Let's break down the problem step by step, as requested: --- # Problem Statement **Gas phase reaction:** \[ A \rightarrow B \] **Given data:** - Rate: \(-r_A = kC_A\) - \( k = k_ \exp\left(\frac{-E}{RT}\right) \) - \( k_ = 5 \times 10^8\, \text{h}^{-1} \) - \( E = 900\, \text{cal/mol} \) - \( \Delta H_R = -85,000\, \text{J/mol} \) - \( C_p = 32\, \text{J/(mol·K)} \) - \( F_{A} = 4\, \text{kmol}/\text{h} \) - \( T_ = 500\, K,\quad T_c = 480\, K \) - \( UA/V = 900\, \text{J}/(\text{h}·\text{m}^3·K) \) **Find:** 1. Condition for hotspot formation 2. Maximum allowable reactor length to avoid internal runaway 3. Sensitivity of exit temperature to inlet temperature --- ## 1. **Condition for Hotspot Formation** A hotspot occurs when heat generated by reaction exceeds the heat removed by cooling. **Heat generated:** \[ \text{Rate of heat generation per volume} = -\Delta H_R \cdot r_A \] **Heat removed:** \[ \text{Rate of heat removal per volume} = UA/V \cdot (T - T_c) \] **Condition for hotspot (runaway):** Set the temperature derivative of heat generated = temperature derivative of heat removed: \[ \frac{d}{dT}\left(-\Delta H_R \cdot k_ C_A \exp\left(\frac{-E}{RT}\right)\right) = UA/V \] Expand \( r_A = k_ C_A \exp\left(\frac{-E}{RT}\right) \): \[ \frac{d}{dT}\left(-\Delta H_R k_ C_A \exp\left(\frac{-E}{RT}\right)\right) = -\Delta H_R k_ C_A \frac{E}{RT^2} \exp\left(\frac{-E}{RT}\right) \] Set equal to heat removal rate: \[ -\Delta H_R k_ C_A \frac{E}{RT^2} \exp\left(\frac{-E}{RT}\right) = UA/V \] **This is the condition for hotspot formation.** --- ## 2. **Maximum Allowable Reactor Length** This is to avoid internal runaway. For a plug flow reactor (PFR) or a differential element, use the energy balance: \[ F_{A} C_p \frac{dT}{dz} = -\Delta H_R (-r_A) - UA/V (T - T_c) \] At the point of maximum temperature (hotspot), \(\frac{dT}{dz} = \): \[ -\Delta H_R (-r_A) = UA/V (T - T_c) \] Solve for the length up to which this holds before the runaway condition above is met. The maximum length corresponds to the critical point where the above hotspot condition is just satisfied. **Steps:** - Set up the energy and mole balance equations. - Integrate up to the point where the hotspot condition is met. - This length is the **maximum allowable reactor length**. **Note:** Actual calculation requires numerical solution, but the condition is: \[ L_{max}: \left[ -\Delta H_R k_ C_A \frac{E}{RT^2} \exp\left(\frac{-E}{RT}\right) = UA/V \right] \] Find \(T_{max}\) from this, then integrate the PFR equations up to that \(T_{max}\) to find \(L_{max}\). --- ## 3. **Sensitivity of Exit Temperature to Inlet Temperature** Sensitivity is given by: \[ \frac{dT_{exit}}{dT_} \] This can be found by differentiating the energy balance with respect to \(T_\), holding other variables constant. For a PFR, the energy balance is: \[ F_{A} C_p \frac{dT}{dz} = -\Delta H_R (-r_A) - UA/V (T - T_c) \] A small change in \(T_\) propagates through the reactor; the sensitivity can be high near the hotspot condition. **Procedure:** - Integrate the energy balance numerically for two slightly different \(T_\) values. - Calculate the difference in \(T_{exit}\) per unit change in \(T_\). --- # **Summary** 1. **Hotspot condition:** \[ -\Delta H_R k_ C_A \frac{E}{RT^2} \exp\left(\frac{-E}{RT}\right) = UA/V \] 2. **Maximum reactor length:** Integrate the energy and mole balances up to the temperature where the above condition is just met. 3. **Sensitivity:** \[ \frac{dT_{exit}}{dT_} \] Determine by integrating the reactor equations for two close values of \(T_\). --- Let me know if you want the numerical solution or further step-by-step integration for any part!