independently for the constraints given by eq 4-11, 4-22 and 4-23. Since verification

of satisfaction for all of these constraints requires either directly or indirectly the

calculation of the pressure losses in the supply and return pipes, we begin by doing

so for each of the pipe segments. The pressure loss in either the supply or return line

is calculated by modifying eq 2-15 slightly so that it applies to each pipe indepen-

dently. The results are

(∆*P*d,s )j = *a */ 2 εb (4 / π)2+*c *(ρ1 *c *)d,s md+*c *L d (5+*b*+*c*)

˙2

(5-8)

j

(∆*P*d,r )j = *a */ 2 εb (4 / π)2+*c *(ρ1 *c *)d,r md+*c *L d (5+*b*+*c*) .

˙2

(5-9)

j

Once the piping pressure losses are known, we can calculate the non-control-

valve pressure losses ∆*P*ncv,i for each consumer using eq 5-7 and sum this with the

minimum control valve pressure loss ∆*P*cvm,i to find the consumer with the highest

value of this sum, our critical consumer. The sum of the pressure losses for this will

eq 4-2. For this consumer the constraint of eq 4-5 will be active, as shown earlier.

Using the value of ∆*P*hp calculated for the critical consumer, we can then calculate

the control valve pressure losses for all of the other consumers using eq 4-2.

With the piping and consumer pressure losses known, we can calculate the abso-

lute pressure level at all nodes in the pipe network with either a maximum absolute

pressure assigned to the supply pipe at the heating plant, or a minimum absolute

pressure assigned to the return pipe at the heating plant. If we set the minimum

pressure level in the return pipe at the heating plant, we can use the constraints of

eq 4-23, 4-24 and 4-25 to guide our choice. Note that when eq 4-23 is evaluated at the

Dependent on the particular parameter values for the problem at hand, one of these

constraints will "dominate" (see Papalambros and Wilde [1988] for concept of

constraint dominance). The cases for constraint dominance are simply as follows. If

eq 4-23 dominates. If

eq 4-24 dominates. If

eq 4-25 dominates.

Alternately, as noted above, we can also assign the maximum absolute pressure

in the supply pipe at the heating plant and use that value to find the other absolute

pressures in the network. The logical choice for the maximum absolute pressure

value in the supply pipe at the heating plant would be the maximum absolute

pressure allowable for the piping system being used *P*max. In most cases the

maximum absolute pressure in the system will occur at the heating plant in the

supply pipe; thus, this is a logical choice. It is possible that this will not be the case,

however. Using eq 4-15 we have shown earlier that the maximum pressure must be

at a nodal point location. In the discussion after eq 4-15, we also developed a

procedure that can be used to minimize the number of nodes at which the absolute

pressure must be calculated. If this procedure is used, we can quickly determine if

the heating plant will be the location of the maximum absolute pressure.

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