Let's solve **Problem 1** step by step, as a helpful tutor would. This involves formulating and solving the mass balance for diffusion and reaction in a spherical catalyst pellet under diffusion-limited, steady-state, first-order reaction. --- # **1. Diffusion and Reaction in a Porous Spherical Catalyst Pellet** ## **Step 1: Write the Mass Balance (Governing Differential Equation)** For a porous spherical pellet of radius \( R \), with a first-order reaction (\( -r_A = k C_A \)), under steady-state and isothermal conditions: ### **Start with the General Mass Balance:** For species A, in spherical coordinates: \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 J_A \right) = \text{rate of disappearance by reaction} \] Where \( J_A \) is the flux by Fick's law: \[ J_A = -D_{eff} \frac{dC_A}{dr} \] \( D_{eff} \) = effective diffusivity So, \[ \frac{1}{r^2} \frac{d}{dr} \left( -D_{eff} r^2 \frac{dC_A}{dr} \right) = -k C_A \] Or, \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dC_A}{dr} \right) - \frac{k}{D_{eff}} C_A = \] --- ## **Step 2: Boundary Conditions** - **At the center (\( r= \)):** symmetry, so \( \frac{dC_A}{dr} = \) - **At the surface (\( r=R \)):** \( C_A = C_{A,s} \) (surface concentration; often assumed equal to the external concentration if external mass transfer is not limiting) --- ## **Step 3: Non-dimensionalize the Equation** Let - \( \rho = \frac{r}{R} \) (dimensionless radius) - \( \phi^2 = \frac{k R^2}{D_{eff}} \) (Thiele modulus squared) So the equation becomes: \[ \frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 \frac{dC_A}{d\rho} \right) - \phi^2 C_A = \] --- ## **Step 4: Solve the Differential Equation** This is a standard ODE with known solution: \[ \frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 \frac{dC_A}{d\rho} \right) - \phi^2 C_A = \] The general solution is: \[ C_A(\rho) = \frac{A \sinh(\phi \rho)}{\phi \rho} + \frac{B \cosh(\phi \rho)}{\phi \rho} \] Apply boundary conditions: ### **At \( \rho = \):** - \( \frac{dC_A}{d\rho} = \) - \( \frac{\sinh(\phi \rho)}{\phi \rho} \) is finite as \( \rho \to \) - \( \frac{\cosh(\phi \rho)}{\phi \rho} \) diverges as \( \rho \to \) - To avoid divergence, set \( B = \) So, \[ C_A(\rho) = C_{A,s} \frac{\sinh(\phi \rho)}{\sinh(\phi) \phi \rho} \] where we have adjusted the constant so that at \(\rho = 1\) (\( r = R \)), \( C_A(1) = C_{A,s} \). --- ## **Step 5: Final Solution (Concentration Profile)** \[ \boxed{ C_A(r) = C_{A,s} \frac{\sinh \left( \phi \frac{r}{R} \right)}{\sinh(\phi) \frac{r}{R}} } \] Where: - \( C_{A,s} \): concentration at the pellet surface - \( \phi = R\sqrt{\frac{k}{D_{eff}}} \) (Thiele modulus) --- ## **Physical Meaning** - The profile depends on the **Thiele modulus** \( \phi \), which expresses the ratio of reaction rate to diffusion rate. - For large \( \phi \): reaction is much faster than diffusion (concentration drops steeply inside the pellet). - For small \( \phi \): diffusion is much faster; concentration varies little across the pellet. --- ## **Summary Table** | Step | Expression/Explanation | |----------------------|--------------------------------------------------------------------------------------------------| | Mass Balance | \( \frac{1}{r^2} \frac{d}{dr} \left(r^2 \frac{dC_A}{dr}\right) - \frac{k}{D_{eff}} C_A = \) | | Boundary Conditions | \( \frac{dC_A}{dr}|_{r=} = \), \( C_A(R) = C_{A,s} \) | | Thiele Modulus | \( \phi = R\sqrt{\frac{k}{D_{eff}}} \) | | Solution | \( C_A(r) = C_{A,s} \frac{\sinh \left( \phi \frac{r}{R} \right)}{\sinh(\phi) \frac{r}{R}} \) | --- **Let me know if you want to see the effectiveness factor derivation or help with the other problems!**

# **Problem 1: Diffusion and Reaction in a Porous Spherical Catalyst Pellet** ## **Step 1: Governing Differential Equation** For a porous spherical catalyst pellet undergoing a first-order reaction, the mass balance in spherical coordinates is given by: \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 J_A \right) = -k C_A \] Where: - \( J_A = -D_{eff} \frac{dC_A}{dr} \) (flux by Fick's law) Substituting \( J_A \): \[ \frac{1}{r^2} \frac{d}{dr} \left( -D_{eff} r^2 \frac{dC_A}{dr} \right) = -k C_A \] This simplifies to: \[ \frac{1}{r^2} \frac{d}{dr} \left( r^2 \frac{dC_A}{dr} \right) - \frac{k}{D_{eff}} C_A = 0 \] ## **Step 2: Boundary Conditions** 1. At the center (\( r = 0 \)): \[ \frac{dC_A}{dr} = 0 \quad \text{(symmetry)} \] 2. At the surface (\( r = R \)): \[ C_A(R) = C_{A,s} \quad \text{(surface concentration)} \] ## **Step 3: Non-dimensionalize the Equation** Let: - \( \rho = \frac{r}{R} \) - \( \phi^2 = \frac{k R^2}{D_{eff}} \) (Thiele modulus squared) The equation becomes: \[ \frac{1}{\rho^2} \frac{d}{d\rho} \left( \rho^2 \frac{dC_A}{d\rho} \right) - \phi^2 C_A = 0 \] ## **Step 4: Solve the Differential Equation** The standard solution is: \[ C_A(\rho) = \frac{A \sinh(\phi \rho)}{\phi \rho} + \frac{B \cosh(\phi \rho)}{\phi \rho} \] ### **Applying Boundary Conditions** 1. As \( \rho \to 0 \), \( \frac{dC_A}{d\rho} \) must be finite; thus, \( B = 0 \). 2. At \( \rho = 1 \) (surface): \[ C_A(1) = C_{A,s} \] Final expression: \[ C_A(\rho) = C_{A,s} \frac{\sinh(\phi \rho)}{\sinh(\phi) \phi \rho} \] ## **Step 5: Final Solution** The concentration profile is: \[ \boxed{C_A(r) = C_{A,s} \frac{\sinh \left( \phi \frac{r}{R} \right)}{\sinh(\phi) \frac{r}{R}}} \] Where: - \( C_{A,s} \): concentration at the pellet surface - \( \phi = R\sqrt{\frac{k}{D_{eff}}} \) (Thiele modulus) ## **Summary** - The governing equation describes the balance between diffusion and reaction. - Boundary conditions ensure the solution is physically relevant. - The final concentration profile relates to Thiele modulus, indicating the diffusion versus reaction rate in the pellet. Let me know if you need further assistance!