Sure! Let's break down your problem **step-by-step** and explain each part. This will help you understand distillation column design for a **ternary mixture** (three components: A, B, C). --- # 1. Problem Overview - **Feed:** 100 kmol/h of a liquid mixture (A, B, C) \( z_A = .45 \) - \( z_B = .35 \) - \( z_C = .20 \) - **Feed State:** Saturated liquid (quality \( q = 1 \)) - **A:** Most volatile component --- # 2. Given Data - **Antoine Equation** for vapor pressure: \[ \log_{10} P_{i,\text{sat}} = A_i - \frac{B_i}{T + C_i} \] - **Activity coefficients (\( \gamma \)) follow Wilson model:** \[ \ln \gamma_i = \ln \left( \sum_j x_j \Lambda_{ij} \right) + 1 - \sum_j \frac{x_j \Lambda_{ji}}{\sum_k x_k \Lambda_{jk}} \] - **Total Pressure:** 1 atm - **Distillate Composition:** \( x_{D,A} = .92 \) - **Reflux ratio:** 1.5 × minimum reflux --- # 3. Tasks Breakdown ## 1. Compute VLE at the Feed Tray - **VLE** (Vapor-Liquid Equilibrium) means finding the composition of vapor and liquid in equilibrium at the feed tray. - **Steps:** 1. **Calculate \( P_{i,\text{sat}} \)** for each component using the Antoine equation at the tray temperature (guess or use bubble point calculation). 2. **Compute activity coefficients (\( \gamma_i \))** using the Wilson model and feed composition. 3. **Raoult’s Law (modified for non-ideal):** \[ y_i P = x_i \gamma_i P_{i,\text{sat}} \] Here, \( x_i \) is feed composition, \( y_i \) is vapor composition, \( P \) is total pressure. ## 2. Estimate Minimum Number of Stages (Fenske Equation) - **Fenske equation** is for minimum stages at total reflux (no product withdrawn): \[ N_{\text{min}} = \frac{\log \left( \frac{X_{D,A}/X_{D,B}}{X_{B,A}/X_{B,B}} \right)} {\log \alpha_{\text{avg}}} \] - \( X_{D,A} \): Mole fraction of A in distillate. - \( X_{B,A} \): Mole fraction of A in bottoms. - \( \alpha_{\text{avg}} \): Average relative volatility (can be estimated from VLE). ## 3. Calculate Minimum Reflux Ratio (Underwood Equations) - **Underwood equations** are used to find minimum reflux: 1. **Solve for Underwood root (\( \theta \)):** \[ \sum_i \frac{q z_i}{\alpha_i - \theta} = 1 \] where \( \alpha_i \) is relative volatility. 2. **Calculate minimum reflux:** \[ \frac{R_{\text{min}}}{R_{\text{min}}+1} = \sum_i \frac{X_{D,i} \alpha_i}{\alpha_i - \theta} \] ## 4. Estimate Actual Stages (Gilliland Correlation) - **Gilliland correlation** relates actual stages (\( N \)) to minimum stages and actual reflux: - Use graphical/empirical correlation (or Eduljee equation): \[ Y = \frac{N - N_{\text{min}}}{N + 1} \] \[ X = \frac{R - R_{\text{min}}}{R + 1} \] - Find \( Y \) for your \( X \), then solve for \( N \). --- # 4. Summary Table | Step | Method | Purpose/Equation | |------|----------------------------|------------------------------------------------------------| | 1 | VLE at Feed Tray | Antoine, Wilson, Modified Raoult's Law | | 2 | Fenske Equation | \( N_{\text{min}} \) at total reflux | | 3 | Underwood Equations | Minimum reflux ratio calculation | | 4 | Gilliland Correlation | Find actual stages using calculated \( N_{\text{min}} \) | --- ## **If you want to see a worked example for any step, let me know!**