Mass transfer solid diffusion control

These expressions can also be used for the case of external mass transfer and solid diffusion control by substituting D, for 8pDpi/( p + ppK)) and/c rp/(ppK)D,i) for the Biot number. 

The cases considered thus far have all been based upon the premise that one process, ash-layer diffusion, surface reaction, or gas-film mass transfer, is rate controlling. However, in some cases, more than one process affects the overall kinetics for the conversion of the solid. This has two implications ... 

According to their analysis, if is zero (practically much lower than 1), then the liquid-film diffusion controls the process rate, while if tfis infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the so-called mechanical parameter represents the ratio of the diffusion resistances (solid and liquid film). The authors did not refer to any assumption concerning the type of isotherm for the derivation of the above-mentioned criterion it is sufficient to be favorable (not only rectangular). They noted that for >1.6, the particle diffusion is more significant, whereas if < 0.14, the external mass transfer controls the adsorption rate. 

In the case of solid diffusion control, even in the absence of agitation where the mass transfer coefficient is at its minimum value, sufficient agitation should be provided in order to avoid the negative effect of the liquid-film resistance. The effect of agitation should be taken into account in both the design and application stage. 

The limiting cases of the analytical solutions for external fluid-film mass transfer controlling (c —> 0) and solid diffusion controlling -> oo ) are the following ... 

Equilibrium partitioning and mass transfer relationships that control the fate of HOPs in CRM and in different phases in the environment were presented in this chapter. Partitioning relationships were derived from thermodynamic principles for air, liquid, and solid phases, and they were used to determine the driving force for mass transfer. Diffusion coefficients were examined and those in water were much greater than those in air. Mass transfer relationships were developed for both transport within phases, and transport between phases. Several analytical solutions for mass transfer were examined and applied to relevant problems using calculated diffusion coefficients or mass transfer rate constants obtained from the literature. The equations and approaches used in this chapter can be used to evaluate partitioning and transport of HOP in CRM and the environment. 

The intraparticle mass transfer mechanism is based on the following linear driving force expression (which assumes solid diffusion control) ... 

Here tm is the mass-transfer time. Only under slow reaction kinetic control regime can intrinsic kinetics be derived directly from lab data. Otherwise the intrinsic kinetics have to be extracted from the observed rate by using the mass-transfer and diffusion-reaction equations, in a manner similar to those defined for catalytic gas-solid reactions. For instance, in the slow reaction regime,... 

In industrial practice, the most convenient way of accounting for mass-transfer effects is to view the penetrable catalyst particle as a pseudo-homogeneous phase. Obstruction of mass transfer by the solid material in the particle then is reflected by an "effective" intraparticle mass-transfer or diffusion coefficient that is appropriately lower than in the contacting fluid. If this approach is taken, two fundamentally different mass-transfer situations appear Mass transfer to and from the particle across an adherent boundary layer is affected by the reaction only in that the latter sets the boundary condition at the particle. Here, mass transfer and reaction are sequential and occur in different parts of the system, and the slower of the two is the bottleneck and dictates the overall rate and its temperature dependence. Within the particle, however, mass transfer and reaction occur simultaneously and in the same volume element. Here, the reaction introduces a source-or-sink term into the basic differential material balance. If the reaction is slow, it alone controls the overall rate and its temperature dependence. If mass transfer is slow, both reaction and mass transfer affect the rate, and the apparent reaction order and activation energy are the arithmetic means of those of reaction and mass transfer. 

Figure 1. Schematic illustration of the various mass transfer and diffusion processes that can control the rates of isotopic partitioning between a fluid (or gas) and a solid. Many of these are described in this review along with rate processes inflnential in controlUng rates in gaseous and aqueous systems. Modified from Manning (1974).

A summary of some of the available solutions for irreversible systems is given in Table 8.2. All solutions assume plug flow. The limiting cases of solid diffusion control and pore diffusion control were solved by Cooper - and by Cooper and Libermann while the solution for the case of combined pore diffusion and external mass transfer resistance was obtained by Weber and Chakravorti using the method of Cooper and Libermann. Weber and... 

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. 

Retention of a given solids particle in the system is on the average veiy short, usually no more than a few seconds. This means that any process conducted in a pneumatic system cannot be diffusion-controlled. The reaction must be mainly a surface phenomenon, or the solids particles must be veiy small so that heat transfer and mass transfer from the interiors are essentially instantaneous. 

The reaction kinetics approximation is mechanistically correct for systems where the reaction step at pore surfaces or other fluid-solid interfaces is controlling. This may occur in the case of chemisorption on porous catalysts and in affinity adsorbents that involve veiy slow binding steps. In these cases, the mass-transfer parameter k is replaced by a second-order reaction rate constant k. The driving force is written for a constant separation fac tor isotherm (column 4 in Table 16-12). When diffusion steps control the process, it is still possible to describe the system hy its apparent second-order kinetic behavior, since it usually provides a good approximation to a more complex exact form for single transition systems (see Fixed Bed Transitions ). 

The linear driving force (LDF) approximation is obtained when the driving force is expressed as a concentration difference. It was originally developed to describe packed-bed dynamics under linear equilibrium conditions [Glueckauf, Trans. Far. Soc., 51, 1540 (1955)]. This form is exact for a nonlinear isotherm only when external mass transfer is controlling. However, it can also be used for nonlinear systems with pore or solid diffusion mechanisms as an approximation, since it provides qualitatively correct results. 

Equation (43) describes the transport-controlled dissolution rate of a solid according to the diffusion layer theory in its simplest form. The mass transfer coefficient here is given by k, = kT = Dlh. 

The performance of a reactor for a gas-solid reaction (A(g) + bB(s) -> products) is to be analyzed based on the following model solids in BMF, uniform gas composition, and no overhead loss of solid as a result of entrainment. Calculate the fractional conversion of B (fB) based on the following information and assumptions T = 800 K, pA = 2 bar the particles are cylindrical with a radius of 0.5 mm from a batch-reactor study, the time for 100% conversion of 2-mm particles is 40 min at 600 K and pA = 1 bar. Compare results for /b assuming (a) gas-film (mass-transfer) control (b) surface-reaction control and (c) ash-layer diffusion control. The solid flow rate is 1000 kg min-1, and the solid holdup (WB) in the reactor is 20,000 kg. Assume also that the SCM is valid, and the surface reaction is first-order with respect to A. 

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. 

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