Let's break down and solve the problem step by step, as requested in the image. --- ## **Step 1: Problem Summary** We have a quaternary mixture (A/B/C/D) with a feed of 20 kmol/hr. Feed composition: zA = .3 zB = .25 zC = .25 zD = .2 There is chemical association in the liquid phase: A + B ⇌ AB (Kanou) Relative volatilities: αA = 5, αB = 3, αC = 1.8, αD = 1 Activity coefficient: lnγi = 2.5(1 - xi)² Targets: Distillate: 99% A Bottoms: 98% D Tasks include: 1. Modify phase equilibrium relations 2. Effective composition due to association 3. Fenske-Underwood-Gilliland with association 4. Actual tray count (Efficiency = 65%) 5. Energy duties 6. Sensitivity to K-values --- ## **Step 2: Modify Phase Equilibrium Relations** For non-ideal mixtures, the phase equilibrium relation is: \[ y_i = K_i x_i \] Where \[ K_i = \frac{\alpha_i P^* \gamma_i}{P} \] But with association (A + B ⇌ AB), mole fractions xA, xB must be corrected for association. --- ## **Step 3: Effective Composition due to Association** If *f* is the fraction of A and B that forms AB, Let xA', xB' = free A and B (not associated), xAB = associated product. Material balances: xA, = xA' + xAB xB, = xB' + xAB xAB = f (assuming 1:1 stoichiometry and same extent for A and B due to the reaction) So, xA' = xA, - xAB xB' = xB, - xAB xAB = min(xA,, xB,) × f The effective (free) composition to be used in equilibrium calculations is xA', xB', xC, xD. --- ## **Step 4: Activity Coefficient Calculation** Given: lnγi = 2.5(1 - xi)² For each component, calculate γi using the current xi (effective composition after association): γi = exp[2.5(1 - xi)²] --- ## **Step 5: Fenske-Underwood-Gilliland Calculation** ### Fenske Equation (Minimum Trays at Total Reflux): \[ N_{min} = \frac{\ln \left( \frac{X_D/(1-X_D)}{X_B/(1-X_B)} \right)}{\ln \alpha_{avg}} \] Where - XD = mole fraction of A in distillate = .99 - XB = mole fraction of A in bottoms = (1 - .98) = .02 (assuming D is bottom product) - α_avg = geometric mean of volatilities (A/D separation) \[ N_{min} = \frac{\ln \left( \frac{.99/.01}{.02/.98} \right)}{\ln 5} \] \[ N_{min} = \frac{\ln \left( 99 / .0204 \right)}{\ln 5} \] \[ N_{min} = \frac{\ln (4852)}{1.609} \] \[ N_{min} = \frac{8.487}{1.609} \approx 5.28 \] --- ### Underwood Equation (Minimum Reflux Ratio): For a quaternary mixture with non-idealities and association, Underwood's method must be modified by using effective (free) compositions and activity coefficients in the K-values. --- ## **Step 6: Actual Tray Count** Gilliland correlation relates actual trays to minimum trays and minimum/actual reflux. Given efficiency = 65%, Actual trays = Theoretical trays / .65 --- ## **Step 7: Energy Duties** Calculate condenser and reboiler duties using enthalpy balances, feed and product compositions, and reflux ratio (obtained from Underwood and Gilliland steps). --- ## **Step 8: Sensitivity to K-values** Perform a sensitivity analysis by varying K-values (affected by activity coefficients and association) and observing the impact on tray count and energy duties. --- ## **Final Answers (Concise Summary)** 1. Modify phase equilibria using effective (free) compositions after accounting for A+B⇌AB association. 2. Calculate effective mole fractions: xA' = xA, - xAB; xB' = xB, - xAB; xAB = f × min(xA,, xB,) 3. Calculate activity coefficients with lnγi = 2.5(1 - xi)² for each component. 4. Apply Fenske equation for minimum stages using corrected α values and target purities. For A/D separation: Minimum stages ≈ 5.3 5. Use Underwood and Gilliland methods (with association-corrected values) for minimum reflux and actual trays. 6. Actual trays = theoretical trays / .65 7. Calculate energy duties from mass and energy balances. 8. Analyze sensitivity by varying K-values and noting the effect on tray count and duties. If you need a worked-out example with numbers for each step (for a specific tray calculation, energy duty, etc.), please specify which step you'd like to see in detail.