Dominant fluid-solid mass transfer

The ratio of timescales for internal diffusion within the catalyst pellet and for external mass transfer is of the order of the mass Biot number given in (3.43e). Since for many practical conditions Bim is reasonably high (10 -10 as estimated in Ref. [72]), it is expected that the external mass transfer dominates over the internal one. If we extend this consideration to the reactor scale, then a very useful pseudo-homogeneous model [51] is obtained. The scaling condition that expresses fast fluid-solid mass transfer compared with other processes (X) in Equation 3.22 is 

Replacing this result in the nondominant terms of Equation 3.58, the following improved expression is obtained  

The fluid mass balance with uniform physical properties becomes 

A perturbation solution for concentration (in the fluid and at the particle surface) can be written as 

Series solutions of the same type can be written for the average concentration. The leading-order problem that can be collected from (3.62) appears at 0(1) and simply expresses 

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Dominant fluid-solid mass and heat transfer... 

Diffusion is ubiquitous in nature whenever there is heterogeneity, there is diffusion. In liquid and gas, flow or convection is often present, which might be the dominant means of mass transfer. However, inside solid phases (minerals and glass), diffusion is the only way of mass transfer. Diffusion often plays a major role in solid-state reactions, but in the presence of a fluid dissolution and recrystallization may dominate. 

Possibilities for a single resistance include a linear rate expression with a lumped parameter mass transfer coefficient based either on the external fluid film or on a hypothetical solid film, depending on which film is controlling the rate of uptake of adsorbate. A quadratic driving force expression, again with a lumped parameter mass transfer coefficient, may be used instead. Alternatively, intraparticle diffusion, if the dominant form of mass transfer, may be described by the general diffusion equation (Pick s second law) with its appropriate boundary conditions, as described in Chapter 4. 

Combined Pore and Solid Diffusion In porous adsorbents and ion-exchange resins, intraparticle transport can occur with pore and solid diffusion in parallel. The dominant transport process is the faster one, and this depends on the relative diffusivities and concentrations in the pore fluid and in the adsorbed phase. Often, equilibrium between the pore fluid and the solid phase can be assumed to exist locally at each point within a particle. In this case, the mass-transfer flux is expressed by ... 

At high velocities where turbulence dominates, the main body of flowing fluid is well mixed in the direction normal to the flow, minor differences in temperature and concentration can be neglected, and the film concept can be applied. This describes the flow as if all gradients for temperature and concentration are in a narrow film along the interface with the solid (Nernst 1904), and inside the film conduction and diffusion are the transfer mechanisms. This film concept greatly simplifies the engineering calculation of heat and mass transfer. 

Component exchange between phases is controlled by mass transfer. Between solid phases, mass transfer is through diffusion where the exchange of components may be used as a geospeedometer (Lasaga, 1983). Convection rather than diffusion may play a dominant role if fluid phases are involved. In reactions between solid and fluid phases, diffusion in the solid phase is usually the slowest step. However, dissolution and reprecipitation may occur and may accomplish the exchange more rapidly than diffusion through the solid phase. 

Step 1. Reactants enter a packed catalytic tubular reactor, and they must diffuse from the bulk fluid phase to the external surface of the solid catalyst. If external mass transfer limitations provide the dominant resistance in this sequence of diffusion, adsorption, and chemical reaction, then diffusion from the bulk fluid phase to the external surface of the catalyst is the slowest step in the overall process. Since rates of interphase mass transfer are expressed as a product of a mass transfer coefficient and a concentration driving force, the apparent rate at which reactants are converted to products follows a first-order process even though the true kinetics may not be described by a first-order rate expression. Hence, diffusion acts as an intruder and falsifies the true kinetics. The chemical kineticist seeks to minimize external and internal diffusional limitations in catalytic pellets and to extract kinetic information that is not camouflaged by rates of mass transfer. The reactor design engineer must identify the rate-limiting step that governs the reactant product conversion rate. 

Adsorption is the loading of solid surfaces with substances present in a surrounding fluid phase or, in other words, it is a surface effect between a solid and a fluid phase. Sometimes molecules of the fluid phase are not only fixed on the surface but can additionally enter the bulk of the nonporous solid phase according to a volume effect. This is called occlusion or absorption. When it is not known which of these two effects is dominant the term sorption is used. Adsorption means the loading of one or several components (adsorptives) on a solid material (adsorbent). The reverse process, e.g., the separation of adsorptives from the surface is called desorption. It is a question of mass transfer in a two-phase system solid/fluid ... 

According to the previous chapters the reader may come to the conclusion that countercurrent columns are dominant as has been shown for mass transfer equipment used in the areas of rectification, absorption, and extraction. However, this is not true because the continuous transport of solid granular material is much more difficult in comparison to a fluid. Therefore, nearly all adsorbers are fixed beds which are operated batchwise. As a rule, at least two fixed beds are installed in continuously operated industrial processes. The first bed is used for the adsorption step whereas in the second the adsorbates is removed or desorbed at the same time. The duty of the two beds is changed when the adsorption capacity is exhausted. Sometimes several beds are arranged to cany out pressurization and depressurization steps. 

Similarly to partially overlapping channels, microchannels with mesh contactors (Figure 7.2h) are used to create the partial contact of fluids. The advantage of these contactors is that both modes of operation, cocurrent and countercurrent, can be apphed. Besides, the flow is stabilized because of the solid support between two fluids. The solid contactors are porous membrane [9, 10] and metal sheets with sieve-like structure [11]. Similarly to parallel flow, the mass transfer in both cases is only by diffusion and the flow is under laminar flow regime dominated by capillary forces. The membrane contactor has the advantage of being flexible with respect to the ratio of two fluids. In addition to flow velocities, the mass transfer is a function of membrane porosity and thickness. In another type of microextractor, two microchaimels are separated by a sieve-like wall architecture to achieve the separation of two continuous phases. However, the hydrodynamics in both types of contactors is more complex because of interfadal support and bursting of fluid... 

If two minerals contact with each other at constant the pressure-temperature condition where two minerals are unstable, reaction occurs between them to form stable mineral. The dominant rate limiting mechanisms are diffusion of aqueous species dissolved from minerals in fluid and dissolution and precipitation reactions. If fluid is not present, diffusion in solid phase occurs. But the rate of diffusion in solid phase is generally very slow. However, at very high temperature and pressure (metamorphic condition) the diffusion in solid phase may control the mass transfer. Reaction-diffusion model is able to be used to obtain the development of reaction zone between two minerals with time. 

Convective heat transfer, or convection, is the transfer of heat from one place to another by the movement of fluids, a process that is essentially the transfer of heat via mass transfer. Bulk motion of fluid enhances heat transfer in many physical situations, such as between a solid surface and the fluid. Convection is usually the dominant form of heat transfer in liquids and gases. Although sometimes discussed as a third method of heat transfer, convection is usually used to describe the combined effects of heat conduction within the fluid (diffusion) and heat transference by bulk fluid flow streaming. 

In this chapter the mass and fluid transfer processes that dominate as a solvent passes over particles in a packed column bed are summarized in both physical and philosophical terms. To introduce some basic terminology and to put us on common ground, the liquid passing through column is referred to as the mobile phase whilst, in most cases, the solid particle is called the stationary phase. 

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