The semi-batch MTSR matches dynamic simulation within 0.03 K when the feed stops at a cooling failure

CheAI Research Inc. | Research article | August 2025

Abstract

1. Introduction

In a semi-batch reactor, one reagent is fed into the other so that the jacket can remove the heat as it is released. The fed reagent that has not yet reacted, the accumulation, can still react and heat the mass if the cooling fails. Stoessel called the maximum temperature of this reaction the MTSR [1]. With the time to maximum rate of a decomposition under adiabatic conditions (TMR_ad) and a maximum technical temperature (MTT), it defines five criticality classes [1]-[3].

Three estimates are in use. The first adds the maximum accumulation times the adiabatic temperature rise to the process temperature [2]-[4]. The second takes the cooling-failure temperature at the time of maximum accumulation [5], and the third its maximum over time [6]-[8]. All three assume that the feed stops when the cooling fails. The sources cited here do not follow the failure with a dynamic model. We asked whether they agree with dynamic runs from the worst failure time. The maximum over time agrees when the feed stops. The shortcuts fall low in the catalysed case, and no accumulation-based estimate bounds a continued feed.

2. Methods

C1 is the generic second-order reaction A + B → C of Ubrich et al. [7], run isoperibolically on 335-375 K and 2-48 h dosing. C2 is the uncatalysed esterification of 2-butanol with propionic anhydride [8], held isothermal on 50-90 °C and 1-20 h. C3 is the same reaction catalysed by sulfuric acid [5], [9], held isothermal on 20-90 °C and 5-180 min. All cases use the hypothetical zero-order decomposition of Benaissa [5], borrowed from a sulfonation. T_D24 is the temperature at which the estimated TMR_ad is 24 h. It is 112.0 °C at a specific heat of 2000 J kg^-1 K^-1, and the classes use the specific heat of the relevant inventory.

A dynamic model integrated the species and energy balances, with a total heat capacity C that grows during the feed. The jacket heat flow became zero at the worst failure time, found by a refined scan. In scenario S1 the feed stops at the failure, and in S2 it continues. The dynamic MTSR is the maximum temperature after the failure with the decomposition off, and a run with it on gives the dynamic TMR_ad.

With n_ac the accumulated amount, the cooling-failure temperature of the cooled run is

T_cf(t) = T(t) + n_ac(t) (-ΔH_r) / C(t).

The batch estimate is T_p + ΔT_ad. F3 is T_p + X_ac,maxΔT_ad, with ΔT_ad in the end-of-feed heat capacity C_end. The Tableau 2.18 definition of Benaissa takes T_cf at maximum accumulation. F4 is the semi-batch MTSR, max_t T_cf. F5 is the Townsend-Tou TMR_ad [10] at the dynamic MTSR. An estimate is non-conservative if its MTSR is more than 0.5 K low, its TMR_ad more than 5 % long, or its class lower. We fixed these thresholds before the runs.

Energy closure stayed below 4.9 × 10^-4 %. The isoperibolic C1 run and the C2 MTSR matched their printed answers. C3 matched all 21 printed worst-case temperatures within 1 K and is a calibrated, approximate benchmark.

3. Results

3.1 With the feed stopped, F4 equals the dynamic MTSR

F4 matched the dynamic MTSR within 0.03 K at all 1875 grid points, and its class agreed everywhere (Figure 1a). With the feed stopped, T_cf is exactly the adiabatic end point of the accumulation, so the residual is numerical. The batch estimate lay 1.5-135 K above the dynamic MTSR.

3.2 With the feed continued, only the batch estimate holds

Continued feeding adds reactive inventory, and F4 and F3 were low by up to 27 K in C1, 106 K in C2 and 135-138 K in C3 (Figure 1b). They were non-conservative at 625, 619 and 625 of the 625 points. Only the batch estimate bounded the S2 MTSR. In C1 and C3 it lay within 2 × 10^-6 K of the dynamic value or above it, because S2 converts the full charge. In C2 its margin was 6.2-27.2 K, because the 2-butanol still enters at 25 °C. With continued feeding, every C2 and C3 point fell in class 5.

3.3 The worst failure comes before the maximum accumulation

At constant reactor temperature, T_cf peaks with the ratio n_ac/C. Because C grows during the feed, this peak comes no later than the peak of n_ac, and earlier when n_ac levels off. F3 adds a second error, because it divides the accumulated heat by C_end. In C3 at 30 °C and 30 min dosing, T_cf peaks at 96.74 °C after about 19 min (Figure 2a-c). At maximum accumulation, about 28 min, it has fallen to 90.82 °C, a timing error of 5.93 K. F3 gives 88.27 °C, a further 2.55 K lower.

Over the C3 grid, F3 was up to 14.8 K low and non-conservative at 420 of 625 points (Figure 2d). The Tableau 2.18 definition, which has only the timing error, was up to 6.7 K low and non-conservative at 329 points. Most of this deficit falls far from a class boundary, and each shortcut lowered the class at one C3 cell. Against the 21 printed values of the source, our model gives a maximum T_cf up to 4.94 K higher. In C1, F3 was 0.02-0.36 K low because the reactor runs above the jacket, which lowered the class at three cells. In C2 the worst failure fell at the maximum accumulation, and F3 and the Tableau 2.18 definition coincided with F4.

3.4 The Townsend-Tou time is short against both grid references

F5 was 0.13-0.94 times the dynamic TMR_ad counted from the failure in S1, and 0.01-0.82 in S2 (Figure 3a). Most of this margin is the climb to the MTSR. Against a decomposition-only run from the same MTSR, F5 was 5.9-8.9 % shorter in S1 and 6.9-9.3 % in S2, as expected for a zero-order decomposition. At four diagnostic points we also measured the time left from MTSR - 0.1 K. F5 exceeded it by 5.02 % at C1, 77.85 °C and 24 h dosing, just beyond the 5 % threshold, and by 4.12 % at a C2 point. At both, the slow synthesis lets the decomposition add about 3 K first.

4. Discussion

If the feed stops when the cooling fails, max_t T_cf equals the dynamic answer in all three windows, provided it uses the actual reactor temperature and total heat capacity. The feed stop must therefore be guaranteed, for example by an interlock between cooling and dosing. Otherwise the evidence points to the batch estimate, the only one that bounded the MTSR with continued feeding.

Where the reaction is fast and the total heat capacity more than doubles over the feed, as in C3, the timing and heat-capacity errors cost up to 14.8 K. Reaction calorimetry measures the inputs of T_cf(t) [11], so its maximum needs no extra run.

F5 at a correct MTSR was conservative against both grid-wide references. The diagnostic points show that it should not be read as a bound on the time left once the MTSR is reached.

5. Limitations

The decomposition is hypothetical for these reactions. A 30 K shift of T_D24 or a different MTT changes most or all cells, so the classes illustrate the method and do not classify these reactions. C1 is also an unnamed generic reaction whose MTT of 363 K is a design limit.

The benchmarks reproduce their sources only in part. C3 heat capacities are back-derived from the source, and C3 matches the printed precision at only 5 of 21 points. Its TMR_ad is 6.6-16.3 % longer than the printed values, and the cause is unresolved. The isothermal C1 values and the C2 rate constants (4.0-5.1 % high) miss their printed values as well.

The model simplifies the physics. Product heat capacities equal those of the reagents they replace, and the sulfuric acid has none. Component heat capacities stay fixed within a run, at the process temperature (base case) or 25 °C (sensitivity case). The model omits bubble points, evaporation, venting and relief, so the decomposition peaks (531 and 603 °C for C2 at its source point) only time the runaway.

The scope is narrow. The maps are at laboratory scale, and a 1 m³ plant check only marks where a plant jacket could hold isothermal operation. The F5 remaining-time comparison rests on four points. The study is not a safety assessment of any plant, process or reaction.

6. Conclusions

With the feed stopped at the cooling failure, max_t T_cf reproduces the dynamic MTSR and its class. Shortcuts based on maximum accumulation underestimated the MTSR in the catalysed case, where the total heat capacity grew fastest during the feed. With continued feeding, only the batch estimate bounds the MTSR. For screening, the maximum over time of T_cf with the actual total heat capacity is the estimate to use, but only where the feed stop at cooling failure is assured.

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