Redundancy-Free Models for Mathematical Descriptions of Three-Phase Catalytic Hydrogenation of Cinnamaldehyde

Abstract

A new approach on how to formulate redundancy-free models for mathematical descriptions of three-phase catalytic hydrogenation of cinnamaldehyde is presented. An automatically created redundant (generalized) model is formulated according to the complete reaction network. Models based on formal kinetics and kinetics concerning the Langmuir-Hinshelwood theory for three-phase catalytic hydrogenation of cinnamaldehyde were investigated. Redundancy-free models were obtained as a result of a step-by-step elimination of model parameters using sensitivity and interval analysis. Starting with 24 parameters in the redundant model, the redundancy-free model based on the Langmuir-Hinshelwood mechanism contains 6 parameters, while the model based on formal kinetics includes only 4 parameters. Due to less degrees of freedom of molecular rotation in the adsorbed state, the probability of a direct conversion of cinnamaldehyde to 3-phenylpropanol according to the redundancy-free model based on Langmuir-Hinshelwood approach is practically negligible compared to the model based on formal kinetics.

1. Introduction

The heterogeneously catalyzed selective hydrogenation of α,β-unsaturated aldehydes is an important reaction for industrial production of fine chemicals. This reaction is particularly common in the production of α,β-unsaturated alcohols, which are used in perfumery and pharmaceutical industries.

In 1934, Horiuti and Polanyi [1] proposed a multistage mechanism for the heterogeneously catalyzed alkenes hydrogenation on metallic catalysts, which is regarded as a classical approach for description of the selective hydrogenation of α,β-unsaturated aldehydes. According to this mechanism, it is postulated that the hydrogenation of unsaturated compounds proceeds via the intermediate formation of semi-hydrogenated intermediates. On this basis, Claus formulated a complex mechanism of metal-catalyzed hydrogenation of crotonaldehyde [2]. He discussed the concept of possible adsorption geometry of the reactant with the metallic surface and provided the formation of semi-hydrogenated intermediates, whereby side reactions were not taken into account.

Heterogeneously catalyzed hydrogenation of high-molecular α,β-unsaturated aldehydes is usually carried out as a three-phase reaction in a gaseous-liquid-solid system, which is conducted in a batch reactor. The reaction products remain in the reactor, increasing the probability of readsorption of the desired products and their further reaction.

In order to evaluate the catalyst performance for the hydrogenation reaction, knowledge about the microkinetics is necessary. For modelling of this reaction, the kinetic approach is often used, which takes into account the adsorption-desorption steps. Neri et al. formulated a kinetic model of liquid-phase hydrogenation of cinnamaldehyde over Ru/Al₂O₃ based on Langmuir−Hinshelwood type rate expressions [3]. Toebes et al. provided modelling of the hydrogenation of cinnamaldehyde over carbon nanofiber-supported platinum catalysts based on the concentration–time experimental results. In agreement with the literature, Langmuir–Hinselwood–Hougen–Watson kinetics were chosen for the modelling [4]. Kinetic description of such reactions can be simplified by assuming that the surface reaction is the rate-determining step (Langmuir–Hinshelwood mechanism). However, if adsorption-desorption processes in the reaction system are very slow, the process limitation are to be described with a more complex model according to Hougen–Watson mechanism [5].

The reaction scheme proposed by Goupil et al. [6] can serve as a good basis for kinetic modelling of the hydrogenation of cinnamic aldehyde (Figure 1). Hydrogenation reaction of CAL to HCOL, COL, and HCAL involves five individual parallel and consecutive reaction steps. Hydrogenation of the C=O group of cinnamic aldehyde (CAL) causes building of cinnamyl alcohol (COL). The C=C group hydrogenation leads to formation of 3-phenylpropanal (HCAL). Further hydrogenation of either COL or HCAL provides formation of 3-phenylpropanol (HCOL) [7,8].

The mechanism of the catalytic hydrogenation of cinnamic aldehyde is quite complex. Reaction rates can be usually determined based on theoretical conceptions about the mechanism of single reaction stages on the catalyst surface. It is extremely important to investigate scientifically based approaches for formulation of redundancy-free models of complex catalytic reactions in order to develop an efficient method of finding sufficient and reliable kinetic schemes. The idea of this paper is to provide a simplified reaction scheme of the three-phase catalytic hydrogenation of cinnamic aldehyde based on the generalized model description (GD), which was already demonstrated for the isomerization of n-decane [9].

2. Results and Discussion

The strategy for formulation of mathematical description of the three-phase catalytic hydrogenation of cinnamic aldehyde was used according to the procedure described in [10]. According to the principle “everyone with everyone”, an automatically created redundant GD was obtained (see Figure 2).

The model structure in Figure 2 can be described by the system of ordinary differential equations:

$$\begin{gathered} \frac{d{C}_A}{dt}=-r_1+r_2-r_7+r_8-r_{11}+r_{12} \\ \frac{d{C}_B}{dt}=r_1-r_2-r_3+r_4-r_5+r_6 \\ \frac{d{C}_C}{dt}=r_3-r_4+r_{11}-r_{12}-r_9+r_{10} \\ \frac{d{C}_D}{dt}=r_5-r_6+r_9-r_{10}+r_7-r_8 \\ {C_A|}_{t=0}=C_{A,0}=47.9\mathrm{mol}/\mathrm{L}, \\ {C_B|}_{t=0}=C_{B,0}=0, \\ {C_C|}_{t=0}=C_{C,0}=0.326\mathrm{mol}/\mathrm{L} \\ {C_D|}_{t=0}=C_{D,0}=1.69\mathrm{mol}/\mathrm{L} \end{gathered}$$

(1)

where: C_i—the concentration of the i-th component in mol/L; r_j—the rate of the j-th reaction mol/s, defined as r_j = k_jC_iⁿⁱ with k_j—reaction rate constant and n_j—reaction order.

The component A represents CAL, B—COL, C—HCOL, and D—HCAL in Figure 1. Experimental kinetic data for the hydrogenation of cinnamic aldehyde on the catalyst Pt₇₀-Fe₃₀/SiO₂ at T = 350 °C are shown in Figure 3 [5].

2.1. Formal Kinetic Approach

In the first step, the ranges of the model parameter values were determined. For this purpose, the GD model calculation at different model parameter intervals was provided. The obtained value of the RSS at k_i ϵ [0, 1], n_i ϵ [0.55, 3] indicated that the reaction is inaccurately described by the model under the chosen conditions (RSS = 1.0924). On the other hand, the parameter values for k₁, k₃, k₆, k₇, k₈, k₁₁, k₁₂ are already at the interval limit. Some values for the determined reaction orders are also at the limit of the selected interval, but its further extension seems unfavorable from a physical-chemical point of view. Therefore, calculation experiments with extended interval limits for k_i were carried out (

Table 1. Solution results of the kinetic inverse problem for the hydrogenation of cinnamic aldehyde based on formal kinetic approach.

Solution Interval	Determined Numerical Values for the Kinetic Constants k and Reaction Orders n	Equation Nr.	RSS
Test run 1
ki ϵ [0,5], ni ϵ [0.55,3]	k1 = 4.999 k2 = 0.00169 k3 = 0.0289 k4 = 4.299 k5 = 0.798 k6 = 4.999 k7 = 4.999 k8 = 4.834 k9 = 4.496 k10 = 0.913 k11 = 1.997 k12 = 0.885 n1 = 1.008 n2 = 2.672 n3 = 0.769 n4 = 2.815 n5 = 0.550 n6 = 0.550 n7 = 2.508 n8 = 1.471 n9 = 2.995 n10 = 0.596 n11 = 2.999 n12 = 2.650	(1)	0.0375
Test run 1a
ki ϵ [0,5], ni ϵ [0.55,3]	k1 = 4.999 k2 = 0.00003 k3 = 0.274 k4 = 0.370 k5 = 0.886 k6 = 4.999 k7 = 4.999 k8 = 2.000 k11 = 2.472 k12 = 3.626 n1 = 1.019 n2 = 2.967 n3 = 0.772 n4 = 0.581 n5 = 0.500 n6 = 0.502 n7 = 2.935 n8 = 1.958 n11 = 2.983 n12 = 2.529	(2)	0.0375
Test run 1b
ki ϵ [0,5], ni ϵ [0.55,3]	k1 = 4.999 k5 = 0.8526 k6 = 4.999 k7 = 4.999 k8 = 4.960 k11 = 0.996 k12 = 0.00009 n1 = 1.1064 n5 = 0.5038 n6 = 0.5000 n7 = 2.3408 n8 = 1.1094 n11 = 0.828 n12 = 2.994	-	0.0402
Test run 1c
ki ϵ [0;5], ni ϵ [0.55,3]	k1 = 4.999 k5 = 9.638·10−9 k7 = 0.804 k11 = 3.30473 k12 = 3.969·10−5 n1 = 0.722 n5 = 2.844 n7 = 2.999 n11 = 2.695 n12 = 2.219	(3)	0.1001
Test run 1d
ki ϵ [0,5], ni ϵ [0.55,3]	k1 = 4.999 k5 = 6.160 × 10−9 k7 = 0.211 k11 = 1.349 k12 = 9.394·10−8 n1 = 0.751 n5 = 1 n7 = 2.913 n11 = 1 n12 = 1	(3)	0.1103
Test run 1e
ki ϵ [0,5], ni ϵ [0.55,3]	k1 = 5 k5 = 1.907·10−12 k7 = 0.143 k11 = 1.133 n1 = 1 n5 = 1 n7 = 1 n11 = 1	(3)	0.1270
Test run 2
ki ϵ [0,100], ni ϵ [0.55,3]	k1 = 6.771 k5 = 3.675 × 10−12 k7 = 0.199 k11 = 1.429 n1 = 1 n5 = 1 n7 = 1 n11 = 1 (the reaction orders n1, n5, n7, n11 were not determined, but assumed to be 1)	(4)	0.0648

, 2nd test run). Many model parameters were still at interval limits, although the value of the RSS was significantly reduced.

For more detailed analysis, Figure 4 provides a comparison of the model parameter values obtained in test runs (Table 1), which were normalized to the upper interval limit. Figure 5 compares the sizes of the corresponding parameter intervals. In the next step, based on the DoE, the following parameters of the optimization method were determined: the number of generated start points (P₁ = 220); the number of target function calculations (P₂ = 500); the number of times the found values are taken as start points to continue the search (P₃ = 5).

Figure 4 clearly shows that the parameter values for k₁, k₆, k₇, k₈ are at the upper limit of the interval in test runs 1 and 2. This means that neither in the parameter range k_i ϵ [0, 1] nor in the interval range k_i ϵ [0, 5] could characteristic parameter values be found. The reaction rate constant k₂ is close to zero in both cases, which means that it can be neglected. The constants k₅, k₉, k₁₀ and the reaction orders n₅ and n₆ could also be excluded. In order to make a conclusion in this context, the extended parameter intervals were also analysed (Figure 5). The maximum intervals have parameters k₄, k₈, k₉, k₁₀, k₁₂, and n₂, n₃, n₄, n₈, n₉, n₁₀, n₁₂, so that they can be assumed to be negligible. The constants k₉ und k₁₀ also take very small values (Figure 4). This means that, in conjunction with the associated reaction orders n₉ und n₁₀, they can also be ignored in the mathematical description of the hydrogenation reaction. Accordingly, the differential equation system (1) can be simplified as follows:

$$\begin{gathered} \frac{d{C}_A}{dt}=-r_1+r_2-r_7+r_8-r_{11}+r_{12} \\ \frac{d{C}_B}{dt}=r_1-r_2-r_3+r_4-r_5+r_6 \\ \frac{d{C}_C}{dt}=r_3-r_4+r_{11}-r_{12} \\ \frac{d{C}_D}{dt}=r_5-r_6+r_7-r_8 \\ {C_A|}_{t=0}=C_{A,0}= 47.9\mathrm{mol}/\mathrm{L}, \\ {C_B|}_{t=0}=C_{B,0}=0, \\ {C_C|}_{t=0}=C_{C,0}= 0.326\mathrm{mol}/\mathrm{L} \\ {C_D|}_{t=0}=C_{D,0}= 1.69\mathrm{mol}/\mathrm{L} \end{gathered}$$

(2)

The parameter values calculated using the simplified model (2) and the corresponding parameter value ranges are summarized in Table 1. Figure 4 and Figure 5 show normalized values of the model parameters and their intervals for the model with 20 parameters (test run 1a).

As a result of test run 1a, the minimum parameter value for the constant k₂ was found. At the same time, the constants k₂–k₄ have the maximum size of the parameter intervals. Therefore, these parameters were excluded in the next model (test run 1b). In the test run 1b (Table 1), the parameter intervals for k₆, k₈ and n₅–n₁₂ reached their maximum value. This shows that these parameters have little effect on the model output. The exclusion of the rate constants k₆, k_8, and the corresponding reaction orders n₆, n₈ was realized in the model with 10 parameters:

$$\begin{gathered} \frac{d{C}_A}{d\tau }=-r_1+r_2-r_7-r_{11}+r_{12} \\ \frac{d{C}_B}{d\tau }=r_1-r_2-r_3+r_4-r_5 \\ \frac{d{C}_C}{d\tau }=r_3-r_4+r_{11}-r_{12} \\ \frac{d{C}_D}{d\tau }=r_5+r_7 \\ {C_A|}_{t=0}=C_{A,0}= 47.9\mathrm{mol}/\mathrm{L}, \\ {C_B|}_{t=0}=C_{B,0}=0, \\ {C_C|}_{t=0}=C_{C,0}= 0.326\mathrm{mol}/\mathrm{L} \\ {C_D|}_{t=0}=C_{D,0}= 1.69\mathrm{mol}/\mathrm{L} \end{gathered}$$

(3)

As expected, the elimination of the model parameters k₆, k₈, n_6, and n₈ leads to a certain decrease of the model accuracy; the RSS increases from 0.0375 to 0.0402.

In the following simplification step of model (3), the test runs 1c, 1d and 1e were performed (Table 1). The simplified, redundancy-free model structures are shown in Figure 6. The numerical values of the reaction orders determined in test runs 1c and 1d were set to 1 in test run 1e as a first approximation. Finally, test run 1e leads to the final version of a redundancy-free model based on the following differential equation system:

$$\begin{gathered} \frac{d{C}_A}{dt}=-r_1+r_7-r_{11} \\ \frac{d{C}_B}{dt}=r_1-r_5 \\ \frac{d{C}_C}{dt}=r_{11} \\ \frac{d{C}_D}{dt}=r_5+r_7 \\ {C_A|}_{t=0}=C_{A,0}= 47.9\mathrm{mol}/\mathrm{L}, \\ {C_B|}_{t=0}=C_{B,0}=0, \\ {C_C|}_{t=0}=C_{C,0}= 0.326\mathrm{mol}/\mathrm{L} \\ {C_D|}_{t=0}=C_{D,0}= 1.69\mathrm{mol}/\mathrm{L} \end{gathered}$$

(4)

To check the performance of the resulting redundancy-free model, the ranges of the parameter values were extended: k_i ϵ [0; 100] and n_i ϵ [0.55, 3]. The inverse kinetic task was solved in the test run 2 (Table 1). The obtained parameter values and their intervals are shown in Figure 3 and Figure 4. Despite an extension of the parameter range, the values remained within the previously determined limits (see Table 1). Since the obtained sizes of the intervals are relatively small, it can be assumed that the remaining parameters are significant.

The quality of obtained models can be verified by information criteria. In

Table 2. Evaluation of the model quality for mathematical description of the hydrogenation of cinnamic aldehyde according to the formal kinetic approach using information criteria.

Model- No.	Test Run	RSS	$\mathbf{log}\left((\mathit{L}(\widehat{\mathit{k},}{\widehat{\mathit{\sigma }}}^{\mathbf{2}}|\mathit{D}\mathit{a}\mathit{t}\mathit{e}\mathit{n})\right)$	N	$\mathit{A}\mathit{I}\mathit{C}$	$\mathit{A}\mathit{I}{\mathit{C}}_{\mathit{c}}$	${{\Delta }}_{\mathit{i}}\mathit{A}\mathit{I}{\mathit{C}}_{\mathit{c}}$	${\mathit{\omega }}_{\mathit{i}}$
1	1	0.0375	67.53	25	43.86	116.08	110.17	0.0000
2	1a	0.0375	67.53	21	35.86	77.86	71.95	0.0000
3	1b	0.0402	66.86	15	23.92	41.06	35.15	0.0000
4	1c	0.1000	58.16	11	16.71	24.96	19.05	0.0000
5	1d	0.1103	57.22	8	10.80	14.91	9.00	0.0063
6	1e	0.1270	55.87	5	4.92	6.50	0.58	0.4247
7	2	0.0648	62.30	5	4.34	5.92	0.00	0.5689

, the characteristic criteria for all model approaches with the respective RSS are compared. Models 6 and 7 with the lowest number of parameters are among the best of the analyzed models (Table 2). They have the best values of the evaluation criteria: AIC = min, AIC_c = min, Δ_i = Δ_min → 0. Models 1 to 5, with Δ_i > 10, are physico-chemically not very useful and do not represent experimentally observed dependencies adequately. The difference between models 6 and 7 is not the number of parameters or model structures, but rather the RSS, which is smaller for model 7 than for model 6 due to the widening of the model parameters range. The weights of the two models, calculated according to (17), are: ω₆ = 0.4247, ω₇ = 0.5689. Model 5 with ω₅ = 0.0063 is not significant. The weights of models 1–4 cannot be taken into account, since ω₁ = ω₂ = ω₃ = ω₄ = 0.

Based on analysis of weight ratios between the models ω₇/ω_j: ω₇/ω₆ = 1.34, ω₇/ω₅ = 89.89, model 7 can be identified as the best model. The slight advantage of model 7 over model 6 is achieved, as already explained, by extending the range of parameter values, since the number of model parameters is not different for both models. Due to a small number of model parameters in both models, they are characterized by a high degree of clarity and, can optimally describe the kinetics of the hydrogenation in agreement with the formal-kinetic model according to (4).

2.2. Kinetic Approach according to Langmuir-Hinshelwood Mechanism

The redundant model (1) of the cinnamic aldehyde hydrogenation was provided according to the Langmuir-Hinshelwood mechanism. The rate of the individual reaction stages was defined by Equation (11). The test runs according to Langmuir-Hinshelwood mechanism are summarized in

Table 3. Solution results of the inverse kinetic task determined for the model according to the Langmuir-Hinshelwood mechanism at P₁ = 220, P₂ = 500, P₃ = 5.

Solution Interval	Determined Numerical Values for the Model Parameters k and n	Equation Nr.	RSS
Test run 1
$k_i^{\prime }$ϵ [0,1], Ki ϵ [0,1]	k1 = 0.999 k2 = 0.256 k3 = 0.0000 k4 = 0.999 k5 = 0.0233 k6 = 0.999 k7 = 0.999 k8 = 0.0000 k9 = 0.300 k10 = 0.999 k11 = 0.999 k12 = 0.655 K1 = 0.999 K2 = 0.0000 K3 = 0.999 K4 = 0.999	(1)	2.8512
Test run 2
$k_i^{\prime }$ϵ [0,5], Ki ϵ [0,5]	k1 = 4.999 k2 = 0.832 k3 = 0.0004 k4 = 0.0937 k5 = 0.0283 k6 = 4.999 k7 = 2.891 k8 = 4.999 k9 = 0.140 k10 = 0.0003 k11 = 1.124 k12 = 0.004 K1 = 4.999 K2 = 1.559 K3 = 2.008 K4 = 4.999	(1)	0.1056
Test run 3
$k_i^{\prime }$ϵ [0,10], Ki ϵ [0,10]	k1 = 9.999 k2 = 0.00000118 k3 = 0.188 k4 = 0.702 k5 = 0.259 k6 = 9.999 k7 = 5.441 k8 = 9.999 k9 = 0.0598 k10 = 1.100 k11 = 1.502 k12 = 0.000006 K1 = 4.238 K2 = 9.999 K3 = 9.997 K4 = 7.701	(1)	0.0177
Test run 4
$k_i^{\prime }$ϵ [0,100], Ki ϵ [0,100]	k1 = 20.819 k2 = 0.00018 k3 = 0.0146 k4 = 14.641 k5 = 0.0354 k6 = 39.731 k7 = 3.633 k8 = 0.095 k9 = 75.125 k10 = 47.151 k11 = 1.739 k12 = 65.005 K1 = 10.942 K2 = 92.385 K3 = 0.0003 K4 = 2.752	(1)	7.965 × 10−4
Test run 5
$k_i^{\prime }$ϵ [0,100], Ki ϵ [0,100]	k1 = 40.708 k2 = 0.0002 k3 = 0.0057 k4 = 0.1396 k5 = 0.0336 k7 = 7.136 k8 = 0.6578 k10 = 1.1538 k11 = 2.7629 K1 = 2.403 K4 = 58.04589	(5)	9.6711·× 10−4
Test run 6
$k_i^{\prime }$ϵ [0,100], Ki ϵ [0,100]	k1 = 91.235 k2 = 5.632·10−7 k3 = 2.795·10−8 k5 = 0.015 k7 = 12.094 k10 = 0.636 k11 = 7.234 K1 = 1.016 K4 = 99.994	(5)	9.7681 × 10−4
Test run 7
$k_i^{\prime }$ϵ [0,100], Ki ϵ [0,100]	k1 = 19.262 k2 = 0.039 k3 = 1.109·10−8 k5 = 6.465 × 10−9 k7 = 0.332 k10 = 67.251 K1 = 13.854	(5)	0.0166
Test run 8
$k_i^{\prime }$ϵ [0,100], Ki ϵ [0,100]	k1 = 19.272 k2 = 0.0392 k3 = 1.364·10−10 k5 = 3.808 × 10−9 k7 = 0.332 K1 = 13.782	(5)	0.0166

.

The first test run provides values for nine model parameters, which are located at the upper range and thus the range is to be extended. After the second test run, five parameters still reach the upper range. For this reason, the parameter range was extended by a further number of times as follows: $k_i^{\prime }$ ϵ [0; 100] and $K_j$ ϵ [0; 100]. As expected, with the extension of the range of the model parameters the value of the objective function RSS significantly decreases (Table 3, Figure 7).

According to the analysis of parameter values and parameter intervals determined in the 4th test run (Figure 7 and Figure 8), the model parameters k₆, k₉, k₁₂, K₂ and K₃ can be excluded from further consideration. Thus, the differential equation system (1) can be simplified:

$$\begin{gathered} \frac{d{C}_A}{dt}=-r_1+r_2-r_7+r_8-r_{11} \\ \frac{d{C}_B}{dt}=r_1-r_2-r_3+r_4-r_5 \\ \frac{d{C}_C}{dt}=r_3-r_4+r_{11}-r_{12}+r_{10} \\ \frac{d{C}_D}{dt}=r_5-r_{10}+r_7-r_8 \\ {C_A|}_{t=0}=C_{A0}= 47.9\mathrm{mol}/\mathrm{L}, \\ {C_B|}_{t=0}=C_{B0}=0, \\ {C_C|}_{t=0}=C_{C0}= 0.326\mathrm{mol}/\mathrm{L} \\ {C_D|}_{t=0}=C_{D0}= 1.69\mathrm{mol}/\mathrm{L} \end{gathered}$$

(5)

The parameter values obtained using the model approach according to (5) and the corresponding parameter value ranges are summarized in Table 3 and Figure 7 and Figure 8. As shown in Figure 7 and Figure 8 for the 5th test run, the rate constants k₄ and k₈ have only a very small influence on the overall result and can therefore be excluded in the 6th test run. After the 6th test run, the model structure can be simplified consequently up to six parameters: k₁, k₂, k₃, k₅, k₇, K₁ (Table 3, 8th test run). However, after model simplification, the RSS increased from 9.6711 × 10⁻⁴ to 0.0166.

Table 4. Evaluation of model quality based on information criteria for the mathematical description of cinnamic aldehyde hydrogenation according to the Langmuir-Hinshelwood approach.

Model- No.	Test Run	${\mathit{F}}^{\mathbf{2}}$	$\mathbf{log}\left((\mathit{L}(\widehat{\mathit{k},}{\widehat{\mathit{\sigma }}}^{\mathbf{2}}|\mathit{D}\mathit{a}\mathit{t}\mathit{e}\mathit{n})\right)$	N	$\mathit{A}\mathit{I}\mathit{C}$	$\mathit{A}\mathit{I}{\mathit{C}}_{\mathit{c}}$	${{\Delta }}_{\mathit{i}}\mathit{A}\mathit{I}{\mathit{C}}_{\mathit{c}}$	${\mathit{\omega }}_{\mathit{i}}$
1	1	2.8512	26.15	17	31.62	55.16	44.90	0.0000
2	2	0.1056	57.64	17	28.76	52.30	42.03	0.0000
3	3	0.0177	74.70	17	27.21	50.75	40.48	0.0000
4	4	7.965 × 10−4	104.33	17	24.52	48.05	37.79	0.0000
5	5	9.6711 × 10−4	102.48	12	14.68	24.75	14.48	0.0006
6	6	9.7681 × 10−4	102.38	10	10.69	17.36	7.09	0.0230
7	7	0.0166	75.31	8	9.15	13.27	3.00	0.1779
8	8	0.0166	75.31	7	7.15	10.26	0.00	0.7985

summarizes the results of the model evaluation of the hydrogenation of cinnamic aldehyde according to the Langmuir-Hinshelwood approach. This analysis confirms that the model 8, obtained in the 8th test run, is characterized by the lowest number of model parameters and the best values of the evaluation criteria: AIC = min, AIC_c = min, Δ_i = Δ_min → 0. Models 1 to 5, for which Δ_i > 10, cannot represent the experimental dependencies in the required way. For these models: ω₁ = ω₂ = ω₃ = ω₄ ≈ ω₄ ≈ 0. The weights of models 6, 7, and 8, calculated according to (17), are: ω₆ = 0.0230, ω₇ = 0.1779, ω₈ = 0.7985. The weight ratios ω₃/ω_j, which indicate how much the respective model contributes to the overall prediction, are ω₈/ω₆ = 34.73 und ω₈/ω₇ = 4.49.

This confirms that model 8 is best suited for the redundancy-free description of the hydrogenation of cinnamic aldehyde based on the Langmuir-Hinshelwood mechanism. The kinetic reaction scheme based on this mechanism is shown in Figure 9. The advantage of model 8 over model 7 is achieved by reducing the number of model parameters and thus simplifying the complexity of the reaction system without changing the sum of squares of errors.

The best formulated redundancy-free models for the hydrogenation reaction based on the formal-kinetic (formal kinetic model 7 (2nd test run) in Table 2) and the Langmuir-Hinshelwood approach (Table 2 and the model 8 (8th test run) are to be compared in terms of their quality and significance. The value of the error square sum of RSS = 0.0648 was determined for the first model and of RSS = 0.0166 for the second model. It means that the model based on the Langmuir-Hinshelwood theory reflects the experimental results better compared to the model based on the formal kinetic approach. On the other hand, it contains 6 model parameters, while the formal kinetic model has only 4 parameters. In contrast, the values of the Akaike criterion for the Langmuir-Hinshelwood model approach are AIC = 7.15 and for the formal kinetic approach AIC = 4.34.

The simplification achieved in the redundancy-free model based on the Langmuir-Hinshelwood approach can be explained by the fact that, in the case of significant adsorption of the starting molecules in the catalyst surface, the degrees of freedom of adsorbed molecules decrease, which reduces their reactivity. In this way, three degrees of freedom of molecular rotation in the gas phase are reduced to one degree of freedom for the adsorbed molecules. The same is characteristic for the degrees of freedom of molecular rotation, the number of which decreases from three in the gas phase to two in the adsorbed state. Consequently, the probability of the direct conversion of cinnamic aldehyde (A) to 3-phenylpropanol (C) according to the redundancy-free model in Figure 9 is practically negligible compared to the model in Figure 6c

3. Materials and Methods

Two different modelling ways were used to describe the three-phase catalytic hydrogenation of cinnamaldehyde: formal kinetics and kinetics concerning the Langmuir-Hinshelwood theory in the mechanism of heterogeneously catalyzed reaction [11].

The introduced approach is based on an automatically created redundant (generalized) model, which is formulated according to the complete reaction network—GD (Figure 2) [10].

There are six main steps to obtain the redundancy-free model (Figure 10) [9]. These steps can be used several times depending on a one-step simplification strategy or step-by-step simplification strategy [10]. For the hydrogenation of cinnamic aldehyde, only step-by-step simplification was applied. The aim of the model simplification is to obtain such a model, which provides minimal difference between the outputs of the redundant and simplified model:

$$\min \left|\mathit{C}-{\mathit{C}}^{\prime }\right|$$

(6)

With conditions

$$\mathit{C}={{\displaystyle \sum }}^{}\mathit{v}·\mathit{r}\left(\mathit{k}, \mathit{n}\right), \mathit{C}\left(0\right)={\mathit{C}}_{\mathbf{0}},$$

(7)

$${\mathit{C}}^{\prime }={{\displaystyle \sum }}^{}\mathit{v}·\mathit{R}·\mathit{r}\left({\mathit{k}}^{\prime }, {\mathit{n}}^{\prime }\right), {\mathit{C}}^{\prime }\left(0\right)={\mathit{C}}_{\mathbf{0}},$$

(8)

$${\displaystyle \sum }_{i=1}^N{R}_i=m, \mathrm{with} R_i=0 \mathrm{or} 1$$

(9)

where $\mathit{v}$ is the matrix of stoichiometric coefficients, $\mathit{r}, \mathit{k},\mathit{C}, \mathit{n}$ are reaction rate, rate constant, concentration and reaction order vectors, $\mathit{R}$ is a matrix with elements equal to 0 or 1, ${\mathit{k}}^{\prime }, {\mathit{C}}^{\prime },{\mathit{n}}^{\prime }$ are rate constants, concentrations and reaction order vectors, for the redundancy-free model, $N$ is the number of equations in the redundant model; $m$ is the number of equations in the simplified (redundant-free) model.

The formal kinetic model is based on the rate law. The reaction rate is assumed proportional to the rate constant $k$ and concentration $C$ of the reaction components:

$$\frac{1}{v_i}\frac{d{C}_i}{dt}=r_j=k_j{\displaystyle \prod }_i{C}_i^{n_i},$$

(10)

where $n_i$ is the reaction order of the component $i$.

In addition to the formal kinetic approach, the procedure according to Langmuir-Hinshelwood also plays an important role in the heterogeneously catalyzed hydrogenation of cinnamic aldehyde. The assumptions of the Langmuir-Hinshelwood mechanism are to be considered:

• Adsorption of organic molecules on the catalyst surface occurs according to Langmuir;
• Hydrogen adsorption does not compete with the adsorption of organic molecules;
• Surface reaction is the rate-determining step;
• Organic molecules are irreversibly hydrogenated;
• Hydrogenation takes place at the so-called active site (single-site model).

As experimental investigations of the three-phase catalytic hydrogenation of cinnamic aldehyde were carried out at constant pressure, it was assumed that the degree of coverage of the active catalyst surface with hydrogen always remains constant during the experiment. This degree of coverage was included in the equation for determining the corresponding rate constant: $k_i^{\prime }=k_i·{\theta }_H$, where ${\theta }_H$—is the degree of surface coverage with hydrogen. The formulated system of differential equations (10) can also be solved under the assumption of the Langmuir-Hinshelwood mechanism, whereby the rate of every reaction stage is defined by the following equation:

$$r_j=\frac{k_i^{\prime }{C}_i{K}_i}{1+{{\displaystyle \sum }}_{j=1}^M{C}_i{K}_i},$$

(11)

It is assumed that the reaction order is equal 1, and a new parameter, the adsorption constant K_j, is introduced.

In the first step, concerning Equations (10) and (11) a vector-type GD, the (redundant) model is formulated. In the second step, the optimization parameters for the formulated system are determined based on design of experiments (DoE). A fine-tuned minimization of the variance between experimental and simulation data is provided for parameter estimation. In the next step, the solution of the problem is estimated. Starting points are generated using uniformly distributed random numbers to provide a global solution of parameter determination by the downhill simplex method. This procedure is repeated several times according to the optimization parameters defined at the previous stage. The algorithm works in the parameter definition area. At each optimization step, the objective function RSS (Residual Sum of Squares) is calculated:

$$RSS={\displaystyle \sum }_{i=1}^N{\displaystyle \sum }_{j=1}^M{\left[C_{ij}-\overline{C_{ij}}\left(k\right)\right]}^2,$$

(12)

where $C_{ij}$ are the experimental concentration values, and $\overline{C_{ij}}\left(k\right)$ are according to model calculated values.

Based on the contribution of a parameter to the final value of the objective function, it is possible to make a statement about its relevance and redundancy freedom. Thus, this simple procedure can also be applied for the evaluation of parameter relevance in the models of complex chemical reactions. In the approach, the uncertainty is defined in the form of an interval in which the parameter changes take place during its determination from different starting points. These parameters have low contribution to the objective function. Therefore, these determined parameter have almost no influence on the model output and can be eliminated at every elimination step. Simultaneously, sensitivity analysis is performed to determine normalized model parameter values and their intervals and to identify the model parameters that have the greatest influence on the model output. Local or global methods of sensitivity analysis can be used for this procedure [12]. Based on the results, model simplification is provided. If the model can be simplified, a new model is formulated, which is then verified by statistical tests, i.e., the Fisher test (F-test), and the Akaike information criterion (AIC).

The information-theoretic approach allows the data-based selection of the “best” model and a ranking of the remaining models in a pre-defined set. Using the Akaike criterion, the Kullback-Leibler discrimination information can be estimated based on the empirical log-likelihood function at its maximum point [13,14]:

$$L=\log \left((L(\widehat{N,}{\widehat{\sigma }}^2|\mathrm{data})\right)=-\frac{n}{2}·\log \left({\widehat{\sigma }}^2\right),$$

(13)

where ${\widehat{\sigma }}^2$—consistent estimation (maximum likelihood method) of the variance of the model random error; $L$—the value of the log likelihood function for the model; $n$—the sample size. This can be determined according to the following equation:

$$AIC=2N-2L.$$

(14)

where $N$—number of parameters, the Akaike criterion indicates that the experimentally determined kinetic relationships are not informative enough. For such cases, Sugiura suggested model comparisons according to the modified Akaike criterion [15,16]:

$$AI{C}_c=AIC+\left[2N\left(N+1\right)\right]/\left(n-N-1\right),$$

(15)

Since the values of the AIC criterion can be compared only with each other, the analysis uses the criteria difference ${{\Delta }}_i$. The larger the numerical value of ${{\Delta }}_i$ is, the less probability is given that the model f(x)_i is the best of all considered models:

$${{\Delta }}_{i}AI{C}_i=AI{C}_i-AI{C}_{min}.$$

(16)

To simplify the discussion about the model’s information reliability, so-called Akaike weights for R models are introduced [17]:

$${\omega }_i=exp\left(-0,5{{\Delta }}_i\right)/\left[{\displaystyle \sum }_{r=1}^{R}exp\left(-0,5{{\Delta }}_r\right)\right].$$

(17)

After this step, the new model is to be checked in order to find if further simplification is possible. If further simplification is not possible, the redundancy-free model is obtained.

4. Conclusions

An approach for redundancy-free model description of the three-phase catalytic hydrogenation of cinnamaldehyde is introduced. Two different modelling ways were investigated and analyzed: formal kinetics and kinetics concerning the Langmuir-Hinshelwood mechanism. The redundancy-free model formulation procedure consists of six steps, which are used several times in a step-by-step simplification strategy.

Model 7 based on formal kinetics (2nd test run) with RSS = 0.0648 and model 8 (8th test run) based on the Langmuir-Hinshelwood mechanism with RSS = 0.0166 were obtained. The model based on the Langmuir-Hinshelwood approach seems to reflect the experimental results better compared to the formal kinetic approach. On the other hand, it contains 6 model parameters, while the formal kinetic model has 4 parameters. The values of the Akaike criterion for the Langmuir-Hinshelwood model approach are AIC = 7.15 and for the formal kinetic approach AIC = 4.34.

Significant adsorption of the starting molecules on the catalyst surface degreases the freedom of the adsorbed molecules, reducing their reactivity in the redundancy-free model based on the Langmuir-Hinshelwood approach. In this way, three degrees of freedom of molecular rotation in the gas phase are reduced to one degree of freedom for adsorbed molecules. The degrees of freedom of molecular rotation decrease from three in the gas phase to two in the adsorbed state. Therefore, the probability of a direct conversion of cinnamic aldehyde (A) to 3-phenylpropanol (C) according to the redundancy-free model based on Langmuir-Hinshelwood approach in Figure 9 is practically negligible compared to the model based on formal kinetics Figure 6c.

The provided models are based on the assumption that the redundancy-free reaction model is obtained more or less automatically and therefore fast. It was shown that the formulation of the mathematical description of detailed chemical kinetics for the simplest boundary conditions without physical transport is possible without great effort. The approach was also tested for other reaction systems. Therefore, the proposed approach could also be of interest in industrial practice.