First-order-reaction Fluxes

All these equations differ from the corresponding equations for diffusion polarization, only in that the equilibrium concentration Cq appears in them instead of bulk concentration Cy. Formally, diffusion can be regarded as a first-order reaction, the fimiting diffusion flux being proportionaf to the first power of concentration. 

In case of Fischer-Tropsch synthesis, we have to consider that the first-order reaction rate constant is related to the concentration in the gas phase (e.g., ce2), and that the diffusive flux in the liquid-filled pores is related to the concentration in the liquid (ce21). Thus, instead of Equation 12.10, we have to use... 

Assuming that no significant in-situ degradation of PCBs occurs (k htm = photo = Kio = 0 ) three elimination pathways remain which, if described in terms of first-order reaction rates, can be directly compared with respect to their relative importance for the elimination of each PCB congener from the water column. As shown by the removal rates listed in Table 23.4, for both compounds the flux to the atmosphere is by far the most important process. Because of its larger Kd value, removal of the heptachlorobiphenyl to the sediments is predicted to be also of some importance. By the way, from this simple model we would expect to find the heptachlorobiphenyl relatively enriched in the sediments compared to the trichloro-biphenyl. We shall see later whether this is true. 

When homogeneous poisoning occurs, since no reaction will be possible on the poisoned fraction ( , say, as shown in Fig. 3.11) of active surface it is reasonable to suppose that the intrinsic activity of the catalyst is in proportion to the fraction of active surface remaining unpoisoned. To find the ratio of activity of the poisoned catalyst to the activity of an unpoisoned catalyst one would compare the stationary flux of reactant to the particle surface in each case. For a first-order reaction... 

At steady-state, the flux of A equals the rate of reaction thus preventing accumulation or depletion. For a simple first-order reaction, the kinetics depend on the surface rate constant, ks, and the concentration of A at the surface ... 

Now consider the first-order reaction in a porous flat plate catalyst pellet so that both external (interphase) and internal (intraphase) transport limitations are encountered. At steady state, the flux of A to the surface of the pellet is equal to the flux entering the pellet ... 

As an example we will consider a catalytic reactor, Fig. 2.58, in which by a chemical reaction between a gas A and its reaction partner R, a new reaction product P is formed. The reaction partner R and the gas A are fed into the reactor, excess gas A and reaction product P are removed from the reactor. The reaction is filled with spheres, whose surfaces are covered with a catalytic material. The reaction between gas A and reactant R occurs at the catalyst surface and is accelerated due to the presence of the catalyst. In most cases the complex reaction mechanisms at the catalyst surface are not known completely, which suggests the use of very simplified models. For this we will consider a section of the catalyst surface, Fig. 2.59. On the catalyst surface x = 0 at steady-state, the same amount of gas as is generated will be transported away by diffusion. The reaction rate is equal to the diffusive flux. In general the reaction rate hA0 of a catalytic reaction depends on the concentration of the reaction partner. In the present case we assume that the reaction rate will be predominantly determined by the concentration cA(x = 0) = cA0 of gas A at the surface. For a first order reaction it is given by... 

The local diffusion flux for the first-order reaction is calculated by the formula... 

Only for an isothermal, first-order reaction where Sa = —k a will the chemical source term in (3.102) be closed, i.e., ++(<+ = h (u,(pa). Indeed, for more complex chemistry, closure of the chemical source term in the scalar-flux transport equation is a major challenge. However, note that, unlike the scalar-flux dissipation term, which involves the correlation between gradients (and hence two-point statistical information), the chemical source term is given in terms of u(x, t) and 0(x, t). Thus, given the one-point joint velocity, composition PDF /u,chemical source term is closed, and can be computed from... 

The initial and boundary conditions are identical to those for Equation 11.37 a = a at X = 0 and a = a/ at x = oo. For a first-order reaction, = —ka, and Equation 11.40 has an analytical solution. Using this solution to find fhe average flux gives... 

Insertion of the concentration, eq. (9.78), into eq. (9.76) gives the final expression for the flux, N la. for first-order reactions... 

Control of Catalyst Location in a CMR - From the previous section, it can be seen that the optimal location or distribution of a catalyst in a membrane can strongly influence results. This question was studied theoretically by Keller et for an isothermal, irreversible first-order reaction in a homogeneous membrane with slab geometry. They enumerated some different criteria of optimality maximize the reaction rate per membrane area minimize the flux of reactant leaving the membrane and maximize the purity at the output side, expressed as the ratio (flux of product)/(flux of reactant). The first optimality... 

Outline of the Theoreyical Model. The main assumptions for the unsteady state dynamics are as follows l) Only polymer moleciiles which are raised into excited state by absorbing UV light (photon flux, no wavelength, X) near the absorption band charasteristic of polymers can participate in photochemical reactions (efficiency, n molar concentration, C ). (2) Photochemical reactions are i) depolymerization of activated polymer molecules (first order reaction,... 

The equations for simultaneous pore diffusion and reaction were solved independently by Thiele and by Zeldovitch [16,17]. They assumed a straight cylindrical pore with a first-order reaction on the surface, and they showed how pore length, diffusivity, and rate constant influenced the overall reaction rate. Their solution cannot be directly adapted to a catalyst pellet, since the number of pores decreases going toward the center and assuming an average pore length would introduce some error. The approach used here is that of Wheeler [18] and Weisz [19], who considered reactions in a porous sphere and related the diffusion flux to the effective diffusivity, Z). The basic equation is a material balance on a thin shell within the sphere. The difference between the steady-state flux of reactant into and out of the shell is the amount consumed by reaction. 

The price that is paid for the greater generality of the models is twofold, however. First, there is the need for two parameters one expressing the surface renewal and one expressing the thickness of the element. Second, thoe is the mathematical complexity of the expression for the flux, N. Is the price worth paying This question can be partly answered by means of Huang and Kuo s application of the film-penetration model to first-order reactions, both irreversible and reversible [32,12]. 

Under stationary conditions, the molar flux from the fluid phase reaching the catalytic surface and the rate of transformation per surface unit must be identical Ji = Vs- It follows that for a first-order reaction... 

Experimental data on the absorption of CO2 in blends of Diethanolamine (DEA) and Piperazine (PZ) reported in the literatureare very little. The finding of additional kinetics is very much valuable. In this work, the CO2 absorption rate into aqueous solutions of mixture DEA and PZ was measured and the experimental results are presented in Fig. 1-4. The absorption flux, Rco2> was analysed using the Eq.2 and Eq.3. The pseudo-first order reaction regime assumption was verified by ealeulating Hatta... 

A gas A contacts a static liquid B in a tall vertical container. When the gas A diffuses, it also reacts (irreversible first-order reaction). Find the concentration profile of A in the liquid as well as its molar flux. 

Equation 7.112 is valid for the molar flux for first-order reactions. For fast first-order reactions, is zero and M is large (M / is the Hatta number). For large M values, we have... 

For reaction kinetics defined in Equation 7.115, for a pseudo-first-order reaction, the same expressions are obtained for the flux and the enhancement factor, Nfj and Ea, as for fast first-order reactions, from Equations 7.117 and 7.119. Equations 7.130 and 7.131 can, therefore, be used for fast pseudo-first-order reactions, but the parameter M is defined as... 

For a general system containing N components in the gas phase, the coupled system of N + I differential equations, Equations 8.110 and 8.18, is solved. The flux N, and the surface concentration c are given in Equations 8.50 and 8.51, respectively. The coupled differential equations must be solved numerically using the tools and methods introduced, for instance, in Appendix 2. For first-order reactions, however, a simplified procedure is possible. 

For an arbitrary kinetic model, the surface concentrations c are solved by Equation 8.130 and the flux thus obtained, N, is inserted into the differential Equation 8.129. This equation is then solved numerically as an initial value problem (Appendix 2). A simplified solution procedure is possible for a first-order reaction. 

The reaction is assumed to be rapid. The expression for the flux of a first-order reaction is (Equation 7.112 in Section 7.2.4)... 

This relationship does not depend on another variable than these three parameters and will be used as a tool to detect conditions in which self-organisation based on proto-metabolic fluxes of energy can take place. Selecting a value of 1 for the transmission coefficient k (meaning that there is no possibility of reverting to the reactants after the system has crossed the transition state), the value of the free energy of activation can then be deduced as a function of the half-life of a first-order (or pseudo-first-order) reaction at different values of temperature (Fig. 8.6). 

Gibbs free energy change on reaction reaction enthalpy enthalpy (specific) enthalpy, Henry s law constant electric current diffusive mass flux conduction heat flux W m reaction velocity thermal conductivity Boltzmann constant chemical equilibrium constant resistance coefficients effective thermal conductivity first-order reaction rate constant characteristic half thickness Lewis number... 

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