where the j subscript is the pipe segment index and ∆Ps&r is the combined pressure
loss of supply and return (N/m2).
The control valve will have varying amounts of pressure drop across it, de-
pending on the consumer's load. The minimum pressure drop for any given flow
rate condition will occur when the control valve is completely open and the flow rate
through the consumer's heat exchanger is at its maximum value--what we have
called the design condition. Thus, we have the following simple constraint for each
control valve in the system
∆Pcv,i ≥ ∆Pcvm,i
(4-5)
where ∆Pcvm is the minimum pressure drop in the control valve (N/m2).
And finally, for the heat exchanger the pressure drop will be related to the flow
rate in a manner very similar to that for the pipes as given by eq 4-4 above. First, let's
assume the following simple form
˙β
∆Phe,i = Ahe,i mi
(4-6)
where Ahe relates the fluid properties and physical properties of the heat exchanger
to the pressure drop and flow rate (kg1β/m s2β) and β is an exponent yielding the
appropriate mass flow rate dependency for the heat exchanger (dimensionless).
In most cases Ahe and β would probably be empirically determined coefficients
and would depend on the type of heat exchanger and its specific design. The basic
example, if a straight section of pipe formed the hydraulic passageway for the heat
exchanger, the form of Ahe based on our previous analysis for pressure loss in pipes
would be
Ahe,sp = (a / 2) (4 / π)2+ c εb e ρh1 he Lhe dhe5+b + c)
-(
- -c
(4-7)
h
e
where the sp subscript denotes straight pipe heat exchanger and the he subscripts
denote conditions within the heat exchanger or physical parameters of the heat
exchanger.
Equation 4-7 assumes the same form of approximation for the friction factor as
was derived for flow in the district heating pipes earlier in Chapter 2. Using this
approximation for the friction factor also determines β from eq 4-6 to be
βsp = 2 + c.
(4-8)
We can now substitute the results for the pressure losses around the district
heating system loop (eq 4-3, 4-4 and 4-6) into our original pressure loss constraint (eq
4-2) to obtain the following constraint
(
)
PPf,i ρ/ mi = ∑ a εb (4 / π)2+c A6, j mj2+cLj dj-(5+b+c) + ∆Pcv,i + Ahe,i mi2+c
(4-9)
˙
˙
˙
j
j
Again, in eq 4-9 the summation over the j pipes only includes those pipes that serve
consumer i.
MAXIMUM ABSOLUTE PRESSURE CONSTRAINTS
Several constraints on the absolute pressure of the water within the system must
be considered. First, we consider the upper limit on pressure that results from the
absolute pressure limits of the piping. This limit will be established by the prevailing
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