Question

Consider a spherical catalyst particle of radius R in a catalytic reactor, where it is submerged in a gas stream containing the reactant A and the product B. In the neighborhood of the surface of the catalyst particle, we presume that the concentration is CAs moles of A per unit volume. Reactant A diffuses through the passages in the catalyst and is converted to B through a first order reaction on the catalytic surfaces inside the particle (rA = -kaCA where a is the available catalytic surface per unit volume). The effective diffusivity for reactant A in the porous medium is known as DA.

Derive a differential equation governing the concentration profile of A in the particle (i.e. in the radial direction). (You should be obtaining a second-order differential equation with variable coefficients.) Write the appropriate boundary conditions and Use the change of variable CA/CAs = (1/ r) f(r) to obtain the following differential equation:

Answer 1

To derive the differential equation governing the concentration profile of A in the particle, we can start by considering Fick's second law of diffusion.

Fick's second law states that the rate of change of concentration with respect to time is proportional to the second derivative of concentration with respect to distance. In this case, the distance is the radial direction within the particle.

Starting with Fick's second law in spherical coordinates, we have:

∂CA/∂t = (1/r^2) ∂/∂r (r^2 D∂CA/∂r)

Since the reaction is assumed to be first-order, we can write the reaction rate as rA = -kaCA, where ka is the rate constant. Substituting this into the equation, we have:

(1/r^2) ∂/∂r (r^2 D∂CA/∂r) = -kaCA

Next, we introduce the change of variable CA/CAs = (1/r) f(r), where CAs is the concentration at the catalyst surface. Differentiating f(r) with respect to r, we get:

∂CA/∂r = (∂/∂r) [(CAs/r)f(r)]

Applying the product rule and simplifying, we obtain:

∂CA/∂r = (-CAs/r^2)f(r) + (∂f/∂r)

Substituting this expression back into the original equation, we have:

(1/r^2) ∂/∂r (r^2 D[(-CAs/r^2)f(r) + (∂f/∂r)]) = -ka(CAs/r)f(r)

Expanding and simplifying, we get:

(1/r^2) ∂/∂r (r^2 D(∂f/∂r)) + kaCAs f(r)/r = 0

This is the derived differential equation governing the concentration profile of A in the particle.

To solve this equation, appropriate boundary conditions need to be specified. These typically include the concentration at the catalyst surface (r = R) and the concentration at the center of the particle (r = 0). The specific boundary conditions will depend on the problem at hand and any additional assumptions or constraints.

It's important to note that the equation obtained is a second-order differential equation with variable coefficients, which may require numerical or analytical methods for solving, depending on the specific conditions and assumptions of the problem.