SECTION VI APPENDIX 7.1 FLUID ENERGY TRANSPORT MATHEMATICAL MODEL The transport system design is performed in a series of stages. This documentation of the calculation follows this sequence. After initial inputs the first major calculation is the determination of the optimum internal diameter distribution in the piping network. For a given straight length of pipe, the pressure drop is given by equation (A-3), and the pump work necessary to overcome fluid friction is described by equation (A-5). The total system required pump work to circulate the fluid is given by equation (A-6). Since the goal is to minimize installed system pipe cost (equation (A-7)) within the constraint of a calculated system pump work, the method of Lagrange multipliers is applied. The Lagrange multiplier given by equation (A-12) is the same for every pipe, which leads directly to equation (A-13) for the required pipe internal diameter distribution. To obtain the pipe inner diameter (D^) described by equation (A-13), it is necessary to select an initial value of internal diameter D-j (for the smallest pipe carrying one basic flow unit). One basic flow unit is the flow required in one collector. The in equation (A-13) is the cost of pipe i, and is a weak function of diameter (to the 1/6 power). After the D^ are available, the pressure drop per unit foot in each line is checked against a predetermined allowable unit pressure drop. Equation (A-3) is used to determine if any diameter is too small and produces an unacceptable pressure drop. If such a pipe diameter is found, then the internal diameter is recalculated (enlarged) directly via equation (A-3).
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