Pipe Network Analysis

Pipe Network Analysis

Introduction

Fig. 5-1 Pipe Network

Using the conservation equations discussed in the previous sections, pipe network analysis seeks to determine the discharge and pressure at every node. To accomplish this, the physical features of the network must be known. These features include pipe diameter, length, and roughness, as well as the location of the reservoir, pumps, pressure reduction valves, and fittings.

Defining A Pipe System

Fig. 5-2 Skeletonized Pipe System

It is not always necessary to include all of the pipes in a network. The pipe network can usually be "skeletonized" to simplify the problem. Some basic rules when skeletonizing a system include:

Lumping houses along one block of a street, or even a small subdivision into a single node

Including only main transmission lines (lines that carry water from supply sources to demand areas)

Considering only components vital to proper operation of the system

Basic Network Elements

Fig. 5-3 Supply Source

Fig. 5-4 "T" Junction

All network analysis is developed around two basic principles, the continuity principle (conservation of mass) and the work-energy principle. The work-energy principle is used to develop "energy loop equations" around independent loops. The continuity principle is used to develop conservation of mass equations at a junction. A junction (or node) is a point where two or more pipes join together. A supply source is a point where the elevation of the energy line, or hydraulic grade, is established. It is the point where flow enters the pipe system. A demand is a point where flow exits the pipe system.

Equation Systems For Steady Flow In Networks

We will focus on two different systems of equations that can be developed to solve pipe network problems. The system name is derived from the variable regarded as the principle unknown in the solution method. The two equation systems are as follows:

Q-equations

The Q-equation system is based on continuity and work-energy principles and assumes discharge or flow as the principle unknowns. To satisfy the continuity principle, the flow into a junction must be equal to the flow out of the junction (Eq. 5-1).

                                    (Eq. 5-1)

where
           QJ_j is the flow out of the junction (demand)
           Q_i is the flow into the junction from pipe i

To satisfy the work-energy principle, the sum of the head loss around each independent loop must be equal to zero.

                                             (Eq. 5-2a)

where
           h_fi is the frictional head loss in pipe i

The head loss can be expressed as a function of discharge via the Hazen-Williams or Manning equation:

                                       (Eq. 5-2b)

where
           h_fi is the frictional head loss
           K and n are constants that depend on the            properties of the pipes
           Q is the discharge

So, in terms of discharge equation 5-2a becomes:

                                         (Eq. 5-2c)

where
           K_i is the K constant of pipe i
           Q_i is the discharge of pipe i
           n is a constant that depends on which equation is            being used (Hazen-Williams or Manning)

H-Equations

The H-equation system is based on the the continuity principle and assumes that head loss is the principle unknowns.

                                    (Eq. 5-3)

where
           QJ_j is the flow out of the junction (demand)
           Q_i is the flow into the junction from pipe i

Using equation 5-3 H-equations are written at each junction.

Pipe Network Analysis

		Introduction
Fig. 5-1 Pipe Network		Using the conservation equations discussed in the previous sections, pipe network analysis seeks to determine the discharge and pressure at every node. To accomplish this, the physical features of the network must be known. These features include pipe diameter, length, and roughness, as well as the location of the reservoir, pumps, pressure reduction valves, and fittings.

		Defining A Pipe System
Fig. 5-2 Skeletonized Pipe System		It is not always necessary to include all of the pipes in a network. The pipe network can usually be "skeletonized" to simplify the problem. Some basic rules when skeletonizing a system include: Lumping houses along one block of a street, or even a small subdivision into a single node Including only main transmission lines (lines that carry water from supply sources to demand areas) Considering only components vital to proper operation of the system

		Basic Network Elements
Fig. 5-3 Supply Source Fig. 5-4 "T" Junction		All network analysis is developed around two basic principles, the continuity principle (conservation of mass) and the work-energy principle. The work-energy principle is used to develop "energy loop equations" around independent loops. The continuity principle is used to develop conservation of mass equations at a junction. A junction (or node) is a point where two or more pipes join together. A supply source is a point where the elevation of the energy line, or hydraulic grade, is established. It is the point where flow enters the pipe system. A demand is a point where flow exits the pipe system.

		Equation Systems For Steady Flow In Networks
		We will focus on two different systems of equations that can be developed to solve pipe network problems. The system name is derived from the variable regarded as the principle unknown in the solution method. The two equation systems are as follows: Q-equations The Q-equation system is based on continuity and work-energy principles and assumes discharge or flow as the principle unknowns. To satisfy the continuity principle, the flow into a junction must be equal to the flow out of the junction (Eq. 5-1). (Eq. 5-1) where QJ_j is the flow out of the junction (demand) Q_i is the flow into the junction from pipe i To satisfy the work-energy principle, the sum of the head loss around each independent loop must be equal to zero. (Eq. 5-2a) where h_fi is the frictional head loss in pipe i The head loss can be expressed as a function of discharge via the Hazen-Williams or Manning equation: (Eq. 5-2b) where h_fi is the frictional head loss K and n are constants that depend on the properties of the pipes Q is the discharge So, in terms of discharge equation 5-2a becomes: (Eq. 5-2c) where K_i is the K constant of pipe i Q_i is the discharge of pipe i n is a constant that depends on which equation is being used (Hazen-Williams or Manning) H-Equations The H-equation system is based on the the continuity principle and assumes that head loss is the principle unknowns. (Eq. 5-3) where QJ_j is the flow out of the junction (demand) Q_i is the flow into the junction from pipe i Using equation 5-3 H-equations are written at each junction.