where (d*P*/d*x*)d is the hydrodynamic pressure gradient (N/m3).

The pressure at any point along a segment of the piping system is simply the

pressure at the inlet to the pipe segment plus the sum of the hydrostatic and

hydrodynamic gradients multiplied by the distance

(4-14)

where *P*x is the pressure at point *x *(N/m2) and *P*I is the pressure at the inlet to the

pipe segment (*x *= 0) (N/m2).

Since (d*P*/d*x*)h and (d*P*/d*x*)d are both independent of *x*, we only need to know

if their sum is positive or negative to determine if the pressure will be higher or lower

than the inlet pressure at the outlet of the pipe segment. We can also easily show by

using monotonicity analysis (Papalambros and Wilde 1988) that the maximum

pressure must occur at either the inlet or the outlet of the pipe section and cannot

occur at an intermediate point. To do so we convert the maximization problem to its

equivalent minimization problem

min. *P*x = {*P*I + [(d*P*/d*x*)h + (d*P*/d*x*)d]*x*}

(4-15)

subject to

(4-16)

(4-17)

The conventions for constraints and the labeling of monotonicity are from

Papalambros and Wilde (1988). If the sum of the gradients is positive, the objective

function is monotonically decreasing in *x *and must therefore be bounded from

above. Constraint *g*2 is the only constraint that bounds the objective from above, so

it must be critical and thus *x = L*. If the opposite is true, the sum of the gradients is

negative, the objective will be monotonically increasing in *x *and must therefore be

bounded below. Constraint *g*1 is the only constraint that bounds the objective from

below, so it must be critical and *x *= 0. If the sum of the gradients is zero, the pressure

will be the same at all points along the pipe segment. Thus, we have shown that the

maximum pressure must always be at one end of the pipe segment and it will only

be necessary for us to ensure that our absolute pressure constraint (eq 4-11) is

satisfied at these points. Remember that in arriving at this result, we assumed that

the pipe segment had a constant slope between end points. If in reality this is not the

case, the pipe segment in question can be broken up into two or more equivalent pipe

segments for applying this constraint.

The number of points at which the maximum absolute pressure constraint must

be checked for satisfaction may possibly be reduced even further if we proceed as

follows.

1. Starting at the heating plant, we proceed along the supply line checking only

the "junction" points as discussed earlier.

2. For any point that is at the same elevation or higher than the upstream point

previously identified as having the maximum pressure, we need not compute the

pressure.

3. When a point is identified that does not meet the above criteria, we proceed by

first computing the sum of the hydrostatic and hydrodynamic gradients. If this

quantity is negative, we do not need to compute the pressure.

4. If the sum of the gradients is positive, we will need to compute the pressure at

this point. To compute the pressure, we first find the elevation difference and

resulting hydrostatic pressure difference between the point in question and the

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