Equations convection-diffusion-reaction

Hence, under the quasi-steady approximation, the movement of the species is dictated by a macroscopic convection-diffusion-reaction equation with an instantaneous adsorption/desorption source term. A notable consequence of the three-scale approach is the double-averaging representation for the partition coefficient A which is defined as... 

One chemical comes in at the left, and another chemical plus a catalyst enters at the top inlet. The concentration distributions are found by solving the convection-diffusion-reaction equations for the two chemicals (A and B) plus the catalyst (C). The system is assumed to be dilute so that the total concentration does not change appreciably. The reaction is taken as ... 

As an alternative to film models, McNamara and Amidon [6] included convection, or mass transfer via fluid flow, into the general solid dissolution and reaction modeling scheme. The idea was to recognize that diffusion was not the only process by which mass could be transferred from the solid surface through the boundary layer [7], McNamara and Amidon constructed a set of steady-state convective diffusion continuity equations such as... 

In Eq. (13), r stands for the production (or consumption) of the species of interest due to a chemical reaction, while in Eq. (12) q represents the heat production, e.g., due to one of more chemical reactions. Equation (13) is often referred to as the Convection-Diffusion-Reaction (CDR) equation. 

A fundamental fuel cell model consists of five principles of conservation mass, momentum, species, charge, and thermal energy. These transport equations are then coupled with electrochemical processes through source terms to describe reaction kinetics and electro-osmotic drag in the polymer electrolyte. Such convection—diffusion—source equations can be summarized in the following general form... 

At the other extreme, it may be argued that the traditional low-dimensional models of reactors (such as the CSTR, PFR, etc.) should be abandoned in favor of the detailed models of these systems and numerical solution of the full convection-diffusion reaction (CDR) equations using computational fluid dynamics (CFD). While this approach is certainly feasible (at least for singlephase systems) due to the recent availability of computational power and tools, it may be computationally prohibitive, especially for multi-phase systems with complex chemistry. It is also not practical when design, control and optimization of the reactor or the process is of main interest. The two main drawbacks/criticisms of this approach are (i) It leads to discrete models of very high dimension that are difficult to incorporate into design and control schemes. 

By examining these characteristic dimensionless numbers, it is possible to appreciate possible interactions of different processes (convection, diffusion, reaction and so on) and to simplify the governing equations accordingly. A typical dimensionless form of the governing equation can be written (for a general variable, (p) ... 

Equation 2.20 is the advection-dispersion (AD) equation. In the petroleum literature, the term convection-diffusion (CD) equation is used, or simply diffusion equation (Brigham, 1974). When a reaction term is included, the term advection-reaction-dispersion (ARD) equation is used elsewhere. When the adsorption term is expressed as a reaction term, the ARD equation is as discussed later in Section 2.4. Several solutions of Eq. 2.20 have been presented in the literature, depending on the boundary conditions imposed. In general, they are various combinations of the error function. When the porous medium is long compared with the length of the mixed zone, they all give virtually identical results. 

As a preliminary test, the governing equation (i.e., convection-diffusion-reaction PDE) is solved using an ODE time integrator for the binary system Including only A and B components. Fig. 2 shows effects of the three kernels on liquid concentrations. Abnormal concentration profiles are found in inactive zones for the conventional rate model (Fig. 2a) and that with the sum kernel (Fig. 2b). The product kernel will play an effective role for multi-component systems. 

As cells are much smaller than the size of the scaffold, we can employ a continuous formulation to describe nutrient or GF transport in the scaffold/tissue scale. In the presence of forced convection (i.e., unidirectional fluid flow through the scaffold or perfusion [9]), the spatiotemporal evolution of the extracellular concentration C x,y,z,t) can be computed by solving a convection-diffusion-reaction problem described by the following partial differential equation ... 

The continuous PDE of Equation 26.1 must be coupled and solved together with the CA model that treats cells as discrete entities. Thus, the migrating and proliferating cells must be considered as moving sinks (and/or sources) for the convection-diffusion-reaction problem and the cell density p j is actually a discontinuous function that is nonzero only in lattice sites occupied by cells ... 

Example Consider the equation for convection, diffusion, and reaction in a tiihiilar reactor. 

The reaction plane model with heterogeneous reactions was discussed at length for acid-base reactions in the previous section. The same modeling technique, of confining the reactions to planes, can be applied to micelle-facilitated dissolution. As with the acid-base model, one starts with a one-dimensional steady-state equation for mass transfer that includes diffusion, convection, and reaction. This equation is then applied to the individual species i, i.e., the solute, s, the micelle, m, and the drug-loaded micelle, sm, to yield... 

For a number of flow situations, the mass-transfer rate can be derived directly from the equation of convective diffusion (see Table VII, Part A). The velocity profile near the electrode is known, and the equation is reduced to a simpler form by appropriate similarity transformations (N6). These well-defined flows, therefore, are being exploited increasingly by electrochemists as tools for the kinetic characterization of electrode reactions. Current distributions at, or below, the limiting current, transient mass transfer, and other aspects of these flows are amenable to analysis. Especially noteworthy are the systematic investigations conducted by Newman (review until 1973 in N7 also N9b, N9c, H6b and references in Table VII), by Daguenet and other French workers (references in Table VII), and by Matsuda (M4a-d). Here we only want to comment on the nature of the velocity profile near the electrode, and on the agreement between theory and mass-transfer experiment. 

In both AEA and RRA, there are inert convective-diffusive regions on the fuel and oxidizer sides of the main reaction regions of the diffusion flame. Conservation equations are written for each of the outer inert regions, and their solutions are employed as matching conditions for the solutions in the inner reaction regions. The inner structure for RRA is more complicated than that for AEA because the chemistry is more complex [53]. The inner solutions nevertheless can be developed, and matching can be achieved. The outer solutions will be summarized first, then the reaction region will be discussed. 

In what follows, the preceding evaluation procedure is employed in a somewhat different mode, the main objective now being to obtain expressions for the heat or mass transfer coefficient in complex situations on the basis of information available for some simpler asymptotic cases. The order-of-magnitude procedure replaces the convective diffusion equation by an algebraic equation whose coefficients are determined from exact solutions available in simpler limiting cases [13,14]. Various cases involving free convection, forced convection, mixed convection, diffusion with reaction, convective diffusion with reaction, turbulent mass transfer with chemical reaction, and unsteady heat transfer are examined to demonstrate the usefulness of this simple approach. There are, of course, cases, such as the one treated earlier, in which the constants cannot be obtained because exact solutions are not available even for simpler limiting cases. In such cases, the procedure is still useful to correlate experimental data if the constants are determined on the basis of those data. 

Modification of the convective-diffusion equation involves addition of terms to take account of the preceding reaction. Thus, for the rotating disc electrode, we obtain... 

Solution proceeds via addition of the two convective-diffusion equations and leads to the result [151], assuming a rapid chemical reaction... 

We have assumed above that the rate constant of the homogeneous reaction, fe2, is infinitely large so that the boundary between the A2 and X-dominated zones is clearly defined. When k2 is smaller, this boundary will be less clear. The modified convective-diffusion equations are difficult to solve and numerical solution is necessary. This has been performed at the RRDE making the assumptions that [221, 222]... 

Needless to say, the assumption of plug flow is not always appropriate. In plug flow we assume that the convective flow, i. e., the flow at velocity qjAt = v that is caused by a compressor or pump, is dominating any other transport mode. In practice this is not always so and dispersion of mass and heat, driven by concentration and temperature gradients are sometimes significant enough to need to be included in the model. We will discuss such a model in detail, not only because of its importance, but also because the techniques used to handle the ensuing boundary value differential equations are similar to those used for other diffusion-reaction problems such as catalyst pellets, as well as for counter-current processes. 

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