PIPE AND PIPE JUNCTION LABELING SCHEME
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 ∆P1,2. 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.
SYSTEM 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-
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