Abstract
Transient mass-transfer phenomena occurring in natural and engineered systems consist of convection, diffusion, and reaction processes. The coupled phenomena can be described by using the unsteady convection-diffusion-reaction (CDR) equation, which is classified in mathematics as a linear, parabolic partial-differential equation. The availability of analytic solutions is limited to simple cases, e.g., unsteady diffusion and steady convective diffusion. The CDR equation has been considered analytically intractable, depending on the initial and boundary conditions. If spatial adsorption and desorption of matter are super-positioned in the CDR equation as sink and source functions, respectively, then the governing equation becomes an unsteady convection-diffusion-reaction-source (CDRS) equation, of which general solutions are unknown. In this study, a general 1D analytic solution of the CDRS equation is obtained by using a one-sided Laplace transform, by assuming constant diffusivity, velocity, and reactivity. This paper also provides a general formalism to derive 1D analytic solutions for Dirichlet/Dirichlet and Dirichlet/Neumann boundary conditions. Derivations of the analytic solutions are found to be straightforward if a combination of the source function and the initial concentration provide a non-zero singularity pole of inverse Laplace transform.
Highlights
Transient mass-transfer phenomena occurring in natural and engineered systems consist of convection, diffusion, and reaction processes
A full mathematical expression of the above four mass transfer mechanisms is known as the convection-diffusion-reaction-source (CDRS) equation, wherein a sink can be expressed as a negative source
Www.nature.com/scientificreports depends on an initial condition (IC) expressed as C(x, t = 0) = CI(x) and two boundary conditions (BCs): one at the inlet, C(x0, t) = C0(t), and the other at the outlet,C(x1, t) = C1(t), where x0 < x1 is given for convenience
Summary
Transient mass-transfer phenomena occurring in natural and engineered systems consist of convection, diffusion, and reaction processes. The conditional expression of Eq (6) has a simpler transient dependence as a product of the steady-state solution and exponential transient decay, but the Fourier series of Eq (7) resolves the discontinuity of an L-shaped initial concentration profile at the inlet boundary by using an infinite series.
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