Synthetic Fuels, Gas-to-Liquids (GTL) & Polymerization Kinetics
Fischer-Tropsch ASF Chain-Growth Alpha & C5+ Selectivity Formula
Calculate Anderson-Schulz-Flory (ASF) Methane C1 Weight Selectivity (%), Liquid+Wax C5+ Selectivity (%), Heavy Paraffin Wax C21+ Fraction (%), and Daily C5+ Syncrude Tonnage (t/d).
A single chain-growth probability alpha locks methane C1, C5+ liquids, and C21+ hydrocracker wax together!
In Gas-to-Liquids (GTL) and Power-to-Liquids e-fuel Fischer-Tropsch reactors converting CO + 2H2 syngas over Cobalt or Iron catalysts, surface methylene (-CH2-) monomers either propagate a growing carbon chain with probability alpha or desorb with probability (1 - alpha). Under the exact Anderson-Schulz-Flory (ASF) weight distribution w_n = n * (1 - alpha)^2 * alpha^(n - 1), summing the series from carbon number N to infinity yields closed-form Cumulative C_N+ Weight Fraction W_(N+) = alpha^(N - 1) * (N - (N - 1) * alpha)! For N = 5, C5+ Liquid & Wax Selectivity is alpha^4 * (5 - 4 * alpha); for N = 21, Heavy Wax C21+ (destined for mild hydrocracking into jet/diesel) is alpha^20 * (21 - 20 * alpha), while unwanted C1 Methane slip is simply (1 - alpha)^2!
Verify on a 500 t/d Total Hydrocarbon Cobalt Slurry-Bed FT Unit—Chain_Growth_Alpha = 0.90 (B2) and Total_Hydrocarbon_Product = 500.0 t/d (B3): Methane C1 Selectivity is (1 - 0.90)^2 * 100 = 1.00 wt% (E2), C5+ Liquid+Wax Selectivity hits (0.90^4 * (5 - 3.60)) * 100 = 91.85 wt% (E3), Heavy Paraffin Wax C21+ reaches (0.90^20 * (21 - 18.0)) * 100 = 36.47 wt% (E4), and Daily C5+ Syncrude Output equals 500 * 0.91854 = 459.27 t/d (E5)!
| ASF Product Distribution KPI (alpha = 0.90 Cobalt Catalyst, 500 t/d FT) | Formula | Verified Output |
|---|---|---|
| Unwanted Methane C1 Slip (wt%) | ((1 - alpha)^2) * 100 | 1.00% (4.00% @ alpha = 0.80) |
| Total C5+ Liquids & Wax (wt%) | (alpha^4 * (5 - 4 * alpha)) * 100 | 91.85% (73.73% @ alpha = 0.80) |
| Heavy Wax C21+ Fraction (wt%) | (alpha^20 * (21 - 20 * alpha)) * 100 | 36.47% (5.76% @ alpha = 0.80) |
| Daily C5+ Syncrude Output (t/d) | Total_HC_t_d * (alpha^4 * (5 - 4 * alpha)) | 459.27 t/d (368.64 @ alpha = 0.80) |
Closed-Form ASF Cumulative C5+ Selectivity Formula
Validate Chain_Growth_Alpha strictly in (0.40, 0.98) and Total_HC_Product > 0!
=LET(al_p, B2, hc_t, B3, IF(OR(al_p <= 0.4, al_p >= 0.98, hc_t <= 0), "Invalid ASF Alpha Input", ROUND(((al_p ^ 4) * (5 - 4 * al_p)) * 100, 2)))High-Temperature FT Catalyst Overheating Check (Set Alpha = 0.80)
Drop Chain_Growth_Alpha B2 from 0.90 down to 0.80 (warmer iron-catalyst regime): Methane C1 Slip E2 quadruples to 4.00 wt%, Heavy Wax C21+ E4 plunges more than 6x from 36.47% to 5.76%, and Daily C5+ Syncrude E5 drops to 368.64 t/d!
=IF(ISTEXT(E3), E3, ROUND(B3 * ((B2 ^ 4) * (5 - 4 * B2)), 2))Try the example
Use the original values or change them to test your understanding.