Mass transfer particle diffusion

Interest extends from transfer to single particles to systems in which the particles are in the form of fixed or fluidised beds. The only case for which there is a rigorous analytical solution is that for heat by conduction and mass transfer by diffusion to a sphere. 

The fact that diffusion models describe a number of chemical processes in solid particles is not surprising since in most cases, mass transfer and chemical kinetics phenomena occur simultaneously and it is difficult to separate them [133-135]. Therefore, the overall kinetics of many chemical reactions in soils may often be better described by mass transfer and diffusion-based models than with simple models such as first-order kinetics. This is particularly true for slower chemical reactions in soils where a fast reaction is followed by a much slower reaction (biphasic kinetics), and is often observed in soils for many reactions involving organic and inorganic compounds. 

At catalytically active centers in the center of carrier particles, external mass transfer (film diffusion) and/or internal mass transfer (pore diffusion) can alter or even dominate the observed reaction rate. External mass transfer limitations occur if the rate of diffusive transport of relevant solutes through the stagnating layer at a macroscopic surface becomes rate-limiting. Internal mass transfer limitations in porous carriers indicate that transport of solutes from the surface of the particle towards the active site in the interior is the slowest step. 

The overall process can be affected by pore diffusion and external mass transfer. Molecular diffusion coefficients DPB may be calculated by Aspen Plus. Effective pore diffusion may be estimated by the relation DP = Dpb( j,/tp) = 0.1 DPE, in which ep is the particle porosity and rp the tortuosity. Furthermore, the Thiele modulus and internal effectiveness can be calculated as ... 

The solution of liquid extract out of solid particles is described according to a shrinking particle model by convective mass transfer and diffusion in the particle structure. 

Single Sphere Model II (Equations 4, 5, 8, 9 and 10 in reference 6) In this model allowance is made for the resistance to mass transfer offered by the surface film surrounding the herb particles. The mass transfer coefficient kf was obtained from correlations proposed by Catchpole et al (8, 9) for mass transfer and diffusion into near-critical fluids. An average of the binary diffusivities of the major essential oil components present was used in calculating kf (these diffusivities were all rather similar because of their similar structures). 

Experimental data have shown that the first two items are factors of only secondary importance under conditions normally existing in commercial operations (73). Thus, conversion is not significantly affected by changing the vapor velocity (by altering the length/diameter ratio of the reactor, at constant volume), but is markedly influenced by temperature. Furthermore, the effect of catalyst particle size on cracking rate is ordinarily less pronounced than would be the case if mass transfer or diffusion were controlling. ... 

Corresponding considerations also hold for mass transfer. The diffusion resistance of the particle can be neglected when compared to that of the gas, and so a unified composition can be assigned to the particle. When only one component A is transferred between a fluid, consisting of components A and B, and the solid S, the calculation can be usefully made using the mass content of the fluid XF = (MA/MB)F. Then, under the assumption of a not too large mass content of component A in the solid, see equation (1.195a)... 

The number of transfer units for each mechanism can be estimated from known parameters and mass transfer correlations (4). For example, for a column with particles 0.01 cm in diameter, a superficial velocity of 0.01 cm/sec, and a solute bulk diffusivity of 7 x 10-7 cm2/sec, the estimated number of transfer units in a packed bed of length L for the four mechanisms, axial dispersion, external fluid film mass transfer, pore diffusion, and solid homogeneous particle diffusion,are... 

In practice, a solid catalyst is most conveniently modeled as a quasi-homo-geneous phase. Even if the catalyst particle is porous, visualize it as a homogeneous, but permeable solid. Mass transfer in its interior is retarded by two effects obstruction of part of the cross-sectional area by the solid material, and diffusion paths that are longer because molecules have to wind their way around the obstructions (tortuosity effect). In the quasi-homogeneous model, the retardation is accounted for by the use of appropriately smaller "effective mass-transfer or diffusion coefficients. 

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. 

Because of the analogy between mass transfer by diffusion and heat transfer by conduction in a boundary layer, correlations for mass transfer and heat transfer to particles are similar. For mass transfer to a single isolated sphere,... 

Whereas, in principle, simple experiments with tracers and one for each solute (explained below) allow the determination of e, e, and the component specific H from the experimentally determined pt <-> the other model parameters cannot be simply extracted from the second moment. Dispersion (D x), liquid film mass transfer (kfiijn), diffusion inside the particles (Dapp,pore)> and adsorption kinetics (kads) contribute in a complex manner jointly to the overall band broadening as described by Ot < (Equation 6.136). Therefore, an independent determination of these four parameters is not possible from Equation 6.136 only. In principle, additional equations could be obtained from higher moments (Kucera, 1965 Kubin, 1965). However, as the effect of detector noise on the accuracy of the moment value strongly increases the higher the order of the moment, a meaningful measurement of the third, fourth, and fifth moments is practically impossible. Equation 6.136 is thus not directly suited for parameter determination, but... 

This rate depends on particle size (pellet) and in this case spherical particles. If there is no mass transfer limitations (diffusion effects), the effectiveness factor t] = 1. However, if diffusion effects take place this factor decreases significantly in a manner that is dependent on the Thiele modulus. The effectiveness factor varies according to Equation 18.24 ... 

As the adsorption in the gas phase is independent of particle size, the intersection remains constant. With decreasing particle size, both the effectiveness factor and the coefficient of mass transfer increase. Consequently, the combined resistance to mass transfer and diffusion decrease [rep). This is shown in Figure 21.5 for both cases. 

The H2S reaction is a typical gas-solid reaction. External mass transfer or diffusion of H2S through the ZnO bed could hmit the reaction rate. Novichinskii et al. [24] reported that flake- or plate-type adsorbents offer lower mass transfer limitations compared with cube- or prism-type materials. Furthermore, an optimum ZnO particle size should be chosen with regard to capacity and pressure difference. 

For reactions in which (i) and (ii) are controlling, the effect of temperature is generally very small and is present only insofar as the kinetic motion of the reactant molecules is influenced by temperature. Similarly, when (i) and (ii) are the controlling steps, particle size exerts a strong effect on reaction rate. Generally, the mass transfer and diffusion are factors of a secondary nature and steps (iii) and (iv) involving activated adsorption and/or the surface reaction are considered as key rate-controlling factors. 

Internal Mass Transfer (Pore Diffusion) The simplified criterion for exclusion of an influence of internal mass transfer is given for spherical particles, a first-order reaction, and the assumption Des= 0.1 Di g by Eq. (4.7.19) ... 

The concentration gradients at the external surface are calculated by the radial concentration profiles of each component in the pellet at r= R. Equation. (6.17.14) then yields the rate of consumption/formation of each compound, which equals the mass transfer by diffusion to/from the outer surface of the catalyst from/into the bulk phase. A at.ex represents the overall external surface of all particles, and is calculated by the mean particle diameter and the number of particles z, which can be expressed by the particle density and total mass of catalyst ... 

In the simplest case of a fixed bed of adsorbent particles, the following mass transport processes are considered axial dispersion in the interparticle fluid phase, fluid-to-particle mass transfer, intrapaitide diffusion, and a first-order, reversible adsorption in the interior of the particle. The last step corresponds to a linear adsorption isotherm with a finite adsorption rate. This assumption includes the case of inflnitdy fast adsorption rate. 

Eqs. (6-16) and (6-18) show that the first moment expression includes only equilibrium parameters and Eqs. (6-17) and (6-19) to (6-22) mean that the contributions of axial dispersion, fluid-to-particle mass transfer, intraparticle diffusion, and adsorption rate to the second central moment are additive. 

In most chemical reactors, ext al mass transfer to die catalyst particles cannot be neglected. For describing the combined effect of external mass transfer, internal diffusion and chemical reaction (for the limitations indicated above) the following expresssion can be used ... 

This implies that the rate of species production by reaction is large compared to the rate of mass transfer by diffusion. The meaning of this condition is that all particles approaching the surface react instantaneously. 

Provided chemical reaction does not occur simultaneously with the diffusion processes in an adsorbent particle, analysis of the response to a pulse input of an adsorbate to a column packed with an adsorbent provides a convenient experimental method of deducing the separate contributions of inter- and intraphase mass transfer and diffusion to the overall resistance to adsorption. This is because each one of the resistances to mass transfer is in series and thus linearly additive. 

Fig. 6. Concentration profiles through an idealized biporous adsorbent particle showing some of the possible regimes. (1) + (a) rapid mass transfer, equihbrium throughout particle (1) + (b) micropore diffusion control with no significant macropore or external resistance (1) + (c) controlling resistance at the surface of the microparticles (2) + (a) macropore diffusion control with some external resistance and no resistance within the microparticle (2) + (b) all three resistances (micropore, macropore, and film) significant (2) + (c) diffusional resistance within the macroparticle and resistance at the surface of the...

External Fluid Film Resistance. A particle immersed ia a fluid is always surrounded by a laminar fluid film or boundary layer through which an adsorbiag or desorbiag molecule must diffuse. The thickness of this layer, and therefore the mass transfer resistance, depends on the hydrodynamic conditions. Mass transfer ia packed beds and other common contacting devices has been widely studied. The rate data are normally expressed ia terms of a simple linear rate expression of the form... 

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