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Raoult's law misplaces the cost-optimal condenser temperature of a 300 bar ammonia synthesis loop by 46-54 °F

We rebuilt a 2019 ammonia-loop design study and re-optimized it with a condenser model calibrated to high-pressure data.

Thinh D. Ho | CheAI Research Inc. | Research article | November 2025

Abstract

A 2019 design study optimized the condenser temperature of a 300 bar ammonia synthesis loop with a Raoult's-law condenser. A Python rebuild recovered all seven of its published results within tolerance, including the 44.40 °F optimum. Against 1925 measurements at 300 atm, Raoult's law predicts 53-58 % too little ammonia in the vapor over liquid ammonia. A Peng-Robinson condenser fitted to the available solubility data moves the optimum 46-54 °F colder and raises the 2019 objective by 4.1-5.0 %. With the reactor capital annualized, the Peng-Robinson optimum is still 58-66 °F colder and its objective 5.9-7.0 % higher. No condenser temperature brings the purge to a 5 ppmv ammonia target. In the model, a water scrubber meets it at its outlet, and with an order-of-magnitude stripper estimate this partial cost adds 0.24-0.37 % to the Peng-Robinson loop objective.

1. Introduction

An ammonia synthesis reactor converts only part of its feed per pass, so the product is condensed out of the reactor effluent and the unconverted gas is recycled [1]. A colder condenser costs more refrigeration but returns less ammonia to the reactor, which lowers the reverse reaction and the reactor volume.

A 2019 design study [2] optimized this trade for a 300 bar loop with an ideal-gas, Raoult's-law condenser and found the minimum at 44.4 °F. Its authors flagged the ideal-gas assumption as inaccurate but did not test it. At 300 bar, the mostly supercritical gas holds more ammonia over the liquid than Raoult's law predicts [3]. The report also attributed a 5 ppm purge limit to federal regulation without examining it. We asked how far a calibrated equation of state moves the optimum and what a 5 ppmv purge target costs. The optimum moves 46-54 °F colder, and in the model a scrubber, not the condenser, meets the target for a small fraction of the loop cost.

2. Methods

We ported the 2019 code to Python twice. A faithful port kept the 2019 numerics, including a 100-point grid, and was scored against the seven published results. A clean version minimized continuously. We then changed its assumptions one at a time.

We built the loop in the DWSIM 9.0.5 simulator [4] with a Peng-Robinson (PR) condenser [5] at the 2019 conditions. The hydrogen-ammonia binary parameter, absent from the simulator, was fitted to the 1925 vapor data at -20 and 15 °C [3, 6]. This best estimate is 0.404. A fit to the liquid hydrogen solubility of the same authors [7] gave 0.224, the range end. The two are alternative calibrations, not a statistical interval. The argon-ammonia parameter was refitted to 25 solubility points [8].

The 2019 objective adds the full reactor capital to the annual cooling bill (a mixed basis). We also report an annualized objective,

Jann = CRF · cVV + cQQyr,

with reactor volume V, annual cooling duty Qyr, cV = $200,000/ft3, cQ = $0.0002/Btu and capital recovery factor CRF = 0.0527 (3.25 %, 30 yr). We costed the purge scrubber with the EPA Air Pollution Control Cost Manual [9] and the stripper to an order of magnitude. For this comparison, production was counted after a 50 bar product let-down, whose vent went to the same scrubber. An independent re-derivation with separate model ports and PR flash codes checked the key numbers.

3. Results

3.1 The rebuild reproduces the 2019 optimum

The faithful port recovered all seven published results within tolerance, including the $217.40 M/yr objective and the 44.40 °F optimum. Numerical shortcuts in the 2019 code moved the optimum by less than 1 °F. An ideal-gas heat capacity in the cooling duty moved it 13.0 °F colder. Adding the latent heat of the condensed ammonia moved it 8.0 °F warmer. The report's "18 ppt" of ammonia in the purge is 18,182 ppmv, or 3,636 times the 5 ppmv target.

3.2 Raoult's law predicts about half the measured vapor ammonia

Raoult's law predicted 58, 53 and 57 % less ammonia in the vapor than measured at -20, 0 and 15 °C (Figure 1). The pure-ammonia vapor pressure was not the cause. A reference equation in its place moved the 2019 optimum by only -1.3 °F. With default parameters, PR overpredicted the vapor ammonia by 27-53 %. No single hydrogen-ammonia parameter fitted both observables of the 1925 study. The vapor fit underpredicted dissolved hydrogen by 35 % and failed the nitrogen solubility check (-11.2 % mean).

Vapor ammonia against temperature for five models and the 1925 data.
Figure 1. Raoult's law predicts about half the measured ammonia in the condenser vapor. Ammonia in the vapor over liquid ammonia under a 3 to 1 hydrogen-nitrogen gas at 300 atm, with data from Larson and Black [3] as reprinted in [6]. The 0 °C value is a lower bound (arrow).

3.3 The calibrated loop moves the optimum colder on both cost bases

With the calibrated PR condenser, the loop optimum moved from 44.40 to -1.56 °F, and the 2019 objective rose 4.12 % to $226.35 M/yr (Figure 2a). The alternative calibration gave -9.71 °F and +4.99 %. Annualizing the capital moved every optimum warmer (Figure 2b), the 2019 model to 127.37 °F and the best-estimate PR loop to 69.26 °F. The PR optima stayed 58.1-65.9 °F colder, with objectives 5.9-7.0 % higher. On the mixed basis, the closure and cooling-duty sensitivities also kept the PR optimum below 44.4 °F.

Loop objective against condenser temperature on the mixed and annualized cost bases.
Figure 2. The calibrated PR loop moves the cost optimum 46-54 °F colder on the 2019 objective (a) and 58-66 °F colder on the annualized objective (b). Dashed lines show the 2019 model, solid lines the simulator loop. The PR loop does not reach 20 % conversion at 140 °F and above.

3.4 A scrubber, not the condenser, meets the purge target

We found no federal ammonia concentration limit for the purge, only a release-reporting quantity [10]. We do not know which source the 2019 authors used. The nearest "5 ppm" is a regional limit for ammonia slip from new selective catalytic reduction units [11], so we used 5 ppmv (dry) as a permit-style design target. Condensation cannot meet it. At the ammonia triple point the PR model gave 925 ppmv and the Raoult model 204 ppmv (Figure 3a). Independent PR implementations gave up to 2,117 ppmv, all far above the target. At the PR optimum the untreated purge carried 16,835 ppmv ammonia. In the model, a packed water scrubber on both streams brought the discharged gas to 5 ppmv. With the stripper and preheater it added $0.548 M/yr, or 0.24 % of the loop objective (0.37 % annualized, Figure 3b). Joint optimization saved at most $3,836/yr, well inside the ±30 % accuracy of the scrubber estimate.

Purge ammonia against condenser temperature, and the added cost of purge treatment.
Figure 3. No condenser temperature meets the 5 ppmv target, and a scrubber meets it for less than 0.5 % of the loop objective. (a) Ammonia in the condenser vapor down to the ammonia triple point. (b) Added annual cost of scrubber, stripper and gas preheat, excluding storage emissions and several equipment items.

4. Discussion

The 2019 answer was internally sound, but its physics and cost basis decided the optimum. The numerics and the vapor-pressure correlation each moved the optimum by about 1 °F or less. The latent heat moved it 8 °F, the heat capacity 13 °F, the phase-equilibrium model about 46 °F and the cost basis about 83 °F.

Gas-phase nonideality and dissolved hydrogen and nitrogen let the vapor hold about twice the ammonia of an ideal solution [3], so the condenser must run colder. The shift depends on the binary parameters, as expected for cubic equations of state [12]. The simulator's default argon parameter made argon about ten times too soluble. With that default, the cost penalty fell from 4.12 to 0.55 %.

Annualizing the capital makes the reactor cheap relative to cooling, so a larger reactor and a warmer condenser win. The emission target barely moves the condenser optimum, but the design must include a scrubber. Design textbooks ask students to check the thermodynamic model first [13]. We propose the case as a staged design exercise, not yet run with students.

5. Limitations

The phase-equilibrium data are thin. One 1925 study supplied the hydrogen-ammonia data. Its vapor and liquid parts need different parameters, and the selected model misses its nitrogen solubility by 11 %. One compilation supplied the argon-ammonia data, applied outside its fitted range of 273-323 K. The vapor fit covers 253-288 K and excludes the annualized PR optimum (293.9 K), and the hydrogen-argon pair was not checked. The ranges are therefore conditional scenarios spanning only the two hydrogen-ammonia calibrations. They exclude uncertainty in kinetics, utility prices and the argon transfer.

The process model is simplified. The reactor is a conversion block sized with the 2019 kinetics, and the cooling duty keeps the 2019 formula and one utility price. The warm annualized optima would allow cooling water for part of it. Neither objective includes compression, heat recovery, feed or labor, so the 2019 objective is not a plant annual cost. Storage emissions are not modeled either. The 50 bar product still holds 21-49 % of its dissolved hydrogen, nitrogen and argon (17-36 % annualized), released with some ammonia at storage.

The emission treatment is costed only in part. The cost excludes let-down valves and lost pressure energy, exchanger and reboiler capital beyond the column proxy, the stripper condenser, handling of the recovered ammonia and pressure-rated construction. Refrigerated recovery of the product flash vapor was not modeled. The stripper was not simulated, and no treated-water residual was computed. Its cost assumes full ammonia recovery, which neither objective credits, and it uses a Henry's law slope extrapolated to 100 °C. The scrubber estimate escalates 1991 costs over 28 years, beyond the five years the manual advises, and its liquid loading lies just outside the flooding correlation range.

6. Conclusions

A Peng-Robinson condenser calibrated to the available high-pressure data moved the optimum 46-54 °F colder and raised the 2019 objective by 4.1-5.0 %. With the capital annualized, the optima were 58-66 °F colder and their objectives 5.9-7.0 % higher. Condensation cannot meet a 5 ppmv ammonia target, and in the model a water scrubber met it for 0.24-0.37 % of the loop objective, over a partial scope. The cost basis and the binary parameters of the phase-equilibrium model decide the condenser temperature far more than the numerical method does.

References

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