Let's break down solve each part-by-step as. --- ##1. Sketch the Evolution of \(_{A1 \) for Non-Ideal Flow Reactor (PFR with High Disp** Given: **First-order reaction** -Non-idealFR:** High dispersion large \( D_L \)axial dispersion) - **eclet number: \( Pe \frac{L}{D_L \), limit \( Pe \ is very small( D_L \ is very large**Gover Equation (with):** \[ \frac{\ C_A}{partial t} u \fracpartial C_A}{\partial x = D_L \{\partial^2 C_A}{\partial x^2} - kC \] **Steady-State (\( \frac C_A}{\partial t}= \)), Boundary Conditions:** - \( x = C_A = C_{A} \) - = L: \) zero gradient (\( \frac{\partial C_A}{\partial} = \)) **When \( D_L \) is very large ( \( Pe \)),** the reactor behaves like a **CSTR** (-mixed): \[ C_{A,\text{ = \frac{C_{A}}{1 + k\tau} \] where \( \tau \frac{V}{Q} \). **Sketch- For large dispersion, the concentration drops smoothly and is almost (similar to a CSTR), not the exponential drop seen in an ideal PFR. --- ## **. Reaction Conversion Efficiency** **Conversion, \( X \\[ X = 1 - \frac{C_{,\text{out}}}{C_{A\For a CSTR: \[ C_{A,\text{out}} = \frac{C_{A}}{1 + k\tau\implies X = 1 - \frac{1 + k\tau} = \frac{k\tau}{1 + k\tau\] **Result:** - As dispersion increases ( ), conversion approaches that of a CSTR. --- ## **3. Rate Law for Autatalytic Reaction** Given: \[ \frac{_A}{dt} = -kC_A(C_{A} + C_A - C_A*) \] where \( C_A^* \) constant. **Integration to Find Residence Time Distribution:** Given: \[ \int_{C_{A^{_{A}} \frac{dC_A}{kC_A(C_{A} + C_{A} - C_A)} -\int_{}^{t} dt \] This requires partial fraction decomposition. Let's simplify denominator: Let \(_A^* = C_{A} \) (only assumed for autocatalytic reactions; adjust if stated otherwise): \[ C_{A} + C_{A} - C_A = 2CA} - C_A \] So denominator is \( C_A (2C_{A} - C) \). Decompose: \[ \frac{1}{C_A (2C_{A} - C_A)} = \frac{A}{C_A} + \fracB}{2C_{A} - C_A} \] Solving for A and B: \[ 1 A(2C_{A} - C_A) + B C_A \] Let \( C_A = \Rightarrow 1 = 2C_{A}A \Rightarrow A = \frac{}{2C_{A}} \) Let \( C_A = 2C_{A} \Rightarrow 1 = B (2CA}) \Rightarrow B = \frac{1}{2C_{A}} \) So: \\int_{C_{A}}^{C_{A}} \left[ \frac{1}{2C_{}}\left( \frac{1}{C_A} + \frac{1}{2C_{A} - C_A} \right)\right] dC_A = -kt\] Integrate: \[ \frac{1}{2C_{A}} \left[ \ln(C) - \ln(2C_{A} - C_A) \right] \Big|_{C_{}}^{C_{A}} = -kt \] \[ \frac{1}{2C_{A}} \left( \ln \frac{C_A}{2C_{A} - C_A} - \ \frac{C_{A}}{2C_{A} - C_{A}} \right) -kt \] \[ \frac{1}{2C_{A}} \left( \ln \frac{C_A}{2C_{A} - C_A} - \ln \frac{C_{A}}{C_{A}} \right) = -kt \] \[ \frac{1}{2_{A}} \ln \frac{C_A}{2C_{A} - C_A} = - \] \[ \ln \frac{C_A}{2C_{A} - C_A} = -2k_{A} t \] \[ \frac{C_A}{2C_{A} - C} = \exp(-2kC_{A} t) \] \[ C_A = (2CA} - C_A) \exp(-2kC_{A} t) \] \[ C_A (1 + \exp(-kC_{A} t)) = 2C_{A} \exp(-2kC_{A} t) \] \[ C_A = \frac{2C_{A} \exp(-2k_{A} t)}{1 + \exp(-2kC_{A} t)} \] --- ## **4. Influence of Volumetric Expansion on Residence TimeGiven: - **Batch system, constant pressure \( P \), temperature \( T \)** - Initial mole fraction of species: \( y_ = .25 \) - Reaction: \( A \rightarrow 3B \- **Total time required for 90% conversion** ### **Step 1: Calculate the Expansion Factor (\( \ \))** For \( A \rightarrow 3B \): \[ \epsilon = \frac{\text{moles of product gas} - \text{moles of reactant gas}}{\textmoles of A reacted}} = \frac{3 - 1}{1} = 2 \] ###Step 2: Total Initial Moles** Let total initial moles = 1 mol Moles of inert = y_ = .25 \) So, moles of A = .75 mol ###Step 3: At 90% Conversion (\( X = .9 \))** Moles of A = \( .9 \times .75 = .675 \) Moles of A left = .75 - .675 = .075 \) Moles of B formed = \( \times .675 = 2.025 \) Total moles at 90% conversion: \[ n_{,90\%} = \text{A left} + \text{B formed} + \text{in} = .075 + 2.025 + .25 = 2.35 \] moles = .75 (A) + .25 (inert) = 1 mol **Expansion factor at 90% conversion:** \[ \text{Expansion factor = \frac{n_{total,90\%}}{n_{total,initial}} = \frac{2.35}{1} = 2.35 \] --- ## **5. Final Answer Summary** - **Spatial profile of \( C_{A1} \):** For high dispersion (low Pe), concentration profile is nearly flat (CSTR-like). - **Conversion efficiency:** \( X = \frac{k\tau}{1 + k\tau} \) - **Integrated rate expression (autocatalytic):** \[ C_A = \frac{2C_{A} \exp(-2kC_{A} t)}{1 \exp(-2kC_{A} t)} \] - **Volumetric expansion at 90% conversion:** **Expansion factor = 2.35** --- ## **Boxed Final Answers** ### **a Conversion for CSTR:** \[ X = \frac{k\tau}{1 + k\tau} \] ### **b) Autocatalytic Batch Solution:** \[ C_A = \frac{2_{A} \exp(-2kC_{A} t)}{1 + \exp(-2k_{A} t)} \] ### **c) Volumetric Expansion Factor for 90% Conversion:** \[ \boxed{2.35} \] Let me know if you need further details or!