Let's break this advanced chemical reactor problem into clear, step-by-step tasks, as requested. We'll focus on **formulating the full mass-energy PDEs**, **non-dimensionalizing**, **identifying key dimensionless groups**, and outlining steps for **analysis** and **numerical computation**. --- # 1. **Formulate the Full Coupled Mass-Energy PDEs** Let: - \( C_{A,g}(z,t) \): Concentration of A in gas phase - \( C_{A,l}(z,t) \): Concentration of A in liquid phase - \( C_{A,s}(z,t) \): Concentration of A in solid catalyst - \( T(z,t) \): Temperature - \( u_g, u_l \): Superficial velocities (gas, liquid) - \( \varepsilon_g, \varepsilon_l, \varepsilon_s \): Volume fractions (gas, liquid, solid) - \( a_{GL}, a_{LS} \): Interfacial areas (gas-liquid, liquid-solid) - \( k_L \): Liquid-side mass transfer coefficient - \( r_s, r_l \): Solid-phase and liquid-phase reaction rates ### **Mass Balances** #### **Gas Phase (A):** \[ \varepsilon_g \frac{\partial C_{A,g}}{\partial t} + u_g \frac{\partial C_{A,g}}{\partial z} = D_{ax,g} \frac{\partial^2 C_{A,g}}{\partial z^2} - a_{GL} k_L (C_{A,g} - C_{A,l}^*) \] where \( C_{A,l}^* = \frac{C_{A,g}}{H(T)} \) (from Henry’s law). --- #### **Liquid Phase (A):** \[ \varepsilon_l \frac{\partial C_{A,l}}{\partial t} + u_l \frac{\partial C_{A,l}}{\partial z} = D_{ax,l} \frac{\partial^2 C_{A,l}}{\partial z^2} + a_{GL}k_L (C_{A,g} - C_{A,l}^*) - a_{LS} k_{LS} (C_{A,l} - C_{A,s}) - r_{l,\text{hom}} \] - \( k_{LS} \): Mass transfer coefficient (liquid-solid) - \( r_{l,\text{hom}} \): Homogeneous reaction in liquid --- #### **Solid Phase (A):** \[ \varepsilon_s \frac{\partial C_{A,s}}{\partial t} = D_e \frac{\partial^2 C_{A,s}}{\partial r^2} + a_{LS}k_{LS} (C_{A,l} - C_{A,s}) - r_{s,\text{het}} \] - \( D_e \): Effective diffusivity in solid - \( r_{s,\text{het}} \): Heterogeneous reaction rate --- ### **Energy Balance (All Phases):** \[ \rho C_p \frac{\partial T}{\partial t} + u_g \rho_g C_{p,g} \frac{\partial T}{\partial z} + u_l \rho_l C_{p,l} \frac{\partial T}{\partial z} = \lambda \frac{\partial^2 T}{\partial z^2} + (-\Delta H_s) r_{s,\text{het}} + (-\Delta H_l) r_{l,\text{hom}} \] - \( \lambda \): Effective thermal conductivity --- # 2. **Non-Dimensionalize the Equations** Define dimensionless variables: - \( \hat{z} = \frac{z}{L} \) - \( \hat{t} = \frac{t u_g}{L} \) - \( \hat{C}_{A,g} = \frac{C_{A,g}}{C_{A,g,}} \), etc. - \( \hat{T} = \frac{T - T_}{T_} \) Key dimensionless groups: - **Peclet number**: \( \mathrm{Pe}_g = \frac{u_g L}{D_{ax,g}} \), \( \mathrm{Pe}_l = \frac{u_l L}{D_{ax,l}} \) - **Damköhler numbers** (solid, liquid): \( \mathrm{Da}_s = \frac{r_{s,\text{het}} L}{u_g C_{A,g,}} \), \( \mathrm{Da}_l = \frac{r_{l,\text{hom}} L}{u_l C_{A,l,}} \) - **Biot numbers** for mass transfer: \( \mathrm{Bi}_{GL} = \frac{a_{GL} k_L L}{u_g} \), \( \mathrm{Bi}_{LS} = \frac{a_{LS} k_{LS} L}{u_l} \) - **Thiele modulus** for effectiveness in catalyst: \( \phi = L \sqrt{ \frac{r_{s,\text{het}}}{D_e} } \) - **Dimensionless heat release**: \( \beta = \frac{(-\Delta H) C_{A,g,}}{\rho C_p T_} \) --- # 3. **Identify When Liquid Phase Reaction Dominates** Compare Damköhler numbers: - If \( \mathrm{Da}_l \gg \mathrm{Da}_s \), and mass transfer is not limiting (large Biot), then the **liquid phase reaction dominates**. - If \( k_L a_{GL} \) (interfacial transfer) is small, **mass transfer limits**. --- # 4. **Predict Thermal Runaway Zones** - **Thermal runaway** occurs if heat generation from reaction exceeds heat removal: - Large **\(\beta\)** and **exothermic reaction** (\( \Delta H < \)) - **Low axial dispersion** (\( D_{ax} \) small), i.e., poor heat conduction - **High Damköhler number** - **Critical temperature** can be found by analyzing the energy balance, seeking points where \( \frac{dT}{dt} > \) grows uncontrollably. --- # 5. **Numerical Solution for Conversion Profiles** - **Discretize** the PDEs (finite difference/volume method). - **March** along reactor length (\( z \)), solve coupled ODEs at each node for \( C_{A,g}, C_{A,l}, C_{A,s}, T \). - Use **initial** and **boundary conditions** (inlet concentrations, temperature). - **Iterate** until steady-state or desired reactor length is reached. --- ## **Summary Table: Key Equations and Steps** | Step | Equation/Parameter | Notes | |------|-------------------|-------| | 1. | Full PDEs (see above) | Mass and energy balances for all phases | | 2. | Non-dimensionalize | Define \( \mathrm{Pe}, \mathrm{Da}, \phi, \beta \) | | 3. | Compare Damköhler numbers | Dominance of liquid vs. solid | | 4. | Energy balance, \(\beta\) | Runaway if heat gen. > removal | | 5. | Discretize, solve numerically | Compute conversion, temp profiles | --- ## **Final Answer (in summary):** 1. **Write coupled PDEs for gas, liquid, and solid phases, including interfacial mass transfer and reaction terms.** 2. **Non-dimensionalize equations, identifying Peclet, Damköhler, Biot, Thiele, and heat release numbers.** 3. **Liquid-phase reaction dominates when its Damköhler number is much larger than others and mass transfer is not limiting.** 4. **Thermal runaway zones occur where exothermic heat release exceeds heat removal—analyzed using the energy PDE and dimensionless heat parameter.** 5. **Numerically solve the equations (e.g., finite difference) for conversion and temperature profiles along the reactor.** --- **If you need help with the explicit non-dimensionalized equations or the actual code for numerical solution, let me know!**