where the *j *subscript is the pipe segment index and ∆*P*s&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

∆*P*cv,i ≥ ∆*P*cvm,i

(4-5)

where ∆*P*cvm 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

∆*P*he,i = *A*he,i mi

(4-6)

where *A*he 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 *A*he 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 *A*he based on our previous analysis for pressure loss in pipes

would be

-(

- -*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

(

)

(4-9)

˙

˙

˙

j

Again, in eq 4-9 the summation over the *j *pipes only includes those pipes that serve

consumer *i*.

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

34