Engineering & Science
Kirkbride Equation: Optimal Distillation Feed-Tray Location (NR / NS)
Locate the optimal feed stage in multicomponent Fenske-Underwood-Gilliland distillation design via C. G. Kirkbride’s rectifying-to-stripping stage ratio N_R / N_S = ((B / D) * (x_HK_F / x_LK_F) * (x_LK_B / x_HK_D)^2)^0.206.
Complete the Fenske-Underwood-Gilliland shortcut sequence with Kirkbride feed-stage placement!
Shortcut multicomponent distillation design relies on four classic equations: Fenske finds minimum stages N_min, Underwood finds minimum reflux R_min, Gilliland finds total theoretical stages N_total, and C. G. Kirkbride (Petroleum Refiner 1944) partitions N_total into Rectifying stages above the feed (N_R) and Stripping stages below the feed (N_S)! Misplacing the feed nozzle by just 4–6 trays wastes reboiler steam by forcing stripping trays to rectify or vice versa.
Kirkbride’s empirical correlation relates stage ratio N_R / N_S to molar bottoms-to-distillate flow ratio (B / D), feed Heavy-Key-to-Light-Key mole ratio (x_HK,F / x_LK,F), and the squared ratio of light-key impurity in bottoms to heavy-key impurity in distillate (x_LK,B / x_HK,D)^2 raised to the 0.206 power: N_R / N_S = ((B / D) * (x_HK_F / x_LK_F) * ((x_LK_B / x_HK_D)^2))^0.206! Across N_total = 28 theoretical trays (excluding reboiler) with B = 60 kmol/h, D = 40 kmol/h, x_HK,F = 0.55, x_LK,F = 0.45, x_LK,B = 0.015, and x_HK,D = 0.020, N_R / N_S = 1.0064—placing N_R = N_total * (ratio / (1 + ratio)) = 14.0 trays above the feed!
| Column Split Scenario (N_total = 28 Stages) | Kirkbride Ratio N_R / N_S | Rectifying Stages N_R | Stripping Stages N_S |
|---|---|---|---|
| Heavy-Rich Feed + Tighter Bottoms Spec | 1.0064 | 14.0 Stages | 14.0 Stages |
| Symmetric Equimolar Split (50/50, Equal Specs) | 1.0000 | 14.0 Stages | 14.0 Stages |
| Zero Flow or Impurity (<= 0 Input) | Singularity Guard | "Invalid Column Specs" | Requires Positive Flows & Specs |
Kirkbride Rectifying-to-Stripping Stage Ratio (N_R / N_S)
Evaluates ((B / D) * (x_HK,F / x_LK,F) * ((x_LK,B / x_HK,D)^2))^0.206.
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 <= 0, B8 <= 0), "Invalid Column Specs", ROUND(((B2 / B3) * (B4 / B5) * ((B6 / B7) ^ 2)) ^ 0.206, 4))Number of Rectifying Stages Above Feed Nozzle N_R
Partitions total column stages B8 via N_total * ((N_R / N_S) / (1 + (N_R / N_S))).
=IF(OR(B2 <= 0, B3 <= 0, B4 <= 0, B5 <= 0, B6 <= 0, B7 <= 0, B8 <= 0), "Invalid Column Specs", ROUND(B8 * ((((B2 / B3) * (B4 / B5) * ((B6 / B7) ^ 2)) ^ 0.206) / (1 + (((B2 / B3) * (B4 / B5) * ((B6 / B7) ^ 2)) ^ 0.206))), 1))Try the example
Use the original values or change them to test your understanding.