Sure! Let's down your catalytic problem step by, using the from your image. --- ## **Given Data** - **Reaction:** \( A \rightarrow B \) (1st order, exothermic) - **Rate Law:** \( -r_A = kC_A \), \( k = k_ e^{-E/RT} \) - **\( k_ \) (pre-exponential factor):** \( 6 \times 10^4 \, \textrm{s}^{-1} \) - **Activation Energy (E):** \( 108 \, \textrm{kJ/mol} \) - **ΔH:** \( -115 \, \textrm{kJ/mol} \) - **Reactor length (L):** \( 15\,\textrm{m} \) - **Reactor diameter (D):** \( 1.2\,\textrm{m} \) - **Bed void fraction (ε):** \( .40 \) - **Catalyst pellet radius (\( r_p \)):** \( 3 \, \textrm{mm} = .003\,\textrm{m} \) - **Effective diffusivity (\( D_e \))**: \( 6.5 \times 10^{-10} \, \textrm{m}^2/\textrm{s} \) - **Pellet thermal conductivity (\( k_s \))**: \( 1.4\,\textrm{W}/\textrm{m}\cdot\textrm{K} \) - **Feed:** \( C_{A} = 3.2\,\textrm{kmol}/\textrm{m}^3 \), \( T_ = 580\,\textrm{K} \), \( u = .55\,\textrm{m/s} \) - **Heat transfer:** \( U = 160\,\textrm{W}/\textrm{m}^2\cdot\textrm{K} \), \( T_c = 575\,\textrm{K} \) --- ## **Your Questions, Step-by-Step** --- ### **1. Derive Interparticle Mass/Energy Balance Equation** #### **Mass Balance (for a single pellet):** For **steady-state** and assuming **isothermal** for now: \[ = D_e \left( \frac{1}{r^2} \frac{d}{dr} \left(r^2 \frac{dC_A}{dr}\right) \right) - kC_A \] where: - \( r \) = radial position in the pellet ( to \( r_p \)) - \( D_e \) = effective diffusivity - \( k \) = rate constant **Boundary conditions:** - At \( r = \): \( \frac{dC_A}{dr} = \) (symmetry) - At \( r = r_p \): \( C_A = C_{As} \) (surface concentration) #### **Energy Balance (with heat of reaction):** \[ = k_s \left( \frac{1}{r^2} \frac{d}{dr} \left(r^2 \frac{dT}{dr}\right) \right) + (-\Delta H) \cdot kC_A \] - \( k_s \) = pellet thermal conductivity - \( -\Delta H \) = heat of reaction --- ### **2. Define Thiele Modulus (including temp. dependence)** **Thiele modulus (\( \phi \))** for 1st order reaction: \[ \phi = r_p \sqrt{\frac{k}{D_e}} \] Since \( k = k_ e^{-E/RT} \): \[ \phi = r_p \sqrt{\frac{k_ e^{-E/RT}}{D_e}} \] So, **Thiele modulus increases with temperature decrease** (since \( k \) drops with \( T \)). --- ### **3. Determine Effectiveness Factor (η) with Heat Effect** For a 1st order, spherical pellet: \[ \eta = \frac{3}{\phi^2} \left( \phi \coth{\phi} - 1 \right) \] With heat effects, both \( k \) and thus \( \phi \) depend on **local pellet temperature**, which may be **higher than the external temperature** due to exothermicity. --- ### **4. Calculate Maximum Catalyst Pellet Temperature** Use the **energy balance** at the center of the pellet (worst case for hot spot): \[ k_s \left.\frac{dT}{dr}\right|_{r=} = \] \[ -k_s \left.\frac{dT}{dr}\right|_{r=r_p} = h (T_s - T_{bulk}) \] Assuming steady-state, the maximum temperature is at the center. For large Thiele modulus and high exothermicity, estimate **maximum temperature rise** using the **Weisz-Prater criterion** or by solving the coupled mass/energy equations. #### **Approximate Solution:** For a first estimation (assuming uniform temperature): \[ q_{gen} = (-\Delta H) \cdot k C_{A,s} \] \[ \text{Temperature rise:} \quad \Delta T = \frac{q_{gen} \cdot r_p^2}{6 k_s} \] --- ### **5. Identify Whether Internal or External Transport Controls Reaction** - **Internal diffusion limited** if \( \eta \ll 1 \) (i.e., large Thiele modulus) - **External mass transfer limited** if surface concentration much less than bulk - **Reaction limited** if \( \eta \approx 1 \) Calculate both **Thiele modulus** (for internal) and **external Biot number**: \[ \text{Bi}_m = \frac{k_f r_p}{D_e} \] Compare relative resistances. --- ## **Summary Table** | Step | Description | Main Equation/Concept | |------|-------------|----------------------| | 1 | Interparticle balances | See mass/energy equations above | | 2 | Thiele modulus | \( \phi = r_p \sqrt{\frac{k_ e^{-E/RT}}{D_e}} \) | | 3 | Effectiveness factor | \( \eta = \frac{3}{\phi^2} (\phi \coth{\phi} - 1) \) | | 4 | Max pellet temperature | \( \Delta T = \frac{(-\Delta H) k C_{A,s} r_p^2}{6 k_s} \) | | 5 | Control regime | Compare \( \eta \), Biot number | --- ## **Let Me Know Which Step You Want to See Solved Numerically!** If you want full worked numbers (e.g., calculate \( \phi \), \( \eta \), etc.), just ask!