Mass transfer pore diffusion

Special forms of equation 8.5-50 arise depending on the relative importance of mass transfer, pore diffusion, and surface reaction in such cases, one or two of the three may be the rate-controlling step or steps. These cases are explored in problem 8-18. The result given there for problem 8-18(a) is derived in the following example. 

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. 

Rate Controlling Step Mass Transfer (pore diffusion)... 

Figure 6.6 Comparison of the chromatogram given by the film mass transfer-pore diffusion model of chromatography with a Gaussian Profile. Dimensionless plot of versus f. Solid line Gaussian profile. Dotted line Carta s solution [34]. (a) Nap = N = 25 theoretical plates, (b) Nap = N = 100. Reprinted by permission of Kluwer Academic Publishing, from S. Golshan-Shirazi and G. Guiochon, NATO ASI Series C, vol 383, 61 (Fig. 4), with kind permission of Springer Science and Business Media.

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... 

The balances of mass of the chemical species i and the terms for the adsorption kinetics (mass transfer, pore diffusion) are listed in Table 9.5-1 for the three systems with Cj as the concentration in the fluid phase and Xj as the mass loading of the adsorbent. J3 denotes the mass transfer coefficient of a pellet and sj, is its internal porosity. The tortuosity factor will be explained later. The derivation of equations describing instationary diffusion in spheres has already been presented in Sect. 4.3.3. With respect to diffusion in macropores it is important to consider that diffusion can take place in the fluid as well as in the adsorbate phase. In Table 9.5-1 special initial and boimdaty conditions valid for a completely unloaded bed (adsorption) or totally loaded bed (desorption) are given. In this section only the model valid for a thin layer in a fixed bed with the thickness dz and the volmne / dz will be derived, see Fig. 9.5-2. 

Equation 21.7 includes the mass transfer, pore diffusion, and chemical reaction. Figure 21.3 allows observing the effect of each step, represented by Equation 21.7. As measure parameters, one has the inverse of the mass on the abscissa and the inverse of the global rate on the ordinate. 

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 parameters obtained through the measurements (and calculations) described in (2)-(4) may then be used for predicting the behavior of the system in any intermediate regime, i.e., where mass transfer, pore diffusion, and chemical kinetics may all play a major role. 

The equilibrium models of nonlinear chromatography assume that there always is an instantaneous equilibrium between the mobile phase and the stationary phase. That model is widely applied for the separation of small molecules, when mass transfer or diffusion in the stagnant pores of the mobile phase does not have a significant impact on the band profile. 

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 various steps in the removal of a gas from air by a porous adsorbent may be confined broadly to the following processes (a) mass transfer or diffusion of the gas to the gross surface (b) diffusion of the gas into or along the surface of the pores of granular adsorbent (c) adsorption on the interior surface of the granules (d) chemical reaction between the adsorbed gas and adsorbent (e) desorption of the product and (/) transfer of the products from the surface to the gas phase. Whether surface reaction or diffusion (mass transfer) to the surface becomes the rate-controlling step will become evident in the analysis of the experimental data with respect to the rate constant. 

Generally, there are four steps are included in the mass transfer mechanisms of the adsorption process. These steps are fluid-film transfer, pore diffusion, surface adhesion, and surface diffusion. The rate of surface adhesion for physical adsorption on the surface of porous adsorbents is very rapid, enough to be assumed instantaneous relative to the other transfer rates [5]. 

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... 

After extracting the kinetic parameters, selected results for CO oxidation over were used to analyze the effect of non-uniform temperature and velocity distributions on the conversion of CO. In order to determine the optimum number of multiple CSTR s to capture the behavior of a PFR, the rate law of Oh and Carpenter (14) for the NO+CO reaction was used to model a monolith channel as a CSTR in series. The results indicated that it was sufficient to use 5 reactors in series to capture the performance of the PFR behavior in the NO+CO reaction The cells of a monolith reactor were taken as independent parallel reactors ignoring the mass transfer and diffusion through the ceramic pores. The axial and radial temperature and velocity profiles collected from the literature(4,5) are used to calculate the... 

The resistance of the vacuum-side boundary layer ( g) is negligible compared to the others. The gas-filled membrane wall offers much smaller mass transfer resistance across the membrane. The resistance of the membrane has dominated overall mass transfer resistance in the past. This was because the permeability of the membrane was low and because the membrane was thick. The small resistance observed across the new-generation membranes is also the consequence of the choice of a hydrophobic fiber. The water does not wet this fiber, so its pores remain filled with nitrogen gas. Diffusion through the nitrogen gas is fast, making the membrane resistance unimportant. This conclusion would be different had we used a hydrophilic fiber. This implies that the key to the mass transfer is diffusion in the liquid. The overall mass transfer coefficients of oxygen were observed to be dominated by the individual mass transfer coefficient in the liquid film (Tai et al., 1994). 

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual particles are difficult to assess because it is impossible to define the shapes of the pores or channels through which mass transfer (qv) has to take place. However, the nature of the diffusional process in a porous soHd could be illustrated by considering the diffusion of solute through a pore. This is described mathematically by the diffusion equation, the solutions of which indicate that the concentration in the pore would be expected to decrease according to an exponential decay function. 

FIG. 16-9 General scheme of adsorbent particles in a packed bed showing the locations of mass transfer and dispersive mechanisms. Numerals correspond to mimhered paragraphs in the text 1, pore diffusion 2, solid diffusion 3, reaction kinetics at phase boundary 4, external mass transfer 5, fluid mixing. 

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 ... 

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 ). 

Combined Intraparticle Resistances When solid diffusion and pore diffusion operate in parallel, the effec tive rate is the sum of these two rates. When solid diffusion predominates, mass transfer can be represented approximately in terms of the LDF approximation, replacing/c in column 2 of Table 16-12 with... 

External Mass Transfer and Intraparticle Diffusion Control With a linear isotherm, the solution for combined external mass transfer and pore diffusion control with an infinite fluid volume is (Crank, Mathematics of Diffusion, 2d ed., Clarendon Press, 1975) ... 

Asymptotic Solution Rate equations for the various mass-transfer mechanisms are written in dimensionless form in Table 16-13 in terms of a number of transfer units, N = L/HTU, for particle-scale mass-transfer resistances, a number of reaction units for the reaction kinetics mechanism, and a number of dispersion units, Np, for axial dispersion. For pore and sohd diffusion, q = / // p is a dimensionless radial coordinate, where / p is the radius of the particle, if a particle is bidisperse, then / p can be replaced by the radius of a suoparticle. For prehminary calculations. Fig. 16-13 can be used to estimate N for use with the LDF approximation when more than one resistance is important. 

Figure 16-27 compares the various constant pattern solutions for R = 0.5. The curves are of a similar shape. The solution for reaction kinetics is perfectly symmetrical. The cui ves for the axial dispersion fluid-phase concentration profile and the linear driving force approximation are identical except that the latter occurs one transfer unit further down the bed. The cui ve for external mass transfer is exactly that for the linear driving force approximation turned upside down [i.e., rotated 180° about cf= nf = 0.5, N — Ti) = 0]. The hnear driving force approximation provides a good approximation for both pore diffusion and surface diffusion. 

FIG. 16-27 Constant pattern solutions for R = 0.5. Ordinant is cfor nfexcept for axial dispersion for which individual curves are labeled a, axial dispersion h, external mass transfer c, pore diffusion (spherical particles) d, surface diffusion (spherical particles) e, linear driving force approximation f, reaction kinetics. [from LeVan in Rodrigues et al. (eds.), Adsorption Science and Technology, Kluwer Academic Publishers, Dor drecht, The Nether lands, 1989 r eprinted with permission.]... 

The rectangular isotherm has received special attention. For this, many of the constant patterns are developed fuUy at the bed inlet, as shown for external mass transfer [Klotz, Chem. Rev.s., 39, 241 (1946)], pore diffusion [Vermeulen, Adv. Chem. Eng., 2, 147 (1958) Hall et al., Jnd. Eng. Chem. Fundam., 5, 212 (1966)], the linear driving force approximation [Cooper, Jnd. Eng. Chem. Fundam., 4, 308 (1965)], reaction kinetics [Hiester and Vermeulen, Chem. Eng. Progre.s.s, 48, 505 (1952) Bohart and Adams, J. Amei Chem. Soc., 42, 523 (1920)], and axial dispersion [Coppola and LeVan, Chem. Eng. ScL, 38, 991 (1983)]. 

Lenhoff, J. Chromatogr., 384, 285 (1987)] or by direct numerical solution of the conservation and rate equations. For the special case of no-axial dispersion with external mass transfer and pore diffusion, an explicit time-domain solution, useful for the case of time-periodic injections, is also available [Carta, Chem. Eng. Sci, 43, 2877 (1988)]. In most cases, however, when N > 50, use of Eq. (16-161), or (16-172) and (16-174) with N 2Np calculated from Eq. (16-181) provides an approximation sufficiently accurate for most practical purposes. 

Pore diffusion limitation was studied on a very porous catalyst at the operating conditions of a commercial reactor. The aim of the experiments was to measure the effective diffiisivity in the porous catalyst and the mass transfer coefficient at operating conditions. Few experimental results were published before 1970, but some important mathematical analyses had already been presented. Publications of Clements and Schnelle (1963) and Turner (1967) gave some advice. 

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