1
Pipe Networks

1.1. Introduction

Pipe networks, which are interconnected and often have a netting structure, are used for transportation or even distribution of fluids from their storage or production areas to various other areas for certain purposes.

These networks are used in everyday life, such as plumbing networks of water transportation for productive applications such as fire extinguishing, networks of natural gas transportation, networks of fluid and gas transportation, and networks of water pipes, wet waste and compressed air.

Some of these networks are simple piping systems equipped with flow adjustment devices, whereas others are complicated, such as fluid distribution networks. Some of the most complicated networks are fire extinguishing networks (because of the fluid used for fire extinguishing), water distribution networks and plumbing networks.

Depending on the complication of mixed networks and the form they take for the feasibility of the distribution service, we distinguish them into the following three categories:

• 1) The tree-system-type pipe networks: these networks are characterized by the presence of a central pipe from which other pipes are branched (pipes of distribution) with a gradual decrease in their cross-section, taking the form of a tree (tree system), as shown in Figure 1.1.
• 2) The grid pattern: during development the pipes are formed into this pattern, resembling a chessboard, which covers the whole area of the distribution, with a decrease in their cross-sections with respect to the distance (grid system), as shown in Figure 1.2.

  Figure 1.1. Tree-system-type pipe network

  Figure 1.2. Grid pattern networks

• 3) The loop pattern: this pattern contains a central pipe, forming loops with smaller pipes at low flow rates (loop system), as shown in Figure 1.3.

Figure 1.3. Loop pattern network

Of the three forms of networks, the loop and grid systems show high reliability due to their flexibility, expandability and ability to offer multiple paths to the fluid.

In general, in all the aforementioned network systems, we distinguish three groups of pipelines:

• 1) Transportation lines: these pipes transfer the fluid from the storage or production area to the distribution area.
• 2) Central pipes: these pipes transfer the fluid to the target area, e.g. transfer of water to a town or village or transfer of natural gas to an industrial installation.
• 3) Supply lines: these pipes of small diameters transfer the fluid from central pipes to the users.

Thus, the entire distribution system consists of pipes, valves and pumps. The fundamental aim of a network system is to supply sufficient amounts of fluid to target areas with desired pressures and flow rates. Therefore, the choice of materials, the diameter of pipes and the formation of pipelines in networks are mostly influenced by the necessity of ensuring sufficient pressures and flow rates, despite installation costs and operations.

1.2. Calculation of pipe networks

Simple pipe networks have procedures for connecting the pipes in a row or in parallel, as described in Chapter 5 of Volume 1 [GEO 18]. However, the same is not true for complicated pipe networks. The schematic representation of a typical plumbing network is shown in Figure 1.4.

Figure 1.4. Schematic representation of a pipe network

The geometric convergence of three or more pipes is called a network node. In technical applications, nodes with more than four branches do not exist. Nodes with three branches (which are most common in practice) are classified into branch nodes (Figure 1.5a) and convergence nodes (Figure 1.5b). In branch nodes, the incoming branch with a flow rate Q is divided into two branches with flow rates Q1 and Q2, while in convergence nodes, two branches with flow rates Q1 and Q2 converge into one branch with a flow rate Q.

When the fluid passes through the node, there is some kind of energy loss, which can be attributed to a decrease in the cross-sectional areas of branches 1 and 2, so that the average velocity of the fluid in all the three branches of the node is approximately the same (V ~ V1 ~ V2), and the walls of the branches are rounded instead of being sharp, to avoid higher energy losses.

Figure 1.5. Pipe network nodes

Minor energy losses at a node can be calculated either by the method of losses coefficient or by the method of equivalent lengths. In Table 1.1, typical values of the losses coefficients for nodes T-90° with constant diameter are given. Furthermore, in Table 1.2, representative values of the equivalent lengths in diameters for tees are given. For a flow in parallel connection, the following two basic rules are applicable:

1) For an incompressible flow in a node, the algebraic sum of flow rates in its branches is zero. This means:

[1.1]

for the flow rates of the k branches of the node. During the addition of flow rates, streams entering the node are considered positive, while those leaving the node are considered negative. By applying this rule at node a, shown in Figure 1.6, we get:

[1.2]

Table 1.1. Coefficient of losses for T-90 nodes

Table 1.2. Equivalent length values

KIND OF APPLIANCE le/d KIND OF APPLIANCE le/d Sudden dilatation (d1/d2 = 1/2) 20 Tee, entrance from main line 20 Sudden systole (d2/d1 = 1/2) 12 Tee, entrance from branch 60 Borda mouth 28 Valves (totally open): Mouth with sharp lips 18 back flow with swing 135 45° curve 16 Angular 145 Standard 90° curve 30 Hydrant 18 90° curve of large radius 20 Butterfly (d = 6in) 20 90° angle 60 Sliding 13 180° curve 65 Spherical 340

Relationship [1.1] is a mathematical expression of the node theorem, also known as the first law of Kirchhoff in the theory of electrical networks

2) For a steady flow in a hydraulic network, for example, in the network shown in Figure 1.6, the total head loss hl between the nodes a and b is the same as the respective head losses, hl,i, in each branch i of the network, which means:

[1.3]

where hl,1, hl,2 and hl,3 are the heads of the energy losses in branches 1, 2 and 3, respectively, for the corresponding flow rates Q1, Q2 and Q3. Equation [1.3] constitutes the mathematical statement of the energy principle of the hydraulic network

Figure 1.6. Nodes and branches of a pipe network

The procedure followed for obtaining the solutions of these problems depends on the type of information asked. Therefore, if the total head loss of the flow is known, then it is easy to calculate individual flow rates Qi and finally their sum.

The reversed problem is solved with successive approximations because the distribution of the flow rate Q in the individual branches of the network is not known. In fact, in the first approximation, we consider zero energy loss at the nodes and, if they are considered important, we add them in the second approximation. In more complex hydraulic networks, the balance sheet method, head losses in loops and flow rates in nodes are applied. It is difficult to obtain the solution of such network problems; it can be obtained with a suitable computer program.

However, in addition to this form of network, there are more complicated distribution networks, particularly the grid and loop patterns. A loop with one inlet and two exits, as shown in Figure 1.7, is impossible to obtain by the methods that we developed in Chapter 5 of Volume 1 [GEO 18].

Figure 1.7. Loop pattern network

Before we develop a method for solving these types of networks, we should state the basic relation of these calculations. Therefore, considering that a network consists of pipes (branches) with nodes and forms closed circuits or loops, we will individually examine the relations of the nodes and the loops.

Viewing a network macroscopically and applying the continuity equation to the network, we have:

[1.4]

and for incompressible fluids,...