Governing Equations for Transport and Reaction

Consider an observed reaction of the form A + B — P + Q occurring in a packed-bed reactor. 

Step 2. Reactant A in the gas phase at position (r, z) has concentration a(r, z). It is transported across a film resistance and has concentration as(r, z) 

FIGURE 10.2 Illustration of pore and film resistances in a catalyst particle. 

The solution to this equation, which is detailed in Section 10.4.1, gives the concentration at position l down a pore that has its mouth located at position (r, z) in the reactor. The reaction rate in Equation (10.3) remains based on the bulk gas-phase volume, not on the comparatively small volume inside the pore. 

Step 4. A reactant molecule is adsorbed onto the internal surface of the catalyst. The adsorption step is modeled as an elementary reaction, the simplest version of which is 

Step 1. The entering gas is transported to point (r, z) in the reactor and reacts with rate Equation 9.1 governs the combination of bulk transport and pseu-dohomogeneous reaction. We repeat it here  

The initial and boundary conditions are given in Chapter 9. The present treatment does not change the results of Chapter 9 but instead provides a rational basis for using pseudohomogeneous kinetics for a solid-catalyzed reaction. The axial dispersion model in Chapter 9, again with pseudohomogeneous kinetics, is an alternative to Equation 10.1 that can be used when the radial temperature and concentration gradients are small. 

The steady-state flux across the interface must be equal to the reaction rate. Thus, for component A, 

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Like the performance of chemical reactors, in which the transport and reactions of chemical species govern the outcome, the performance of electronic devices is determined by the transport, generation, and recombination of carriers. The main difference is that electronic devices involve charged species and electric fields, which are present only in specialized chemical reactors such as plasma reactors and electrochemical systems. Furthermore, electronic devices involve only two species, electrons and holes, whereas 10-100 species are encountered commonly in chemical reactors. In the same manner that species continuity balances are used to predict the performance of chemical reactors, continuity balances for electrons and holes may be used to simulate electronic devices. The basic continuity equation for electrons has the form... 

During the last 30 years, the measurement of the impedance of an electrode has become a technique widely used for investigating numerous interfacial processes. The interpretation of this quantity is based on models obtained from the equations governing the coupled transport and kinetic processes, which may include heterogeneous and/or homogeneous reaction steps. Although these models are able to... 

Figure2 Ohnishi et al. (1985) and Chijimatsu et al. (2000)) and reactive-mass transport model (inside the box named Chemical in Figure 2). This is a system of governing equations composed of Equations (l)-(9), which couple heat flow, fluid flow, deformation, mass transport and geochemical reaction in terms of following primary variables temperature T, pressure head y/, displacement u total dissolved concentration of the n master species C< > and total dissolved and precipitated concentration of the n" master species T,. Here we set master species as the linear independent basis for geochemical reactions, and speciation in solution and dissolution/precipitation of minerals are calculated by a series of governing equations for geochemical reaction. Now we adopt equilibrium model for geochemical reaction (Parkhurst et al. (1980)), mainly because of reliability and abundance of thermodynamic data for geochemical reaction.

Equations (4)-(9) are governing equations for chemical process as follows. Equation (4) is for mass transport, and the governing equation of mass conservation of each master species through mass transport. A series of Equations (5)-(9) are for geochemical reaction, and the governing equations of electrical neutrality of conservation of electrons of mass conservation of each master species through geochemical reaction of mass action for each mineral and of mass action for each aqueous species, respectively. 

In an effective properties model, the porous microstructures of the SOFC electrodes are treated as continua and microstructural properties such as porosity, tortuosity, grain size, and composition are used to calculate the effective transport and reaction parameters for the model. The microstmctural properties are determined by a number of methods, including fabrication data such as composition and mass fractions of the solid species, characteristic features extracted from micrographs such as particle sizes, pore size, and porosity, experimental measurements, and smaller meso- and nanoscale modeling. Effective transport and reaction parameters are calculated from the measured properties of the porous electrodes and used in the governing equations of the ceU-level model. For example, the effective diffusion coefficients of the porous electrodes are typically calculated from the diffusion coefficient of Eq. (26.4), and the porosity ( gas) and tortuosity I of the electrode ... 

Besides the resuspension of particles, the perfect sink model also neglects the effect of deposited particles on incoming particles. To overcome these limitations, recent models [72, 97-99] assume that particles accumulate within a thin adsorption layer adjacent to the collector surface, and replace the perfect sink conditions with the boundary condition that particles cannot penetrate the collector. General continuity equations are formulated both for the mobile phase and for the immobilized particles in which the immobilization reaction term is decomposed in an accumulation and a removal term, respectively. Through such equations, one can keep track of the particles which arrive at the primary minimum distance and account for their normal and tangential motion. These equations were solved both approximately, and by numerical integration of the governing non-stationary transport equations. 

The equations (23) to (27) can only be solved numerically. However such a numerical solution gives less insight in the factors governing the transport and conversion processes. Therefore we consider another approach. In this approach, the transport and conversion of component A are calculated under the assumption that no reaction of ozone with component B takes place. The enhancement factor for mass transfer of ozone, EA, can now be given by the equation ... 

The basic strategy in the application of electroanalytical methods in studies of the kinetics and mechanisms of reactions of radicals and radical ions is the comparison of experimental results with predictions based on a mechanistic hypothesis. Thus, equations such as 6.28 and 6.29 have to be combined with the expressions describing the transport. Again, we restrict ourselves to considering transport governed only by linear semi-infinite diffusion, in which case the combination of Equations 6.28 and 6.29 with Fick s second law, Equation 6.18, leads to Equations 6.31 and 6.32 (note that we have now replaced the notation for concentration introduced in Equation 6.18 earlier by the more usual square brackets). Also, it is assumed here that the diffusion coefficients of A and A - are the same, i.e. DA = DA.- = D. 

In this chapter you will learn that proper assessment of mass transport controlled corrosion reactions requires knowledge of the concentration distribution of the reacting species in solution, certain properties of the electrolyte, and the geometry of the system. A rigorous calculation of mass transport controlled reaction rates requires detailed information concerning these parameters. Fortunately, many of the governing equations have been solved for several well-defined geometries. 

Optimum temperatures are obtained from the analytical expressions describing the optimum conditions, along with the derivative of the centerline reactant fraction with respect to the deposition modulus. Using an analytical solution to provide the derivative of the centerline reactant fraction, a closed-form implicit expression for the optimum temperature is obtained for the special case of no normalized reaction yield. For the more general case, this derivative is computed from numerical solutions to the equations governing transport and deposition. Optimum temperatures are presented graphically for a very wide range of the normalized preform thickness and normalized reaction yield. 

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