Let's break down your problem and address each task **step-by-step**. This is a **ternary distillation** problem with components A (most volatile), B, and C. ### **Given Data** - Feed: 100 kmol/h, saturated liquid (q = 1) - Feed mole fractions: \( z_A = .45, \; z_B = .35, \; z_C = .20 \) - Total pressure: 1 atm - Distillate composition: \( x_{D,A} = .92 \) - Reflux ratio: 1.5 × minimum reflux - Antoine equation for each component: \[ \log_{10} P_i^{sat} = A_i - \frac{B_i}{T+C_i} \] - Activity coefficients: **Wilson model** \[ \ln \gamma_i = \ln\left(\sum_j x_j \Lambda_{ji}\right) + 1 - \sum_j \frac{x_j \Lambda_{ij}}{\sum_k x_k \Lambda_{kj}} \] - Tasks: 1. Compute VLE at the feed tray 2. Estimate minimum number of stages (Fenske) 3. Calculate minimum reflux ratio (Underwood) 4. Estimate actual stages (Gilliland) --- ## **1. Compute VLE at the Feed Tray** **Step 1: Calculate K-values (using Wilson model & Antoine equation)** **a. Calculate \( P_i^{sat} \) for each component using the Antoine equation.** You need \( A_i, B_i, C_i \) values for each component and the tray temperature. Assume temperature or iterate to find it (not provided, so let's denote as T). **b. Calculate Wilson activity coefficients, \( \gamma_i \), using feed composition.** - Use given Wilson model equation and feed mole fractions. **c. Calculate K-values:** \[ K_i = \frac{\gamma_i P_i^{sat}}{P} \] where \( P = 1 \) atm. --- ## **2. Estimate Minimum Number of Stages (Fenske Equation)** \[ N_{min} = \frac{\log \left[ \frac{(x_{D,A}/x_{D,B})}{(x_{B,A}/x_{B,B})} \right]}{\log \alpha_{AB,avg}} \] - \( x_{D,A} = .92 \), \( x_{D,B} = 1 - x_{D,A} - x_{D,C} \) (assume \( x_{D,C} \) small/negligible for approximation) - \( x_{B,A}, x_{B,B} \) from bottoms (need to estimate or assume a value for significant separation, say \( x_{B,A} \approx .01 \)) - \( \alpha_{AB,avg} \) is the average relative volatility at the top and bottom. --- ## **3. Calculate Minimum Reflux Ratio (Underwood Equations)** **Underwood equation:** \[ \sum_{i} \frac{q z_i}{\alpha_i - \theta} = 1 \] where \( \alpha_i = K_i/K_{ref} \), \( q \) is feed condition, and \( \theta \) is solved iteratively. Once \( \theta \) is found, \[ R_{min} = \sum_{i} \frac{x_{D,i} (\alpha_i)}{\alpha_i - \theta} - 1 \] --- ## **4. Estimate Actual Stages (Gilliland Correlation)** Given: - \( N_{min} \) (from Fenske) - \( R_{min} \) (from Underwood) - Actual \( R = 1.5 R_{min} \) **Gilliland correlation** (graphical or empirical correlation): \[ Y = \frac{N - N_{min}}{N + 1} \] \[ X = \frac{R - R_{min}}{R + 1} \] - Use empirical charts or Eduljee's equation to relate X and Y, then solve for actual stages \( N \). --- # **Summary and Final Answer Format** Since the precise values for Antoine constants and Wilson parameters are not given, **here is the structured solution template**: --- ## **Step 1: VLE at Feed Tray** 1. **Calculate \( P_i^{sat} \) for A, B, C at assumed/estimated T using Antoine equation.** 2. **Compute Wilson activity coefficients \( \gamma_i \) using feed compositions.** 3. **Calculate K-values:** \[ K_i = \frac{\gamma_i P_i^{sat}}{P} \] --- ## **Step 2: Minimum Number of Stages** \[ N_{min} = \frac{\log \left[ \frac{(x_{D,A}/x_{D,B})}{(x_{B,A}/x_{B,B})} \right]}{\log \alpha_{AB,avg}} \] --- ## **Step 3: Minimum Reflux Ratio** 1. **Solve Underwood equation for \( \theta \).** 2. **Calculate \( R_{min} \):** \[ R_{min} = \sum_{i} \frac{x_{D,i} (\alpha_i)}{\alpha_i - \theta} - 1 \] --- ## **Step 4: Actual Number of Stages** 1. **Calculate \( X = \frac{R - R_{min}}{R + 1} \).** 2. **Use Gilliland correlation to find \( Y \).** 3. **Solve for actual stages \( N \):** \[ Y = \frac{N - N_{min}}{N + 1} \] --- ### **NOTE** If you provide the Antoine constants, Wilson parameters, or a temperature, I can plug in the numbers for a complete numerical solution. --- **Would you like to proceed with assumed values for missing data, or do you have the required constants?**