A model and topological analysis procedures for a pipeline network of variable connectivity

doi:10.1016/j.advengsoft.2012.02.013

Advances in Engineering Software

Volume 48, June 2012, Pages 40-51

https://doi.org/10.1016/j.advengsoft.2012.02.013 Get rights and content

Abstract

In the paper, a model is firstly formulated for a pipeline network of variable connectivity in the terminology of graph theory. Through analyzing the topological changes of the pipeline network caused by connecting or disconnecting a pipe, several procedures are then proposed to construct the incidence matrix and fundamental circuit matrix of a graph directly from those of its parent graph or graphs without the time-consuming inversion of the corresponding incidence matrix. Thirdly, the proposed model and topological analysis procedures are used to establish a dynamic solver for a tank farm together with the chord flow method of Rahal (1995) [3]. Finally, the dynamic solver is applied to a tank farm of liquor for verifying the model and procedures proposed in this paper.

Highlights

► Propose a general framework for modeling pipeline networks of variable connectivity. ► Develop several procedures for efficient identification of a network’s topology. ► Build a high-speed dynamic solver for a tank farm. ► Apply the dynamic solver to the simulation of a tank farm.

Introduction

Pipeline networks are the major facilities of water, air and heating supply systems. They are also important integral parts of large-scale chemical, petrochemical, petroleum refining and processing plants and various tank farms. Their safe and efficient operation has a great significance for both themselves and other pieces of equipment connected by them. For design and operation studies with simulation, many efforts such as Martinez-Benet and Puigjaner [1], Nielsen [2], Rahal [3], Álvarez et al. [4], and Ivanov and Bournaski [5], have been devoted to the development of steady state and dynamic mathematical models for municipal water distribution networks. To get the pressure and flow dynamics in a pipeline network, the pseudo-steady state models for slow transients, the water hammer models for rapid transients, or models hybrid of the above two kinds are employed. These models involve sets of rigid differential equations which are usually difficult to be solved.

In modeling the dynamics of chemical processes or tank farms, pipeline networks are different from those in water distribution systems: (1) the scale of networks is usually much smaller; (2) the dynamics of networks is neglected since the overall dynamics is usually governed by equipment units other than the networks themselves. Therefore, a static solver for a pipeline network can be used as the dynamic simulator for the corresponding chemical process or tank farm with the pressures at nodes corresponding to tanks or other liquid accumulators changing with time and operations according to the level equations. In composing the static solver of this paper, the chord flow method of Rahal will be employed for its high execution speed, low demand for memory, and insensitivity to initial values.

Such a static or dynamic solver works only for a pipeline network or sub-network which can be expressed by a connected graph or subgraph. In dynamic simulation, however, the connectivity of a network may be changed due to connecting and disconnecting pipes by the operations of their affiliated valves, switches or pumps. A network may split into sub-networks with smaller scales, and sub-networks may split further or merge into another sub-network with a larger scale. That is, the overall graphical topology of a pipeline network changes, and more than one connected subgraphs are produced. To apply Rahal’s method to such a pipeline network, topological analysis is necessary to identify the involved connected sub-networks and the corresponding incidence matrices and fundamental circuit matrices.

In this paper, we use the following strategy to analyze the topology of a pipeline network of variable connectivity: (1) operations of connecting or disconnecting pipes are treated one by one. (2) For each operation, procedures are suggested to derive the incidence matrix and the fundamental circuit matrix of a graph or subgraph directly from those of its parent subgraphs or graph, without the time-consuming inversion of the incidence matrix for determining the fundamental circuits.

The rest of this paper is organized as follows: in Section 2, a general framework is constructed for modeling a pipeline network of variable connectivity. In Section 3, procedures are proposed for analyzing the topological changes of a pipeline network caused by the connection or disconnection of a pipe. In Section 4, a static solver based on the chord flow method of Rahal is detailed for a connected sub-network. Based the topological analysis procedures and the static solver, a dynamic solver is developed for a tank farm in Section 5. Section 6 is a demonstration for verifying the proposed modeling framework, procedures for topological analysis, and dynamic solver. Conclusions are drawn in Section 7.

Section snippets

Model for a pipeline network of variable connectivity

By a pipeline network of variable connectivity, we mean primarily a pipeline network of which the connections of the pipes can be changed by connecting and disconnecting pipes through the operations of their affiliated valves, switches or pumps, though connectivity also is a measure of the fluid-passing capability of a pipe related to the opening degree(s) of some valve(s). Obviously, the connectivity of such a network is certain at each instant of time. Therefore, such a pipeline network is

Incidence matrix

The incidence matrix and the supplemented incidence matrix are used to express the connection of pipes in a network. They have a typical element $A_{ij} = Γ_{ij} = \{\begin{matrix} 1 & if N_{i} is the source node of B_{j} \\ - 1, & if N_{i} is the destination node of B_{j} \\ 0, & if N_{i} and B_{j} are not connected \end{matrix}$

At this stage, consider all the pipes in the network are connected, namely, all the valves and switches are open and all the pumps run, and all the pseudo-arcs are included. Then the incidence matrix and the supplemented incidence matrix are respectively

Static solver for a connected pipeline network

Without loss of generality, we take graph G defined previously as an exmple to state the solution method for connected graphs and subgraphs. The system equations for graph G includes: (i) the mass balance about each of all the n nodes: $AQ = d$ with $Q = {Q_{k_{1}}, Q_{k_{2}}, \dots, Q_{k_{b + r - 1}}}^{T}$ $d = {d_{i_{1}}, d_{i_{2}}, \dots, d_{i_{n - 1}}}^{T}$ and (ii) the pressure balance around each of all the (b-n + r) fundamental circuits: $C Δ P = 0$ with $Δ P = {Δ P_{k_{1}}, Δ P_{k_{2}}, \dots, Δ P_{k_{b + r - 1}}}^{T}$

It is noted that the flowrates along b arcs and (r − 1) pseudo-arcs (Q) are uniquely determined by (

Dynamic solver for a tank farm

As stated in Section 2, a tank farm can be viewed as a pipeline network with the pressures of the fixed head nodes specified externally or calculated by Eqs. (2), (3). Therefore, the dynamic model for a tank farm is a system of differential and algebraic equations (DAEs). Assume that the volume of the involved tanks be much greater than the space enclosed by the pipes, we have the following strategy for solving the dynamic model of a tank farm:

Step 1. Specify the density and viscosity of the

Simulation of a liquor tank farm

As a demonstration, a liquor tank farm shown in Fig. 7 is simulated with the dynamic solver as described in the above section. In Fig. 7, there are four tanks (T-01–04) with the same cross area of 28.26 m² and initial levels respectively of 7, 9, 11 and 13 m, two feed pumps (P01A,B) and four discharge pumps (P02A–D), four PI regulators (FC02A–D), and four control valves (V1–4). The four PI regulators have the same proportional f_p = 1, integral time τ_I = 50Δt (Δt is the time step of simulation, s),

Conclusions

In summary, we have the concluding remarks below:

(1)
A model is established for a pipeline network of variable connectivity.
(2)
Topological analysis procedures are constructed for deriving the incidence matrix and fundamental circuit matrix of a graph or subgraph directly from those of its parent subgraphs or graph, without the time-consuming inversion of the corresponding incidence matrix.
(3)
By neglecting the dynamics of the involved pipeline network, a dynamic solver for a tank farm is formulated based

Acknowledgements

The work of this paper is supported by China National Hi-tech (863) Project “Enterprise-Wide Simulation and Optimization Environment for Process Industry” (Project Number: 2007AA04Z191).

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