On solving the chlorine transport model via Laplace transform

Abstract

This paper analyzes the two-dimensional chlorine-transport model in pipes. The studied model is in the form of a second-order partial differential equation with a set of boundary conditions. Obtaining exact solution for the current model is a challenge due to the nature of the involved boundary conditions, especially, when applying the Laplace transform. However, such difficulties are solved via implementing the method of residues. The exact solution is obtained in terms of the Bessel functions. The expression for a dimensionless cup-mixing average concentration is also derived analytically. The proposed approach is validated via numerical examples for comparing the results with those in the literature. The present analysis/approach is effective/straightforward and can be further applied on other similar models under different boundary conditions.

Closed-form asymptotic solution for the transport of chlorine concentration in composite pipes

“…As a special case of a concern, we study a somewhat similar case of [2][3][4], where a solid pipe was examined, that is, the radius of the media lies in 0 r b. In this regard, the obtained solution in equation (22) becomes…”

“…Moreover, concerning the level of chlorine for public safety, it was estimated that 'the percentage of chlorine concentration that ensures the avoidance of public health risks has been determined as 0.2 mgl − 1. Accordingly, the successful management of ensuring the quality of drinking water requires adherence to the aforementioned chlorine concentration and ensuring that this percentage does not rise above the specified limit' [2]. For more on the industrial relevance of chlorine, one may read the recent work by Aljohani et al [3], where various uses of chlorine were stated, including its great applications in water sciences, insecticides and drugs manufacturing, and textile industries to mention just a few.…”

“…The just stated model in equations (1)-( 3) was initially studied in [4] through the application of separation of variable method, and in [1,2] using the combination of Laplace transform method and the complex analysis' residue approach via the use of root-finding numerical schemes, and then in [3] where yet the application of separation of variable method is sought through the use of Mittag-Leffler functions after the incorporation of variable-order partial derivative. However, away from incorporating numerical tools, the present study makes consideration of the same formulation of chlorine transport in a single-layered pipe and further extends it to the case of bi-layered cylindrical pipe, having considered a composite conducting cylindrical pipe with perfect interfacial conditions imposed in-between the two layers.…”

A Simple Approach for Explicit Solution of The Neutron Diffusion Kinetic System

“…Besides, several authors ([3]- [8]) implemented different analytical and numerical methods to treating the system (1)- (5). The LT method was widely applied to solve various physical models [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. However, it requires massive calculations as seen in Refs.…”

A Novel Ansatz Method for Solving the Neutron Diffusion System in Cartesian Geometry

“…Although the LT method was shown as an effective and efficient approach to solve several scientific models [9][10][11][12][13][14][15][16][17][18][19][20][21][22][23], it needs massive computational calculations when applied on the present system, especially when using the residues method as can be observed in Khaled [2], Bakodah and Ebaid [16], and Ebaid et al [20]. However, researchers in the field of Mathematics and Physics would prefer simple but effective methods to solve the governing system of the studied phenomena.…”