Transfer, external mass

We first consider the situation where reaction occurs only on the external surface of a pellet. The simplest example is that of a nonporous pellet so that only the external surface is reactive. 

Reactant molecules must get to the surface to find the catalytic sites on which to react. In steady state there cannot be any accumulation on the surface, and therefore we require that 

For a first-order irreversible reaction on a single spherical pellet of radius R the total rate of reaction on the external surface of the pellet is 

4jt RhrnAiCAb - Cas) = R r = 47t R k CAs This says simply that the rate of reaction on the pellet with surface area 4-ti R is equal to the rate of mass transfer from the bulk fluid to the surface. We call /CmA the mass transfer coefficient of reactant species A and note that it has the dimensions of lengthAime, the same dimensions as k . 

It is a fundamental fact that in steady state nothing is accumulating acywhere around the catalyst and therefore these rates must be exactly equal. 

The Sherwood number is blended with the Reynolds and Schmidt numbers to form a jn factor, 

One of the correlations between the jo factor and the Reynolds number is based on the work of Thodos et al. and can be expressed as 

For ideal gases, the concentration and partial pressure are related and the mass transfer formula is often written 

The 7/) factor for which the same curve gives a reasonable correlation is now 

Thus the rate of transfer per unit reactor volume can be written either as 

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External Mass Transfer and Intrapaiticle Diffusion Control. 16-29... 

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. 

External mass-transfer coefficients for particles suspended in agitated contactors can be estimated from equations in Levins and Glastonbury [Trans. Instn. Chem. E/ig., 50, 132 (1972)] and Armenante and Kirwan [Chem. Eng. ScL, 44, 2871 (1989)]. 

TABLE 16-9 Recommended Correlations for External Mass Transfer Coefficients in Adsorption Beds (Re = evdp/v. Sc = v/D)... 

FIG. 16"10 Sherwood mimher correlations for external mass-transfer coefficients in packed beds for e = 0.4 (adapted from Siiziild, gen. refs.). 

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 eqm-librium 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 sys-... 

Solutions are provided for external mass-transfer control, intraparticle diffusion control, and mixed resistances for the case of constant Vj and F, out = 0- The results are in terms of the fractional... 

FIG. 16-14 Constant separation factor batch adsorption curves for external mass-transfer control with an infinite fluid volume and n j = 0. 

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) ... 

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. 

For other mechanisms, the particle-scale equation must be integrated. Equation (16-140) is used to advantage. For example, for external mass transfer acting alone, the dimensionless rate equation in Table 16-13 would be transformed into the ( — Ti, Ti) coordinate system and derivatives with respect to Ti discarded. Equation (16-138) is then used to replace cfwith /ifin the transformed equation. Furthermore, for this case there are assumed to be no gradients within the particles, so we have nf=nf. After making this substitution, the transformed equation can be rearranged to... 

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)]. 

The simplest isotherm is /if = cf corresponding to R = 1. For this isotherm, the rate equation for external mass transfer, the linear driving force approximation, or reaction kinetics, can be combined with Eq. (16-130) to obtain... 

Isocratic Elution In the simplest case, feed with concentration cf is apphed to the column for a time tp followed by the pure carrier fluid. Under trace conditions, for a hnear isotherm with external mass-transfer control, the linear driving force approximation or reaction kinetics (see Table 16-12), solution of Eq. (16-146) gives the following expression for the dimensionless solute concentration at the column outlet ... 

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. 

Correlations of heat and mass-transfer rates are fairly well developed and can be incorporated in models of a reaction process, but the chemical rate data must be determined individually. The most useful rate data are at constant temperature, under conditions where external mass transfer resistance has been avoided, and with small particles... 

The relationship between adsorption capacity and surface area under conditions of optimum pore sizes is concentration dependent. It is very important that any evaluation of adsorption capacity be performed under actual concentration conditions. The dimensions and shape of particles affect both the pressure drop through the adsorbent bed and the rate of diffusion into the particles. Pressure drop is lowest when the adsorbent particles are spherical and uniform in size. External mass transfer increases inversely with d (where, d is particle diameter), and the internal adsorption rate varies inversely with d Pressure drop varies with the Reynolds number, and is roughly proportional to the gas velocity through the bed, and inversely proportional to the particle diameter. Assuming all other parameters being constant, adsorbent beds comprised of small particles tend to provide higher adsorption efficiencies, but at the sacrifice of higher pressure drop. This means that sharper and smaller mass-transfer zones will be achieved. 

In general, the concentration of the reactant will decrease from CAo in the bulk of the fluid to CAi at the surface of the particle, to give a concentration driving force of CAo - CAi)-Thus, within the pellet, the concentration will fall progressively from CAi with distance from the surface. This presupposes that no distinct adsorbed phase is formed in the pores. In this section the combined effects of mass transfer and chemical reaction within the particle are considered, and the effects of external mass transfer are discussed in Section J 0.8.4. 

When there is an external mass transfer resistance, the value of C i (the concentration at the surface of the particle) is less than that in the bulk of the fluid (Cao) and will not be known. However, if the value of the external mass transfer coefficient is known, the mass transfer rate from the bulk of the fluid to the particle may be expressed as ... 

External mass transfer resistance for particle 644 Extruders 306... 

A final, obvious but important, caution about catalyst film preparation Its thickness and surface area Ac must be low enough, so that the catalytic reaction under study is not subject to external or internal mass transfer limitations within the desired operating temperature range. Direct impingement of the reactant stream on the catalyst surface1,19 is advisable in order to diminish the external mass transfer resistance. 

IV. External mass transfer limitations. Symptoms Rate insensitivity to temperature, rate variation with flowrate. Remedy Decrease operating temperature, force reactants to directly impinge on the catalyst surface. 

Checking the absence of external mass transfer limitations is a rather easy procedure. One has simply to vary the total volumetric flowrate while keeping constant the partial pressures of the reactants. In the absence of external mass transfer limitations the rate of consumption of reactants does not change with varying flowrate. As kinetic rate constants increase exponentially with increasing temperature while the dependence of mass transfer coefficient on temperature is weak ( T in the worst case), absence... 

Most of the actual reactions involve a three-phase process gas, liquid, and solid catalysts are present. Internal and external mass transfer limitations in porous catalyst layers play a central role in three-phase processes. The governing phenomena are well known since the days of Thiele [43] and Frank-Kamenetskii [44], but transport phenomena coupled to chemical reactions are not frequently used for complex organic systems, but simple - often too simple - tests based on the use of first-order Thiele modulus and Biot number are used. Instead, complete numerical simulations are preferable to reveal the role of mass and heat transfer at the phase boundaries and inside the porous catalyst particles. 

Figure 8.9 External mass transfer resistance—xylose hydrogenation and isomerization to xylitol and by-products on sponge Ni (based on the results of Mikkola et al. [22]).

Traditionally, an average Sherwood number has been determined for different catalytic fixed-bed reactors assuming constant concentration or constant flux on the catalyst surface. In reality, the boundary condition on the surface has neither a constant concentration nor a constant flux. In addition, the Sh-number will vary locally around the catalyst particles and in time since mass transfer depends on both flow and concentration boundary layers. When external mass transfer becomes important at a high reaction rate, the concentration on the particle surface varies and affects both the reaction rate and selectivity, and consequently, the traditional models fail to predict this outcome. 

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