To determine the minimum number of theoretical trays needed in the distillation column, we can use the Fenske-Underwood-Gilliland FUG method. This method involves three main steps: finding the minimum reflux ratio Rm , finding the actual reflux ratio R , and calculating the number of theoretical trays N .1. Calculate the minimum reflux ratio Rm using the Underwood equation:Rm = - 1 / - xD/xW Where is the relative volatility, xD is the mole fraction of ethanol in the distillate, and xW is the mole fraction of ethanol in the waste.Assuming a relative volatility of 2.3 for ethanol and water, and the desired distillate composition of 90% ethanol xD = 0.9 , we can calculate xW using the feed composition of 40% ethanol xF = 0.4 :xF = xD * D / xD * D + xW * W Assuming a basis of 100 mol feed, D + W = 100, and D = xD * 100 / xD + xW xF = xD * xD * 100 / xD^2 * 100 + xW * xW * 100 0.4 = 0.9 * 0.9 * 100 / 0.9^2 * 100 + xW^2 * 100 Solving for xW, we get xW 0.1.Now, we can calculate the minimum reflux ratio Rm :Rm = 2.3 - 1 / 2.3 - 0.9/0.1 Rm 1.132. Calculate the actual reflux ratio R :Given that the reflux ratio is 1.5, R = 1.5.3. Calculate the number of theoretical trays N using the Gilliland correlation:N = [ R - Rm / R + 1 ]^0.5 * Nmin - 1 + 1Where Nmin is the minimum number of theoretical trays, which can be estimated using the Fenske equation:Nmin = log[ xD * 1 - xF / xF * 1 - xD ] / log Nmin = log[ 0.9 * 1 - 0.4 / 0.4 * 1 - 0.9 ] / log 2.3 Nmin 3.5 rounding up to 4 Now, we can calculate the number of theoretical trays N :N = [ 1.5 - 1.13 / 1.5 + 1 ]^0.5 * 4 - 1 + 1N 5.5 rounding up to 6 Therefore, the minimum number of theoretical trays needed in the distillation column to achieve the desired composition of 90% ethanol is 6.