where (dP/dx)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
Px = PI + [(dP/dx)h + dP/dx)d]x
(4-14)
where Px is the pressure at point x (N/m2) and PI is the pressure at the inlet to the
pipe segment (x = 0) (N/m2).
Since (dP/dx)h and (dP/dx)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. Px = {PI + [(dP/dx)h + (dP/dx)d]x}
(4-15)
subject to
g1 = x ≤ 0
g1(x )
(4-16)
g2 = x L ≤ 0
g2(x+) .
(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 g2 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 g1 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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