Diffusion and Reaction Flashcards
(18 cards)
What is the effective diffusivity?
The pores in the pellet are not straight and cylindrical; rather, they are a series of tortuous, interconnecting paths of pore bodies and pore throats with varying cross-sectional areas. It would not be fruitful to describe diffusion within each and every one of the tortuous pathways individually; consequently, we shall define an effective diffusion coefficient so as to describe the average diffusion taking place at any position r in the pellet. We shall consider only radial variations in the concentration; the radial flux War will be based on the total area (voids and solid) normal to diffusion transport (4pir^2) rather than void area alone. This basis for War is made possible by proper definition of the effective diffusivity De.
What does the effective diffusivity account for?
The effective diffusivity accounts for the fact that:
- Not all of the area normal to the direction of the flux is available (i.e., void) for the molecules to diffuse.
- The paths are tortuous.
- The pores are of varying cross-sectional areas.
Write down the equation for De.
An equation that relates De to either the bulk or the Knudsen diffusivity is
De = DA thetap sigma/ tau
tau = tortuosity = actual distance a molecule travels between two points/ shortest distance between those two points
theta p = pellet porosity = volume of void space / total volume (voids and solids)
sigma = constriction factor
What does the constriction factor account for?
The constriction factor accounts for the variation in the cross-sectional area that is normal to diffusion. It is a function of the ratio of maximum to minimum pore areas. When the two areas A1 and A2, are equal the constriction factor is unity, and when Beta= 10, the construction factor is approximately 0.5.
Example 12-1
Solve Example 12-1
Derive the differential equation describing diffusion and reaction.
Derivation
Write the equation in dimensionless form with the appropriate transformations.
Equations
What is the Thiele modulus?
The square root of the coefficient theta n is called the Thiele modulus. The Thiele modulus will always contain a subscript m which will distinguish this symbol from the symbol of porosity, theta, defined in Chapter Four, which has no subscript. The quantity theta n^2 is a measure of the ratio of a surface reaction rate to a rate of diffusion through the catalyst pellet.
theta n^2 = kn rhoc Cas^n R /De (Cas -0)/R = a surface reaction rate/ a diffusion rate.
When the Thiele modulus is large, internal diffusion usually limits the overall rate of reaction, when theta n is small, the surface reaction is usually rate-limiting.
Derive the solution to the differential equation for a first-order reaction.
Derivation
Describe and sketch the concentration profiles for different values of the Thiele modulus.
The figure sketched shows the concentration profile for three different values of the Thiele modulus. Small values of the Thiele modulus indicate surface reaction controls, and a significant amount of the reactants diffuses well into pellet interior without reacting. Large values of the Thiele modulus indicate that the surface reaction is rapid and that the reactant is consumed very close to the external pellet surface and very little penetrates the interior of the pellet. Consequently, if the porous pellet is to be plated with a precious metal catalyst (e.g., Pt), it should be only plated in the immediate vicinity of the external surface when large values of theta n characterize the diffusion and reaction. That is, it would be a waste of precious metal to plate the entire pellet when internal diffusion is limiting because the reacting gases are consumed near the outer surface. Consequently, the reacting gases would never contact the center portion of the pellet.
What is the internal effectiveness factor?
The magnitude of the effectiveness factor (ranging from 0 to 1) indicates the relative importance of diffusion and reaction limitations. The internal effectiveness factor is defined as
N = actual overall rate of reaction/ rate of reaction that would result if entire surface were exposed to the external pellet surface conditions, CAs, Ts
Derive the effectiveness factor for a first order reaction by working with reaction rates (moles per unit time)
Derivation
What will a plot of the effectiveness factor as a function of the Thiele modulus show?
Plot the effectiveness factor as a function of the Thiele modulus. The figure shows N as a function of theta for a spherical catalyst pellet for zero-, first- and second order reactions.
We observe that as the particle diameter becomes very small, theta decreases so that the effectiveness factor approaches 1 and the reaction is surface-reaction-limited. On the other hand, when theta is large (=30), the internal effectiveness factor N is small, and the reaction is diffusion limited within the pellet. Consequently, factors influencing the rate of mass transport will have a negligible effect on the overall reaction rate.
Give approximations for the internal effectiveness expression.
For large values of the Thiele modulus (theta 1> 20), the effectiveness factor can be written as
N = 3/theta = 3/R (De /k1 Sa rho c) ^ (1/2)
If theta 1>2
N= 3/theta 1^2 (theta 1 -1)
Therefore, to increase the overall rate of reaction -ra’: (1) decrease the radius R (make the pellet smaller); (2) increase the temperature; (3) increase the concentration; and (4) increase the internal surface area
Give the expression of the Thiele modulus for reactions of order n.
For reactions of order n, we have from
theta n^2 = kn Sa rhoc R^2 CAs^n-1/ De
For large values of the Thiele modulus, the effectiveness factor is
N = (2/n+1) ^ (1/2) 3/R (De/kn Sa rhoc)^(1/2) CAs ^ (1-n)/2
Consequently, for reaction orders greater than 1, the effectiveness factor decreases with increasing concentration at the external pellet surface.
What happen to the effectiveness factor for exothermic reactions?
The above discussion of effectiveness factors is valid only for isothermal conditions. When a reaction is exothermic and non-isothermal, the effectiveness factor can be significantly greater than one. Values of N greater than one occur because the external surface temperature of the pellet is less than the temperature inside the pellet where the exothermic reaction is taking place. Therefore, the rate of reaction inside the pellet is greater than the rate at the surface. Thus, because the effectiveness factor is the ratio of the actual reaction rate to the rate at surface conditions, the effectiveness factor can be greater than one, depending on the magnitude of the parameters beta and gamma.
What are the parameters Beta and Gamma?
The parameter gamma is sometimes referred to as the Arrhenius number and the parameter Beta represents the maximum temperature difference that could exist in the pellet relative to the surface temperature.
gamma = E/RT
beta = T max - T/ Ts = DHrxDeCa/ kr Ts
The Thiele modulus is evaluated at the external surface temperature.
