Pore length

As expected, the polarization parameter, = x x r is added to the pore length, / (see Section 2.2.5). The polarization resistanee is dependent on the current density [Eq. (2-35)]. For pure activation polarization, it follows from Eq. (2-45) ... 

For the simplest one-dimensional or flat-plate geometry, a simple statement of the material balance for diffusion and catalytic reactions in the pore at steady-state can be made that which diffuses in and does not come out has been converted. The depth of the pore for a flat plate is the half width L, for long, cylindrical pellets is L = dp/2 and for spherical particles L = dp/3. The varying coordinate along the pore length is x ... 

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). 

FIG. 20 Effective diffusion coefficients using Michaels model [241], Eq. (43), versus porosity for various ratios of pore lengths. 

Influence on Electrolyte Conductivity In porous separators the ionic current passes through the liquid electrolyte present in the separator pores. Therefore, the electrolyte s resistance in the pores has to be calculated for known values of porosity of the separator and of conductivity, o, of the free liquid electrolyte. Such a calculation is highly complex in the general case. Consider the very simple model where a separator of thickness d has cylindrical pores of radius r which are parallel and completely electrolyte-filled (Fig. 18.2). Let / be the pore length and N the number of pores (all calculations refer to the unit surface area of the separator). The ratio p = Ud (where P = cos a > 1) characterizes the tilt of the pores and is called the tortuosity factor of the pores. The total pore volume is given by NnrH, the porosity by... 

For Ca < 0.1 in Figure 7 the critical capillary pressure is also independent of the initial film thickness. In this case, the hydrodynamic resistance to fluid filling or draining is small enough that the film reaches the periodic steady state in less than half a pore length. Figure 7 confirms the trend observed by Khatib, Hirasaki and Falls that P falls with increasing flow rate (5). c... 

The occurrence of pore diffusion can usually be determined by simply grinding the catalyst into smaller and smaller particles. If the rate per gram of catalyst increases as the particles become smaller and smaller, then pore diffusion is likely to be occurring. This effect is due to the fact that the pore lengths are decreased by the catalyst particles being ground into smaller and smaller pieces. Eventually, the pores become short enough that the reactants can readily diffuse in and out of them faster than the chemical reaction occurs on the surface. 

To develop analytical models for processes employing porous catalysts it is necessary to make certain assumptions about the geometry of the catalyst pores. A variety of assumptions are possible, and Thomas and Thomas (15) have discussed some of these. The simplest model assumes that the pores are cylindrical and are not interconnected. Develop expressions for the average pore radius (r), the average pore length (L), and the number of pores per particle (np) in terms of parameters that can be measured in the laboratory [i.e., the apparent particle dimensions, the void volume per gram (Vg), and the surface area per gram (Sg). ... 

To evaluate the average pore length, it is necessary to recognize that the porosity sp will represent not only the volumetric void fraction but also at any cross section the fraction of the area occupied by the pore openings. If the average open area associated with each pore is assumed to be 7cr2, the definition of the porosity indicates that... 

The geometry of the pore structure makes it impossible to determine accurately the effective length of the diffusion path. Interconnections within the pore structure, the tortuous character of individual pores, and variations in cross-sectional area along the pore length all contribute to the difficulty of the task. 

T is the absolute temperature M is the molecular weight AC is the concentration difference over the length of the pore AX is the pore length... 

If a material balance is written over the differential element of pore length shown in Figure 12.1, one finds that at steady state ... 

The numerator of the right side of this equation is equal to the chemical reaction rate that would prevail if there were no diffusional limitations on the reaction rate. In this situation, the reactant concentration is uniform throughout the pore and equal to its value at the pore mouth. The denominator may be regarded as the product of a hypothetical diffusive flux and a cross-sectional area for flow. The hypothetical flux corresponds to the case where there is a linear concentration gradient over the pore length equal to C0/L. The Thiele modulus is thus characteristic of the ratio of an intrinsic reaction rate in the absence of mass transfer limitations to the rate of diffusion into the pore under specified conditions. 

This value is considerably higher than the experimental value (0.17) obtained from rate measurements on different size particles, but several factors may be invoked to explain the inconsistency. There will be a distribution of both pore radii and pore lengths present in the actual catalyst rather than uniquely specified values. Alumina catalysts often have a bimodal pore-size distribution. Our estimate of an apparent first-order rate constant using the method outlined above will be somewhat in error. The catalyst surface may not be equally active throughout if selective deactivation has taken place and the peripheral region is less active than the catalyst core. Other sources of error are the... 

For a second-order reaction, a material balance on a differential element of pore length leads to the following differential equation... 

H(hydrogen)-mordenite catalyst The crystallites were approximate parallelepipeds, the long dimension of which was assumed to be the pore length. Their analysis was based on straight, parallel pores in an isothermal crystallite (2 faces permeable). They measured (initial) rates of dehydration of methanol (A) to dimethyl ether in a differential reactor at 101 kPa using catalyst fractions of different sizes. Results (for two sizes) are given in the table below, together with... 

For the special case of straight pores growing orthogonal to the electrode surface forming a flat interface to the bulk, the pore length l becomes equivalent to the layer thickness D. Equation (6.1) then also defines the growth rate of the whole porous layer rPS. The growth rate rPS of a porous layer depends on several... 

It should be emphasized that n.. and JPS, and therefore c and T, refer to the condition at the pore tip. The dissolution valence and the temperature can be assumed to be independent of pore depth. This is not the case for the HF concentration c. Because convection is negligible in macropores, the mass transport in the pore occurs only by diffusion. A linear decrease in HF concentration with depth and a parabolic growth law for the pores according to Pick s first law is therefore expected, as shown in Fig. 9.18 a. The concentration at the pore tip can be calculated from the concentration in the bulk of the electrolyte c, the pore length l, the diffusion coefficient DHf (Section 1.4) and the flow of HF molecules FHf. which is proportional to the current density at the pore tip ... 

Fig. 9.20 The maximum pore length for cylindrical pores that can be achieved without a degeneration of the pore tips is shown as a function of the electrolyte concentration at RT (circles). The shaded area indicates the degeneration depth for a decrease or increase in pore diameter versus pore depth. After [Lei 7],...

Fig. 10.13 The measured transmittance of macropore arrays of pore diameter d and pore length I, with the pore axis parallel to the light beam. The data are corrected for porosity (p = l). The transmittance of a single...

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