Constrained Optimization MCP Server

Optimize a distillation column design to minimize total annual cost while meeting product purity specifications. Use HiGHS solver. **Feed:** * 100 kmol/h Component A (light, volatile) * 60 kmol/h Component B (heavy, less volatile) * 80°C, 1.5 bar The design variables to optimize are: 1. **Number_of_Trays** (N): Integer between 10-50 trays 2. **Reflux_Ratio** (RR): Real number between 0.5-5.0 3. **Feed_Tray_Location** (NF): Integer between 5 and min(45, N-5) (tray number from bottom) 4. **Column_Diameter** (D): Real number between 0.8-3.0 meters The constraints are: 1. Product Purity Requirements - Distillate purity of Component A: x_A_dist ≥ 0.95 (at least 95% pure) - Bottoms purity of Component B: x_B_bot ≥ 0.90 (at least 90% pure) 2. Operating Constraints - Maximum pressure drop across column: ΔP ≤ 5.0 kPa - Minimum tray efficiency: η ≥ 0.60 - Maximum vapor velocity (flooding constraint): v_vapor ≤ 2.5 m/s - Feed tray location must be between 5 and min(45, N-5) (not too close to ends) 3. Mass Balance Constraints - Total material balance: F = D + B (Feed = Distillate + Bottoms) - Component A balance: F × z_A = D × x_A_dist + B × x_A_bot - Component B balance: F × z_B = D × x_B_dist + B × x_B_bot - Feed composition: z_A = 100/160 = 0.625, z_B = 60/160 = 0.375 - Reflux relationship: L = RR × D (where L is liquid reflux) 4. Physical Constraints - Minimum reflux ratio constraint: RR ≥ RR_min (typically 1.2 × minimum theoretical) - Column diameter sizing based on vapor flow: D ≥ sqrt(4×V_vapor/(π×v_max×ρ_vapor)) - Tray spacing and column height: H = N × tray_spacing (assume 0.6 m per tray) - Vapor density approximation: ρ_vapor ≈ 2.0 kg/m³ (at operating conditions) 5. Utility Constraints - Maximum steam flow for reboiler: Steam ≤ 500 kg/h - Maximum cooling water flow for condenser: CW ≤ 10,000 kg/h The objective function is to minimize total annual cost: Total_Cost = Capital_Cost + Operating_Cost Where: - **Capital_Cost** = Column_Cost + Tray_Cost - Column_Cost = 5000 × H ($/m of height) - Tray_Cost = 1500 × N ($/tray) - **Operating_Cost** = Steam_Cost + Cooling_Water_Cost - Steam_Cost = 0.02 × Steam_Flow × 8760 ($/year) - Cooling_Water_Cost = 0.001 × CW_Flow × 8760 ($/year) For this binary distillation optimization, use these approximations: 1. **Minimum trays** (Fenske equation): N_min = log((x_A_dist/(1-x_A_dist)) × ((1-x_A_bot)/x_A_bot)) / log(α_AB) - Assume relative volatility α_AB = 2.5 (Component A relative to Component B) 2. **Minimum reflux ratio** (Underwood method): RR_min = (x_A_dist - x_A_feed)/(x_A_feed - x_A_bot) - Where x_A_feed = z_A = 0.625 (liquid composition of A in feed) 3. **Vapor flow rate**: V = D × (RR + 1) (kmol/h) 4. **Steam requirement**: Steam ≈ 2.0 × B × MW_avg (kg/h, where MW_avg ≈ 50 kg/kmol) 5. **Cooling water requirement**: CW ≈ 12 × D × MW_avg (kg/h) 6. **Component recovery constraints**: - Recovery of A in distillate: R_A = (D × x_A_dist)/(F × z_A) ≥ 0.90 - Recovery of B in bottoms: R_B = (B × x_B_bot)/(F × z_B) ≥ 0.85 Solve this optimization problem and provide: 1. Optimal values for all design variables 2. Total annual cost breakdown 3. Key performance metrics (recovery rates, energy consumption) As markdown tables.