Spatially varied flow equations

The momentum and energy principles are equally applicable in the derivation of spatially varied flow equations. By applying the energy principle and introducing a slope of energy head Se, the resulting spatially varied flow equation takes a different form of several terms containing the slope and the effect of added or extracted discharge from the equation derived by applying the momentum principle and using the frictional slope Sf. The two slopes, however, become identical if V = V* = V0 and if the flow is steady. The spatially varied flow equation is the same for flow with either increasing or decreasing discharge. In some cases of steady flow, however, the condition V = V* = V0 occurs in flow with decreasing discharge and V* = 0 occurs in flow with increasing discharge; the resulting equations will differ in the coefficient of the term containing the effect of the added or extracted discharge, as shown by equations 5 and 16.