Isothermal pellet

For an irreversible first order surface reaction with Vj = -1, we obtain for the 

A similar development can be observed for irreversible th order reactions. At steady state follows  

The second Damkohler number, Dali, is defined as the ratio between the characteristic mass transfer time = / k Up) and the characteristic reaction time, t, = l/(/r,c -i). 

With Equation 2.133 we find for the external effectiveness under isothermal conditions  

With increasing intrinsic reaction rate (increasing Dall) the observed rate constant approaches the volumetric mass transfer coefficient the 

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For an Isothermal pellet with pores sufficiently small that Knudsen diffusion controls, the flux relations are required to take the form (8.1), which can be written... 

The parameter y reflects the sensitivity of the chemical reaction rate to temperature variations. The parameter represents the ratio of the maximum temperature difference that can exist within the particle (equation 12.3.99) to the external surface temperature. For isothermal pellets, / may be regarded as zero (keff = oo). Weisz and Hicks (61) have summarized their numerical solutions for first-order irreversible... 

The conclusion, therefore, is that when parallel competing first-order reactions occur in isothermal pellets with large pores the intrinsic selectivity is unaffected. However, in Izurge pellets with small pores, the selectivity reduces to the square root of the value for the unimpeded reaction. Thus for large 0... 

The effects of non-uniform distribution of the catalytic material within the support in the performance of catalyst pellets started receiving attention in the late 60 s (cf 1-4). These, as well as later studies, both theoretical and experimental, demonstrated that non-uniformly distributed catalysts can offer superior conversion, selectivity, durability, and thermal sensitivity characteristics over those wherein the activity is uniform. Work in this area has been reviewed by Gavriilidis et al. (5). Recently, Wu et al. (6) showed that for any catalyst performance index (i.e. conversion, selectivity or yield) and for the most general case of an arbitrary number of reactions, following arbitrary kinetics, occurring in a non-isothermal pellet, with finite external mass and heat transfer resistances, the optimal catalyst distribution remains a Dirac-delta function. 

The nondimensional parameter /) (positive for exothermic reactions) is a measure of nonisothermal effects and is called the heat generation function. It represents the ratio between the rate of heat generation due to the chemical reaction and the heat flow by thermal conduction. Nonisothermal effects may become important for increasing values of /3, while the limit (3 - 0 represents an isothermal pellet. Table 9.1 shows the values of [3 and some other parameters for exothermic catalytic reactions. For any interior points within the pore where the reactant is largely consumed, the maximum temperature difference for an exothermic reaction becomes... 

A final topic we shall mention concerning the deactivation of isothermal pellets is the effect of nonuniform distribution of the active ingredient within the pellet. It has been known for sometime that this is possible - wittingly or unwittingly. This is particularly so for metals supported on porous oxides, e.g., alumina. We will take one simple example, the analysis of Shadman-Yazdi and Petersen [7] for self-fouling by a series mechanism in which the concentration of catalytic material was highest on the exterior of the pellet and fell to zero at the center according to the equation... 

An early normalization of the Thiele modulus for an isothermal pellet and arbitrary kinetics was given by R. B. Bird, W. E. Stewart, and E. N. Lightfoot on pages 335-41 of their Notes on Transport Phenomena, the precursor to their well-known Transport Phenomena (New York John Wiley Sons, Inc., 1960). Slightly more general forms—all of them equivalent—have been given independently and almost simultaneously in ... 

Naphtalene 104 Natural gas 342-364 Nitrogen 94-100, 327-328 Non-isothermal pellet 148-151, 153-162 reactor 414... 

FIGURE 5.1 Single isothermal pellet with external resistances only. 

FIGURE 5.2 Supply and consumption functions for the non-porous isothermal pellet with non-monotonic kinetics. 

Single reaction in an isothermal pellet. This case was further divided into a number of special cases. 

If the experimental results of Kehoe and Butt [114] in Figs. 3.7.a-2 and 3.7.a-3 are studied, note that the external heat transfer resistance can be appreciable, and, especially for the isothermal pellet, must be considered. The same mass and heat balance equations (3.7.a-l, 2) [or (3.7.a-8,9)] are used, but surface boundary conditions expressed in terms of the finite external heat and mass transfer resistances are used. 

Internal and External Mass Transport in Isothermal Pellets... 

To calculate ripom, the mass and heat balances must be solved simultaneously. Analytical and numerical solutions are given by Petersen (1962), Tinkler and Pigford (1961), Carberry (1961), Tinkler and Metzner (1961), and Weisz and Hicks (1962). The behavior of a non-isothermal pellet in the regime of pore diffusion limitation is governed by the Thiele modulus (f> (related to Tsurface)> the Prater number and the Arrhenius number /int ... 

From a qualitative analysis of the competition between reaction and diffusion in an isothermal pellet one easily recognizes that the parameter governing the steady state behavior of the pellet is the ratio between time constants for diffusion and reaction,i.e.,T(j/xr-If Tr the reaction rate is much slower than the diffusion rate the concentration profile inside the pellet is then almost flat and equal to the external surface concentration.The effectiveness factor is around unity. However,when the concentration inside the pe-... 

For the absence of concentration gradients in an isothermal pellet (Weisz and Prater 1954) ... 

Only isothermal pellets and internal effectiveness factors have been treated so far. Limitations on internal heat transfer, external mass transfer, and external heat transfer can all affect the reaction rate. Consider a pellet placed in a fluid medium. Assuming constant physical properties, one-dimensional mass and heat balances yield for a slab-like pellet ... 

It is clear from the foregoing discussion that the general problem of diffusion-reaction for the overall effectiveness factor is quite involved. Fortunately, however, the physical and chemical processes at work under realistic conditions favor isothermal pellets and negligible external mass transfer resistances. A more detailed examination of this is in order. Combining Eqs. 4.32 and 4.33 results in ... 

The reactor point effectiveness can now be readily determined from the overall pellet effectiveness developed earlier together with the known time dependence of the change of active surface area. Under the assumptions of negligible external mass transfer resistance and an isothermal pellet, the reactor point effectiveness is simply the pellet effectiveness multiplied by (ks/ki,), where ks and kj, are the rate constants evaluated at the pellet surface and bulk-fluid temperatures ... 

These equations are the steady-state version of Eqs. 9.6 through 9.15 with the assumptions of negligible axial dispersion, negligible external mass transfer resistance, isothermal pellets, and constant v and physical properties. Note that the external heat transfer resistance is still present as given by Eq. 9.36. Needless to... 

The basic approximations made in arriving at the reactor point effectiveness are (1) isothermal pellet, (2) negligible external mass transfer resistance, and (3) estimation of the pellet center concentration by a simple relationship when the reaction is not severely diffusion-limited. The first two approximations are quite adequate in view of the fact that the mass Biot number is of the order of hundreds under realistic reaction conditions. Both theoretical and experimental justifications for these approximations have been given in Chapter 4. The first approximation will be relaxed when reactions affected by pore-mouth poisoning are considered since a definite temperature gradient then exists within the pellet. An additional approximation is the representation of the difference between the Arrhenius exponentials evaluated at the pellet surface and the bulk-fluid temperatures by a linear rela-... 

Under the conditions of negligible external mass transfer resistance and an isothermal pellet, Eqs. 10.32, 10.37, and 10.38 can be removed from consideration for the case of uniform deactivation. However, Eq. 10.32 still needs to be retained in a limited form to account for the temperature drop in the deactivated outer shell in the case of shell-progressive deactivation. 

The fact that the Biot number of mass (a measure of the ratio of internal diffusion to external mass transfer resistance) is much larger than unity, implies that the major resistance lies in the internal diffusion process. A simple analysis can be made to assess the relative importance of internal and external mass transport processes now that the pellet can be considered isothermal. For an isothermal pellet, Eq. 4.32 can be written as ... 

This section develops generalized internal effectiveness factors (Aris 1965 Bischoff 1965 Petersen 1965), generalized in the sense that they are applicable to arbitrary kinetics. For a slab-like, isothermal pellet, Eq. 4.21 reduces to ... 

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