# Trickle-Bed Reactor: Mass and Energy BalancesThis problem involves a trickle-bed reactor with coupled heterogeneous (gas-solid) and homogeneous (liquid-phase) reactions, gas-liquid mass transfer, and heat effects. Below is a systematic derivation and analysis. --- ## 1. **Governing Equations: Full Coupled Mass-Energy PDEs** ### **Assumptions** - 1D plug flow with axial dispersion (z coordinate) - Pseudo-steady state in catalyst pores - Isothermal or non-isothermal as indicated - Constant physical properties unless affected by temperature --- ### **A. Gas Phase (\(C_{A,g}\))** #### **Species Balance (A in Gas)** \[ \frac{\partial C_{A,g}}{\partial t} + u_g \frac{\partial C_{A,g}}{\partial z} = D_{ax,g} \frac{\partial^2 C_{A,g}}{\partial z^2} - a_{GL} k_{GL}(C_{A,g} - \frac{C_{A,l}}{H(T)}) - a_{LS} r_s \] - \(u_g\): Superficial gas velocity - \(a_{GL}\): Gas-liquid interfacial area - \(k_{GL}\): Gas-liquid mass transfer coefficient - \(H(T)\): Henry's constant (temperature dependent) - \(r_s = k_s C_{A,g}\): Heterogeneous reaction rate (per catalyst area) - \(a_{LS}\): Liquid-solid area #### **Energy Balance (Gas)** \[ \rho_g C_{p,g} \left( \frac{\partial T_g}{\partial t} + u_g \frac{\partial T_g}{\partial z} \right) = \lambda_g \frac{\partial^2 T_g}{\partial z^2} - a_{GL} h_{GL} (T_g - T_l) - a_{LS} r_s (-\Delta H_s) \] - \(h_{GL}\): Gas-liquid heat transfer coefficient --- ### **B. Liquid Phase (\(C_{A,l}, C_{B,l}, C_{C,l}\))** #### **Species Balance (A in Liquid)** \[ \frac{\partial C_{A,l}}{\partial t} + u_l \frac{\partial C_{A,l}}{\partial z} = D_{ax,l} \frac{\partial^2 C_{A,l}}{\partial z^2} + a_{GL} k_{GL}(C_{A,g} - \frac{C_{A,l}}{H(T)}) - a_{LS} r_l \] - \(r_l = k_l C_{A,l} C_{B,l}\): Homogeneous reaction rate #### **Species Balance (B in Liquid)** \[ \frac{\partial C_{B,l}}{\partial t} + u_l \frac{\partial C_{B,l}}{\partial z} = D_{ax,l} \frac{\partial^2 C_{B,l}}{\partial z^2} - a_{LS} r_l \] #### **Species Balance (C in Liquid)** \[ \frac{\partial C_{C,l}}{\partial t} + u_l \frac{\partial C_{C,l}}{\partial z} = D_{ax,l} \frac{\partial^2 C_{C,l}}{\partial z^2} + a_{LS} r_l \] #### **Energy Balance (Liquid)** \[ \rho_l C_{p,l} \left( \frac{\partial T_l}{\partial t} + u_l \frac{\partial T_l}{\partial z} \right) = \lambda_l \frac{\partial^2 T_l}{\partial z^2} + a_{GL} h_{GL} (T_g - T_l) - a_{LS} r_l (-\Delta H_l) \] --- ### **C. Catalyst Phase (Porous Pellet, internal)** #### **Diffusion-Reaction (A in Pellet)** \[ = D_e \nabla^2 C_{A,p} - r_s \] - \(D_e\): Effective diffusivity in pellet Boundary conditions: - At pellet surface: \(C_{A,p} = C_{A,g}\) - At pellet center: \(\frac{\partial C_{A,p}}{\partial r} = \) --- ## 2. **Non-Dimensionalization** Let: - \(z^* = \frac{z}{L}\): Axial position - \(C^* = \frac{C}{C_{A}}\): Dimensionless concentration - \(T^* = \frac{T}{T_{ref}}\): Dimensionless temperature Key dimensionless groups: - **Peclet number (gas):** \[ Pe_g = \frac{u_g L}{D_{ax,g}} \] - **Peclet number (liquid):** \[ Pe_l = \frac{u_l L}{D_{ax,l}} \] - **Damköhler number (heterogeneous):** \[ Da_s = \frac{a_{LS} k_s L}{u_g} \] - **Damköhler number (homogeneous):** \[ Da_l = \frac{a_{LS} k_l C_{B} L}{u_l} \] - **Mass transfer Biot number:** \[ Bi_{GL} = \frac{a_{GL} k_{GL} L}{u_g} \] - **Heat generation numbers:** \[ \Theta_s = \frac{a_{LS} k_s C_{A} (-\Delta H_s) L}{\rho_g C_{p,g} u_g T_{ref}} \] \[ \Theta_l = \frac{a_{LS} k_l C_{A} C_{B} (-\Delta H_l) L}{\rho_l C_{p,l} u_l T_{ref}} \] - **Dimensionless Henry's constant:** \(H^*(T^*)\) --- **Example: Non-dimensional mass balance for gas A:** \[ Pe_g \frac{\partial C_{A,g}^*}{\partial z^*} = \frac{\partial^2 C_{A,g}^*}{\partial {z^*}^2} - Bi_{GL} \left(C_{A,g}^* - \frac{C_{A,l}^*}{H^*(T^*)}\right) - Da_s C_{A,g}^* \] Similar scaling applies to other equations. --- ## 3. **Condition for Liquid-Phase Reaction Dominance** Liquid-phase homogeneous reaction dominates if its overall rate is much higher than the heterogeneous rate: \[ \frac{r_l}{r_s} = \frac{k_l C_{A,l} C_{B,l}}{k_s C_{A,g}} \] - If \(Da_l \gg Da_s\) - If \(C_{B,l}\) is high and/or \(k_l\) is large - If gas-liquid mass transfer and dissolution are not limiting **Dominance criterion:** \[ a_{LS} k_l C_{B,l} \gg a_{LS} k_s \frac{C_{A,g}}{C_{A,l}} \] or, dimensionlessly, \[ Da_l \gg Da_s \] --- ## 4. **Prediction of Thermal Runaway Zones** Thermal runaway occurs where the rate of heat generation exceeds removal. This is often where: \[ \text{Heat generated by reactions} > \text{Heat removed by conduction/advection} \] - **Runaway is likely if:** - Damköhler numbers are high (\(Da_s, Da_l\) large) - Peclet numbers are low (poor heat removal) - Heat transfer to surroundings is poor (low \(h_{GL}\), low \(\lambda\)) - Exothermic reaction: negative \(\Delta H\) - **Zones:** Typically in regions where reactant concentrations and temperatures are highest (often the reactor entrance for exothermic reactions). --- ## 5. **Numerical Solution for Conversion Profiles** ### **Discretization Approach** - Discretize PDEs along \(z\) (finite difference or finite volume) - Time-marching if required (implicit/explicit) - Use coupled iteration for temperature and all concentrations - Boundary conditions: specified inlet concentrations, temperature ### **Typical Conversion Profile Calculation** 1. Set up grid over reactor length 2. Initialize \(C_{A,g}, C_{A,l}, C_{B,l}, T\) at \(z=\) 3. At each grid point, compute source/sink terms: - Heterogeneous and homogeneous reaction rates - Mass and heat transfer rates 4. March forward in \(z\) (steady-state) or time (transient) 5. At outlet, calculate: - \(X_{A,g} = 1 - \frac{C_{A,g}(L)}{C_{A,g}()}\) - \(X_{A,l} = 1 - \frac{C_{A,l}(L)}{C_{A,l}()}\) **Parameter values are substituted as needed for actual profiles.** --- ## 6. **Summary Table of Key Parameters** | Symbol | Value/Expression | Units | Description | |------------|---------------------------------------------------|---------------|-----------------------| | \(a_{GL}\) | 300 | m²/m³ | Gas-liquid area | | \(a_{LS}\) | 600 | m²/m³ | Liquid-solid area | | \(k_s\) | \(8 \times 10^4 e^{-95000/(RT)}\) | m/s | Hetero. rate constant | | \(k_l\) | \(5 \times 10^6\) | m³/(mol·s) | Homo. rate constant | | \(D_e\) | \(6 \times 10^{-10}\) | m²/s | Pellet diffusivity | | \(H\) | \(140 \exp[.02(T - 300)]\) | - | Henry's constant | | \(D_{ax,g}\)| \(1.2 \times 10^{-4}\) | m²/s | Gas dispersion | | \(D_{ax,l}\)| \(8 \times 10^{-5}\) | m²/s | Liquid dispersion | | \(\Delta H_s\)| -110 | kJ/mol | Hetero. heat of rxn | | \(\Delta H_l\)| -75 | kJ/mol | Homo. heat of rxn | --- ## 7. **Further Notes** - For numerical simulation, use implicit methods for stability due to stiffness from fast reactions and heat effects. - Include temperature dependence of kinetic parameters and Henry's constant. - For thermal runaway, monitor temperature rise along \(z\); runaway occurs if temperature rapidly increases in a zone. --- ## 8. **References** - Levenspiel, O. "Chemical Reaction Engineering" - Froment, G. F., & Bischoff, K. B. "Chemical Reactor Analysis and Design" --- **[No image is included as the problem is text-based. If a schematic is needed, describe it as a packed bed with gas and liquid co-flow, catalyst particles, and interfacial mass/heat transfer.]**