Search

This paper discusses a possible generalization of the transport model describing the chlorine concentration decay in pipes. The proposed generalized model is governed by a second-order fractional partial differential equation. The exact solution of the generalized model is obtained via the Laplace transform method and the method of residues. The exact solution reduces to the corresponding published one as the fractional order α tends to one. Analytical expression for the dimensionless cup-mixing average concentration is deduced. Influences of various parameters on the behavior of the dimensionless cup-mixing average concentration are discussed. It is shown that the physical interpretation of the dimensionless cup-mixing average concentration in view of the fractional calculus is completely different than its interpretation in the classical calculus.

This paper re-investigates the mathematical transport model of chlorine used as a water treatment model, when a variable order partial derivative is incorporated for describing the chlorine transport system. This model was introduced in the literature and governed by a fractional partial differential equation (FPDE) with prescribed boundary conditions. The obtained solution in the literature was based on implementing the Laplace transform (LT) combined with the method of residues and expressed in terms of regular exponential functions. However, the present analysis avoids such a method of residues, and thus a new analytical solution is introduced in this paper via Mittag-Leffler functions. Therefore, an effective approach is developed in this paper to solve the chlorine transport model with non-integer order derivative. In addition, our results are compared with several studies in the literature in case of integer-order derivative and the differences in results are explained.