Engineering & Science
McCabe-Thiele Distillation Operating Lines & Feed q-Line Intersection Formula
Calculate the McCabe-Thiele rectifying operating-line slope R / (R + 1), intercept xD / (R + 1), and exact singularity-free feed q-line intersection coordinates (x_int, y_int) valid for ALL thermal qualities q (including q = 1!).
Avoid division-by-zero at saturated liquid feed q = 1 with the exact (R + q) algebra!
Warren McCabe and Ernest Thiele (1925) revolutionized binary distillation by plotting constant-molar-overflow material balances on an x-y vapor-liquid equilibrium diagram! Above the feed tray, the Rectifying Operating Line obeys y = (R / (R + 1)) * x + (xD / (R + 1)), where R = L / D is reflux ratio and xD is distillate purity! Meanwhile, feed thermal condition q (fraction of feed entering as liquid: q = 1 boiling liquid, q = 0 saturated vapor, 0 < q < 1 two-phase flash) defines the Feed q-Line y = (q / (q - 1)) * x - (xF / (q - 1))!
Notice what happens if you code the textbook q-line slope q / (q - 1) directly into a spreadsheet at the most common industrial case q = 1.0 (saturated liquid feed): 1 / (1 - 1) explodes with #DIV/0!! Multiplying the intersection equation through by (R + 1) * (q - 1) cancels the singularity algebraically and yields the universal intersection coordinates: x_int = ((xF * (R + 1)) + (xD * (q - 1))) / (R + q) and y_int = ((R * x_int) + xD) / (R + 1)—working seamlessly at q = 1.0 (giving x_int = 0.4500, y_int = 0.6200 at R = 2.0, xD = 0.96, xF = 0.45) AND at 50% flashed feed q = 0.50 (x_int = 0.3480, y_int = 0.5520)!
| Feed Thermal State q (R = 2.0, xF = 0.45, xD = 0.96) | Rectifying Slope R/(R+1) | Intersection Liquid x_int | Intersection Vapor y_int |
|---|---|---|---|
| Saturated Boiling Liquid (q = 1.00) | 0.6667 | 0.4500 (Vertical at xF!) | 0.6200 |
| 50% Vaporized Flash Feed (q = 0.50) | 0.6667 | 0.3480 | 0.5520 |
| Saturated Dew-Point Vapor (q = 0.00) | 0.6667 | 0.1950 | 0.4500 (Horizontal at xF!) |
Singularity-Free Feed q-Line & Rectifying Line Intersection x_int
Evaluates ((xF * (R + 1)) + (xD * (q - 1))) / (R + q) without dividing by (q - 1).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= B3, B4 >= 1, (B2 + B5) <= 0), "Invalid Column Inputs", ROUND(((B3 * (B2 + 1)) + (B4 * (B5 - 1))) / (B2 + B5), 4))Corresponding Intersection Vapor Mole Fraction y_int on Rectifying Line
Evaluates y_int = ((R * x_int) + xD) / (R + 1) to anchor the bottom of the rectifying line and top of the stripping line.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= B3, B4 >= 1, (B2 + B5) <= 0), "Invalid Column Inputs", ROUND(((B2 * (((B3 * (B2 + 1)) + (B4 * (B5 - 1))) / (B2 + B5))) + B4) / (B2 + 1), 4))Try the example
Use the original values or change them to test your understanding.