2. Results and Discussion
shows the concentration of jasmin aldehyde as a function of reaction time for the four evaluated reaction temperatures and the condition (B
= 2 mol·mol−1
. For the lowest temperature it can be concluded that ethanol was the best solvent, i.e., the highest jasmin aldehyde yield was obtained. For example, after 4 h of reaction the yield was 30% higher than in the case that other alcohol solvents are used.
At 100 °C ethanol competes with methanol at higher reaction times, but overall ethanol is still a good choice in order to have the highest concentration of desired reaction product. For higher reaction temperatures, methanol and ethanol showed comparable results. Overall, it could be observed that n-propanol and n-butanol have the lowest yield for product E1, so it can be concluded that methanol or ethanol are advisable to be used in aldol condensation reactions.
From Figure 1
it can also be deduced that equilibrium was faster reached for higher reaction temperatures. At 140 °C, equilibrium was reached after ~1 h, whereas for the lowest temperature the equilibrium was not established after 8 h. Regarding the calculation of the average activity coefficient, see Section 3.2
, the subsequent spread will be more pronounced for lower temperatures, since the variation of the actual value versus reaction time was more pronounced.
Based on the given simulations, a temperature of 140 °C is not advisable since E1 is obtained in concentrations lower than 1 μM. For the other (lower) temperatures, a concentration of ~1.10 μM is obtained. The author realizes that the difference is not that pronounced and that the results were in fact the consequence of the ballpark values for the kinetic parameters, but it is important to spot the relation between the temperature and equilibrium: The higher the former, the sooner the latter and, hence, no improvement in yield is possible. For the other initial conditions, (B/H)0 = 1 mol·mol−1 and (B/H)0 = 0.5 mol·mol−1, lower concentrations were obtained, so this is also a point for intensive process improvement.
shows the product distribution for E1
versus reaction time. It was observed that a better selectivity was obtained for alcohols with a higher carbon number. However, the biggest difference was noticed for methanol versus ethanol. The application of n-propanol and n-butanol versus ethanol was a marginal gain and, therefore, possible spread on experimental data will probably result in very similar results. Again, it appears that ethanol is the best solvent for the given aldol condensation reaction, envisaging an optimal yield regarding jasmin aldehyde.
It can also be recognized from Figure 2
that for an excess in benzaldehyde, (B
= 2 mol·mol−1
, the product distributions were closer to each other, compared to an equimolar initial start of the experiment, i.e., (B
= 1. A second observation is that for the equimolar initial condition, a maximal value for the product distribution was obtained. For an excess of heptanal, (B
= 0.5 mol·mol−1
, similar graphs were obtained, however at lower values for SE1
, and, therefore, they are not shown in the manuscript.
In this respect the reader is kindly reminded at this stage in the discussion of the results that, in case of not updating the activity coefficients, the same output would be produced. In other words, possible optimization points for maximal output were overlooked in that regard. This points out the importance of updating the activity coefficients with time, i.e., with varying composition of the reaction mixture.
plots byproduct E2
concentration versus reaction time at 80 °C. It can be clearly observed that for a higher carbon number of the alcohol as solvent, the contribution of the byproduct was less for the same initial conditions, and this perception was more pronounced for higher (B
values. For the sake of completeness, at higher temperatures similar trends were obtained as explained for 80 °C; however, higher E2
concentration values together with lower E1
product distribution were noted.
a–d describes the concentration of benzaldehyde versus reaction time for T = 80 °C and 120 °C. At higher temperatures a long reaction time did not result in an increase of E1
production. Figure 4
e,f depicts the benzaldehyde conversion and it is clear that methanol as solvent resulted in the lowest conversion results. At 80 °C, a conversion of ~0.48 mol·mol−1
was obtained for ethanol and the order was ethanol > n-propanol ≈ n-butanol > methanol, whereas at a higher temperature, e.g., at 120 °C, only ~ 0.37 mol·mol−1
was noted. In the latter case, the data overlap more or less, so a distinct ranking could not be made.
In the presented work, a lot of simulations were performed and in order not to overload the manuscript with data, only exemplary excerpts of activity coefficient results are reported. Figure 5
gives the activity coefficient for byproduct E2
versus reaction time with (B
= 2 mol·mol−1
. The activity coefficient with methanol as solvent was ~2.20. This means that the ‘chemical concentration’ was 2.2 times higher than the physical concentration and this has a pronounced effect on concentration profiling or parameter estimation. It can also be spotted that for an increasing carbon number of the alcohol solvent, the activity codefficient decreased. This appears to contradict the previously made conclusions for Figure 4
. However, inspection of Figure 6
reveals that the decrease in activity coefficient for E2
, was compensated for by the increase of activity coefficient for W. In other words, the product of aE2
, both occurring in reaction rate (5), was increasing for an increasing carbon number in the alcohol solvent (not shown in the manuscript). This product influences the backward step in the equilibrium and, hence, this explains why for n-butanol a lower contribution of E2
was observed, compared to the use of methanol.
Regarding the temperature dependency of the activity coefficients, average values were plotted according to the following relation, lnγ = a + b/T, in order to check the value of b. In other words, if for example b was found to be 5 kJ·mol−1, the corresponding activation energy for the specific reaction would be estimated with at least a deviation of 5 kJ·mol−1, and the activity coefficients would appear together with the true kinetic parameters in the reaction rate expressions. The author is currently working on experimental data for which b values bigger than 5 kJ·mol−1 were found, having significant implications on the reported kinetics (details cannot be given, since the results are not yet published). In this work, however, the variations in activity coefficients versus temperature were not that pronounced, i.e., it was less than 1.5 kJ·mol−1, and this was just one of the reasons why this test case was developed: It can be expected that in the case of higher variations, the implications on the kinetic parameters and the corresponding parameter estimation procedure would be far more pronounced. This should be a caveat for future work.
The product of activity coefficients, in the case of ethanol as solvent, are reported in Table 1
, i.e., these values appear in the bimolecular reaction rates. It can be noticed that the correction for activity reached almost a factor 10 for γE2
at the lowest reaction temperature. The desired reaction had a forward activity product of ~2.6, whereas the undesired reaction path showed a factor of ~1.8. In general, the product values, including water, showed the highest correction for activity in the liquid phase aldol condensation reaction. Also, the factor b, see above, reached values of 2.4 ± 0.3 kJ·mol−1
for the backward reaction involving E2
Taking a closer look at the core business of this manuscript, in silico experimental data were used in an in-house written parameter estimation routine, see Section 3.2
, with the chemical reaction Equations (4) and (5), in which first step concentrations were used, i.e., all activity coefficients γ were set to unity by default. As expected, the variations in activation energy values for the two forward and backward reactions were not that pronounced, see entry ‘Without correction’ in Table 2
, Table 3
, Table 4
and Table 5
, since a rather weak temperature dependency was observed for γ, the given test case of aldol condensation. On the other hand, huge deviations were observed in the pre-exponential factors. For example, with methanol as solvent, pre-exponential factor k2,∞
deviated ~31 times from the original value, see Table 2
. The same parameter, k2,∞
, showed significant deviation for solvents ethanol and n-propanol. n-Butanol gave a deviation of ~ 20. Parameter k4,∞
deviated ~10 times for ethanol as solvent, whereas deviations lower than factor 6 were obtained for n-propanol and n-butanol.
In a second step, average activity coefficients were used, i.e., for the whole concentration profile only one value was implemented. The average values for γ were obtained using the in silico experimental data and point values for the activity coefficients, based on the experimental composition, acquired at equidistant time points; these average values distinctly differ from unity and range between 1 and 6.
The results of the subsequent parameter estimation are reported in Table 2
, Table 3
, Table 4
and Table 5
with entry ‘With correction’. The reader can notice that most deviations were linked to the pre-exponential factors and that, after correction with the average activity coefficients, a better estimate for these factors was obtained. A slight improvement for the activation energies was witnessed. Especially, for E4
with n-butanol as solvent a good improvement, going from 59.4 ± 6.2 kJ·mol−1
to 70.0 ± 2.0 kJ·mol−1
(true value was 70 kJ·mol−1
) was observed. In the case of methanol the activation energy improved from 60.1 ± 4.4 kJ·mol−1
to 65.9 ± 3.5 kJ·mol−1
. For ethanol and n-propanol no specific improvement was observed and the slight overestimation of E4
can be explained by the earlier reported correction of 2.4 kJ·mol−1
Arrhenius diagrams from the isothermal parameter estimation procedure with n-butanol as solvent are given in Figure 7
. It was observed that, in the case when no correction was used for the activity coefficients, still very satisfactory Arrhenius diagrams were obtained. Including the correction, however, yielded a significantly different position with a smaller spread of the isothermal estimates. For the backward reaction coefficients, k2
, the difference in position was the most pronounced. Similar diagrams and corresponding conclusions were obtained for the three other alcohol solvents.
The corresponding residual sum of squares (RSSQ) for the calculated concentrations is reported in Table 2
, Table 3
, Table 4
and Table 5
. Although the estimated parameters were more in line with actual kinetics when the average value for γ was implemented, only a minimal difference in RSSQ has been noted. If these values are used in the conventional model discrimination tools, such as the F test or the so-called ‘likelihood ratio’ [3
], the model with implementation and the model without would not be statistically different. However, it was shown that the ‘true’ kinetic parameters are better approached with the implementation of the average activity coefficients. This signifies a second caveat: In the interpretation of RSSQ for model discrimination, the inspection of the obtained kinetic parameters and the corresponding comparison to typical values is advised and the nonideal liquid properties should be incorporated.
It has to be added that the choice of solvent in this in silico manuscript was only reflected in a variation of activity coefficients. If the choice of another solvent results in a real change in reaction mechanism [24
], then of course the correction for the solvent type will not do the trick since fundamental chemical reactions or transition states are altered by this choice.
So far, in silico data were presented and, therefore, it is of interest to link the reported results with some literature data. Some excellent references on experimental work can be mentioned [12
]. In this report, two examples are addressed where alcohols were also used as solvent.
First, when the selection of solvents as pure alcohol is enlarged to include binary mixtures of water-ethanol mixtures, Cueto et al. [27
] described an enhancement of furfural-cyclopentanone aldol condensation as they observed that reaction coefficients increased for an increasing water content. A plausible reason for this is the stabilization of the formed enolate ion by water during the aldol condensation reaction.
Starting from the same kinetic coefficient, ktrue
, the forward reaction rate is written as r
) × CB
and, hence, if authors report concentration dependencies for the description of their variations via continuity equations, the reported kinetic coefficient is the product of the true coefficient and the corresponding activity coefficients. Figure 8
shows the product of activity coefficients in both forward and backward reactions (4) and (5) for a varying (EtOH/W)0
ratio. Inspection of these results showed that as the ratio (EtOH/W)0
decreases, the product of activity coefficients significantly increases. This corresponds to the aforementioned experimental results and, hence, it is clear that the inclusion of activity coefficients in parameter estimation is a necessary requirement for correct data interpretation. It is also remarkable that these products can vary from a factor of ~2 to ~21.
Secondly, solvent effects are also reported in liquid-phase hydrogenation reactions, e.g., Ramos et al. observed a pronounced solvent effect on the conversion of 4-(2-furyl)-3-buten-2-one over Pt/TiO2
catalyst regarding the use of protic and aprotic solvents [30
]. The former solvents were varied as methanol, ethanol, 1- and 2-propanol, and pentanol, for which ethanol performed significantly better than methanol; and 1- and 2-propanol resulted in a marginal improvement of the conversion versus time profile. This is in line with the conclusion that ethanol had the best conversion and yield values towards the desired product in the present study.
The major part of this work addresses alcohols as solvents. In order to investigate the general character of this work, a brief case study is mentioned where toluene and 1, 4-dioxane (non-polar), dimethylformamide (DMF) (polar aprotic), and ethanol and water (polar protic) were compared regarding activity coefficient values, and the corresponding conversion and product selectivity.
The results for the activity coefficient product B*H
are given in Figure 9
a,b. For the forward reaction the order was: Water > ethanol > 1, 4-dioxane ≈ toluene > DMF. This explains the order of initial conversion profile, see Figure 9
c. The equilibrium conversion when water was solvent was already established after 2 h; the other solvents required a longer reaction time. The conversion at 8 h reaction time can be ranked as ethanol > 1, 4-dioxane > water > toluene ≈ DMF.
After ~2 h of reaction, ethanol performed the best to convert benzaldehyde, and together with the highest selectivity during the whole reaction time, see Figure 9
d, these simulations confirm that ethanol is the best solvent to perform the given aldol condensation reaction. With DMF as solvent, the selectivity was significantly lower, compared to the others.
As a closing remark, the author would like to state that the simulation results are not elaborated in terms of physical interpretation of the activity coefficients. This contribution merely indicates their effect on product distributions and corresponding parameter estimation results, starting from the same kinetics. In other words, sometimes different kinetic parameters are reported in literature for the same reaction with different solvents. This work indicates that it would be a good practice to use all experimental data simultaneously regarding parameter estimation, together with the varying nonideal liquid properties depending on the composition of the liquid phase. In this respect, the parameter estimation with a synchronous update of the activity coefficients yields to original parameter value, see Table 6
, within 3% error. The runtime for this procedure was significantly higher than the case for average values. These results are omitted from the manuscript, since this was to be expected.