Explicit Rate Expression

While the above formal rate expression is adequate for numerical computation of the rate from numerically calculated resistances, it is more desirable to obtain, if possible, an explicit rate expression in terms of the terminal species composition. This is accomplished as follows. 

Under the QSS assumption, the overall rate equation may be substantially simplified. Since 59, ii2, 5m and Sn have been eliminated from the mechanism, there now remain only 3 linearly independent RRs in Table 3, namely, RRn, RRm and RRvi- Thus, the QSS conditions now provide 

Using these relations in Eq. (45) and keeping in mind that the affinities along all RRs are equal, after some algebra, we obtain 

This is precisely the same result as that obtained earlier based on a different approach [14], 

The error in the conversion of CO provided by this overall rate equation is virtually zero as compared with the exact microkinetic model, which points to the robustness of the reaction network analysis approach presented here. 

---

For the system (2.36), in the limit e —> 0, the term (l/sjkfx) becomes indeterminate. For rate-based chemical and physical process models, this allows a physical interpretation in the limit when the large parameters in the rate expressions approach infinity, the fast heat and mass transfer, reactions, etc., approach the quasi-steady-state conditions of phase and/or reaction equilibrium (specified by k(x) = 0). In this case, the rates of the fast phenomena, as given by the explicit rate expressions, become indeterminate (but, generally, remain different from zero i.e., the fast reactions and heat and mass transfer do still occur). 

As several workers have shown (for example, Ref. 6), it is possible to avoid any explicit rate expression for the overall decomposition of a paraffin. One defines all the larger number of free radical and molecular equations of significance, develops an appropriate computer program and adjusts, within credible limits, Arrhenius parameters of the various rate expressions to fit an available body of rate and yield data. 

There are currently two available different ways in which one might use the predicted kinetic information on elementary reaction steps 1) the conventional Langmuir-Hinshelwood-Hougen-Watson (LHHW) approach [3], in which an explicit rate expression might be derived based on the common, but rather arbitrary. 

In this way, a temperature-explicit rate expression can be obtained as ... 

The generahzed rate laws serve as a substitute for the actual mechanism, which might be either unknown or too complex to derive explicit rate expressions. The apparent danger of Eq. (6.138) and similar approaches is that not being based on a biochemical mechanism, it can give an infinite value of rates when, for example, the substrate or product concentrations are set to zero. 

C THE REACTION IS NOW CALCULATED USING THE EXPLICIT FORM OF C THE RATE EXPRESSION GIVEN IN THE PAPER C... 

Before returning to the non-BO rate expression, it is important to note that, in this spectroscopy case, the perturbation (i.e., the photon s vector potential) appears explicitly only in the p.i f matrix element because this external field is purely an electronic operator. In contrast, in the non-BO case, the perturbation involves a product of momentum operators, one acting on the electronic wavefimction and the second acting on the vibration/rotation wavefunction because the non-BO perturbation involves an explicit exchange of momentum between the electrons and the nuclei. As a result, one has matrix elements of the form (P/ t)Xf > in the non-BO case where one finds lXf > in the spectroscopy case. A primary difference is that derivatives of the vibration/rotation functions appear in the former case (in (P/(J.)x ) where only X appears in the latter. 

This reaction is complex even though it has a stoichiometric equation and rate expression that could correspond to an elementary reaction. Recall the convention used in this text when a rate constant is written above the reaction arrow, the reaction is assumed to be elementary with a rate that is consistent with the stoichiometry according to Equation (1.14). The reactions in Equations (2.5) are examples. When the rate constant is missing, the reaction rate must be explicitly specihed. The reaction in Equation (2.6) is an example. This reaction is complex since the mechanism involves a short-lived intermediate, B. 

In Eq. (1.5) the surface coverage is given by 9c, and 9c is related to parameter X of Eq. (1.7). Equation (1.5) can be rewritten to show explicitly its dependence on gas-phase concentration. Equation (1.17a) gives the result. This expression can be related to practical kinetic expressions by writing it as a power law as is done in Eq. (1.18b). Power-law-type rate expressions present the rate of a reaction as a function of the reaction order. In Eq. (1.17b) the reaction order is m in H2 and —n in CO. 

In its application to specific kinetics studies this general procedure may take on a variety of forms that are minor modifications of that outlined above. One modification does not require an explicit assumption of the form of 0(Q) including numerical values of the orders of the reaction with respect to the various species, but merely an assumption that the rate expression is of the following form. 

In the examples in Sections 7.1 and 7.2.1, explicit analytical expressions for rate laws are obtained from proposed mechanisms (except branched-chain mechanisms), with the aid of the SSH applied to reactive intermediates. In a particular case, a rate law obtained in this way can be used, if the Arrhenius parameters are known, to simulate or model the reaction in a specified reactor context. For example, it can be used to determine the concentration-(residence) time profiles for the various species in a BR or PFR, and hence the product distribution. It may be necessary to use a computer-implemented numerical procedure for integration of the resulting differential equations. The software package E-Z Solve can be used for this purpose. 

This simple form for the burning rate expression is possible because the equations are developed for the conditions in the gas phase and the mass burning rate arises explicitly in the boundary condition to the problem. Since the assumption is made that no radiation is absorbed by the gases, the radiation term appears only in the boundary condition to the problem. 

The liquid bulk is assumed to be at chemical equilibrium. Contrary to gas-liquid systems, for vapour-liquid systems it is not possible to derive explicit analytical expressions for the mass fluxes which is due to the fact that two or more physical equilibrium constants m, have to be dealt with. This will lead to coupling of all the mass fluxes at the vapour - liquid interface since eqs (15c) and (19) have to be satisfied. For the system described above several simulations have been performed in which the chemical equilibrium constant K = koiAo2 and the reaction rate constant koi have been varied. Parameter values used in the simulations are given in Table 5. The results are presented in Figs 9 and 10. 

No explicit mathematical model of the method was presented. However, a short descriptive model was outlined For each run, the average ignition and combustion rate (expressed as weight of fuel ignited or burnt per unit bed area and unit time) were calculated by determining the time taken for the ignition front to pass down through the bed and the completion of burn-out, respectively. No discussion is presented about limitations and assumptions of the method. 

The essential difference between the homogeneous model and the heterogeneous one is that the latter model takes into account the fact that the diffusion of the absorbed component alternately occurs through continuous- and dispersed phases in the liquid boundary layer at the gas-hquid interface. The mass transport through this heterogeneous phase is a nonUnear process, one can get explicit mathematical expression for the absorption rate only after its simpHfica-tion. 

In the Surface Chemkin formalism, surface processes are written as balanced chemical reactions governed by the law of mass-action kinetics. The framework was developed to provide a very general way to describe heterogeneous processes. In this section many of the standard surface rate expressions are introduced. The connection between these common forms and the explicit mass-action kinetics approach is shown in each case. 

Obviously, the faradaic impedance equals the sum of the two contributions f ct, the charge transfer resistance, and Zw = aco-1/2 (1 — i), the Warburg impedance. Again, the meaning of the parameters Rct and a is still implicit at this stage of the treatment and explicit expressions have to be deduced from an explicit rate equation, e.g. the expressions given in eqns. (51). 

Under some circumstances there will be a resistance to the transport of material from the bulk fluid stream to the exterior surface of a catalyst particle. When such a resistance to mass transfer exists, the concentration CA of a reactant in the bulk fluid will differ from its concentration CAi at the solid-gas interface. Because CAi is usually unknown it is necessary to eliminate it from the rate equation describing the external mass transfer process. Since, in the steady state, the rates of all of the steps in the process are equal, it is possible to obtain an overall rate expression in which CM does not appear explicitly. 

This sort of analysis is very important in the formulation of the steady state approximation, developed to deal with kinetic schemes which are too complex mathematically to give simple explicit solutions by integration. Here the differential rate expression can be integrated. The differential and integrated rate equations are given in equations (3.61)—(3.66). 

Phenol, here denoted as reactant A. Since the second reactant, formaldehyde, is fed to the reactor in large excess, its concentration can be assumed as a constant during the reaction thus, it does not explicitly appear in the rate expressions and has not been considered in the reduced kinetic models. 

A solution methodology of the above, a nonlinear differential equation, will be shown. In essence, this solution method serves the mass-transfer rate and the concentration distribution in closed, explicit mathematical expression. The method can be applied for Cartesian coordinates and cylindrical coordinates, as will be shown. For the solution of Equation 14.2, the biocatalytic membrane should be divided into M sublayers, in the direction ofthe mass transport, that is perpendicular to the membrane interface (for details see e.g., Nagy s paper [40]), with thickness of A8 (A8 = 8/M) and with constant transport parameters in every sublayer. Thus, for the mth sublayer ofthe membrane layer, using dimensionless quantities, it can be obtained ... 

It should be understood that this rate expression may in fact represent a set of diffusion and mass transfer equations with their associated boundary conditions, rather than a simple explicit expression. In addition one may write a differential heat balance for a column element, which has the same general form as Eq. (17), and a heat balance for heat transfer between particle and fluid. In a nonisothermal system the heat and mass balance equations are therefore coupled through the temperature dependence of the rate of adsorption and the adsorption equilibrium, as expressed in Eq. (18). 

On the basis of the principle of independency of the rates of elementary acts of chemical reactions, equations (2.5i)-(2.62) are assumed to be independent of each other. Therefore, the increases of layer thicknesses can explicitly be expressed from these equations as follows ... 

The V-B coupling Hamiltonian to first order in the three HOD dimensionless normal coordinates is Hv b = —2, c], l , where F, is the inter-molecular force due to the solvent exerted on the harmonic normal coordinate, evaluated at the equilibrium position of the latter. This force obviously depends on the relative separations of all molecules, and on their relative orientations. In the most rigorous quantum description of rotations, this term would depend on the excited molecule rotational eigenstates and of the solvent molecules. Instead rotation was treated classically, a reasonable approximation for water at room temperature. With this form for the coupling, the formal conversion of the Golden Rule formula into a rate expression follows along the lines developed by Oxtoby (2,53), with a slight variation to maintain the explicit time dependence of the vibrational coordinates (57),... 

We can avoid these conversion factors that relate the different rate expressions, by defining the rate in terms of an equivalent concentration instead of the molar concentration. If X is the number of equivalents per liter that reacted in a time t, then dX/dt is a convenient expression of the reaction rate. However, the definition of equivalent must be made explicit. 

The total number of adsorption sites on the surface now appears explicitly in the rate expression for this elementary adsorption step. Since the site balance is the same as before (Equation 5.2.5), the equilibrium adsorption isotherm can be calculated in the manner described above ... 

With C representing the total concentration, + Cg, the rate expression can be rewritten in terms of C and C, thus eliminating the explicit dependence on C ... 

---