Pores diffusion

Diffusion of molecules to be adsorbed in the macropore of the porous body is easily understood by describing the flux using partial pressure or concentration of the species in the fluid phase in the pore. 

The effective diffusion coeflicient in the particle. Dp, is considered to be proportional to the diffusivity in the bulk phase. A, when macropore diffusion is dominant and 

The diffusibility, i, may be determined from a detailed structure or configurations of pore network but actual pore structure is usually quite complicated and in most cases only simplified considerations are made. 

One of the simplest models of a porous body is a packed bed of particles. Typical measured diffusibilities for packed beds of particles are shown in Fig. 4.1. The broken line in the figure shows the diffusibility for the system with dispersed inactive particles (spherical) determined from the 

Source O Hoogschagen (1955), Cume (I960), J Suzuki and Smith (1972), Porous particle, Suzuki and Smith (1972), A Wooding (1959), 

Pick s laws are the mathematical basis for the description of diffusion. 

The particle flow density is proportional to the concentration gradient (Equation 2.1-19)  

Da = diffusion coefficient for spatially isotropic and isothermal diffusion of A. 

This describes the change in concentration with space and time due to diffusion (Equation 2.1-20)  

Next we will discuss the individual diffusion mechanisms. 

The most important mass transfer limitation is diffusion in the micropores of the catalyst. A simplified model of pore diffusion treats the pores as long, narrow cylinders of length The narrowness allows radial gradients to be neglected so that concentrations depend only on the distance I from the mouth of the pore. Equation (10.3) governs diffusion within the pore. The boundary condition at the mouth of the pore is 

An analytical solution is possible when the reaction is first order e.g., a reaction of the form A — P with adsorption as the rate-controlling step. Then Equation (10.3) becomes 

This gives the concentration profile inside the pore, a l). The total rate of reaction within a pore can be found using the principle of equal rates. The reaction rate within a pore must equal the rate at which reactant molecules enter the pore. Molecules enter by diffusion. The flux of reactants molecules diffusing into a pore of diameter dp re equals the reaction rate. Thus, 

The ratio of actual rate to intrinsic rate is the effectiveness factor  

It depends only on J sJkj A, which is a dimensionless group known as the Thiele modulus. The Thiele modulus can be measured experimentally by comparing actual rates to intrinsic rates. It can also be predicted from first principles given an estimate of the pore length =2 . Note that the pore radius does not enter the calculations (although the effective diffusivity will be affected by the pore radius when dpore is less than about 100 run). 

Above we considered a porous catalyst particle, but we could similarly consider a single pore as shown in Fig. 5.36. This leads to rather similar results. The transport of reactant and product is now determined by diffusion in and out of the pores, since there is no net flow in this region. We consider the situation in which a reaction takes place on a particle inside a pore. The latter is modeled by a cylinder with diameter R and length L (Fig. 5.36). The gas concentration of the reactant is Cq at the entrance of the pore and the rate is given by 

If we take a thin slice of the cylinder of thickness dx we can write an expression for the transport of mass through this slice at steady state. What goes in either comes out or reacts, i.e. 

(40) the time has been omitted as it is a steady state solution, and we have introduced the Thiele modulus for a pore. We solve the differential equation with the following constraints  

The reader may verify by insertion in the differential equation above that this is indeed a solution that fulfils the constraints. 

We shall now introduce an efficiency factor, which is again defined as the ratio of the conversion in the pore with and without mass transport limitation  

The basic problem is that the rate is a function of position X within the peUef while we need an average concentration to insert into our reactor mass balances. We first have to solve for the concentration profile Ca (x) and then we eliminate it in terms of the concentration at the surface of the pellet C/4s and geometry of the pellet. We wiU find that we can represent the reaction as 

We next need to solve this second-order differential equation subject to the boundary conditions 

We now have found Ca(t), but this isn t useful yet because Ca is a function of position X within the pore. To find the average rate within the pore we must integrate the rate along the length of the pore 

We want to compare this to the rate in the pore if the concentration remained at Cas, 

The ratio of these rates (actual rate/ideal rate) is the fraction by which the rate is reduced by pore diffusion limitations, which we call rj, and this is the definition of the effectiveness factor. 

---

For adsorption from the vapor phase, Kmay be very large (sometimes as high as 10 ) and then clearly the effective diffusivity is very much smaller than the pore diffusivity. Furthermore, the temperature dependence of K follows equation 2, giving the appearance of an activated diffusion process with... 

Fig. 15. Theoretical breakthrough curves for a nonlinear (Langmuir) system showing the comparison between the linear driving force (—), pore diffusion (--------------------), and intracrystalline diffusion (-) models based on the Glueckauf approximation (eqs. 40—45). From Ref. 7.

Successful operation of the gaseous diffusion process requires a special, fine-pored diffusion barrier, mechanically rehable and chemically resistant to corrosive attack by the process gas. For an effective separating barrier, the diameter of the pores must approach the range of the mean free path of the gas molecules, and in order to keep the total barrier area required as small as possible, the number of pores per unit area must be large. Seals are needed on the compressors to prevent both the escape of process gas and the inflow of harm fill impurities. Some of the problems of cascade operation are discussed in Reference 16. 

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. 

Pore Diffusion When flmd transport through a network of fluid-filled pores inside the particles provides access for solute adsorption sites, the diffusion fliix can be expressed in terms of a pore diffusion coefficient D as ... 

For adsorbent materials, experimental tortuosity factors generally fall in the range 2-6 and generally decrease as the particle porosity is increased. Higher apparent values may be obtained when the experimental measurements are affected by other resistances, while v ues much lower than 2 generally indicate that surface or solid diffusion occurs in parallel to pore diffusion. 

Ruthven (gen. refs.) summarizes methods for the measurement of effective pore diffusivities that can be used to obtain tortuosity factors by comparison with the estimated pore diffusion coefficient of the adsorbate. Molecular diffusivities can be estimated with the methods in Sec. 6. 

SoUd Diffusion In the case of pore diffusion discussed above, transport occurs within the fluid phase contained inside the particle here the solute concentration is generally similar in magnitude to the external fluid concentration. A solute molecule transported by pore diffusion may attach to the sorbent and detach many times along its... 

The diffusion coefficient in these phases D,j is usually considerably smaller than that in fluid-filled pores however, the adsorbate concentration is often much larger. Thus, the diffusion rate can be smaller or larger than can be expected for pore diffusion, depending on the magnitude of the flmd/solid partition coefficient. 

Alternate driving force approximations, item 2B in Table 16-12, for solid diffusion, and item 3B in Table 16-12, for pore diffusion, provide somewhat more accurate results in constant pattern packed-bed calculations with pore or solid diffusion controlling for constant separation factor systems. 

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... 

Mechanism 1. External film 2. Solid diffusion 3. Pore diffusion 4. Reaction kinetics... 

When pore diffusion predominates, use of column 3 in Table 16-12 is prefer le, with /c replacing k. ... 

For a linear isotherm tij = KjCj), this equation is identical to the con-seiwation equation for sohd diffusion, except that the solid diffusivity D,i is replaced by the equivalent diffusivity = pDj,i/ p + Ppi< ). Thus, Eqs. (16-96) and (16-99) can be used for pore diffusion control with infinite and finite fluid volumes simply by replacing D,j with D. When the adsorption isotherm is nonhnear, a numerical solution is... 

FIG. 16-18 Constant separation factor batch adsorption curves for pore diffusion control with an infinite fluid volume. X is defined in the text. 

In the irreversible limit, the sohidon for combined external resistance and pore diffusion with infinite fluid volume is (Yagi and Knnii) ... 

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. 

The design approach is particularly feasible for those reactions in which chemical and pore diffusion rates are most important. For flow related phenomena semi-empirical, dimensionless correlations must be relied on. Therefore in this book scale-up will be used in the more general sense with the airri of using methods that are fundamentally based wherever feasible. 

The work of Thiele (1939) and Zeldovich (1939) called attention to the fact that reaction rates can be influenced by diffusion in the pores of particulate catalysts. For industrial, high-performance catalysts, where reaction rates are high, the pore diffusion limitation can reduce both productivity and selectivity. The latter problem emerges because 80% of the processes for the production of basic intermediates are oxidations and hydrogenations. In these processes the reactive intermediates are the valuable products, but because of their reactivity are subject to secondary degradations. In addition both oxidations and hydrogenation are exothermic processes and inside temperature gradients further complicate secondary processes inside the pores. 

For the effective diffusivity in pores, De = (0/t)D, the void fraction 0 can be measured by a static method to be between 0.2 and 0.7 (Satterfield 1970). The tortuosity factor is more difficult to measure and its value is usually between 3 and 8. Although a preliminary estimate for pore diffusion limitations is always worthwhile, the final check must be made experimentally. Major results of the mathematical treatment involved in pore diffusion limitations with reaction is briefly reviewed next. 

Treatment of thermal conductivity inside the catalyst can be done similarly to that for pore diffusion. The major difference is that while diffusion can occur in the pore volume only, heat can be conducted in both the fluid and solid phases. For strongly exothermic reactions and catalysts with poor heat conductivity, the internal overheating of the catalyst is a possibility. This can result in an effectiveness factor larger than unity. 

Inert gas pressure, temperature, and conversion were selected as these are the critical variables that disclose the nature of the basic rate controlling process. The effect of temperature gives an estimate for the energy of activation. For a catalytic process, this is expected to be about 90 to 100 kJ/mol or 20-25 kcal/mol. It is higher for higher temperature processes, so a better estimate is that of the Arrhenius number, y = E/RT which is about 20. If it is more, a homogeneous reaction can interfere. If it is significantly less, pore diffusion can interact. 

Figure 6.3.2 shows the feed-forward design, in which acrolein and water were included, since previous studies had indicated some inhibition of the catalytic rates by these two substances. Inert gas pressure was kept as a variable to check for pore diffusion limitations. Since no large diffusional limitation was shown, the inert gas pressure was dropped as an independent variable in the second study of feed-back design, and replaced by total pressure. For smaller difftisional effects later tests were recommended, due to the extreme urgency of this project. 

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. 

There are data showing that at the same contact time, but different linear velocities, there is no difference in the performance of a carbon system. It is obvious then that the effect of linear velocity on the diffusion through the film around the particle and the ratio of the magnitude of the film diffusion to the pore diffusion are the factors that determine the effects, if any, that occur. Therefore, the linear velocity cannot be ignored completely when evaluating a system. Systems at the higher linear velocity (LV) treat more liquid per volume of carbon at low-concentration levels and the mass-transfer zone (MTZ) is shorter. 

Pore diffusivity The ability of a material to diffuse gas through its pores, trapping the contaminants. 

---