The unknown parameters in a pipe network can be element head losses, element flows and element resistances. For network analysis to be feasible, the number of unknown parameters and their distribution over the entire network must be such that an adequate number of independent equations can be formulated. If the unknown parameters are concentrated in one part of the network and known parameters in the other, it is not possible to determine all the unknown parameters uniquely. Therefore, the number and distribution of the unknown parameters should obey certain rules so that the network analysis is feasible. Determination of these rules is necessary for the meaningful and effective modelling of network problems.
In this chapter, first a critical review of past solvability rules is done and, then, necessary and sufficient conditions for the existence and uniqueness of a solution for a arbitrary network comprising of different types of elements are developed.
4.1 A Critical Review of Past Solvability Rules
Shamir and Howard (1968) may have been the first to consider the problem of solvability of pipe networks. They suggested some rules for the existence and uniqueness of solution of networks. These rules (rules no. 1) are as follows:
(1) The total number of the unknowns must be equal to the total number of the nodes in the network.
(2) At least one nodal head must be known.
(3) A node having an unknown consumption must be connected to at least one other node with a known consumption.
(4) A subsystem consisting of a pipe with unknown resistance and its two end nodes, must have no more than one additional unknown – one of the heads or consumptions at the two end nodes.
(5) Considering any node, at least one of the following must be unknown : the consumption at the node, the head at the node itself or at any adjacent node, or the resistance of a pipe connected at the node.
These rules were developed heuristically by considering different combinations of unknowns, and no rigorous mathematical proof was provided for their validity. This set of conditions is necessary to guarantee that the jacobian matrix for the nodal method of analysis is non-singular.
These rules lack generality in the sense that they do not incorporate elements type 1, 2 and 3.
Though nodes type 1, 2 and 3, generating pseudo-elements type 1, 2 and 3 respectively, are considered while driving these rules, it is observed that presence of elements type 1, 2 and 3 in the network makes the system more complicated. Moreover, it is found that even for simple networks these rules are incorrect and insufficient. Consider a pipe network (a) as shown in Fig. 4.1 consisting of four nodes with specified heads and unknown outflows. This network violates rule 3 but possesses a unique solution. Flow in each pipe can be uniquely determined by knowing heads at its two ends and, then, continuity equations at each node uniquely defines the outflow.
As an another example, consider a network (b) shown in Fig. 4.1. This network violates rule 4 of Shamir and Howard as the pipe with unknown resistance has two unknowns at its two end nodes. However, the network possesses a unique solution. Flow in pipe between nodes type 3 and 1 is uniquely determined knowing the head loss. From continuity at node type 3, flow and, thus, head loss in pipe between node type 3 and 2 is uniquely determined. This gives head at node type 2 uniquely. From continuity at node type 2, flow in pipe type 4 is uniquely determined and the difference of head at node type 2 and 1 gives the head loss in this pipe.
In general, it is observed that rules presented by Shamir and Howard are incorrect, insufficient and are not general. For a network consisting of different types of elements as described in previous chapter, these rules fails to provide necessary and sufficient conditions for the existence and uniqueness of a solution.
Legend :
1 - Node with known head 2 - Node with known outflow
3 - Node with known head and outflow 4 - Pipe with unknown resistance 5 - Pipe with known resistance R - Reservoir with known head 1
1 1
1
1
R R
5 5
5 5
5
5 4 2
3
(a) (b)
Fig. 3.1 Networks violating solvability rules no. 1 but having unique solutions
Using graph theory, Gofman and Rodeh (1972) also developed certain rules for solvability of networks. They termed pipes with unknown characteristics as head generators. Their rules (rules no. 2) are as follows :
(1) The number of nodes with known heads must exceed the number of head generators by one.
(2) For a given network G, there exists a connected network G’ such that:
a. All head generators and nodes with known heads belong to G’.
b. G’ is loopless.
c. A path in G’ connecting two nodes of known nodal heads contains at least one head generator.
These rules are also not correct and lacks generality as that of rules presented by Shamir and Howard. Consider a pipe network (a) as shown in Fig. 4.2. This network violates the above rules in the sense that there is a loop made by nodes with known head and head generator.
However, network possesses unique solution. Flows in elements type 5 are uniquely determined by knowing heads at its two ends and continuity equations at nodes uniquely defines outflows and flow in element type 4.
Consider another example of network (b) as shown in Fig. 4.2. This network obeys all the rules of Gofman and Rodeh but does not possess a unique solution. Knowing the head loss in type 3 element and head at its upstream node, head at node type 2 is uniquely determined which in turn uniquely defines the flow in pipe connecting this node with reservoir node. This makes the case for possible violation of continuity at node type 2 unless the sum of the flow in the connecting pipes is equal to the specified outflow. In any case, flow in other two pipes can not be determined.
Rules presented by Gofman and Rodeh are also not correct and complete.
(a) (b)
Fig. 4.2 (a) Network violating solvability rules no. 2 but having unique solution (b) Network obeying solvability rules no. 2 but possess no solution
Network solvability rules are also presented by Bhave (1990). These rules were also developed heuristically without any proof and are as follows (rules no. 3):
(1) The total number of unknowns must be equal to the total number of nodes in the network.
(2) At least one nodal head must be known and one nodal flow unknown.
(3) Considering any node, at least one of the following must be unknown : the flow at the node, the head at the node itself or at any adjacent node, or the resistance of the pipe connected to the node.
(4) Each pipe with an unknown resistance must lie on an independent path connecting two nodes of known heads, and each such path must not contain more than one pipe with an unknown resistance. All these paths must form branching configurations without any loops.
Network (b) in Fig. 4.2 obeys the rules presented by Bhave but as shown earlier do not possess a unique solution. Network (a) in Fig. 4.3 also obeys Bhave’s rules but it can be shown that it possess non-unique solution. Network (b) in Fig. 4.3 also possess non-unique solution for some combination of pipe resistances. In network (b) of Fig. 4.3, if resistances of all the four type 5 pipes are same, it will have infinite solution.
Legend :
1 - Node with known head 2 - Node with known outflow
3 - Node with known headand outflow 4 - Pipe with unknown resistance 5 - Pipe with known resistance R - Reservoir with known head
3 3
3
R
5
5 5 5
5 3
4
4 4 4
1 1
Fig. 4.3 Networks obeying rules no. 3 but having non-unique solutions
In general, it is observed that solvability rules present in the literature are not correct, incomplete and lacks generality and, thus, do not guarantee a unique solution.
Determination of necessary and sufficient conditions for the existence and uniqueness of a solution is must for effective modelling of pipe networks. In the next section, these conditions have been derived for an arbitrary network consisting of elements type 1, 2, 3, 4 and 5.
4.2 Theorems Defining Necessary and Sufficient Conditions for the Existence and Uniqueness of a Solution
For an arbitrary network comprising of different elements as described in chapter 2, a unique solution of a network problem may exist only if the following necessary and sufficient conditions are satisfied.
First Necessary Condition