Diffusion flux concentrated sources, equations

Central differences are typically used when the flux is predominately due to a diffusive term, where the gradients on both sides of the control volume are important. An exphcit equation is one where the unknown variable can be isolated on one side of the equation. To get an explicit equation in a computational routine, we must use the flux and source/sink quantities of the previous time step to predict the concentration of the next time step. 

In this section, we consider the singularly perturbed diffusion equation when linear combinations of the solution and its diffusion flux are given on the domain boundary. Such boundary conditions make it possible to realize any of the boundary conditions considered in Sections II and III. Moreover, concentrated sources act inside the domain. These sources lead to the appearance of interior layers. Thus, in addition to the computational problems accompanying the solution of the boundary value problems in Sections II and III, there arise new problems due to the presence of these interior layers. 

Thus, for singularly perturbed diffusion equations with mixed boundary conditions and concentrated sources, we have constructed new finite difference schemes that enable us to find solutions and normalized diffusion fluxes to an e-uniform accuracy. 

Equations (10) are generally valid for both liquid and gas phases if reactions take place there. They represent nothing but a differential mass balance for the film region with the account of the source term due to the reaction. To link this balance to the process variables like component concentrations, some additional relationships - often called constitutive relations (see Ref. [16]) - are necessary. For the component fluxes Ni, these constitutive relations result from the multicomponent diffusion description (Eqs. (1), (2)) for the source terms, from the reaction kinetics description. The latter strongly depends on the specific reaction mechanism [27]. The reaction rate expressions lli usually represent nonlinear dependencies on the mixture composition and temperature of the corresponding phase. 

These equations repeat those previously set down. Flete, u is the kinematic viscosity, and a is the thermal diffusivity. The subscripts have been dropped in the convective diffusion equation, and D can be the binary diffusion coefficient, the effective electrolytic diffusion coefficient, or the diffusion coefficient of the fth species. The molar concentration is to be interpreted in the same context. In the energy equation, sometimes referred to as the heat conduction equation in the form written, heat flux due to interdiffusion and due to viscous dissipation have been neglected as small. Heat sources are also absent. 

Equations (6) and (7) were solved with two sets of boundary conditions. The first set was source limited , i.e., disassociation rate-controlled and the second was flux limited , i.e., the concentration at the interface S was equal to an equilibrium value. The functions fi and f2 were assumed to be unity, Le., concentration-independent diffusion coefficients were used. The multi-phase Stefan problem was solved numerically [44] using a Crank-Nicholson scheme and the predictions were compared to experimental data for PS dissolution in MEK [45]. Critical angle illumination microscopy was used to measure the positions of the moving boundaries as a function of time and reasonably good agreement was obtained between the data and the model predictions (Fig. 4). 

Abstract In this chapter, an exothermic catalytic reaction process is simulated by using computational mass transfer (CMT) models as presented in Chap. 3. The difference between the simulation in this chapter from those in Chaps. 4,5, and 6 is that chemical reaction is involved. The source term in the species conservation equation represents not only the mass transferred from one phase to the other, but also the mass created or depleted by a chemical reaction. Thus, the application of the CMT model is extended to simulating the chemical reactor. The simulation is carried out on a wall-cooled catalytic reactor for the synthesis of vinyl acetate from acetic acid and acetylene by using both c — Sc model and Reynolds mass flux model. The simulated axial concentration and temperature distributions are in agreement with the experimental measurement. As the distribution of lx shows dissimilarity with Dj and the Sci or Pri are thus varying throughout the reactor. The anisotropic axial and radial turbulent mass transfer diffusivities are predicted where the wavy shape of axial diffusivity D, along the radial direction indicates the important influence of catalysis porosity distribution on the performance of a reactor. 

Partial differential equations [12]-[13] can be solved for a defined set of boundary and initial conditions. For sample flow that starts at t=0, the conditions ([14]-[18]) can be applied. Equations [14] and [15] define the zero concentration of free a and captured y analytes, respectively, before introducing a sample to the flow cell. Equation [16] ensures that there are no sources of the analyte a in the gel at time t = 0, eqn [17] defines the analyte flux through the inner hydrogel interface (i.e., analyte cannot pass through the solid support on which gel is attached), and eqn [18] describes that the analyte flux through the outer interface (with the sample) is driven by the mass diffusion rate k. ... 

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