Network Boundary Conditions : Types of Elements in a Network Graph

2

1 4 R1

R : Reservoir V : Valve ST : Surge Tank

ST

V1

V2 V3 V4

A B C

D

E F G

H I

J

Number along the edges/nodes are element/node numbers

(a) (b)

Fig. 3.1 (a) A pipe network; (b) Directed graph of pipe network

3.2 Network Boundary Conditions : Types of Elements in a Network Graph

Hydraulic pipe networks consist of several types of components such as reservoir, surge tank, valve, pipe, pump etc. In the directed graph of a network each component is either an element or a node. In the graph, a node is characterised by a particular head, H and an outflow discharge, Q. Similarly, an element is associated with a particular head loss, h and a discharge, q.

Mathematically, a node or an element is equivalent to a boundary where a special boundary condition or specification is imposed. Depending upon whether the head loss and/or discharge variable is a known quantity in an element, possible boundary conditions in an element can be categorised into six categories called, hereinafter, element types. Similarly, depending upon whether the head and/or outflow discharge is a known quantity at a node, possible boundary conditions at a node can be categorised into six categories called, hereinafter, node types. However, by joining all the nodes to a selected reference node of known head with pseudo-elements, all nodal boundary conditions are transformed into element boundary conditions. Joining of nodes with the reference node by pseudo-elements

makes an extended network, which is closed. It is important to note that only those nodes, which have network components or a boundary condition, are joined with the reference node.

Junction nodes with no boundary condition need not be joined to the reference node.

Hence, an arbitrary network comprising of different types of components or boundary conditions is represented by six types of elements. These six element types are as follows:

Element Type 1 or Pressure Constrained Element

Type 1 or pressure constrained elements are those which have a specified or known head loss and an unknown arbitrary discharge through it. Relationship between head loss and discharge through this element is unknown. Characteristic of type 1 element is shown in Fig. 3.2. Head loss and discharge in this element are represented by h1* and q1 respectively.

In unsteady flow problems, this element has a specified or known time history of head loss, h1*(t), and unknown discharge variation, q1(t), through it. (Asterix (*) denotes the known value of head loss.)

All elements joining reservoir nodes having known heads or nodes with specified head with the datum or reference node are type 1 elements. Network components, which are to be designed for a specified head loss, are also type 1 elements. For example, a pipe with unknown diameter with specified head loss, a valve with unknown opening with specified head loss etc.

Time

Head Loss

Fig. 3.2 Characteristic of element type 1

Element Type 2 or Flow Constrained Element

Type 2 elements are those which have a specified discharge and an unknown arbitrary head loss through it. In this element also, relationship between head loss and discharge is

unknown. Characteristic of this element is shown in Fig. 3.3. Head loss and discharge through this element are represented by h2 and q2* respectively.

In unsteady flow problems, these elements have specified time history of discharge, q2*(t), and unknown head loss variation, h2(t).

All elements joining nodes having known outflows or inflows with the reference node are of type 2. Network components which are to be designed for a specified discharge are type 2 elements such as a pipe with unknown diameter to be designed for a specified discharge, designing a valve opening for the specified discharge through it etc.

Time

Discharge

Fig. 3.3 Characteristic of element type 2

Element Type 3 or Pressure-Flow Constrained Element

Type 3 or pressure-flow constrained elements are those which have both specified or known head loss and discharge through it. Relationship between head loss and discharge through this element is, thus, known. Characteristic of type 3 element is shown in Fig. 3.4. Head loss and discharge in this element are represented by h3* and q3* respectively. In unsteady flow problems, this element has a specified or known time history of both head loss, h3*(t), and discharge, q3*(t), through it.

All network components with known relationship between head loss and discharge through it and specified head loss or discharge, are of type 3. Elements joining nodes having specified both head and outflow (or inflow) with the reference node are type 3 elements.

Head Loss, Discharge

Time q

h

Fig. 3.4 Characteristic of element type 3

Element Type 4 or Open Element

Type 4 or open elements are those that have both unknown head loss and discharge through it. Relationship between head loss and discharge through this element is also unknown. Head loss and discharge in this element are represented by h4 and q4 respectively.

In unsteady flow problems, this element has an unspecified or unknown time history of both head loss, h4(t), and discharge, q4(t), through it.

Network components, which are to be designed and may have any arbitrary head loss and discharge through it, are of type 4.

Element Type 5

Type 5 elements are those that have both, unknown head loss and unknown discharge through it, though the relationship between head loss and discharge through this element is known. Characteristic of type 5 element is shown in Fig. 3.5. Head loss and discharge in this element are represented by h5 and q5 respectively.

In unsteady flow problems, this element has an unknown time history of both head loss, h5(t), and discharge, q5(t), through it. Type 5 element can be described through following relation:

(t)) (h q (t) q or (t)) (q h (t)

h₅ = _{5 5 5} = _{5 5} (3.1)

All network components with known physical parameters having unknown head loss and discharge are type 5 elements, such as, an existing pipe with known diameter and other physical parameters, a surge tank with known area, a pump with known characteristic curve, a valve with known area of opening etc.

Head Loss

Head Loss

Discharge

Fig. 3.5 Characteristic of element type 5

Element Type 6 or Ideal Check Valve

Type 6 elements represent ideal check valve conditions. For an ideal check valve if there is a flow through it, there is no head loss and if no flow passes through it, head loss across the valve exists. Hence,

q6 > 0 , h6 = 0 and when q6 = 0 , h6 < 0 (3.2) and thus,

0 h

q_{6 6} = (3.3)

Characteristic of element type 6 is shown in Fig. 3.6.

Fig. 3.6 Characteristic of element type 6

Discharge