Reaction kinetics diffusion equation

The reaction-diffusion equation describes how the local concentration of the reactant A varies within some infinitesimal volume element at some point in the reaction zone. For the above reaction kinetics, this equation will have the following form ... 

If the reaction mechanism is known, a transfer function may be calculated using the kinetics of the reactions, the diffusion equations, and so on. This transfer function couples the input from a system to the output. It includes all possible variables and is directly fitted to the measurements. In this way direct information on reaction parameters is obtained from the impedance. If the reaction mechanism is known and the system is not too complex, this method is favorable [92]. 

Figure 10 shows that Tj is a unique function of the Thiele modulus. When the modulus ( ) is small (- SdSl), the effectiveness factor is unity, which means that there is no effect of mass transport on the rate of the catalytic reaction. When ( ) is greater than about 1, the effectiveness factor is less than unity and the reaction rate is influenced by mass transport in the pores. When the modulus is large (- 10), the effectiveness factor is inversely proportional to the modulus, and the reaction rate (eq. 19) is proportional to k ( ), which, from the definition of ( ), implies that the rate and the observed reaction rate constant are proportional to (1 /R)(f9This result shows that both the rate constant, ie, a measure of the intrinsic activity of the catalyst, and the effective diffusion coefficient, ie, a measure of the resistance to transport of the reactant offered by the pore stmcture, influence the rate. It is not appropriate to say that the reaction is diffusion controlled it depends on both the diffusion and the chemical kinetics. In contrast, as shown by equation 3, a reaction in solution can be diffusion controlled, depending on D but not on k. 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

In the kinetics of formation of carbides by reaction of the metal widr CH4, the diffusion equation is solved for the general case where carbon is dissolved into tire metal forming a solid solution, until the concentration at the surface reaches saturation, when a solid carbide phase begins to develop on the free surface. If tire carbide has a tirickness at a given instant and the diffusion coefficient of carbon is D in the metal and D in the carbide. Pick s 2nd law may be written in the form (Figure 8.1)... 

The simple pore structure shown in Figure 2.69 allows the use of some simplified models for mass transfer in the porous medium coupled with chemical reaction kinetics. An overview of corresponding modeling approaches is given in [194]. The reaction-diffusion dynamics inside a pore can be approximated by a one-dimensional equation... 

In a conventional relaxation kinetics experiment in a closed reaction system, because of mass conservation, the system can be described in a single equation, e.g., SCc(t) = SCc(0)e Rt where R = ((Ca) + ( C b)) + kh- The forward and reverse rate constants are k and k t, respectively. In an open system A, B, and C, can change independently and so three equations, one each for A, B, and C, are required, each equation having contributions from both diffusion and reaction. Consequently, three normal modes rather than one will be required to describe the fluctuation dynamics. Despite this complexity, some general comments about FCS measurements of reaction kinetics are useful. 

The experimental and simulation results presented here indicate that the system viscosity has an important effect on the overall rate of the photosensitization of diary liodonium salts by anthracene. These studies reveal that as the viscosity of the solvent is increased from 1 to 1000 cP, the overall rate of the photosensitization reaction decreases by an order of magnitude. This decrease in reaction rate is qualitatively explained using the Smoluchowski-Stokes-Einstein model for the rate constants of the bimolecular, diffusion-controlled elementary reactions in the numerical solution of the kinetic photophysical equations. A more quantitative fit between the experimental data and the simulation results was obtained by scaling the bimolecular rate constants by rj"07 rather than the rf1 as suggested by the Smoluchowski-Stokes-Einstein analysis. These simulation results provide a semi-empirical correlation which may be used to estimate the effective photosensitization rate constant for viscosities ranging from 1 to 1000 cP. 

In the biomedical literature (e.g. solute = enzyme, drug, etc.), values of kf and kr are often estimated from kinetic experiments that do not distinguish between diffusive transport in the external medium and chemical reaction effects. In that case, reaction kinetics are generally assumed to be rate-limiting with respect to mass transport. This assumption is typically confirmed by comparing the adsorption transient to maximum rates of diffusive flux to the cell surface. Values of kf and kr are then determined from the start of short-term experiments with either no (determination of kf) or a finite concentration (determination of kT) of initial surface bound solute [189]. If the rate constant for the reaction at the cell surface is near or equal to (cf. equation (16)), then... 

If Da = 1 is defined as the transition between diffusionally controlled and kinetically controlled regimes, an inverse relationship is observed between the particle diameter and the system pressure and temperature for a fixed Da. Thus, for a system to be kinetically controlled, combustion temperatures need to be low (or the particle size has to be very small, so that the diffusive time scales are short relative to the kinetic time scale). Often for small particle diameters, the particle loses so much heat, so rapidly, that extinction occurs. Thus, the particle temperature is nearly the same as the gas temperature and to maintain a steady-state burning rate in the kinetically controlled regime, the ambient temperatures need to be high enough to sustain reaction. The above equation also shows that large particles at high pressure likely experience diffusion-controlled combustion, and small particles at low pressures often lead to kinetically controlled combustion. 

The variation of efficiencies is due to interaction phenomena caused by the simultaneous diffusional transport of several components. From a fundamental point of view one should therefore take these interaction phenomena explicitly into account in the description of the elementary processes (i.e. mass and heat transfer with chemical reaction). In literature this approach has been used within the non-equilibrium stage model (Sivasubramanian and Boston, 1990). Sawistowski (1983) and Sawistowski and Pilavakis (1979) have developed a model describing reactive distillation in a packed column. Their model incorporates a simple representation of the prevailing mass and heat transfer processes supplemented with a rate equation for chemical reaction, allowing chemical enhancement of mass transfer. They assumed elementary reaction kinetics, equal binary diffusion coefficients and equal molar latent heat of evaporation for each component. 

The equations used in these models are primarily those described above. Mainly, the diffusion equation with reaction is used (e.g., eq 56). For the flooded-agglomerate models, diffusion across the electrolyte film is included, along with the use of equilibrium for the dissolved gas concentration in the electrolyte. These models were able to match the experimental findings such as the doubling of the Tafel slope due to mass-transport limitations. The equations are amenable to analytic solution mainly because of the assumption of first-order reaction with Tafel kinetics, which means that eq 13 and not eq 15 must be used for the kinetic expression. The different equations and limiting cases are described in the literature models as well as elsewhere. 

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... 

The mathematical difficulty increases from homogeneous reactions, to mass transfer, and to heterogeneous reactions. To quantify the kinetics of homogeneous reactions, ordinary differential equations must be solved. To quantify diffusion, the diffusion equation (a partial differential equation) must be solved. To quantify mass transport including both convection and diffusion, the combined equation of flow and diffusion (a more complicated partial differential equation than the simple diffusion equation) must be solved. To understand kinetics of heterogeneous reactions, the equations for mass or heat transfer must be solved under other constraints (such as interface equilibrium or reaction), often with very complicated boundary conditions because of many particles. 

In what I regard as the world of change (essentially chemical kinetics and dynamics), there are three central equations. One is the form of a rate law, v = /[A],[B]...), and all its implications for the prediction of the outcome of reactions, their mechanisms, and, increasingly, nonlinear phenomena, and the other closely related, augmenting expression, is the Arrhenius relation, k = Aexp(-EJRT), and its implications for the temperature-dependence of reaction rates. Lurking behind discussions of this kind is the diffusion equation, in its various flavors starting from the vanilla dP/dt = -d2P/dl2 (which elsewhere I have referred to as summarizing the fact that Nature abhors a wrinkle ). 

CO oxidation activity of (Ceo.g.Lao.OOpgs and (Ce0.8,Zr0.2)O2 heated at 1,000°C are shown in Fig. 8. The activity of Ce02 is extremely improved by the addition of La and Zr into Ce02. The reaction kinetics is controlled by the diffusion of lattice oxygen and is described by the following equation ... 

By contrast, when both the reactive solute molecules are of a size similar to or smaller than the solvent molecules, reaction cannot be described satisfactorily by Langevin, Fokker—Planck or diffusion equation analysis. Recently, theories of chemical reaction in solution have been developed by several groups. Those of Kapral and co-workers [37, 285, 286] use the kinetic theory of liquids to treat solute and solvent molecules as hard spheres, but on an equal basis (see Chap. 12). While this approach in its simplest approximation leads to an identical result to that of Smoluchowski, it is relatively straightforward to include more details of molecular motion. Furthermore, re-encounter events can be discussed very much more satisfactorily because the motion of both reactants and also the surrounding solvent is followed. An unreactive collision between reactant molecules necessarily leads to a correlation in the motion of both reactants. Even after collision with solvent molecules, some correlation of motion between reactants remains. Subsequent encounters between reactants are more or less probable than predicted by a random walk model (loss of correlation on each jump) and so reaction rates may be expected to depart from those predicted by the Smoluchowski analysis. Furthermore, such analysis based on the kinetic theory of liquids leads to both an easy incorporation of competitive effects (see Sect. 2.3 and Chap. 9, Sect. 5) and back reaction (see Sect. 3.3). Cukier et al. have found that to include hydrodynamic repulsion in a kinetic theory analysis is a much more difficult task [454]. 

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