Let's solve the problem step by step. --- ## **Step 1: Basis and Stream Compositions** **Basis:** 100 mol/s feed - Feed: 80% pentane (\( C_5H_{12} \)), 20% hexane (\( C_6H_{14} \)) - Feed at 30°C \[ \text{Feed pentane: } 100 \times .8 = 80 \text{ mol/s} \] \[ \text{Feed hexane: } 100 \times .2 = 20 \text{ mol/s} \] --- ## **Step 2: Reboiler Operation** At 65°C, the feed is partially vaporized so that the two streams (liquid and vapor) are in equilibrium: - **Bottom liquid**: 65% pentane, taken as product - **Let the amount of liquid product = \( L \) mol/s** - **Let the amount of vapor leaving = \( V \) mol/s** Material balances for pentane and hexane: ### **Total balance:** \[ L + V = 100 \] ### **Pentane balance:** \[ .65L + y_P V = 80 \] Where \( y_P \) is the mole fraction of pentane in the vapor. ### **Hexane balance:** \[ .35L + y_H V = 20 \] Where \( y_H \) is the mole fraction of hexane in the vapor. But, since vapor and liquid must sum to 1 in each stream: \[ y_P + y_H = 1 \] **We need equilibrium data (K-values)** for pentane and hexane at 65°C. Let's estimate using Raoult's Law: \[ y_i = K_i x_i \] Where \( K_i = \frac{P_i^*}{P_{tot}} \). Let’s assume atmospheric pressure (\( P_{tot} = 760 \) mmHg). For pentane at 65°C: \( P^*_{C_5H_{12}} = 2327 \) mmHg For hexane at 65°C: interpolate or use the given values (at 70°C: 674.758 mmHg, at 65°C slightly lower; let's estimate \( \approx 600 \) mmHg). \[ K_{C_5H_{12}} = \frac{2327}{760} = 3.06 \] \[ K_{C_6H_{14}} = \frac{600}{760} = .79 \] Let \( x_{P} = .65 \) and \( x_{H} = .35 \): \[ y_P = K_P x_P = 3.06 \times .65 = 1.99 \quad (\text{but can't exceed 1, so normalize}) \] \[ y_H = K_H x_H = .79 \times .35 = .277 \] Sum: \( 1.99 + .277 = 2.267 \) Normalize: \[ y_P = \frac{1.99}{2.267} = .878 \] \[ y_H = \frac{.277}{2.267} = .122 \] Now, use balances: \[ \text{Pentane:} \quad .65L + .878V = 80 \] \[ \text{Hexane:} \quad .35L + .122V = 20 \] \[ L + V = 100 \] Solve for \( L \) and \( V \): From hexane equation: \[ .35L + .122V = 20 \] \[ .35L + .122(100 - L) = 20 \] \[ .35L + 12.2 - .122L = 20 \] \[ .228L = 20 - 12.2 = 7.8 \] \[ L = \frac{7.8}{.228} \approx 34.21 \] \[ V = 100 - 34.21 = 65.79 \] --- ## **Step 3: Heat Required to Vaporize Pentane and Hexane in Reboiler** The vapor stream has \( V \) mol/s, composition: \( y_P = .878 \), \( y_H = .122 \). \[ \text{Moles of pentane vaporized: } V \times y_P = 65.79 \times .878 = 57.8 \text{ mol/s} \] \[ \text{Moles of hexane vaporized: } V \times y_H = 65.79 \times .122 = 8.03 \text{ mol/s} \] ### **Enthalpy Calculations** **Heat required, \( Q_{reb} \):** Each component heated from 30°C to 65°C, then vaporized at 65°C. \[ Q_{reb} = n_{P,vap} \left[ C_{p,\ell,P} (65-30) + \Delta H_{vap,P} \right] + n_{H,vap} \left[ C_{p,\ell,H} (65-30) + \Delta H_{vap,H} \right] \] Given: - \( C_{p,\ell,P} = 155.4 \) J/mol·°C - \( \Delta H_{vap,P} = 25,770 \) J/mol - \( C_{p,\ell,H} = 216.3 \) J/mol·°C - \( \Delta H_{vap,H} = 28,850 \) J/mol Calculate: \[ Q_{reb,P} = 57.8 \left[ 155.4 \times 35 + 25,770 \right] \] \[ = 57.8 \left[ 5,439 + 25,770 \right] \] \[ = 57.8 \times 31,209 = 1,804,780 \text{ J/s} \] \[ Q_{reb,H} = 8.03 \left[ 216.3 \times 35 + 28,850 \right] \] \[ = 8.03 \left[ 7,571 + 28,850 \right] \] \[ = 8.03 \times 36,421 = 292,350 \text{ J/s} \] \[ \boxed{ Q_{reb,total} = 1,804,780 + 292,350 = 2,097,130 \text{ J/s} = 2,097 \text{ kJ/s} } \] --- ## **Step 4: Heat Released by the Cooler** The product gas (after separation, at 700°C) is cooled to 100°C. - Find the moles and composition of product gas (from reactor and separator). - Calculate enthalpy drop for each component from 700°C to 100°C. **[This depends on combustion and separation details, which are not fully given here. If you want, I can proceed with an assumption, or you can provide more info.]** --- ## **Step 5: Dew Point Temperature of Product Gas** This requires the composition of the product gas and vapor pressure data. This will also depend on the result of the reactor/separator step. --- ### **Summary of Answers (so far)** #### **a. Heat Required to Vaporize in Reboiler** \[ \boxed{2,097 \text{ kJ/s}} \] #### **b. Heat Released by Cooler** Requires additional combustion and flow data from the reactor/separator step. #### **c. Dew Point Temperature** Requires product gas composition. --- **If you'd like to proceed with (b) and (c), please provide any missing details or let me know if you'd like to make assumptions!**