Mass transport Laplace transform

A most convenient way to solve the differential equations describing a mass transport problem is the Laplace transform method. Applications of this method to many different cases can be found in several modern and classical textbooks [21—23, 53, 73]. In addition, the fact that electrochemical relationships in the so-called Laplace domain are much simpler than in the original time domain has been employed as an expedient for the analysis of experimental data or even as the basic principle for a new technique. The latter aspect, especially, will be explained in the present section. 

The occurrence of partial differential equations in electrochemistry is due to the variation of concentration with distance and with time, which are two independent variables, and are expressed in Fick s second law or in the convective-diffusion equation, possibly with the addition of kinetic terms. As in the resolution of any differential equation, it is necessary to specify the conditions for its solution, otherwise there are many possible solutions. Examples of these boundary conditions and the utilization of the Laplace transform in resolving mass transport problems may be found in Chapter 5. 

Since the surface concentration of O becomes zero (Eq. 37), the rate of the reduction of O will be mass-transport-controlled or rather diffusion-controlled, as the migration and convection terms can be neglected. Thus, the experiment can be described by use of Tick s first and second laws (Eqs. 38 and 34, respectively). The substrate O is the only species initially present in the cell (Eq. 35) and because the electrode area A is small compared to the cell volume V in electroanalytical experiments, the bulk concentration can be assumed to be unchanged during the experiment (Eq. 36). Should a preparative conversion of O to R be the goal, a large A/V ratio would have been desirable. Equations 34-38 can be solved with the Laplace transformations this affords the Cottrel equation (Eq. 39). 

D Me-S surface alloy and/or 3D Me-S bulk alloy formation and dissolution (eq. (3.83)) is considered as either a heterogeneous chemical reaction (site exchange) or a mass transport process (solid state mutual diffusion of Me and S). In site exchange models, the usual rate equations for the kinetics of heterogeneous reactions of first order (with respect to the species Me in Meads and Me t-S>>) are applied. In solid state diffusion models, Pick s second law and defined boundary conditions must be solved using Laplace transformation. 

Sudicky, E. A., and R.G. McLaren. 1992. The Laplace transform Galerkin technique for large-scale simulation of mass transport in discretely fractured porous formations. Water Resour. Res. 28 499-514. 

A discrete view of the transport considers either random trajectories of discrete particles, or random streamtubes of tracer "parcels" carried by a constant flow rate through a network of fractures. For pulse injection of mass Af, we compute the tracer discharge as J(t) = My(t,T), the Laplace transform of y is... 

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