Convection-Diffusion Equation with Reactions

This section is devoted to probabilistic solutions of reaction-diffusion equations in terms of functional integrals. We will not attempt to cover the general theory and all relevant equations. Our purpose is to discuss the main ideas and principal results and give illustrating examples involving typical equations. The reader interested in the general theory and all mathematical details will find a comprehensive treatment of the subject in Freidlin s book [141], 

So far we have discussed the probabilistic solution of the convection-diffusion equation only. There are various directions in which a probabilistic approach to PDFs can be extended and generalized. The first direction is to extend it to the case where chemical reactions are taken into account. The next direction would be to allow the velocity field v and the diffusion matrix D to depend on both space x and time t. Another direction for generalization is to analyze initial-boundary problems. 

We start with the one-dimensional reaction-diffusion equation 

If the mesoscopic density of particles obeys the integro-differential equation 

If the process X t) is a symmetric a-stable Ldvy motion 5 (t) on R, then the formula (3.329) provides the solution to the Cauchy problem 

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T.4.2 Nonstationary Convection-Diffusion Equation with Reactions... 

Convective diffusion equation with a volume reaction. If a volume (homogeneous) chemical reaction proceeds in the bulk of flow, the convective diffusion... 

The phenomena and processes described can be modeled by convective diffusion equations with chemical reactions. In the simplest model, we may apply these equations in a cylindrical capillary and by means of a capillary model to a porous medium. Assuming dilute solutions, rapid chemical reactions, the double-layer thickness to the soil pore radius and the Peclet number based on the pore radius both small, the overall transport rate for the ith species in a straight cylindrical capillary is... 

The advection—diffusion equation with a source term can be solved by CFD algorithms in general. Patankar provided an excellent introduction to numerical fluid flow and heat transfer. Oran and Boris discussed numerical solutions of diffusion—convection problems with chemical reactions. Since fuel cells feature an aspect ratio of the order of 100, 0(100), the upwind scheme for the flow-field solution is applicable and proves to be very effective. Unstructured meshes are commonly employed in commercial CFD codes. 

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. 

Measurements of the rate of deposition of particles, suspended in a moving phase, onto a surface also change dramatically with ionic strength (Marshall and Kitchener, 1966 Hull and Kitchener, 1969 Fitzpatrick and Spiel-man, 1973 Clint et al., 1973). This indicates that repulsive double-layer forces are also of importance to the transport rates of particulate solutes. When the interactions act over distances that are small compared to the diffusion boundary-layer thickness, the rate of transport can be computed (Ruckenstein and Prieve, 1973 Spiel-man and Friedlander, 1974) by lumping the interactions into a boundary condition on the usual convective-diffusion equation. This takes die form of an irreversible, first-order reaction on tlie surface. A similar analysis has also been performed for the case of unsteady deposition from stagnant suspensions (Ruckenstein and Prieve, 1975). 

The Eulerian (bottom-up) approach is to start with the convective-diffusion equation and through Reynolds averaging, obtain time-smoothed transport equations that describe micromixing effectively. Several schemes have been proposed to close the two terms in the time-smoothed equations, namely, scalar turbulent flux in reactive mixing, and the mean reaction rate (Bourne and Toor, 1977 Brodkey and Lewalle, 1985 Dutta and Tarbell, 1989 Fox, 1992 Li and Toor, 1986). However, numerical solution of the three-dimensional transport equations for reacting flows using CFD codes are prohibitive in terms of the numerical effort required, especially for the case of multiple reactions with... 

The heat-loss term is analogous to that appearing in equation (B-52) and is assumed to be nonnegative and to vanish at the initial temperature (t = 0). The heat-loss distribution, k (t), is a critical factor in the structure of the nonadiabatic flame. It is found that if k (t) is of order unity or larger over a range of of order unity, then flame-structure solutions do not exist with the present formulation therefore, at first let us assume that k t) is of order p everywhere. In this case, upstream from the reaction zone in equation (10) there is a convective-diffusive balance with negligible loss in the first approximation, and the approximate solution... 

In this paper we combine the approach of [6], which consists in solving the equations for the electric fields in the anode, cathode and the electrolyte under steady state conditions, with our own approximation of the electrochemical reaction and the transport of reactants. We solve a 2D problem for the Laplace equation coupled with a system of the convection-diffusion equations through use of the boundary conditions. Therefore om problem becomes non-stationary. We study the time period of about one horn and observe the formation of the C02 boundary layer and the variation of the Galvani potential caused by it. 

To increase the electrolyte conductivity, an additional ionic component that does not participate in the electrochemical reactions is often added to a solution. This nom-eactive component is called a supporting electrolyte or indifferent electrolyte. In the presence of a supporting electrolyte, there is a lowering of the electric field in solution, due to the electrolyte s high conductivity. Transport of the minor ionic species in solution is due primarily to diffusion and convection, in accordance with Equation (26.54) with VO = 0. Also, in the presence of a supporting electrolyte, the convective diffusion equation for a minor component in solution is written as... 

We now turn to the second criterion, in particular bearing in mind the criticism, alluded to above, about the difficulty associated with the theoretical description of processes at non-uniformly accessible electrodes. Again, we will compare and contrast the channel electrode and the RDE. Now the theoretical description of electrode reactions involves, typically, the solution of perhaps several coupled steady-state convective-diffusion equations of the form... 

DIMENSIONLESS FORM OF THE GENERALIZED MASS TRANSFER EQUATION WITH UNSTEADY-STATE CONVECTION, DIFFUSION, AND CHEMICAL REACTION... 

Hence, the dimensionless quantity that appears on the left side of the following mass transfer equation with convection, diffusion, and chemical reaction,... 

The Reynolds number in microreaction systems usually ranges from 0.2 to 10. In contrast to the turbulent flow patterns that occur on the macroscale, viscous effects govern the behavior of fluids on the microscale and the flow is always laminar, resulting in a parabolic flow profile. In microfluidic reaction systems, where the characteristic length is usually greater than 10 pm, a continuum description can be used to predict the flow characteristics. This allows commercially written Navier-Stokes solvers such as FEMLAB and FLUENT to model liquid flows in microreaction channels. However, modeling gas flows may require one to take account of boundary sUp conditions (if 10 < Kn < 10 , where Kn is the Knudsen number) and compressibility (if the Mach number Ma is greater than 0.3). Microfluidic reaction systems can be modeled on the basis of the Navier-Stokes equation, in conjunction with convection-diffusion equations for heat and mass transfer, and reaction-kinetic equations. 

Here, p is mass density and yk th mass fraction, t is time and div the divergence operator v is local mass flow velocity (vector) and jk the it-th molecular diffusion flux vector, added to the term pykV representing the convection of particles Ck by the motion of a material element as a whole. So the instantaneous local change (increase) of the Ck-concentration (mass per unit volume) equals minus the amount that escapes from a volume element (the divergence term) plus the amount produced by chemical reactions. Physically, the balance makes sense if we know how the flux jk depends on the gradients (most simply by Pick s law), and how the rates of possible reactions depend on the local state of the element. If also the latter information is available then the balance takes the form of convective diffusion equation, possibly with chemical reactions. [If we have no information on the reaction rates, the w -terms can be eliminated from Eqs. (C.2) by an algebraic transformation in the same manner as in Chapter 4 indeed, it is sufficient to substitute for W, in (4.3.2) and to define the components of column vector n as follows from (C.2).] Observe finally that we have... 

The conservation of species is actually the law of conservation of mass applied to each species in a mixture of various species. The fluid element, as described in Sections 6.2.1.1 through 6.2.1.3, does not comprise of pure fluid with only one species, such as water, but of many species forming a multicomponent mixture. This law is mathematically described by the continuity equation for species, also known as the species equation, advection-diffusion equation, or convection-diffusion equation. If the species equation additionally includes a reaction term, it is known as the reaction-diffusion-advection equation. 

The efficiency of solution-phase (two aqueous phase) enzymatic reaction in microreactor was demonstrated by laccase-catalyzed l-DOPA oxidation in an oxygen-saturated water solution, and analyzed in a Y-shaped microreactor at different residence times (Figure 10.24) [142]. Up to 87% conversions of l-DOPA were achieved at residence times below 2 min. A two-dimensional mathematical model composed of convection, diffusion, and enzyme reaction terms was developed. Enzyme kinetics was described with the double substrate Michaelis-Menten equation, where kinetic parameters from previously performed batch experiments were used. Model simulations, obtained by a nonequidistant finite differences numerical solution of a complex equation system, were proved and verified in a set of experiments performed in a microreactor. Based on the developed model, further microreactor design and process optimization are feasible. 

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