless zst = compressibility factor at standard conditions, zst ≈ 1 Greek Symbols

α = coefficient, dimensionless β = coefficient,dimensionless ∆max = maximum variation of the friction factor, dimensionless ∆P = pressure drop, Pa ε = wall roughness, m η = efficiency factor, dimensionless µ = gas dynamic viscosity, Pa s ν = gas cinematic viscosity, m2/s ρ = gas density, kg/m3 τ = shear stress, Pa
Subscripts air air app apparent avg average cg city gas cr critical ent entrance st standard sta standard for air w wall 1 relative to thegeneric point 1 2 relative to the generic point 2

C, C’ = generic constant d = gas relative density, dimensionless D = internal diameter of pipe, m E = potential energy term, Eq.(34), Pa2 f = Darcy friction coefficient, dimensionless g = gravitational acceleration, m/s2 H = height of points 1 and 2, m K = constant, dimensionless L = pipe length, m M = molecular mass, kg/kmol n =exponent for the gas flow rate (range of values between 1.74 and 2) P, P’ = absolute pressure, Pa P1 = absolute pressure at pipe entrance, Pa P2 = absolute pressure at pipe exit, Pa Pavg = flow average pressure, Pa Pst =standard pressure, 1.01325×105 Pa & Qst = volume gas flow rate at standard conditions, m3/s R = flow resistance per unit length of pipe R = universal gas constant, 8314.41 J/(kmol K)Re = Reynolds number of the gas flow, dimensionless T = absolute temperature, K Tavg = flow average temperature, K
Paper accepted April, 2006. Technical Editor: Clovis R. Maliska.

General Equation for Steady-State Flow
Considering the momentum equation applied to a portion of pipe of length dx, inside which flows a compressible fluid with an average velocity u, for example natural gas,assuming steady state conditions where ρ is the gas density, p is the gas absolute static pressure, A is the area of the pipe cross section (πD2/4) and dH represents a variation in high, the resultant differential equation is,
u du + dP + g dH + f dx u 2 =0 D 2 (1)


In the above equation f is the Darcy friction coefficient which is related with the wall shear stress by means of, f = 8 τw ρ u2...
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