Chemical, Process & Catalysis Engineering
Differential Batch Distillation Formula: Remaining Still Pot Moles W & Composite Purity x_D
Calculate remaining kettle liquid W (mol), boiled-off overhead distillate D = F - W, and pooled composite distillate purity x_D,avg in single-stage constant-alpha batch distillation via Lord Rayleigh’s (1902) integrated VLE material balance.
Track progressive pot depletion and pooled overhead purity during batch solvent recovery!
In pharmaceutical, essential-oil, and specialty-chemical plants where small multi-product campaigns use batch kettle stills without reflux, boiling vapor y = alpha * x / (1 + (alpha - 1) * x) continuously strips the more volatile component out of the pot, causing liquid mole fraction in the still to drop from initial x_F down to final bottom target x_W! Integrating dW / W = dx / (y - x) at constant relative volatility alpha > 1 gives Lord Rayleigh’s (1902, Phil. Mag.) exact analytical relation:
LN(F / W) = (1 / (alpha - 1)) * LN((x_F * (1 - x_W)) / (x_W * (1 - x_F))) + LN((1 - x_W) / (1 - x_F))! Exponentiating yields Remaining Kettle Moles W = F * EXP(-LN_F_over_W), Boiled-Off Overhead Moles D = F - W, and Overall Component Material Balance gives Pooled Composite Distillate Mole Fraction x_D,avg = (F * x_F - W * x_W) / (F - W)!
| Batch Still Stage (Charge F = 100 kmol, x_F = 0.50, alpha = 2.50) | Formula | Evaluated Value |
|---|---|---|
| Target Final Still-Pot Composition x_W | Specified Stripping Endpoint | x_W = 0.2000 |
| Logarithmic Boil-Down Ratio LN(F / W) | Rayleigh Constant-alpha Integral | 1.3942 -> W = 24.80 kmol |
| Collected Overhead Distillate D = F - W | 100 - 24.80 | 75.20 kmol |
| Pooled Receiver Purity x_D,avg | (F * x_F - W * x_W) / (F - W) | 0.5990 (59.90 mol%) |
Remaining Kettle Still Moles W After Stripping x_F Down to x_W
Evaluates F * EXP(-((1 / (alpha - 1)) * LN((x_F * (1 - x_W)) / (x_W * (1 - x_F))) + LN((1 - x_W) / (1 - x_F)))) across initial charge F (B2 kmol), initial mole fraction x_F (B3), final pot mole fraction x_W (B4 < x_F), and relative volatility alpha > 1 (B5).
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B4 <= 0, B4 >= B3, B5 <= 1), "Requires F > 0, 0 < x_W < x_F < 1, and alpha > 1.0", ROUND(B2 * EXP(-((1 / (B5 - 1)) * LN((B3 * (1 - B4)) / (B4 * (1 - B3))) + LN((1 - B4) / (1 - B3)))), 2))Pooled Composite Overhead Distillate Mole Fraction x_D,avg
Applies exact component conservation (F * x_F - W * x_W) / (F - W) to determine accumulated receiver purity.
=IF(OR(B2 <= 0, B3 <= 0, B3 >= 1, B4 <= 0, B4 >= B3, B5 <= 1), "Requires F > 0, 0 < x_W < x_F < 1, and alpha > 1.0", ROUND((B2 * B3 - (B2 * EXP(-((1 / (B5 - 1)) * LN((B3 * (1 - B4)) / (B4 * (1 - B3))) + LN((1 - B4) / (1 - B3))))) * B4) / (B2 - (B2 * EXP(-((1 / (B5 - 1)) * LN((B3 * (1 - B4)) / (B4 * (1 - B3))) + LN((1 - B4) / (1 - B3)))))), 4))Try the example
Use the original values or change them to test your understanding.