Steady-State Pipeline Network Problems Formulation

Steady-State Pipeline Network Problems Formulation 

A pipeline network is a collection of elements such as pipes, compressors, pumps, valves, regulators, heaters, tanks, and reservoirs interconnected in a specific way. The behavior of the network is governed by two factors (i) the specific characteristics of the elements and (ii) how the elements are connected together. The first factor is determined by the physical laws and the second by the topology of the network. 

Description and Characterization of Flow Networks 1. Graphs and Digraphs 

The mathematical abstraction of the topology of a pipeline network is called a graph which consists of a set of vertices (sometimes also referred to as nodes, junctions, or points) 

In this abstraction each edge corresponds to a pipeline network element and each vertex corresponds to a junction connecting two or more elements. It is often convenient to refer to the formal definition of a graph G as the sets 

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II. Steady-State Pipeline Network Problems Formulation.127... 

Equations (16) and (17) show the important link between fundamental cycles and cut-sets of a graph. Thus, a spanning tree provides a convenient starting point for formulating a consistent set of governing equations for steady-state pipeline network problems. 

We are now in a position to formulate the steady-state pipeline network problem based on the laws governing the behavior of the network and its elements. As it turns out, there is more than one way of formulating the problem, and since the computational efforts required for the solution are unequal, it behooves us to examine the ramifications of these formulations. 

In our treatment so far we have dwelt on the description, formulation, and specification of steady-state pipeline network problems. As we stated at the... 

Under all but laminar flow conditions, the steady-state pipeline network problems are described by mixed sets of linear and nonlinear equations regardless of the choice of formulations. Since these equations cannot be solved directly, an iterative procedure is usually employed. For ease of reference let us represent the steady-state equations as... 

In the treatment of steady-state pipeline network problems so far we have tacitly assumed that there is a unique solution for each problem. For certain types of networks the existence of a unique solution can indeed be rigorously established. The existence and uniqueness theorems for formulation C were proved by Duffin (DIO) and later extended by Warga (Wl). In Warga s derivation the governing relation for each network element assumes the form,... 

In Section II,C we have deliberately chosen a simple set of problem specifications for our steady-state pipeline network formulation. The specification of the pressure at one vertex and a consistent set of inputs and outputs (satisfying the overall material balance) to the network seems intuitively reasonable. However, such a choice may not correspond to the engineering requirements in many applications. For instance, in analyzing an existing network we may wish to determine certain input and output flow rates from a knowledge of pressure distribution in the network, or to compute the parameters in the network element models on the basis of flow and pressure measurements. Clearly, the specified and the unknown variables will be different in these cases. For any pipeline network how many variables must be specified And what constitutes an admissible set of specifications in... 

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