CHAPTER 4: CONSTRAINTS ON SYSTEMS WITH
MULTIPLE CONSUMERS AND PIPES
All district heating systems, with the exception of pure transmission systems, will
have multiple consumers and pipes. If each pipe were independent of the others, it
would be possible to apply the procedure developed earlier to each pipe and create
a complete design in that way. Of course each pipe segment does not operate
independently of the others and the system can thus not be designed completely in
that way for all but the most trivial cases. Many constraints are imposed on the
design by the physical process involved in the network, the consumer's require-
ments and physical limitations of the piping. As we shall see, many of these
constraints will be inactive at the optimum system design and thus they can be
relaxed. Our task is then to formulate these constraints into mathematical expres-
sions, and then identify those that must be active and use this information to develop
a solution methodology. All of this must be done with the minimum amount of
computational effort so as not to render the method intractable for large networks,
which often have hundreds or thousands of piping segments.
SYSTEM CONSTRAINTS
Constraints on the design of a heat distribution system originate from limitations
imposed in several distinct areas. Before we begin to formulate constraints into a
form suitable for inclusion in our problem, let's consider where and why these
constraints arise. The source of constraints can be grouped into three basic catego-
ries:
1. Physical limitations of the piping systems.
2. Fluid dynamic and thermodynamic considerations for the network, con-
sumers, and heat source.
3. Requirements dictated by the consumer's equipment or processes.
In some instances considerations from each of these categories are coupled
together into a single constraint or set of constraints. Thus, as we formulate the
constraints below, we will address considerations from each of the categories above
and their interaction.
DIFFERENTIAL PRESSURE CONSTRAINTS
A very important set of constraints on the system arises from requirements for the
pressure difference between supply and return. At the consumer this differential
pressure must maintain a minimum level to ensure adequate flow through the
consumer's heat exchanger. This pressure differential is consumed in both the heat
exchangers and control valves. In the heat exchanger, the pressure losses are caused
by fluid dynamic friction. In the control valve, the pressure losses are introduced by
a throttling process used to control the flow rate through the heat exchanger and
thus control its output. In the supply piping between the heat source and the
consumer, pressure losses occur due to friction. Similarly, in the return line from the
consumer back to the heat source, pressure losses also occur. There is then, in effect,
a requirement at each point in the piping network for a given differential between
supply and return pressure necessary to overcome downstream losses, including
those in the return system. Ultimately, at the heating plant pumps must be used to
provide the total differential pressure needed downstream of that point. In theory
it's possible that pumps can be placed anywhere in the system or even dispersed
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