Let's address each problem **step-by-step** as requested, with explanations and the final answers. --- ### 1. **Analytical Solution for the E-curve in a Laminar Flow Reactor** Given equation: \[ \frac{\partial C}{\partial \theta} = \frac{1}{Pe} \frac{\partial^2 C}{\partial x^2} - \frac{\partial C}{\partial x} \] We are to find the E-curve \(E(\theta)\) in terms of the vessel Peclet number \(Pe\). **Step 1:** Recognize that the E-curve is the exit age distribution function, related to the concentration profile at the outlet. **Step 2:** The boundary conditions for open-open (Danckwerts) are: At \(x = \) (inlet): \(Pe \frac{\partial C}{\partial x} + C = \) At \(x = 1\) (outlet): \(\frac{\partial C}{\partial x} = \) **Step 3:** The Laplace transform (with respect to θ) of the PDE gives an ODE for \(\bar{C}(x, s)\): \[ s\bar{C} - C_ = \frac{1}{Pe} \frac{d^2 \bar{C}}{dx^2} - \frac{d \bar{C}}{dx} \] **Step 4:** Solve for \(\bar{C}(x, s)\) subject to boundary conditions, and evaluate at \(x = 1\) for the outlet. **Step 5:** The Laplace transform solution for the E-curve is: \[ \bar{E}(s) = \frac{Pe \left[ (1+\beta) e^{\gamma} - (1-\beta) e^{-\gamma} \right] }{ \left[ (Pe+1+\beta) e^{\gamma} - (Pe+1-\beta)e^{-\gamma} \right] } \] where \[ \gamma = \frac{1}{2} \sqrt{Pe^2 + 4Pe s} \] \[ \beta = \frac{Pe}{2} \] **Step 6:** The time-domain solution (the E-curve) is: \[ E(\theta) = \frac{Pe}{2\pi} \int_{-\infty}^{\infty} \frac{e^{(i\omega)\theta}}{1 + (Pe/2)[1 - \sqrt{1 + 4i\omega/Pe}]} d\omega \] Alternatively, for the open-open boundary, the E-curve is commonly given by: \[ E(\theta) = \frac{Pe}{4\pi \theta^{3/2}} \exp\left[ -\frac{Pe(1-\theta)^2}{4\theta} \right], \quad \theta > \] **Final Answer:** The analytical solution for the E-curve in terms of Peclet number \(Pe\) is: \[ \boxed{ E(\theta) = \frac{Pe}{4\pi \theta^{3/2}} \exp\left[ -\frac{Pe(1-\theta)^2}{4\theta} \right] } \] --- ### 2. **Shrinking Core Model: Integrated Expression for Conversion \(X_B\)** The model states that when diffusion through the ash layer controls the rate: \[ t/\tau = 1 - 3(1-X_B)^{2/3} + 2(1-X_B) \] **Step 1:** Recognize this formula directly from the shrinking core model for diffusion control. **Step 2:** \(t\) is the time, \(\tau\) is the time required for complete conversion. **Final Answer:** \[ \boxed{ \frac{t}{\tau} = 1 - 3(1-X_B)^{2/3} + 2(1-X_B) } \] --- ### 3. **Selectivity Parameter \(S_{D,U}\) in Parallel Reactions** For a parallel reaction network \(A \rightarrow D\) (desired), \(A \rightarrow U\) (undesired): \[ S_{D,U} = \frac{k_D}{k_U} = \exp\left[\frac{E_U - E_D}{RT}\right] \] **Step 1:** Arrhenius law: \(k = k_ \exp(-E/RT)\) **Step 2:** Selectivity is the ratio of rate constants: \[ S_{D,U} = \frac{k_D}{k_U} = \frac{k_{D} \exp(-E_D/RT)}{k_{U} \exp(-E_U/RT)} \] Assuming \(k_{D} \approx k_{U}\), then: \[ S_{D,U} = \exp\left(\frac{E_U - E_D}{RT}\right) \] **Step 3:** If \(E_U > E_D\), selectivity decreases as temperature increases. **Step 4:** The optimum temperature for a PFR is where selectivity is maximized, generally at the lowest allowable temperature. **Final Answer:** \[ \boxed{ S_{D,U} = \frac{k_D}{k_U} = \exp\left(\frac{E_U - E_D}{RT}\right) } \] --- ### 4. **Autocatalytic Reactions and Optimal Recycle Ratio \(R_{opt}\)** For \(A + R \rightarrow R + R\) in a mixed flow reactor, rate is \(r \propto C_A C_R\). **Step 1:** For a differential CSTR at the maximum rate point, you want to maximize the rate with respect to recycle ratio for a fixed conversion. **Step 2:** The optimal recycle ratio for a fixed conversion is found by setting \(\frac{dr}{dR} = \). **Step 3:** Maximum rate occurs when the reactor behaves as a differential CSTR (very high recycle), i.e., the composition in the reactor is nearly constant and close to the feed. **Final Answer:** \[ \boxed{ The optimal recycle ratio for maximum rate in an autocatalytic reaction is infinite, meaning the reactor should operate as a differential CSTR at the maximum rate point. } \] --- **Summary Table:** | Problem | Final Answer | |---------|--------------| | 1. E-curve | \(E(\theta) = \frac{Pe}{4\pi \theta^{3/2}} \exp\left[ -\frac{Pe(1-\theta)^2}{4\theta} \right]\) | | 2. Shrinking Core Model | \(\frac{t}{\tau} = 1 - 3(1-X_B)^{2/3} + 2(1-X_B)\) | | 3. Selectivity | \(S_{D,U} = \exp\left(\frac{E_U - E_D}{RT}\right)\) | | 4. Autocatalytic Recycle | Optimal recycle ratio is infinite (differential CSTR) |