With diffusional resistance

The commonest multiple step control mechanism in use is that of diffusion to the surface of the catalyst combined with one of the adsorption or surface reaction steps. Mass transfer by diffusion is proportional to the difference between partial pressures in the bulk of the gas and at the catalyst surface, 

The mass transfer coefficient can be found along with other constants from appropriate rate data, or it can be evaluated from an independent known correlation of mass transfer data, of which several are available. In most of 

When kj is known, the result can be arranged in a form that is linear in the unknown constants, 

Note that when k1 is large, the last equation with surface reaction alone controlling. 

Substitution of the last two equations into Eq (v) will find function of only measurable quantities and various constants. 

---

Overall Rate Equations with Diffusional Resistances. 23-52... 

OVERALL RATE EQUATIONS WITH DIFFUSIONAL RESISTANCES... 

Overall rate equations with diffusional resistances... 

The effectiveness factor is defined as the ratio of the rate of reaction with diffusional resistances over the rate of reaction at bulk conditions. It follows that the effectiveness factor is not constant along the reactor length and therefore has to be calculated at each axial point along the reactor length. In the case of a two dimensional model, the effectiveness factor radial variations should also be calculated. It is important to realize that if the catalyst pellet effectiveness factor is different from unity, then the packed bed reactor model must be described by a heterogeneous model, and pseudo-homogeneous models cannot be used except in a few very special cases as discussed earlier. 

We now turn to the task of converting the concentration profile (Equation 4.16h), into the more useful effectiveness factor E, which had been defined as the ratio of the reaction rate with diffusional resistance to the reaction rate without diffusional resistance ... 

One procedure makes use of a box on whose silk screen bottom powdered desiccant has been placed, usually lithium chloride. The box is positioned 1-2 mm above the surface, and the rate of gain in weight is measured for the film-free and the film-covered surface. The rate of water uptake is reported as u = m/fA, or in g/sec cm. This is taken to be proportional to - Cd)/R, where Ch, and Cd are the concentrations of water vapor in equilibrium with water and with the desiccant, respectively, and R is the diffusional resistance across the gap between the surface and the screen. Qualitatively, R can be regarded as actually being the sum of a series of resistances corresponding to the various diffusion gradients present ... 

The mesopores make some contribution to the adsorptive capacity, but thek main role is as conduits to provide access to the smaller micropores. Diffusion ia the mesopores may occur by several different mechanisms, as discussed below. The macropores make very Htde contribution to the adsorptive capacity, but they commonly provide a major contribution to the kinetics. Thek role is thus analogous to that of a super highway, aHowkig the adsorbate molecules to diffuse far kito a particle with a minimum of diffusional resistance. 

As illustrated ia Figure 6, a porous adsorbent ia contact with a fluid phase offers at least two and often three distinct resistances to mass transfer external film resistance and iatraparticle diffusional resistance. When the pore size distribution has a well-defined bimodal form, the latter may be divided iato macropore and micropore diffusional resistances. Depending on the particular system and the conditions, any one of these resistances maybe dominant or the overall rate of mass transfer may be determined by the combiaed effects of more than one resistance. 

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...

For catalytic investigations, the rotating basket or fixed basket with internal recirciilation are the standard devices nowadays, usually more convenient and less expensive than equipment with external recirculation. In the fixed basket type, an internal recirculation rate of 10 to 15 or so times the feed rate effectively eliminates external diffusional resistance, and temperature gradients. A unit holding 50 cm (3.05 in ) of catalyst can operate up to 800 K (1440 R) and 50 bar (725 psi). 

Use of Operating Curve Frequently, it is not possible to assume that = 0 as in Example 2, owing to diffusional resistance in the liquid phase or to the accumulation of solute in the hquid stream. When the back pressure cannot be neglected, it is necessary to supplement the equations with a material balance representing the operating line or curve. In view of the countercurrent flows into and from the differential section of packing shown in Fig. 14-3, a steady-state material balance leads to the fohowing equivalent relations ... 

The partial pressures in the rate equations are those in the vicinity of the catalyst surface. In the presence of diffusional resistance, in the steady state the rate of diffusion through the stagnant film equals the rate of chemical reaction. For the reaction A -1- B C -1-. . . , with rate of diffusion of A limited. 

Although they are termed homogeneous, most industrial gas-phase reactions take place in contact with solids, either the vessel wall or particles as heat carriers or catalysts. With catalysts, mass diffusional resistances are present with inert solids, the only complication is with heat transfer. A few of the reactions in Table 23-1 are gas-phase type, mostly catalytic. Usually a system of industrial interest is liquefiea to take advantage of the higher rates of liquid reactions, or to utihze liquid homogeneous cat ysts, or simply to keep equipment size down. In this section, some important noncatalytic gas reactions are described. 

Decompositions may be exothermic or endothermic. Solids that decompose without melting upon heating are mostly such that can give rise to gaseous products. When a gas is made, the rate can be affected by the diffusional resistance of the product zone. Particle size is a factor. Aging of a solid can result in crystallization of the surface that has been found to affect the rate of reaction. Annealing reduces strains and slows any decomposition rates. The decompositions of some fine powders follow a first-order law. In other cases, the decomposed fraction x is in accordance with the Avrami-Erofeyev equation (cited by Galwey, Chemistry of Solids, Chapman Hall, 1967)... 

When the product layer is porous the reaction will continue but at decreasing rate as the diffusional resistance increases with increasing conversion. Then,... 

Thus, two factors may be pointed out that determine the possibility of obtaining high yields of crystalline polyethylene on a solid catalyst with no diffusional restriction (1) the splitting up of the catalyst into small particles (< 1000 A), possible when using supports with low resistance to breaking (2) the formation of polymer grains with polydispersed porosity. 

A simple rectifying column consists of a tube arranged vertically and supplied at the bottom with a mixture of benzene and toluene as vapour. At the top a condenser returns some of the product as a reflux which flows in a thin film down the inner wall of the tube. The tube is insulated and heat losses can be neglected. At one point in the column the vapour contains 70 mol% benzene and the adjacent liquid reflux contains 59 moi% benzene. The temperature at this point is 365 K. Assuming the diffusional resistance to vaponr transfer to be equivalent to the diffusional resistance of a stagnant vapour layer 0.2 mm thick, calculate the rate of interchange of benzene and toluene between vapour and liquid. The molar latent heats of the two materials can be taken as equal. The vapour pressure of toluene at 365 K is 54.0 kN/nt2 and the diffusivity of the vapours is 0.051 cm2/s... 

The effectiveness factor rj, defined in equation 8.5-5, is a measure of the effectiveness of the interior surface of the particle, since it compares the observed rate through the particle as a whole with the intrinsic rate at the exterior surface conditions the latter would occur if there were no diffusional resistance, so that all parts of the interior surface were equally effective (at cA = cAs). To obtain T], since all A entering the particle reacts (irreversible reaction), the observed rate is given by the rate of diffusion across the permeable face at z = 0 ... 

In Figure 9.1(c), the opposite extreme case of a very porous solid B is shown. In this case, there is no internal diffusional resistance, all parts of the interior of B are equally accessible to A, and reaction occurs uniformly (but not instantaneously) throughout the particle. The concentration profiles are flat with respect to radial position, but cB decreases with respect to time, as indicated by the arrow. This model may be called a uniform-reaction model (URM). Its use is equivalent to that of a homogeneous model, in which the rate is a function of the intrinsic reactivity of B (Section 9.3), and we do not pursue it fiirther here. 

Equimolal concentrations of A and B are charged to the reactor and the density remains constant. Laboratory tests were made with a rotating basket reactor in which diffusional resistances were effectively eliminated, with the tabulated results. 

It is to be carried out with a porous catalyst with external diffusional resistance with the rate relation... 

A fluidized bed reactor with two equal stages is used for a reaction with external diffusional resistance. The rate equation is... 

A second order gas phase reaction, 2A => 2B, -is conducted under flow conditions with a porous catalyst. Diffusional resistance is present both externally and within the pores. Inlet partial pressure of the reactant is 2 atm and the temperature is 2k73 K. Conversion is to be 80%. Other data are, particle radius R = 0.1 cm... 

The design of fixed-bed ion exchangers shares a common theory with fixed-bed adsorbers, which are discussed in Chapter 17. In addition, Thomas(14) has developed a theory of fixed-bed ion exchange based on equation 18.21. It assumed that diffusional resistances are negligible. Though this is now known to be unlikely, the general form of the solutions proposed by Thomas may be used for film- and pellet-diffusion control. 

---