throughout. In practice this is not done very frequently, owing to the practical
considerations of monitoring, controlling and maintaining the pumps as well as
availability of power for them. Here, we will assume that all pumps are located at
one central heating plant, although some very interesting optimal design questions
arise if this limitation is removed, as we will see later.
We can write the constraint that arises from all of these differential pressure re-
quirements easily by summing the pressure drops and increases around the system.
Since the entire district heating system, consisting of the heating plant, the piping
system and the consumer, forms an essentially closed loop, the pressure losses and
increases around this loop must sum to zero. Thus, we have the following result
∆Php = ∑ (∆Ps + ∆Pr ) + ∆Pcv + ∆Phe
(4-2)
pipes
where ∆Php = pressure increase across the pump (N/m2)
∆Ps = pressure drop in the supply piping (N/m2)
∆Pr = pressure drop in the return piping (N/m2)
∆Pcv = pressure drop in the consumer control valves (N/m2)
∆Phe = pressure drop in the consumer heat exchangers (N/m2).
Each consumer will have at least one segment of the piping system that is not
shared with any other consumers. In addition, all consumers will have their own
control valve and heat exchanger. Thus, we will have one of these equations for each
consumer, each one representing a constraint on the design. Therefore, the summation
in eq 4-2 must be conducted over only the pipes that serve the consumer in question.
The pressure losses given above will vary with the flow rate in the system. In
many cases, flow rates in district heating networks are modulated over the course
of the year as a means of meeting varying loads. Flow can be modulated either by
using variable speed pumps or using what is called a "shunt" at the heating plant.
The shunt simply diverts a fraction of the flow from the pump back to its inlet. The
pressure increase across the pump is reduced as is the flow rate into the network.
Regardless of how it is done, if flow modulation is used, we must ensure that the
constraint of eq 4-2 is not only satisfied for each consumer, but in addition we must
also determine that this will be the case for all load (i.e., flow) conditions encoun-
tered. However, we will show later, after some other necessary constraints have
been introduced, that satisfying this set of constraints for only one load condition
will be sufficient, if we assume that all consumers have loads that vary in the same
manner. This is a reasonable assumption as long as the primary loads are space and
hot water heating, as is the case for most hot water based systems. Steam systems,
which are not addressed by this work, often have much larger fractions of industrial
and absorption air conditioning loads; thus, this assumption might not be reason-
able for them.
In addition to the above equality constraint (eq 4-2), we have both equality and
inequality constraints on each of the quantities appearing within that constraint. At
this point we will formulate each of these additional constraints.
At the heating plant, the pressure increase by the pump must be related to the
pumping power attributable to the consumer in question. This results in the
following expression
∆Php= PPf,i ρ/ m i
˙
(4-3)
where the i subscript is the consumer index. The combined pressure loss of the
supply and return piping is simply given by
2+c
∆Ps + ∆Pr = ∆Ps&r = a εb (4 / π)
A6, j mj2+cLj d -(5+b+c)
˙
(4-4)
j
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