CHAPTER 7: CONCLUSIONS AND RECOMMENDATIONS
SUMMARY
In Chapter 2 we found a suitable method for determining the optimal size for a
single pipe, independent of any others. This method was developed to be as simple
as possible yet complete and accurate enough for design calculations. The method
is general enough to allow for any set of economic or physical parameter values. In
addition, any form of load management, i.e., temperature or flow modulation, or
both, can be accommodated by the integral form of the coefficients in the cost
equation. A new approximation was developed for the friction factor. The form of
this expression was a simple power function of the Reynolds number and the
relative pipe roughness. This form allowed us to easily incorporate it into the head
loss equation without additional complication or rendering the result implicit. We
made use of geometric programming theory to identify a lower bounding problem
that can be used to provide us with a very good first estimate of our solution and a
global lower bound on cost. At the end of Chapter 2, an example is presented that
shows a 17% saving in life cycle cost over a design based on a common rule of thumb.
In Chapter 3 we looked at the heat consumers and the effect that they have on the
piping system. We developed a new model for the consumer's heat exchanger that
uses the geometric mean temperature difference as an approximation for the
logarithmic mean temperature difference. This allowed us to develop an explicit
expression for return temperature, a result not possible when using the logarithmic
mean temperature difference. We conducted a complete error analysis on the
geometric mean approximation and our new consumer model based on it. This
analysis confirmed that the resulting error from this model was acceptable for
design purposes and much less than the error resulting from using the arithmetic
mean temperature difference as an approximation of the logarithmic mean tempera-
ture difference, as has been suggested by others. We integrated our new consumer
model into our single pipe model of Chapter 2 and for a sample case determined
what effect the addition of the consumer has on the integral coefficients of the cost
equation. At the end of Chapter 3, we reworked the example of Chapter 2, including
the effects of the new consumer model.
In Chapter 4 we developed the constraints for systems with multiple pipes and
consumers. Both absolute and differential pressure constraints were derived. By
using the monotonicity of the hydrodynamic and hydrostatic pressure gradients,
we were able to easily show that the maximum pressure within a pipe segment must
occur at one of the end points. We then developed a strategy to allow for constraint
satisfaction at all points implicitly without considering every point in the system.
In Chapter 5 we briefly reviewed general methods for constrained nonlinear
optimization. For various reasons these alternatives are all abandoned in favor of the
approach taken. Subsequently, our general solution strategy is developed for
systems with multiple pipes and consumers. The method makes use of the solution
to the problem, unconstrained by the network constraint requirements, as a starting
point for the constrained solution. Monotonicity analysis was then used to prove
activity of some of the constraints and thus simplify the problem. In addition, the
concept of constraint dominance is used to reduce the number of constraints that
must be considered. Before proceeding with the problem solution, a brief graphical
analysis verified that we only needed to provide for constraint satisfaction at the
maximum load condition to ensure satisfaction at all other load conditions. The
resulting reduced problem was then used as a starting point for two methods
proposed to find a solution to the constrained problem with continuous values for
some of the pipe diameters. Finally, the branch-and-bound technique is explained
and then shown to be suitable for finding a design with discrete values for all the
pipe diameters.
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