Pore diffusion resistance negligible

Figure 8.10(b) shows a plot of if/ = cAlcAs as a function of z, the fractional distance into the particle, with the Thiele modulus cj) as parameter. For = 0, characteristic of a very porous particle, the concentration of A remains the same throughout the particle. For (f> = 0.5, characteristic of a relatively porous particle with almost negligible pore-diffusion resistance, cA decreases slightly as z —> 1. At the other extreme, for = 10, characteristic of relatively strong pore-diffusion resistance, cA drops rapidly as z increases, indicating that reaction takes place mostly in the outer part (on the side of the permeable face) of the particle, and the inner part is relatively ineffective. 

Figure 18.7 Shows the limits for negligible and for strong pore diffusion resistance.

Let rA be the rate of transfer of A into the catalyst pellet had the resistance to internal pore diffusion been negligible. If the resistance to internal pore diffusion is neglected, then the concentration of A at all the active sites within the pores inside the catalyst pellet will be the same as the concentration of A at the surface of the catalyst pellet, that is, Q = Qs for all values ofx,-Lspecific reaction rate of A is the same at all locations within the catalyst pellet and is equal to (-ri) = RCas- So,... 

In this case. Equation 4.226 derived for packed bed catalytic reactor is applicable after suitable modification of certain terms appearing in that equation. The effectiveness factor q = 1 as the catalyst particle is very small in size and the resistance to internal pore diffusion is negligible. Taking the catalyst particles to be spherical in shape with radius R, Lq = R/3 and the equation for bed height Z reduces to... 

By using Eq. (7), they showed that in the combxastion of char resistance d je to pore diffusion is negligible for teraperatxires below 576 0. 

Ma et al. [104] attributed a decrease in diffusivity with an increase in initial concentration to pore diffusion effects. Because zeolites are bi-dispersed sorbents, both surface and pore diffusions may dominate different regions. In micropores, surface diffusion may be dominant, while pore diffusion may be dominant in macropores. This, therefore, supports the use of a lumped parameter (De). To explore further the relative importance of external mass transfer vis-a-vis internal diffusion, Biot number (NBl — kf r0/De) was considered. Table 9 summarizes the NBi values for the four initial concentrations. The NBi values are significantly larger than 100 indicating that film diffusion resistance was negligible. 

For small values mL < 0.3 in (2.377) we obtain rjp > 0.97, which lies close to 1. This implies that the composition of the reaction partner A hardly changes over the length of the pore. The resistance to diffusion is negligible in comparison to the other resistances. A small value of mL = k1/DeS L indicates a small pore, a slow reaction or rapid diffusion. 

It is evident from the foregoing discussion that the effective diffusivity cannot be predicted accurately for use under reaction conditions unless surface diffusion is negligible and a valid model for the pore structure is available. The prediction of an effective thermal conductivity is even more difficult. Hence sizable errors are frequent in predicting the global rate from the rate equation for the chemical step on the interior catalyst surface. This is not to imply that for certain special cases accuracy is not possible (see Sec. 11-10). It does mean that heavy reliance must be placed on experimental measurements for effective diffusivities and thermal conductivities. Note also from some of the examples and data mentioned later that intrapellet resistances can greatly affect the rate. Hence the problem is significant. 

If we plot the LHS of eq. (14.3-9) versus (e /u), we could obtain the axial dispersion from the slope and the intercept of such plot in 8f + 8, . We note that the intercept is corresponding to infinite velocity at which the contribution of the film mass transfer resistance is negligible, and hence the intercept is simply 8, from which the pore diffusivity can be extracted ... 

Boundary layers next to the two sides of the membrane can substantially contribute to the overall mass transfer resistance molecular diffusion across the boundary layer is often the rate-limiting step. Resistance to mass transfer within the pores of the membrane results from the transfer of momentum, or from the collision of diffusing molecules with other molecules (molecular resistance) or with the membrane itself (Knudsen resistance). In the same figure, resistance related to surface diffusion is also shown in MD this mechanism is present but generally its influence is negligible because the surface diffusion area is small in comparison to the pore area. Surface diffusion resistance may however be important in the development of hydrophilic membranes. 

The value of O increases with an increase in resistance to internal pore diffusion. Also, the larger the size of the catalyst pellet, the larger is the value of O. The value of the Thiele modulus can be taken as negligible for smaller-sized catalyst particles. 

One may check if sheet s internal transport resistance can be attributed to diffusion within the fibers or the void space by changing sheet thickness while the external resistance is rendered insignificant. Under this condition, if the sheet s pore space diffusion resistance is negligible, dynamics would be independent of sheet thickness thus providing a critical test of the hypothesis that it is only radial fiber diffusion that is important and not sheet transverse diffusion that is important. 

In many SCR applications, the thickness and effective diffusivity of the active layer may not allow the simplifying assumption of negligible internal diffusion resistance. In these cases, a more detailed approach which models mass transfer both in the gas phase and in the washcoat/active volume pores is needed. 

Up to now, we have only considered the thermodynamics of adsorption. However, in technical processes - for example, in a fixed bed - adsorption is a transient process until a particle or a zone of a fixed bed has reached the equilibrium loading. The intrinsic chemical process of adsorption can be regarded as instantaneous, and the mass transfer to and into the porous adsorbent determines how fast the equilibrium is reached. Experience teaches that the mass transfer resistance by film diffusion is mostly negligible, and so the adsorption is governed by pore diffusion (Topic 3.3.4). Pore diffusion can roughly be divided into macropore and micropore diffusion, for example, zeolites have macropores as well as micropores in the small crystallites, which form a sub-structure in a partide. The... 

Mass transfer resistances lead to a lower effective rate compared to the intrinsic chemical reaction, but may also significantly change the selectivity of parallel and consecutive reactions. In the following, this is discussed for two first-order reactions occurring in series or parallel, for simplification, the influence of external mass transfer is only discussed for a non-porous catalyst (to exclude pore diffusion), and the effect of pore diffusion is examined for a negligible influence of external mass transfer. Other more complicated cases are treated elsewhere (Baerns et al, 2006 Levenspiel, 1999 froment and Bischoff, 1990). 

In simple cases, only one of the above-mentioned transport steps has an influence on the effective rate (e.g., pore diffusion of reactant A or B). The reaction system may also be simplified if the concentration of the dissolved gas is much higher than of the liquid reactant (or vice versa). Then the concentration of either A or B can be regarded as constant throughout the whole liquid phase (including the liquid filled pores), and at least all mass transfer resistances of one reactant are negligible. 

Example 4.2.2 Calculate the time required to bring about the complete gasification of a graphite particle under conditions where external mass transport and pore diffusion present a negligible resistance to the progress of reaction. 

If the chemical reaction step presents a negligible resistance to the progress of reaction, compared with the resistance due to pore diffusion, the overall rate is controlled by diffusion. The reaction occurs in a narrow zone separating the unreacted core and the completely reacted layer, where the... 

As a approaches zero, pore diffusion presents a negligible resistance to the progress of reaction. Under this condition the reactant concentration is uniform throughout the pellet and is equal to that in the bulk (ij/ = 1). Therefore, is independent of rj. From the result obtained in Section 4.3.1, we thus... 

In this section we have presented and solved the BVPs associated with the diffusion and reaction that take place in the pores of a porous catalyst pellet. The results were expressed graphically in terms of the effectiveness factor rj versus the Thiele modulus d> for two cases One with negligible external mass and heat transfer resistances, i.e., when Sh and Nu —> oo, and another with finite Sh and Nu values. This problem is very important in the design of fixed-bed catalytic reactors. The sample results presented here have shown that for exothermal reactions multiple steady states may occur over a range of Thiele moduli d>. Efficient numerical techniques have been presented as MATLAB programs that solve singular two-point boundary value problems. 

However, nonmonotonic kinetics alone will not produce multiplicity. Such kinetics have to be coupled with a diffusion process, either in the form of a mass-transfer resistance between the catalyst pellet surface and the bulk gas, or within the pores of the pellets. If the flow conditions and the catalyst pellets size are such that diffusional resistances between the bulk gas phase and the catalytic active centers are negligible and the... 

In gas-liquid, liquid-liquid, or hquid-gas-liquid contactors there is no convective flow of any phase across the membrane. Mass transfer occurs only by diffusion across the immobilized phase in the pores. The direction of mass transfer of any molecular species depends on the concentration driving force maintained across the membrane for that species. The presence of the stationary phase in the membrane pore creates an extra diffusional mass transfer resistance. However, it can be shown that in many cases the membrane resistance is negligible, and that in most cases the high active mass transfer area created inside a membrane contactor more than compensates for any additional mass transfer resistance [4—5]. 

Since convective mass transfer is negligible in porous catalytic pellets, it is reasonable to let /i = 0 in (21-47) and add the diffusivities inversely. If the pore diameter is much smaller than the mean free path of the gas, then colhsions with the walls are more frequent than collisions with other molecules, and Knudsen diffusion provides the dominant resistance. If the pores are much larger than the mean free path, then collisions with other molecules are more frequent than colhsions with the walls of the channel, and ordinary molecular diffusion provides the dominant resistance. Usually, one arrives at a sum of resistauces in series by following the trajectory of a single gas molecule within a catalytic pore. It is important to emphasize that one obtains the correct result by tracking a single molecule only if there are no other pathways by which diffusion supplies the... 

The three-halves power of dimensionless temperature in the expression for eA( ) is based on the temperature dependence of gas-phase ordinary molecular diffusion coefficients when the catalytic pores are larger than 1 p.m. In this pore-size regime, Knudsen diffusional resistance is negligible. The temperature dependence of the collision integral for ordinary molecular diffusion, illustrated in Bird et al. (2002, pp. 526, 866), has not been included in ea) ). The thermal energy balance given by equation (27-28), which includes conduction and interdiffu-sional fluxes, is written in dimensionless form with the aid of one additional parameter,... 

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