Inverse Modeling of Chlorine Concentration in Pipe Networks under Dynamic Condition

The Surface Water Treatment Rule under the Safe Drinking Water Act and its amendments require that the water utilities maintain a detectable disinfectant residual throughout the distribution system at all times. A new computer model is presented to directly calculate the chlorine concentrations needed at the source(s) to have specified residuals at given locations in a pipe network in unsteady flow conditions by using an inverse method. Unlike the forward solution method, which calculates the source chlorine concentration by a trial-and-error procedure, the inverse solution technique presented in this paper directly calculates the source concentration required to meet a specified value at a particular point in the network. The dynamic flows and pressures are first calculated by solving the governing equations by an implicit finite-difference scheme subject to appropriate boundary conditions. For chlorine concentrations at various pipe sections and at each node, the one-dimensional transport equation is modified for the advection dominated flow. This modified transport equation is then discretized by using the four-point implicit finite-difference scheme and solved simultaneously together with the junction mass balance equations. A first-order reaction rate for chlorine decay is assumed and a flow-weighted averaging procedure is used to calculate the complete mixing of chlorine at the junctions. For the verification of the inverse method, the computed results are compared with those determined by direct simulations. An excellent agreement is observed between the inverse method and direct simulation technique.