APPENDIX B – MOC and its Struggles to Compete with WPM
Surge Analysis and the Wave Plan Method
Supplementary Material: Example Problems and Solutions
Appendix B – Problem 12
B.12 Equations B.17 and B.18 are compatibility equations used in second-order MOC approximation. They were derived based on the assumption that Q (more specifically Q2) in the integral terms of Eqs. B.9 and B.10 (i.e., the baseline compatibility equations in integral form) is a constant (as it is in first-order approximation) but which is set equal to the average of the value of the known flowrate at the starting point of the pipe section and the unknown flowrate at the ending point of the pipe section (see Section B.6). For example, Q is assumed to be a constant and equal to0.5(QL,t+ QC,t+∆t) while integrating along the left characteristic line.
a. Derive second-order compatibility equations in finite difference form with a slightly different assumption: Q = 0.5((QL,t + QC,t )0.5+ QC,t+∆t) along the left characteristic line and Q = 0.5((QR,t + QC,t )0.5+ QC,t+∆t) along the right characteristic line. Are the resulting equations more accurate than those (Eqs. B.17 and B.18) obtained using earlier assumptions?
b. If the head and flowrate on the on the left side of grid point C at time t are 50m and 0.5 m3/s, on the right side are 101.6692m and 0.4 m3/s, and at the center are 49.8678m and 0.5 m3/s, compute the head and flow value at grid point C at time t+Δt using second-order MOC equations derived in part a.
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