The second step in the calculation is to determine the pipe thickness required for each pipe. This is done in the standard manner using information from standard references, as described in computation section 7.1.2. From the pipe internal diameter distribution calculated in computation section 7.1.1 and the pipe thicknesses, the pipe outer diameter distribution is obtained. This is a necessary prerequisite in determining the required insulation thickness. The last major step in the system design is the determination of the cost optimized insulation thickness distribution. The mathematical derivations and equations are presented in section 7.1.3. For convenience the insulation installed cost is assumed proportional to the installed volume. The system insulation cost is modeled by equation (A-20),and the system heat loss from transport piping is given by equation (A-19). Calcium silicate is the commercial insulation considered, and the temperature dependency of the thermal conducti- vity can vary if necessary (Ki is the thermal conductivity of the ith line). The cost optimization procedure is similar to that employed for the internal diameter distribution calculation, previously discussed. The system heat equation and system insulation cost equation are each differentiated with respect to insulation thickness for the ith line. The Lagrange multiplier is formed and the insulation thickness on every pipe is related to the insulation thickness on any given pipe by the insulation cost minimization criterion expressed in equation (A-23). The Newton iteration method is applied to generate the appropriate thickness distribution from a given input. The insulation thickness distribution criterion (equation (A-23)) readily accommodates temperature differences in lines, and generally a cooler line will have less insulation than a warmer line. The optimum design for an entire system requires two separate decoupled calculations, and a parametric study is performed to locate the optimum system
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