Haber-Bosch Ammonia Synthesis Recycle, Purge & Conversion Formula

Why 18% single-pass equilibrium conversion still achieves >90% overall yield via recycle—and why trace Argon forces a purge!

Thermodynamic equilibrium at 450°C and 150–200 bar limits stoichiometric N2 + 3H2 -> 2NH3 single-pass conversion across an iron catalyst bed to X_sp = 15%–20%. Condensing liquid NH3 and recycling unreacted N2 + 3H2 lifts overall conversion above 90%, EXCEPT trace non-reactive inerts (Argon from air separation and CH4 slip from reforming, mole fraction y_i0 in fresh makeup gas) would accumulate infinitely unless bled through a Purge Stream! At steady state, every kmol/hr of inert entering MUST exit in the purge (Purge_Total = (Fresh_Feed * y_i0) / y_loop_inert), dragging Purge_Reactants = Purge_Total * (1 - y_loop_inert) of valuable H2/N2 out to fuel gas! Learners often mistakenly multiply Fresh_Feed * X_sp to estimate plant output, underpredicting NH3 production 5-fold!

Try building a 1,000 kmol/hr Makeup Syngas Loop first—Fresh_Makeup_Feed = 1,000 kmol/hr (B2), Makeup_Inert_MolFrac = 0.01 (1.0 mol% Ar/CH4 = 10 kmol/hr inerts & 990 kmol/hr stoichiometric N2+3H2, B3), Max_Loop_Inert_MolFrac = 0.10 (10.0 mol% loop ceiling, B4 -> Total Purge = 10 / 0.10 = 100.00 kmol/hr in E2, losing 90 kmol/hr reactants), and Single_Pass_Conversion X_sp = 0.18 (18.0%, B5): Overall Reactant Conversion hits (990 - 90) / 990 * 100 = 90.91% (E3), Ammonia Produced equals (900 / 2) * 17 / 1,000 = 7.65 t/hr (E4), and Combined Reactor Inlet Reactants equal 900 / 0.18 = 5,000.00 kmol/hr (E5)! In a shared Arte Sheets process workbook, a `suggester` can propose lowering Max_Loop_Inert_MolFrac in Suggest Mode while asking a hint-first AI prompt ("How do I express steady-state inert balance so purge reactant loss updates overall conversion before sizing the recycle compressor?") so `contributor` engineers inspect cell dependencies before accepting.

Inert-Controlled Recycle Purge & Overall Synthesis Conversion Formula

Validate Fresh_Feed > 0, 0 < Feed_Inert_y0 < Loop_Inert_y < 0.50 (loop inerts must exceed feed inerts!), and Single_Pass_Conv in (0.02, 0.60]!

=LET(fr_f, B2, yi_0, B3, yi_l, B4, xs_p, B5, IF(OR(fr_f <= 0, yi_0 <= 0, yi_l <= yi_0, yi_l >= 0.5, xs_p <= 0.02, xs_p > 0.6), "Invalid Loop Input", LET(rc_f, fr_f * (1 - yi_0), pg_t, (fr_f * yi_0) / yi_l, pg_r, pg_t * (1 - yi_l), ROUND(((rc_f - pg_r) / rc_f) * 100, 2))))

Over-Purging Catalyst Protection Tradeoff Check (Set Max Loop Inert = 0.05)

Tighten Max_Loop_Inert_MolFrac B4 from 0.10 down to 0.05 to boost catalyst partial pressure: Total Purge doubles to 200.00 kmol/hr (190 kmol/hr reactants vented), slashing Overall Reactant Conversion E3 from 90.91% to 80.81% and Ammonia Output E4 from 7.65 t/hr to 6.80 t/hr!

=IF(ISTEXT(E3), E3, LET(rc_f, B2 * (1 - B3), pg_r, ((B2 * B3) / B4) * (1 - B4), ROUND((((rc_f - pg_r) / 2) * 17) / 1000, 2)))

Further reading

• Arte Sheets Lesson — Steam Methane Reforming (SMR) & Water-Gas Shift H2 Yield
• Arte Sheets Lesson — PEM Water Electrolyzer Faradaic Hydrogen & LHV Efficiency