Let's break down and solve the problem step by step as per the tasks listed. ### Problem Summary You have a quaternary mixture (A/B/C/D). Feed: 20 kmol/hr Feed mole fractions: - zA = .3 - zB = .25 - zC = .25 - zD = .2 **Chemical association in liquid phase:** - A + B ⇌ AB (Kassoc) **Relative volatilities:** - αA = 5 - αB = 3 - αC = 1.8 - αD = 1 **Activity coefficient:** - ln γi = 2.5 (1 - xi)² **Targets:** - Distillate: 99% A - Bottoms: 98% D **Tasks:** 1. Modify phase equilibrium relations 2. Effective composition due to association 3. Fenske-Underwood-Gilliland with association 4. Actual tray count (Eff = 65%) 5. Energy duties 6. Sensitivity to Kassoc --- ### 1. Modify Phase Equilibrium Relations Because of A-B association, free A and free B are reduced in the liquid phase. The effective mole fraction for each component changes. The phase equilibrium for each component is given by: \( y_i = K_i x_i \gamma_i \) where \( y_i \) is the vapor phase mole fraction, \( x_i \) the liquid phase mole fraction, \( \gamma_i \) the activity coefficient, and \( K_i \) the distribution coefficient (which includes relative volatility). For associating components (A and B), only the free form is in equilibrium with the vapor. If \( x_{A,free} \) is the mole fraction of free A, and similarly for B: \( y_A = K_A x_{A,free} \gamma_A \) \( y_B = K_B x_{B,free} \gamma_B \) --- ### 2. Effective Composition Due to Association Let \( x_{AB} \) = mole fraction of associated AB in the liquid. The association equilibrium is: A + B ⇌ AB Let \( x_{A,free} \) and \( x_{B,free} \) be free A and B. Total A: \( x_A = x_{A,free} + x_{AB} \) Total B: \( x_B = x_{B,free} + x_{AB} \) The equilibrium constant Kassoc: \( K_{assoc} = \frac{x_{AB}}{x_{A,free} x_{B,free}} \) You have three equations for A, B, and AB: 1. \( x_A = x_{A,free} + x_{AB} \) 2. \( x_B = x_{B,free} + x_{AB} \) 3. \( x_{AB} = K_{assoc} x_{A,free} x_{B,free} \) Solve these for \( x_{A,free} \), \( x_{B,free} \), and \( x_{AB} \), given the total \( x_A \) and \( x_B \) and a value for Kassoc. --- ### 3. Fenske-Underwood-Gilliland with Association **Fenske equation** (for minimum stages, Nmin): \( N_{min} = \frac{\ln \left( \frac{X_{D,A}/X_{B,A}}{X_{D,D}/X_{B,D}} \right)}{\ln \alpha_{avg}} \) Where \( \alpha_{avg} \) is the average relative volatility between key components (A and D here). **Underwood equation** (for minimum reflux): Because association changes the effective composition, use \( x_{A,free} \) and \( x_{B,free} \) in the Underwood equations. **Gilliland correlation** (for actual stages): Given overall efficiency (65%), the actual number of stages: \( N_{actual} = \frac{N_{theoretical}}{.65} \) --- ### 4. Actual Tray Count From Gilliland's method, estimate the tray count as above. --- ### 5. Energy Duties Calculate condenser and reboiler duties using mass and energy balances. Use the total vapor and liquid flow rates, compositions, and enthalpy data (if available). --- ### 6. Sensitivity to Kassoc Vary Kassoc and repeat the calculations for modified phase equilibrium, effective composition, tray count, and energy to assess the impact. --- ### Final Answer (Condensed Summary) To solve this, you must: First, for given feed compositions, write the association equilibrium equations for A, B, and AB in the liquid phase and solve for free and associated mole fractions using Kassoc. Next, use the modified compositions to set up the phase equilibrium relations, incorporating the activity coefficients using the provided formula. Input the effective free mole fractions into the Fenske, Underwood, and Gilliland equations to calculate the minimum and actual number of trays, remembering to adjust for the 65% tray efficiency. Finally, compute energy duties from the determined flow rates and compositions, and repeat the steps for different Kassoc values to determine sensitivity. This sequence addresses each task in order, using chemical equilibrium, thermodynamics, and distillation design equations.