Particle transport processes boundary conditions

In step (1), the solution of equation 9.1-18 requires two boundary conditions, each of which can be expressed in two ways one of these ways introduces die other two rate processes, equating the rate of diffusion of A to the rate of transport of A at the particle surface (equation 9.1-11), and also the rate of diffusion at the core surface to the rate of reaction on the surface (9.1-20), respectively. Thus,... 

In Section III the encounter theory was applied to test particle-bath particle interactions to yield, with additional assumptions, the test particle transport projjerties. In Section IV the theory is applied to pair dissociation dynamics. This is just the inverse process to particle encounter and reaction, and the two are related by the equilibrium constant. This illustrates an advantage of the stochastic encounter theory of Section II. The use of the potential with a transition state (as shown in Fig. 1) partitions conhgura-tion space uniquely into bound pairs and free pairs such that the equilibrium constant is trivially evaluated. This overcomes many of the problems associated with diffusion-based theories in which dubious boundary conditions must be used to mimic chemical reaction and the possibility of redissociation. 

The next important advance in the theory, and the one that provided the foundation for all later work in this field, was made by Boltzmann, who in 1872 derived an equation for the time rate of change of the distribution function for a dilute gas that is not in equilibrium—the Boltzmann transport equation. (See Boltzmann and also Klein. " ) Boltzmann s equation gives a microscopic description of nonequilibrium processes in the dilute gas, and of the approach of the gas to an equilibrium state. Using the Boltzmann equation. Chapman and Enskog derived the Navier-Stokes equations and obtained expressions for the transport coefficients for a dilute gas of particles that interact with pairwise, short-range forces. Even now, more than 100 years after the derivation of the Boltzmann equation, the kinetic theory of dilute gases is largely a study of special solutions of that equation for various initial and boundary conditions and various compositions of the gas.t... 

However, the diffusion through the pores occurs simultaneously with the chemical reaction at the pore walls. There is an important similarity between this situation and the simultaneous reaction and diffusion in liquids in two phase systems. The porous solid can be considered as a quasi homogeneous phase, the concentration of the reactant(s) at the outer particle surface is a boundary condition for both the external and internal transport processes, that can be considered as processes in series. 

Solution of the particle concentration profile in the particle concentration boundary layer from in the feed suspension liquid to the concentration on top of the cake (and equal to the concentration in the cake) requires consideration of the particle transport equation in the boundary layer. We will proceed as follows. We will first identify the basic governing differentied equations and appropriate boundary conditions (Davis and Sherwood, 1990) and then identify the required equations for an integral model and list the desired solutions from Romero and Davis (1988). However, we will first simplify the population balance equation (6.2.51c) for particles under conditions of steady state 8n rp)/dt = O), no birth and death processes (B = 0 = De), no particle growth (lf = 0) and no particle velocity due to external forces Up = 0), namely... 

Gas-solid reactions between a fluid and a solid are important in a number of applications such as coal gasification, metallic ore processing, and catalyst regeneration. They are related in many aspects to the gas-solid catalytic reactions we have treated in developing the concepts of catalytic effectiveness, but differ in the very important aspect that the solid itself (in the form of a porous matrix) is one of the reactants. Since the solid phase itself is involved in reaction, often conditions of diffusion/ reaction change with time of reaction and the overall process is an unsteady-state one. As with effectiveness factors, many variants on a theme can be envisioned, i.e., is the reaction fast or slow, does the particle porosity (hence D ff) change with reaction, are boundary layer transport effects of importance, etc. We will present in some detail the developments of Wen concerning these questions [C.Y. Wen, Ind. Eng. Chem., 60, 34 (1968) H. Ishida and C.Y. Wen, Amer. Inst. Chem. Eng. J., 14, 311 (1968)]. 

A range of membrane processes are used to separate fine particles and colloids, macromolecules such as proteins, low-molecular-weight organics, and dissolved salts. These processes include the pressure-driven liquid-phase processes, microfiltration (MF), ultrafiltration (UF), nanofiltration (NF), and reverse osmosis (RO), and the thermal processes, pervaporation (PV) and membrane distillation (MD), all of which operate with solvent (usually water) transmission. Processes that are solute transport are electrodialysis (ED) and dialysis (D), as well as applications of PV where the trace species is transmitted. In all of these applications, the conditions in the liquid boundary layer have a strong influence on membrane performance. For example, for the pressure-driven processes, the separation of solutes takes place at the membrane surface where the solvent passes through the membrane and the retained solutes cause the local concentration to increase. Membrane performance is usually compromised by concentration polarization and fouling. This section discusses the process limitations caused by the concentration polarization and the strategies available to limit their impact. 

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