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

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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