Networks of pipelines for gas with nonconstant compressibility factor: stationary states

Abstract

For the management of gas transportation networks, it is essential to know how the stationary states of the system are determined by the boundary data. The isothermal Euler equations are an accurate pde-model for the gas flow through each pipe. A compressibility factor is used to model the nonlinear relationship between density and pressure that occurs in real gas in contrast to ideal gas. The gas flow through the nodes is governed by algebraic node conditions that require the conservation of mass and the continuity of the pressure. We examine networks that are described by arbitrary finite graphs and show that for suitably chosen boundary data, subsonic stationary states exist and are uniquely determined by the boundary data. Our construction of the stationary states is based upon explicit representations of the stationary states on each single pipe that can easily be evaluated numerically. We also use the monotonicity properties of these states as functions of the boundary data.

Appendix

For the convenience of the reader Tables 1, 2, 3, 4, 5, 6 and 7 contain the data used in the numerical examples presented in Figs. 8 and 9. The constant $\alpha ^e$ in the compressibility factor model is computed via (see e.g., Schmidt et al. 2014)

$$\begin{aligned} \alpha ^e=\frac{0.257}{p_c}-0.533\frac{T_c}{p_cT^e}. \end{aligned}$$

We have used the formula of Chen (s. Cerbe 2008), which is an explicit estimate for the Colebrook–White law, to calculate realistic values for $\theta ^e=\frac{\lambda ^e_{\text {fric}}}{D^e}$. The equations for the Reynolds number and the friction factor read as follows:

$$\begin{aligned} Re^e&:=\frac{q^eD^e}{\eta _v},\\ \frac{1}{\sqrt{\lambda _{\text {fric}}^e}}&=-2\log _{10}\left( \frac{\frac{k^e}{D^e}}{3.7065}-\frac{5.0425}{Re^e}\log _{10}\left[ \frac{\left( \frac{k^e}{D^e}\right) ^{1.1098}}{2.8257}+\frac{5.8506}{(Re^e)^{0.8981}}\right] \right) . \end{aligned}$$

Since for the stationary states $q^e$ is constant along each pipe, this leads to constant friction factors $\lambda ^e_{\text {fric}}$ along each pipe. In all examples, the dynamic viscosity is $\eta _v:=11.9\times 10^{-6}~\mathrm{kg\,m}^{-1}~\mathrm{s}^{-1}.$ Note that our analysis does not directly cover the dependency of the friction factor on $q^e,$ but it poses no problems for numerical calculations. For trees, there is no difference between a flow-dependent and a flow-independent friction law. Most of the results presented in this article only rely on the monotonicity property $\partial _{q_e} p^e<0$ (see Remark 2 for a sufficient condition to ensure this property).

Table 1 Values for single pipe examples

Table 2 Global values for the tree network example

Table 3 Pipe data for the tree network example

Table 1 contains the constants used for the single pipe computations in Figs. 1 and 2. Table 2 contains the globally used constants and Table 3 the constants for each pipe for the example of a tree network depicted in Fig. 5. The boundary values for the tree example are presented in Table 4. The globally used constants for the diamond graph (see Figs. 8, 9) are contained in Table 5. The pipe data for this example is presented in Table 6 and its boundary flow values in Table 7.

Table 4 Given boundary data for the tree network example

Table 5 Global values for the diamond graph example

Table 6 Pipe data for the diamond graph example

Table 7 Given boundary data for the diamond graph example