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This paper presents the development of source-term representations and associated time-dependent compartmental mass solutions for the case of a consumer product being continuously applied to a surface. The product is assumed to be applied to the surface at a constant rate. The approach is to break up the application into many differential areas (“squares”), which instantly begin emitting once applied and then emit material at: (1) a constant rate, (2) an exponentially decreasing rate, or (3) a two-component exponential rate. This behavior has been modeled numerically, and examination of the resulting source dynamics leads to closed-form algebraic representations of these driving functions. Analytical derivations are also presented for these source behaviors. These functions are compared to the numerical approximations, for increasing numbers of the “squares.” Analytical solutions for the time-dependent concentrations of a pollutant emitted by the applied product are also developed, using Laplace transforms, for one- and two-compartment systems, when driven by each of these three closed-form source behaviors. These analytical results agree with those obtained via purely numerical (Runge-Kutta) solutions for the systems.
mathematical modeling, source terms, compartmental modeling, continuous application, analytical solutions, time-dependent concentrations, linear systems
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