Let's address each part step by step: --- ## 1. **Energy Balance for Adiabatic CSTR** ### **Step 1: Energy Balance Equation** For an **adiabatic CSTR**, the energy balance can be written as: \[ \text{(Rate of energy in by feed)} + \text{(Rate of energy generated by reaction)} = \text{(Rate of energy out by flow)} \] Since it's adiabatic, **no heat is lost or gained** by the reactor. ### **Step 2: Expressing the Terms** - **Energy in by feed:** \( F C_p T_ \) - **Energy out by effluent:** \( F C_p T \) - **Energy generated by reaction:** \( -\Delta H_{Rxn} \cdot r V \) Where: - \( F \) = molar flow rate - \( C_p \) = heat capacity - \( T_ \) = inlet temp, \( T \) = outlet temp - \( -\Delta H_{Rxn} \) = heat of reaction (exothermic if negative) - \( r \) = reaction rate per volume - \( V \) = reactor volume ### **Step 3: Write the Balance** \[ F C_p (T - T_) = -\Delta H_{Rxn} \cdot r V \] ### **Step 4: Substitute Arrhenius Law** If \( r = k(T) C_A \), with \( k(T) = k_ e^{-E/RT} \): \[ F C_p (T - T_) = -\Delta H_{Rxn} \cdot k_ e^{-E/RT} C_A V \] ### **Step 5: Given Equation** The question provides: \[ X(-\Delta H_{Rxn}) = C_{p} (1 + \epsilon X)(T - T_) \] - \( X \) = conversion - \( \epsilon \) = change in moles per mole of A reacted ### **Step 6: Multiple Steady States** - **Heat Generation Curve (\( G(T) \)):** \( G(T) = -\Delta H_{Rxn} \cdot r V \) - **Heat Removal Curve (\( R(T) \)):** \( R(T) = F C_p (T - T_) \) Plot both vs \( T \) — intersection points are steady states. Multiple intersections = multiple steady states (possible ignition/extinction). --- ## 2. **Catalyst Deactivation in Packed Bed Reactor** Given: \( \frac{da}{dt} = -k_d a^2 \) ### **Step 1: Separate Variables** \[ \frac{da}{a^2} = -k_d dt \] ### **Step 2: Integrate** \[ \int \frac{da}{a^2} = -k_d \int dt \] \[ - \frac{1}{a} = -k_d t + C \] At \( t = \), \( a = 1 \): \[ -1 = C \implies C = -1 \] So, \[ - \frac{1}{a} = -k_d t - 1 \] \[ \frac{1}{a} = 1 + k_d t \] ### **Step 3: Linearize and Find \( k_d \) from Data** Plot \( \frac{1}{a} \) vs \( t \), slope = \( k_d \). | t (hr) | a | 1/a | |--------|------|------| | | 1 | 1 | | 2 | .83 | 1.20 | | 4 | .71 | 1.41 | | 6 | .62 | 1.61 | | 8 | .55 | 1.82 | | 10 | .50 | 2.00 | | 15 | .40 | 2.50 | | 20 | .33 | 3.03 | Slope (\( k_d \)) can be estimated as: \[ k_d \approx \frac{3.03 - 1}{20 - } = \frac{2.03}{20} = .102~\text{hr}^{-1} \] **Final Answer:** **Decay constant \( k_d \approx .10~\text{hr}^{-1} \)** --- ## 3. **Shrinking Core Model (SCM) Conversion-Time Relationship** ### **Step 1: Model Statement** Non-catalytic reaction: \( A(g) + bB(s) \to \text{Products} \) Assume **diffusion through ash layer controls** (shrinking core radius = \( r_c \)). ### **Step 2: Rate Expression** \[ \text{Diffusive Flux} = \frac{4\pi r_c D_e (C_{A,b} - C_{A,s})}{\left(\frac{1}{r_c} - \frac{1}{R}\right)} \] But for time to reach a given conversion, use the standard result: \[ t = \frac{\rho_B R}{b C_{A,b} k_D} \left[1 - 3(1 - X_B)^{2/3} + 2(1 - X_B)\right] \] ### **Step 3: Final Answer (as requested)** For **diffusion control**, the time for a specific conversion \( X_B \) is: \[ t = t_{f,1}[1 - 3(1 - X_B)^{2/3} + 2(1 - X_B)] \] --- ### **Summary of Answers** 1. **Adiabatic CSTR energy balance:** \[ X(-\Delta H_{Rxn}) = C_{p} (1 + \epsilon X)(T - T_) \] Combine with Arrhenius law and plot heat generation/removal vs \( T \) to show multiple steady states. 2. **Catalyst decay:** \[ \frac{1}{a} = 1 + k_d t, \quad k_d \approx .10~\text{hr}^{-1} \] 3. **SCM conversion-time relation (diffusion control):** \[ t = t_{f,1}[1 - 3(1 - X_B)^{2/3} + 2(1 - X_B)] \]