Transient and Compressible Flows in Pipeline Networks

Since the derivation of governing equations may be found elsewhere (S6, W6), we need only state them briefly in order to establish our notation and to point out some further simplifications. Let c be the isothermal sonic velocity, then the equation of state in terms of compressibility factor Z is 

The physical significance of the first term is obvious. The second term is the inertia term. The third term accounts for gravitational effects it vanishes for horizontal pipelines. The last term represents the frictional losses. 

In terms of transient behaviors, the most important parameters are the fluid compressibility and the viscous losses. In most field problems the inertia term is small compared with other terms in Eq. (128), and it is sometimes omitted in the analysis of natural gas transient flows (G4). Equations (123) and (128) constitute a pair of partial differential equations with p and W as dependent variables and t and x as independent variables. The equations are hyperbolic as shown, but become parabolic if the inertia term is omitted from Eq. (128). As we shall see later, the hyperbolic form must be retained if the method of characteristics (Section V,B,1) is to be used in the solution. 

Before we move on to a discussion of the methods of solution, we may note the two special cases of this general formulation. The special case of 

This case was studied by Nahavandi and Catanzaro (Nl) using an explicit forward difference formula with Ax set equal to the pipe length and the time step dictated by stability considerations. 

Making use of this relationship allows us to express the continuity equation as 

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