Analytical solutions of TCE transport with convergent reactions

In this paper, we present analytical solutions for reactive transport with convergent reactions using the singular value decomposition approach. We consider a reaction network in which a compound reacts to form multiple daughter products, which further react to a single granddaughter. This reaction scheme is illustrated by the dechlorination of trichloroethylene (TCE), which reacts to form three daughter products, cis-dichloroethylene (cis-DCE), trans-dichloroethylene (trans-DCE), and 1,1-dichloroethylene (1,1-DCE). All three of these daughter products will further react to produce the same granddaughter, vinyl chloride (VC). In achieving the analytical solution, all reactions are assumed to be first-order. Because the partial differential equation describing the final product (granddaughter) concentration, is coupled by the three daughter concentrations, previously published methods for decoupling partial differential equations are not applicable. Instead, we conduct the singular value decomposition process analytically and transform the system of transport equations with convergent reactions into orthogonal (independent) equations, for which analytical solutions are available. The solutions derived in this manuscript are then compared with numerical solutions and the performance of the reactive transport system is analyzed.