Let's proceed step-by-step for each task: --- # 1. Non-dimensionalize Both Equations Let’s define dimensionless variables: - Radial position: \(\xi = \frac{r}{R}\), where \(R = 4 \text{ mm} = .004\) m - Concentration: \(C_A^* = \frac{C_A}{C_{A,b}}\) - Temperature: \(\theta = \frac{T - T_b}{T_b}\) Now, let’s rewrite the equations in terms of these variables. ## Mass Balance (Dimensionless) \[ \frac{1}{r^2}\frac{d}{dr}\left(r^2 D_{eff} \frac{dC_A}{dr}\right) - k_ e^{-E/RT} C_A = \] Substitute and use chain rule: \[ \frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2 \frac{dC_A^*}{d\xi}\right) - \frac{R^2}{D_{eff}} k_ e^{-E/R(T_b(1+\theta))} C_A^* = \] ## Energy Balance (Dimensionless) \[ \frac{1}{r^2}\frac{d}{dr}\left(r^2 k_{th} \frac{dT}{dr}\right) - (-\Delta H) k_ e^{-E/RT} C_A = \] This becomes: \[ \frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2 \frac{d\theta}{d\xi}\right) + \frac{R^2}{k_{th} T_b} (-\Delta H) k_ e^{-E/R(T_b(1+\theta))} C_{A,b} C_A^* = \] --- # 2. Define Thiele Modulus, Biot Numbers, and Exothermicity Parameter **Thiele modulus, \(\phi\):** \[ \phi^2 = \frac{R^2 k_ e^{-E/RT_b}}{D_{eff}} \] **Biot number for mass transfer, \(Bi_m\):** \[ Bi_m = \frac{k_g R}{D_{eff}} \] **Biot number for heat transfer, \(Bi_h\):** \[ Bi_h = \frac{h R}{k_{th}} \] **Exothermicity parameter, \(\beta\):** \[ \beta = \frac{(-\Delta H) C_{A,b}}{k_{th} T_b} \] --- # 3. Solve for Coupled \(C_A(r)\) and \(T(r)\) Profiles The equations are coupled and nonlinear due to the exponential temperature dependence. Analytical solutions are not generally possible; numerical methods (finite difference or shooting method) are used. The equations to solve are: \[ \frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2 \frac{dC_A^*}{d\xi}\right) - \phi^2 e^{-\gamma \theta} C_A^* = \] \[ \frac{1}{\xi^2}\frac{d}{d\xi}\left(\xi^2 \frac{d\theta}{d\xi}\right) + \beta \phi^2 e^{-\gamma \theta} C_A^* = \] where \(\gamma = \frac{E}{R T_b}\). Boundary conditions (dimensionless): At \(\xi = \): \(\frac{dC_A^*}{d\xi} = , \frac{d\theta}{d\xi} = \) At \(\xi = 1\): \[ - \frac{dC_A^*}{d\xi} = Bi_m (1 - C_A^*(1)) \] \[ - \frac{d\theta}{d\xi} = Bi_h \theta(1) \] --- # 4. Determine Effectiveness Factor Including Heat Effects The effectiveness factor \(\eta\) is defined as: \[ \eta = \frac{\text{Actual overall rate}}{\text{Rate if entire pellet at } C_{A,b}, T_b} \] \[ \eta = \frac{3 \int_{}^{1} e^{-\gamma \theta(\xi)} C_A^*(\xi) \xi^2 d\xi}{e^{-\gamma \cdot } \cdot 1} \] Numerically integrate the numerator after solving for \(C_A^*(\xi)\) and \(\theta(\xi)\). --- # 5. Find Maximum Internal Temperature The maximum temperature occurs at the center due to symmetry and is given by: \[ T_{max} = T_b (1 + \theta()) \] Solve the profiles numerically to find \(\theta()\). --- # 6. Determine Criterion for Internal Thermal Runaway Thermal runaway occurs when heat generated exceeds heat removed. The critical parameter is often the Frank-Kamenetskii parameter: \[ \delta = \frac{R^2}{k_{th}} k_ C_{A,b} \frac{E}{R T_b^2} e^{-E/R T_b} (-\Delta H) \] Runaway occurs if \(\delta > \delta_{cr}\), where \(\delta_{cr}\) is a critical value (typically around .88 for a sphere). --- # 7. Identify Controlling Regime: Film, Diffusion, or Kinetic Compare resistances: - If \(Bi_m \ll 1\), film (external mass transfer) controlled. - If \(\phi \ll 1\), kinetic controlled. - If \(\phi \gg 1\) and \(Bi_m \gg 1\), internal diffusion controlled. Calculate: \[ Bi_m = \frac{.02 \times .004}{5 \times 10^{-6}} = 16 \] \[ \phi^2 = \frac{(.004)^2 \times 1 \times 10^{-8} \times e^{-90000/(8.314 \times 500)}}{5 \times 10^{-6}} \] Numerically, \(e^{-21.65} \approx 3.97 \times 10^{-10}\), so \[ \phi^2 \approx \frac{1.6 \times 10^{-5} \times 3.97 \times 10^{-10}}{5 \times 10^{-6}} \approx 1.27 \times 10^{-9} \] So, \(\phi \ll 1\), thus the reaction is kinetically controlled. --- **Final Answers (Summary):** 1. The equations are non-dimensionalized using \(\xi, C_A^*, \theta\). 2. Thiele modulus, Biot numbers, and exothermicity parameter are defined as above. 3. The coupled equations are solved numerically for profiles. 4. Effectiveness factor is computed from the integral of the dimensionless profiles. 5. Maximum temperature is at the center, \(T_{max} = T_b (1+\theta())\). 6. Runaway criterion is based on the Frank-Kamenetskii parameter. 7. Given parameters, the regime is kinetic control (\(\phi \ll 1\)).

# Step-by-Step Solution ## 1. Non-dimensionalize the Equations Define dimensionless variables: - Radial position: \(\xi = \frac{r}{R}\) where \(R = 0.004 \text{ m}\) - Concentration: \(C_A^* = \frac{C_A}{C_{A,b}}\) - Temperature: \(\theta = \frac{T - T_b}{T_b}\) Rewrite the mass and energy balance equations using these variables. ## 2. Define Key Parameters - **Thiele Modulus (\(\phi\))**: \[ \phi^2 = \frac{R^2 k e^{-E/(RT_b)}}{D_{eff}} \] - **Biot Number for Mass Transfer (\(Bi_m\))**: \[ Bi_m = \frac{k_g R}{D_{eff}} \] - **Biot Number for Heat Transfer (\(Bi_h\))**: \[ Bi_h = \frac{h R}{k_{th}} \] - **Exothermicity Parameter (\(\beta\))**: \[ \beta = \frac{(-\Delta H) C_{A,b}}{k_{th} T_b} \] ## 3. Solve for Concentration and Temperature Profiles The equations are coupled and typically solved using numerical methods. The profiles will depend on the dimensionless forms of the governing equations. ## 4. Effectiveness Factor The effectiveness factor (\(\eta\)) is calculated as: \[ \eta = \frac{3 \int_0^1 e^{-\gamma \theta(\xi)} C_A^*(\xi) \xi^2 d\xi}{e^{-\gamma \cdot } \cdot 1} \] ## 5. Maximum Internal Temperature The maximum internal temperature can be found at the center: \[ T_{max} = T_b (1 + \theta) \] ## 6. Criterion for Internal Thermal Runaway The criterion for thermal runaway involves the Frank-Kamenetskii parameter: \[ \delta = \frac{R^2}{k_{th}} k C_{A,b} \frac{E}{R T_b^2} e^{-E/(RT_b)} (-\Delta H) \] Runaway occurs if \(\delta > \delta_{cr}\). ## 7. Controlling Regime Determine whether the reaction is controlled by: - Film diffusion - Internal diffusion - Kinetics Calculate the relevant numbers (Thiele modulus and Biot numbers) to identify the controlling regime. ## Final Answers 1. Non-dimensional equations are established. 2. Key parameters are defined. 3. Concentration and temperature profiles are solved numerically. 4. Effectiveness factor is derived from profiles. 5. Maximum temperature is identified at the center. 6. Thermal runaway criterion is based on \(\delta\). 7. The reaction is kinetically controlled if \(\phi \ll 1\).