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![Fig. 1: Gas pipeline network in Europe [2].](profile/Jeroen-Stolwijk/publication/275828182/figure/fig1/AS:294589563981825@1447247033603/Gas-pipeline-network-in-Europe-2_Q320.jpg)
The presented work contains both a theoretical and a statistical error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations or the temperature dependent algebraic model. These equations are obtained by performing several simplifications of the full Euler equations, which model the gas flow through a pipeline...
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... plays a crucial role in the energy supply of Europe and the world. It is sufficiently and readily available, is traded, and is storable. After oil, natural gas is the second most used energy supplier in Germany, with a total share of 22.3 % of the energy consumption in 2013 [1]. The large European pipeline network that is used for the transportation of natural gas is depicted in Fig. 1. The high and probably increasing demand for gas calls for a mathematical modeling, simulation, and optimisation of the gas transport through the pipeline network. In this work the gas flow through a pipeline is considered as a one-dimensional problem, where the variable x runs along the length of the pipe. This flow is modelled using the Euler equations, which are a system of nonlinear hyperbolic partial differential equations. It describes the behaviour of compressible, non viscous fluids and ...
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... Furthermore, it is analyzed in subsection 5.3 under which condition the temperature dependent model can safely be simplied to the isothermal algebraic model. Further details and examples can be found in [43]. ...
... ,(39),(43), (46) to calculate the individual condition numbers (26) of the components p i R and q i L of the solutions x i in (38), (45) with respect to perturbations in the uncertain data. Using the parameter values in (37), (44) we obtain the results presented in ...
In the simulation and optimization of natural gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization, and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. Moreover, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.
... So we see that in this case our product solutions reduce to exponential travelling waves (as functions of (μ t + βx)) and thus the type of the solutions changes completely due to the change in the model. An error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations, is given in [16]. This is a stationary model, so the solutions are of a different type than the instationary states that we have considered in this paper. ...
The one–dimensional isothermal Euler equations are a well-known model for the flow of gas through a pipe. An essential part of the model is the source term that models the influence of gravity and friction on the flow. In general the solutions of hyperbolic balance laws can blow-up in finite time. We show the existence of initial data with arbitrarily large –norm of the logarithmic derivative where no blow up in finite time occurs. The proof is based upon the explicit construction of product solutions. Often it is desirable to have such analytical solutions for a system described by partial differential equations, for example to validate numerical algorithms, to improve the understanding of the system and to study the effect of simplifications of the model. We present solutions of different types: In the first type of solutions, both the flow rate and the density are increasing functions of time. We also present a second type of solutions where on a certain time interval, both the flow rate and the pressure decrease.
