Gradients inside the catalyst particle

For these gradients, continuity requires that everything that reacted inside, had to diffiise through the outside layer of the catalyst. 

Concentration gradient inside the catalyst particle. The continuity statement, at the catalyst surface, is similar to Pick s first law for diffiasion. The reaction rate is equal to the diffusion rate at the outside layer of the catalyst 

This expressed with the measured rate becomes  

The r is the observable rate calculated from the outside balance. Therefore everything in Dan is measurable but the Dc. If no reliable value exists for this, then from the relationship 

Temperature gradient in the catalyst particle. Continuity in the outermost layer of the catalyst requires that all the heat generated inside has to cross this layer. The continuity statement in the outermost layer is now similar to Fourier s lav/ for thermal conduction. 

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An experimental test to verify the absence of significant concentration gradients inside the catalyst particle is based on the inverse proportional relation between the effectiveness factor and the particle diameter for strong internal diffusion limitations. Hence, a measured rate that is independent of the particle size indicates that internal diffusion limitations can be neglected. Care should be taken to avoid artefacts. External heat transfer effects also depend on particle size and for exothermic reactions might balance the internal diffusion effects. Furthermore, if the catalyst particle consists of a support with a non-uniformly distributed active phase, crushing and sieving to obtain smaller particles is hazardous. 

The concentration gradients inside the catalyst particles and fine solid deposit (Figure 5.7c and d) are given by the equations describing the simultaneous mass transport and reaction within the catalyst particles and the diffusion within the solid deposit considering that the reactants diffusion occurs in the liquid phase by virtue of completely internally wetted catalysts ... 

TEMPERATURE GRADIENTS INSIDE THE CATALYST PARTICLES IN BENZENE HYDROGENATION... 

Diffusion resistance in the catalyst notable Plug flow or axial dispersion neither radial concentration nor temperature gradients in the reactor concentration and temperature gradients inside the catalyst particles Plug flow or axial dispersion radial concentration and temperature gradients in the reactor concentration and temperature gradients inside the catalyst particles... 

The reaction rate —rA now can be written as i x (—r ), where —r is shorthand for the rate that would exist if there were no gradients inside the catalyst particle. 

Consider a case where the resistance to intranal diffusion is very large and consequently the concentration gradient inside the catalyst particle is very steep. For this situation,... 

The temperature gradients inside the catalyst pellet turned out to be negligible, clearly less than 1 K for realistic approximations for AH and heat conductivity. Thus, the final calculations were made by the isothermal model by simply assuming the same temperature inside the catalyst particle as in the liquid phase. 

In the preceding sections, (S.3.b,c) no attention was given to situations where the reaction components encounter important transport resistances inside the catalyst particle. In Qiapter 3 it was shown how concentration gradients then build up in the particle, even when the latter is isothermal. In the steady state a feed component A then has a descending concentration profile from the surface towards the center... 

When the resistance to mass and heat transfer inside the catalyst particle is important the rate of reaction is not uniform throughout the particle. The set of Eqs. 11.8.a-l to 11.8.a-4 of Sec. 11.8.a then no longer adequately describes the system. It has to be completed with equations describing the concentration and temperature gradients inside the particle, so that the complete set may be written ... 

However, usually catalysts are porous materials to strongly increase the active reaction area per unit of reactor volume. Therefore, the intra-particle gradients, i.e., the mass and heat transfer inside the catalyst particle, have to be considered as well. Fluid phase equations are the same as Eqs. 4.10, 4.11, while solid phase equations have to be modified as follows ... 

Balance Equations 5.154 through 5.157 are strictly considered as valid for the pseudoho-mogeneous model, in other words, cases in which neither concentration nor temperature gradients appear in the catalyst particle. In case diffusion inside the catalyst particles—or in the fluid film surrounding the particle— is notable, the term Rj —AHtj) in the energy... 

Concentration and temperature gradients inside a catalyst particle can influence the rate of reaction, i.e., the apparent catalyst activity. They can also influence the product distribution, i.e., the apparent catalyst selectivity. First, let s deal with the reaction rate. 

Furthermore, there are several unique aspects to the behavior of the effectiveness factor with highly exothermic, gas-phase reactions. First, ri can be greater than unity. Under some circumstances, the increase in reaction rate due to the higher temperature inside the catalyst particle can more than offset the decrease in reaction rate caused by the lower concentration of reactants. When this occurs, the actual reaction rate is larger than the rate with no gradients, leading to r > 1. 

Now consider a situation where the resistance to internal transport is significant and internal gradients do exist. The concentration profile of A inside the catalyst particle will be the same as for a single, first-order reaction and wUl be simUar to those shown in previous... 

If the heat of reaction is essentially zero, heat transfer to and from the catalyst particle is not required in order for the reaction to take place at steady state. There will be no temperature gradients, either inside the catalyst particle or through the boundary layer. Therefore, the only external transport step that affects the reaction rate is mass transfer through the boundary layer. 

Temperature gradient normal to flow. In exothermic reactions, the heat generation rate is q=(-AHr)r. This must be removed to maintain steady-state. For endothermic reactions this much heat must be added. Here the equations deal with exothermic reactions as examples. A criterion can be derived for the temperature difference needed for heat transfer from the catalyst particles to the reacting, flowing fluid. For this, inside heat balance can be measured (Berty 1974) directly, with Pt resistance thermometers. Since this is expensive and complicated, here again the heat generation rate is calculated from the rate of reaction that is derived from the outside material balance, and multiplied by the heat of reaction. 

Step 3. Transport within a catalyst pore is usually modeled as a one-dimensional diffusion process. The pore is assumed to be straight and to have length The concentration inside the pore is ai =ai(l,r,z) where I is the position inside the pore measured from the external surface of the catalyst particle. See Figure 10.2. There is no convection inside the pore, and the diameter of the pore is assumed to be so small that there are no concentration gradients in the radial direction. The governing equation is an ODE. 

From a molecular point of view inside a catalyst particle, diffusion may be considered to occur by three different modes molecular, Knudsen, and surface. Molecular diffusion is the result of molecular encounters (collisions) in the void space (pores) of the particle. Knudsen diffusion is the result of molecular collisions with the walls of the pores. Molecular diffusion tends to dominate in relatively large pores at high P, and Knudsen diffusion tends to dominate in small pores at low P. Surface diffusion results from the migration of adsorbed species along the surface of the pore because of a gradient in surface concentration. 

Another aspect concerns catalyst particles with intraparticle temperature gradients. In general the temperature inside a catalyst pellet will not be uniform, due to the heat effects of the reaction occurring inside the catalyst pellet. The temperature inside the catalyst can be related to the concentration with (see for example [4]) ... 

As discussed in chapter 5, diffusion through catalyst pores represents a resistance to mass and heat transfer, which gives rise to concentration and temperature gradients within the catalyst pellet. This causes the rate of reaction in the solid phase to be different from that if the bulk phase conditions prevail inside the particle, and the rate of reaction should be integrated along the radius of the pellet to get the actual rate of reaction. 

The first boundary condition follows from symmetrical reasons. In practice, the effective heat conductivity of the catalyst, Tie, is often so high that the temperature gradients inside the particle are minor. On the other hand, there often emerges a temperature gradient in the fluid film around the catalyst particle, since the thermal conduction of the fluid is limited. The energy balance of the fluid film is reduced to... 

Equations 5.3.b-9 and 5.3.b-ll or 5.3.b-12 form a set of simultaneous equations that clearly shows that the coking of the catalyst not only depends on the mechanism of coking, but also on the composition of the reaction mixture. Consequently, even under isothermal conditions, the coke is not uniformly deposited in a reactor or inside a catalyst particle whenever there are gradients in concentration of reactants and products. This important conclusion will be quantitatively developed in a later section. 

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