Before proceeding further with the development of our solution technique, we

first need to develop a methodology for identifying consumers, pipe junctions and

pipe segments. We would like this system to be as simple and intuitive as possible,

yet sufficiently general so as to be easily extendible to much larger networks. A

method that meets these requirements is a simple identification number for each

"node." A node can be any one of the following items within the pipe network.

1. A source node where a net inflow of heat occurs, i.e., the heating plant.

2. A sink node where a net outflow of heat occurs, i.e., a consumer.

3. A pipe junction node where no net inflow or outflow of heat occurs.

Note that there are at least a couple of special cases of the pipe junction node that

might be of interest: a storage node where heat could be stored for release at later

times, and a "junction" node with only two pipe segments connected. The latter

could be simply a transition in pipe size or an intermediate pumping station for

instance. These special cases would be of interest for advanced system optimization

studies but are beyond the scope here.

The number of a node does not necessarily need to be assigned in any particular

fashion. They could be assigned sequentially from the plant or some consumer, or

in no particular sequence at all. In fact, alphanumeric characters could be used for

identification. The point is that the assigned identification characters have no

significance relative to one another, other than being unique to the node in question.

With an identification system established for our nodes, we need to establish the

identity of the pipes connecting these. The simple convention we will adopt is to use

the node numbers on either end of the pipe segment to identify the pipe segment that

connects them. For example, the pressure loss in the pipe segment between nodes

1 and 2 would be written as ∆*P*1,2. We will establish the convention of letting the first

node number in the pair be the upstream node in the supply line, with the second

node being the downstream node, again in the supply line. For the return line of the

same pipe segment, the convention will be established by the supply line, i.e., the

first node number in the pair will be the downstream node in the return line and the

second node will be the upstream node. Note that a system segment, as we have

currently defined it, can not have any intermediate nodes within it.

Now we are ready to begin the development of our solution method. As always,

we start by determining our objective function.

The objective function for an entire system of pipes will include the sum of the

individual objective functions for each pipe segment. We must also include the cost

of pumping energy dissipated at the consumer and the capital cost of the pumps

needed to generate this pumping energy. At first it might seem unnecessary to

include costs associated with the consumer in our objective function when in fact

there are no decisions to be made about the consumer's equipment. However,

constraints that the consumer places on the system will require that these costs be

included in order to achieve an optimal design that does not violate these con-

straints.

Additional costs would also need to be included if we were to expand the

objective of our design. For example, if we wished to determine an optimal

operational strategy for the system, as well as a design, it would be necessary to

include some additional costs in the objective function. These would be the costs of

generating the heat ultimately supplied to the consumer. Another example of an

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