Mass transfer diffusion through porous solids

Aside from cavitation, the enhanced mass transfer rates in acoustic fields can be attributed to plug flow of capillary liquid as well as to enhanced dispersion of the liquid and vapor moisture due to alternating compression and expansion cycles, which result in reduced viscosity of the liquid-vapor mixture. In fact, a substantial increase in the amount of liquid diffusing through porous solids has been noted in the presence of ultrasound (Fairbanks and Chen, 1969 Woodford and Morrison, 1969 Kuznetsov and Subbotina, 1965). The enhanced diffusion appears to be of the directional type since mass transfer was hindered when ultrasound irradiation was opposed to the direction of diffusive flow (Kuznetsov and Subbotina, 1965). 

In connection with multiphase diffusion another poorly understood topic should be mentioned—namely, the diffusion through porous media. This topic is of importance in connection with the drying of solids, the diffusion in catalyst pellets, and the recovery of petroleum. It is quite common to use Fick s laws to describe diffusion through porous media fJ14). However, the mass transfer is possibly taking place partly by gaseous diffusion and partially by liquid-phase diffusion along the surface of the capillary tubes if the pores are sufficiently small, Knudsen gas flow may prevail (W7, Bl). 

To obtain numerically the mass transfer coefficient, a porous medium is stochastically constructed in the form of a sphere pack. Specifically, the representation of the biphasic domains under consideration is achieved by the random deposition of spheres of radius Rina box of length L. The structure is digitized and the phase function (equal to zero for solid and unity for the pore space) is determined in order to obtain the porosity and to solve numerically the convection- diffusion problem. The next for this purpose is to obtain the detailed flow field in the porous domain through the solution of the Stokes equations ... 

The mechanism of solute transfer to the porous solid includes diffusion through the fluid film around the particle and diffusion inside the pores to internal adsorption sites. The actual process of adsorption is practically instantaneous and equilibrium is assumed to exist between the surface and the fluid at each point inside the particle. The transfer process is approximated using an overall volumetric mass-transfer coefficient (Kca) and an overall driving force ... 

In industrial operations, adsorption is accomplished primarily on the surfaces of internal passages within small porous particles. Three basic mass transfer processes occur in series (1) mass transfer from the bulk gas to the particle surface, (2) diffusion through the passages within the particle, and (3) adsorption on the internal particle surfaces. Each of the processes depends on the system operating conditions and the physical and chemical characteristics of the gas stream and the solid adsorbent. Often, one of the transfer processes will be significantly slower than the other two and will control the overall transfer rate. The other process will operate nearly at equilibrium. 

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual panicles arc difficult to assess because il is impossible to define the shapes of the pores or channels through which mass transfer has to take place. However, the nature of the diffusional process in a porous solid could be illustrated by considering the diffusion of solute through a pore. This is described mathematically hy the diffusion equation, the... 

Two other crucial factors are mass transfer and heat transfer. In Chapter 3 we assumed that the reactions were homogeneous and well stirred, so that every substrate molecule had an equal chance of getting to the catalytic intermediates. Here the situation is different. When a molecule reaches the macroscopic catalyst particle, there is no guarantee that it will react further. In porous materials, the reactant must first diffuse into the pores. Once adsorbed, the molecule may need to travel on the surface, in order to reach the active site. The same holds for the exit of the product molecule, as well as for the transfer of heat to and from the reaction site. In many gas/solid systems, the product is hot as it leaves the catalyst, and carries the excess energy out with it. This energy must dissipate through the catalyst particles and the reactor wall. Uneven heat transfer can lead to hotspots, sintering, and runaway reactions. 

As shown in Fig. 3, the overall resistance to the reaction is considered to consist of three steps (i) gas phase mass transfer (ii) diffusion through the porous reacted zone (in reduction the solid products are generally smaller in volume than the reactants), and (iii) chemical reaction at the boundary. Additional assumptions usually made are... 

The sources of band broadening of kinetic origin include molecular diffusion, eddy diffusion, mass transfer resistances, and the finite rate of the kinetics of ad-sorption/desorption. In turn, the mass transfer resistances can be sorted out into several different contributions. First, the film mass transfer resistance takes place at the interface separating the stream of mobile phase percolating through the column bed and the mobile phase stagnant inside the pores of the particles. Second, the internal mass transfer resistance controls the rate of mass transfer between this interface and the adsorbent surface. It is composed of two contributions, the pore diffusion, which is molecular diffusion taking place in the tortuous, constricted network of pores, and surface diffusion, which takes place in the electric field at the liquid-solid interface [60]. All these mass transfer resistances, except the kinetics of adsorption-desorption, depend on the molecular diffusivity. Thus, it is important to study diffusion in bulk liquids and in porous media. 

In the general case where the active material is dispersed through the pellet and the catalyst is porous, internal diffusion of the species within the pores of the pellet must be included. In fact, for many cases diffusion through catalyst pores represents the main resistance to mass transfer. Therefore, the concentration and temperature profiles inside the catalyst particles are usually not flat and the reaction rates in the solid phase are not constant. As there is a continuous variation in concentration and temperature inside the pellet, differential conservation equations are required to describe the concentration and temperature profiles. These profiles are used with intrinsic rate equations to integrate through the pellet and to obtain the overall rate of reaction for the pellet. The differential equations for the catalyst pellet are two point boundary value differential equations and besides the intrinsic kinetics they require the effective diffusivity and thermal conductivity of the porous pellet. 

Kr restrictive factor for diffusion of liquids in porous solids dimensionless. kx convective mass-transfer coefficient for diffusion of A through stagnant B in dilute liquid-phase solution with driving force in terms of mole fractions mol/m2-s. 

Physical transport processes can play an especially important role in heterogeneous catalysis. Besides film diffusion on the gas/liquid boundary there can also be diffison of the reactants (products) through a boundary layer to (from) the external surface of the solid material and additionally diffusion of them through the porous interior to from the active catalyst sites. Heat and mass transfer processes influence the observed catalytic rates. For instance, as discussed previously the intrinsic rates of catalytic processes follow the Arrhenius... 

SCFs have relatively low viscosity and high diffusivity, and they can penetrate into porous solid materials more effectively than liquid solvents and may render much faster mass transfer resulting in faster extractions. In SFE, a fresh fluid is continuously forced to flow through the samples therefore, it can provide quantitative or complete extraction [38, 51]. Moreover, SFE may allow direct coupling with a chromatographic method, which can be a useful to extract and directly quantify the desired compounds. 

Mass transfer of gas through a porous membrane can involve several processes depending on the pore stmcture and the solid [1]. There are four different mechanisms for the transport Poiseuille flow Knndsendiflusion partial condensation/capillaiy diffusion/selective adsorption and molecular sieving [2, 3]. The transport mechanism exhibited by most of carbon membranes is the molecular sieving mechanism as shown in Fig. 2.1. The carbon membranes contain constrictions in the carbon matrix, which approach the molecular dimensions of the absorbing species [4],... 

In this chapter, we will primarily focus on fluid flow, heat, and mass transport through gas flow channels and in solid porous electrodes, and its effect on the mass transfer loss. Solid-phase diffusion, charge transport in electrolyte membrane, and ohmic loss will be discussed in Chapter 7. Water transport will also be discussed in Chapter 7. 

Heterogeneous catalytic reactions in supercritical solvents Obviously, a solid catalyzed reaction takes place only on the active sites of the porous catalyst with the implication of some mass and heat transport steps prior to and after the reaction. The first step is the diffusion of the reactants through the film surrounding the catalyst particle to the external surface of the catalyst, followed by diffusion of the reactants into the catalyst pore to the active site in the pores. These steps are limited by the dif-fusivity and viscosity of the reactants. In the case of a supercritical fluid phase reaction, the diffusivity is higher than the liquid diffusivity, viscosity is less than the liquid viscosity and therefore, the rate of transfer to the active site will be higher. After the adsorption, reaction and desorption steps, the products have to diffuse out of the pore, and again... 

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