Let's address each part **step by step**, as requested: --- ## **1. Non-Isothermal Batch Reactor: Rate of Disappearance** Given: \[ - r_A = -\frac{dN_A}{dV} = kC_{A}(1-X)\exp\left(\frac{E}{R}\left(\frac{1}{T_}-\frac{1}{T}\right)\right) \] This equation expresses the rate of disappearance of reactant **A** in a non-isothermal batch reactor. - \( N_A \): moles of A - \( V \): reaction volume - \( k \): rate constant at reference temperature \( T_ \) - \( C_{A} \): initial concentration of A - \( X \): conversion - \( E \): activation energy - \( R \): gas constant - \( T \): temperature (which varies with time) **Explanation:** This equation accounts for the temperature dependence of the rate via the exponential Arrhenius term. --- ## **2. Design Equation for a CSTR** **Steady-state mole balance:** \[ V = \frac{F_{A} - F_A}{-r_A} \] where - \( F_{A} \): inlet molar flow rate of A - \( F_A \): outlet molar flow rate of A - \( -r_A \): rate of disappearance of A For a first-order reaction (\( -r_A = k C_A \)), and \( C_A = C_{A}(1-X) \): \[ V = \frac{F_{A}X}{k C_{A}(1-X)} \] --- ## **3. Calculate the CSTR Volume** **Given:** - \( F_{A} = 15.5 \) mol/s - \( X = .85 \) - \( k = .25 \) s\(^{-1}\) - \( C_{A} = 2. \) mol/L **Plug into the equation:** \[ V = \frac{F_{A}X}{k C_{A}(1-X)} \] **Substitute values:** \[ V = \frac{15.5 \times .85}{.25 \times 2. \times (1-.85)} \] Calculate denominator: \[ .25 \times 2. \times (1-.85) = .25 \times 2. \times .15 = .5 \times .15 = .075 \] Calculate numerator: \[ 15.5 \times .85 = 13.175 \] Therefore, \[ V = \frac{13.175}{.075} = 175.67 \text{ L} \] **Final answer:** \[ \boxed{V = 175.7 \text{ L}} \] --- ## **4. Plug Flow Reactor (PFR) Rate Profile** **PFR Differential Design Equation:** \[ \frac{dF_A}{dV} = -r_A = -kC_{A}(1-X) \left( \frac{P}{P_} \right)^\alpha \] Where: - \( \left( \frac{P}{P_} \right) \): accounts for pressure drop (with \( \alpha \) as reaction order with respect to pressure) - For gas-phase decompositions, this term adjusts for concentration changes due to pressure drop along the reactor. For **non-isothermal** or **non-ideal** reactors, use the Ergun equation to relate pressure drop to position in the reactor. --- ## **5. Levenspiel Relationship for Two Mixed Flow Reactors in Series** **Total Volume for Two Reactors in Series:** \[ V_{\text{total}} = V_1 + V_2 = \frac{F_{A}}{-r_{A1}}(X_1 - X_) + \frac{F_{A}}{-r_{A2}}(X_2 - X_1) \] Where: - \( X_ = \) (initial conversion) - \( X_2 \): final conversion - \( X_1 \): intermediate conversion that minimizes total volume **Given:** - \( X_2 = .8 \) - \( -r_{A1} = 1. \) mol/L/s - \( -r_{A2} = .5 \) mol/L/s - \( F_{A} = 2.5 \) L/s \(\times\) \( 1. \) mol/L \( = 2.5 \) mol/s \[ V_{\text{total}} = \frac{2.5}{1.} (X_1) + \frac{2.5}{.5} (.8 - X_1) = 2.5X_1 + 5(.8 - X_1) \] \[ = 2.5X_1 + 4. - 5X_1 = 4. - 2.5X_1 \] **To minimize \( V_{\text{total}} \), maximize \( X_1 \) (since the coefficient is negative):** - But \( X_1 \leq X_2 = .8 \) **So, minimum volume at \( X_1 = .8 \):** \[ V_{\text{total}} = 4. - 2.5 \times .8 = 4. - 2. = 2. \text{ L} \] **Intermediate conversion \( X_1 \) that minimizes the volume:** - The minimum is at the maximum allowed value, \( X_1 = .8 \). ### **Final Answers** #### **CSTR Volume:** \[ \boxed{V = 175.7 \text{ L}} \] #### **Minimum Total Volume for Two CSTRs:** \[ \boxed{V_{\text{total}} = 2. \text{ L at } X_1 = .8} \] --- Let me know if you'd like a specific sketch or further clarifications!