A variable chlorine decay rate modeling of the Matsapha town water network was developed based on initial chlorine dosages. The model was adequately described by a second order rate function of the chlorine decay rate with respect to the initial chlorine dose applied. Simulations of chlorine residuals within the Matsapha water distribution network were run using the EPANET 2.0 program at different initial chlorine dosages and using the variable decay rate as described by the second order model. The measurement results indicated that the use of constant decay rate tended to underestimate chlorine residuals leading to potentially excess dosages with the associated chemical cost and side effects. The error between the two rate models varied between 0% and 15%. It is suggested that the use of water quality simulation programs such as EPANET be enhanced through the extension programs that accommodate variable rate modeling of chlorine residuals within distribution systems.
A model for relating the time-dependent variable rate of reaction to the decay of chlorine residual in water is developed based on the initial chlorine dose, molar concentrations of reactants, and the rate constant itself. The mathematical model, while retaining its second order nature, simplifies the solution as the residual chlorine and aggregate parameters such as molar concentration of reactants can be estimated. The model is based on molar-averaged reaction rates involving arithmetic and harmonic means of reactants that eliminate the individual reaction rates that are difficult to determine. Part of the mathematical assumption used in the derivation of the equations using molar averaging is tested for its validity through theoretical as well as Monte Carlo simulation of the error term over wide ranges of assumed reaction rates and molar concentration of reactants. The second-order variation of the rate of reaction with respect to the initial chlorine concentration has been verified through experimental tests of bulk chlorine decay carried out at different chlorine doses.
An iterative formula based on Newton's Method alone is presented for the iterative solutions of equations that ensures convergence in cases where the traditional Newton Method may fail to converge to the desired root. In addition, the method has super quadratic convergence of order 2.414 (i.e., 1+ √ ). Newton method is said to fail in certain cases leading to oscillation, divergence to increasingly large number or off-shooting away to another root further from the desired domain or off shooting to an invalid domain where the function may not be defined. In addition when the derivative at the iteration point is zero, Newton method stalls. In most of these cases, hybrids of several methods such as Newton, bisection and secant methods are suggested as substitute methods and Newton method is essentially blended with other methods or altogether abandoned. This paper argues that a solution is still possible in most of these cases by the application of Newton Method alone without resorting to other methods and with the same computational effort (two functional evaluations per iteration) like the traditional Newton method. In addition, the proposed modified formula based on Newton method has better convergence characteristics than the traditional Newton method.
Abstract. Monitoring of chlorine residual in water distribution systems is necessary not only for ensuring potable water quality but also prevent emergence of disinfection by-products due to excess chlorination. Modelling work for chlorine residual was carried out for water supply distribution network of a town using both second order and first order reaction rate models. For the development of the model, the bulk reaction decay rate was determined in the laboratory using bottle testing while the wall decay rate was determined by calibration of the water quality model using field residual chlorine concentration measurements. The model results show that there is no significant difference in the residual chlorine between the two models or the cost saving that result in terms chlorine usage for the range of initial chlorine dosages anticipated. Constant rate chlorine model is more conservative and offers additional safety in terms of chlorine residual present. Significant differences only occur at excess chlorine residual concentration within the distribution system above the intended maximum residual to be attained. Further research that relates the chlorine dose with the water quality characteristics is necessary to make a more general evaluation.
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