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

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

_{2}O

_{3}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].

## 2. Results and Discussion

_{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

_{j}C

_{i}

^{ni}with k

_{j}—reaction rate constant and n

_{j}—reaction order.

_{70}-Fe

_{30}/SiO

_{2}at T = 350 °C are shown in Figure 3 [5].

#### 2.1. Formal Kinetic Approach

_{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 k

_{1}, k

_{3}, k

_{6}, k

_{7}, k

_{8}, k

_{11}, k

_{12}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, 2nd test run). Many model parameters were still at interval limits, although the value of the RSS was significantly reduced.

_{1}= 220); the number of target function calculations (P

_{2}= 500); the number of times the found values are taken as start points to continue the search (P

_{3}= 5).

_{1}, k

_{6}, k

_{7}, k

_{8}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 k

_{2}is close to zero in both cases, which means that it can be neglected. The constants k

_{5}, k

_{9}, k

_{10}and the reaction orders n

_{5}and n

_{6}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 k

_{4}, k

_{8}, k

_{9}, k

_{10}, k

_{12}, and n

_{2}, n

_{3}, n

_{4}, n

_{8}, n

_{9}, n

_{10}, n

_{12}, so that they can be assumed to be negligible. The constants k

_{9}und k

_{10}also take very small values (Figure 4). This means that, in conjunction with the associated reaction orders n

_{9}und n

_{10}, they can also be ignored in the mathematical description of the hydrogenation reaction. Accordingly, the differential equation system (1) can be simplified as follows:

_{2}was found. At the same time, the constants k

_{2}–k

_{4}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 k

_{6}, k

_{8}and n

_{5}–n

_{12}reached their maximum value. This shows that these parameters have little effect on the model output. The exclusion of the rate constants k

_{6}, k

_{8,}and the corresponding reaction orders n

_{6}, n

_{8}was realized in the model with 10 parameters:

_{6}, k

_{8}, n

_{6,}and n

_{8}leads to a certain decrease of the model accuracy; the RSS increases from 0.0375 to 0.0402.

_{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.

_{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: ω

_{6}= 0.4247, ω

_{7}= 0.5689. Model 5 with ω

_{5}= 0.0063 is not significant. The weights of models 1–4 cannot be taken into account, since ω

_{1}= ω

_{2}= ω

_{3}= ω

_{4}= 0.

_{7}/ω

_{j}: ω

_{7}/ω

_{6}= 1.34, ω

_{7}/ω

_{5}= 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

_{6}, k

_{9}, k

_{12}, K

_{2}and K

_{3}can be excluded from further consideration. Thus, the differential equation system (1) can be simplified:

_{4}and k

_{8}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: k

_{1}, k

_{2}, k

_{3}, k

_{5}, k

_{7}, K

_{1}(Table 3, 8th test run). However, after model simplification, the RSS increased from 9.6711 × 10

^{−4}to 0.0166.

_{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: ω

_{1}= ω

_{2}= ω

_{3}= ω

_{4}≈ ω

_{4}≈ 0. The weights of models 6, 7, and 8, calculated according to (17), are: ω

_{6}= 0.0230, ω

_{7}= 0.1779, ω

_{8}= 0.7985. The weight ratios ω

_{3}/ω

_{j}, which indicate how much the respective model contributes to the overall prediction, are ω

_{8}/ω

_{6}= 34.73 und ω

_{8}/ω

_{7}= 4.49.

## 3. Materials and Methods

- 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).

_{j}, is introduced.

_{i}is the best of all considered models:

## 4. Conclusions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Simplified reaction scheme of the hydrogenation of cinnamic aldehyde according to Goupil [6].

**Figure 2.**The redundant (generalized) model for the hydrogenation of cinnamaldehyde as a basis for the formulation of the redundancy-free model.

**Figure 3.**Experimental data of the hydrogenation of cinnamic aldehyde (Catalyst Pt

_{70}-Fe

_{30}/SiO

_{2}, T = 350 °C) [5].

**Figure 4.**Parameter values for the models describing the cinnamic aldehyde hydrogenation in test runs 1, 1a and 2 for different solution conditions.

**Figure 6.**Simplified, redundancy-free models for the hydrogenation of cinnamic aldehyde formulated from a redundant model for the mathematical description of the reaction kinetics: (

**a**) Test run 1(

**c,b**) Test run 1(

**d,c**) Test run 1(

**e**).

**Figure 9.**The simplified, redundancy-free model of the cinnamic aldehyde hydrogenation according to the Langmuir-Hinshelwood approach based on data from Table 4, 8th test run.

**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 | |||

k_{i} ϵ [0,5],n _{i} ϵ [0.55,3] | k_{1} = 4.999 k_{2} = 0.00169 k_{3} = 0.0289 k_{4} = 4.299 k_{5} = 0.798 k_{6} = 4.999 k_{7} = 4.999 k_{8} = 4.834 k_{9} = 4.496 k_{10} = 0.913 k_{11} = 1.997 k_{12} = 0.885 n_{1} = 1.008 n_{2} = 2.672 n_{3} = 0.769 n_{4} = 2.815 n_{5} = 0.550 n_{6} = 0.550 n_{7} = 2.508 n_{8} = 1.471 n_{9} = 2.995 n_{10} = 0.596 n_{11} = 2.999 n_{12} = 2.650 | (1) | 0.0375 |

Test run 1a | |||

k_{i} ϵ [0,5],n _{i} ϵ [0.55,3] | k_{1} = 4.999 k_{2} = 0.00003 k_{3} = 0.274 k_{4} = 0.370 k_{5} = 0.886 k_{6} = 4.999 k_{7} = 4.999 k_{8} = 2.000 k_{11} = 2.472 k_{12} = 3.626 n_{1} = 1.019 n_{2} = 2.967 n_{3} = 0.772 n_{4} = 0.581 n_{5} = 0.500 n_{6} = 0.502 n_{7} = 2.935 n_{8} = 1.958 n_{11} = 2.983 n_{12} = 2.529 | (2) | 0.0375 |

Test run 1b | |||

k_{i} ϵ [0,5],n _{i} ϵ [0.55,3] | k_{1} = 4.999 k_{5} = 0.8526 k_{6} = 4.999 k_{7} = 4.999 k_{8} = 4.960 k_{11} = 0.996 k_{12} = 0.00009 n_{1} = 1.1064 n_{5} = 0.5038 n_{6} = 0.5000 n_{7} = 2.3408 n_{8} = 1.1094 n_{11} = 0.828 n_{12} = 2.994 | - | 0.0402 |

Test run 1c | |||

k_{i} ϵ [0;5],n _{i} ϵ [0.55,3] | k_{1} = 4.999 k_{5} = 9.638·10^{−9} k_{7} = 0.804 k_{11} = 3.30473 k_{12} = 3.969·10^{−5} n_{1} = 0.722 n_{5} = 2.844 n_{7} = 2.999 n_{11} = 2.695 n_{12} = 2.219 | (3) | 0.1001 |

Test run 1d | |||

k_{i} ϵ [0,5],n _{i} ϵ [0.55,3] | k_{1} = 4.999 k_{5} = 6.160 × 10^{−9} k_{7} = 0.211 k_{11} = 1.349k _{12} = 9.394·10^{−8} n_{1} = 0.751 n_{5} = 1 n_{7} = 2.913 n_{11} = 1 n_{12} = 1 | (3) | 0.1103 |

Test run 1e | |||

k_{i} ϵ [0,5],ni ϵ [0.55,3] | k_{1} = 5 k_{5} = 1.907·10^{−12} k_{7} = 0.143 k_{11} = 1.133 n_{1} = 1 n_{5} = 1 n_{7} = 1 n_{11} = 1 | (3) | 0.1270 |

Test run 2 | |||

k_{i} ϵ [0,100],n _{i} ϵ [0.55,3] | k_{1} = 6.771 k_{5} = 3.675 × 10^{−12} k_{7} = 0.199 k_{11} = 1.429 n_{1} = 1 n_{5} = 1 n_{7} = 1 n_{11} = 1 (the reaction orders n_{1}, n_{5}, n_{7}, n_{11} were not determined, but assumed to be 1) | (4) | 0.0648 |

**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 |

**Table 3.**Solution results of the inverse kinetic task determined for the model according to the Langmuir-Hinshelwood mechanism at P

_{1}= 220, P

_{2}= 500, P

_{3}= 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], K _{i} ϵ [0,1] | k_{1} = 0.999 k_{2} = 0.256 k_{3} = 0.0000 k_{4} = 0.999 k_{5} = 0.0233 k_{6} = 0.999 k_{7} = 0.999 k_{8} = 0.0000 k_{9} = 0.300 k_{10} = 0.999 k_{11} = 0.999 k_{12} = 0.655 K_{1} = 0.999 K_{2} = 0.0000 K_{3} = 0.999 K_{4} = 0.999 | (1) | 2.8512 |

Test run 2 | |||

${k}_{i}^{\prime}$ϵ [0,5], K _{i} ϵ [0,5] | k_{1} = 4.999 k_{2} = 0.832 k_{3} = 0.0004 k_{4} = 0.0937 k_{5} = 0.0283 k_{6} = 4.999 k_{7} = 2.891 k_{8} = 4.999 k_{9} = 0.140 k_{10} = 0.0003 k_{11} = 1.124 k_{12} = 0.004 K_{1} = 4.999 K_{2} = 1.559 K_{3} = 2.008 K_{4} = 4.999 | (1) | 0.1056 |

Test run 3 | |||

${k}_{i}^{\prime}$ϵ [0,10], K _{i} ϵ [0,10] | k_{1} = 9.999 k_{2} = 0.00000118 k_{3} = 0.188 k_{4} = 0.702 k_{5} = 0.259 k_{6} = 9.999 k_{7} = 5.441 k_{8} = 9.999 k_{9} = 0.0598 k_{10} = 1.100 k_{11} = 1.502 k_{12} = 0.000006 K_{1} = 4.238 K_{2} = 9.999 K_{3} = 9.997 K_{4} = 7.701 | (1) | 0.0177 |

Test run 4 | |||

${k}_{i}^{\prime}$ϵ [0,100], K _{i} ϵ [0,100] | k_{1} = 20.819 k_{2} = 0.00018 k_{3} = 0.0146 k_{4} = 14.641 k_{5} = 0.0354 k_{6} = 39.731 k_{7} = 3.633 k_{8} = 0.095 k_{9} = 75.125 k_{10} = 47.151 k_{11} = 1.739 k_{12} = 65.005 K_{1} = 10.942 K_{2} = 92.385 K_{3} = 0.0003 K_{4} = 2.752 | (1) | 7.965 × 10^{−4} |

Test run 5 | |||

${k}_{i}^{\prime}$ϵ [0,100], K _{i} ϵ [0,100] | k_{1} = 40.708 k_{2} = 0.0002 k_{3} = 0.0057 k_{4} = 0.1396 k_{5} = 0.0336 k_{7} = 7.136 k_{8} = 0.6578 k_{10} = 1.1538 k_{11} = 2.7629 K_{1} = 2.403 K_{4} = 58.04589 | (5) | 9.6711·× 10^{−4} |

Test run 6 | |||

${k}_{i}^{\prime}$ϵ [0,100], K _{i} ϵ [0,100] | k_{1} = 91.235 k_{2} = 5.632·10^{−7} k_{3} = 2.795·10^{−8} k_{5} = 0.015 k_{7} = 12.094 k_{10} = 0.636 k_{11} = 7.234 K_{1} = 1.016 K_{4} = 99.994 | (5) | 9.7681 × 10^{−4} |

Test run 7 | |||

${k}_{i}^{\prime}$ϵ [0,100], K _{i} ϵ [0,100] | k_{1} = 19.262 k_{2} = 0.039 k_{3} = 1.109·10^{−8} k_{5} = 6.465 × 10^{−9} k_{7} = 0.332 k_{10} = 67.251 K_{1} = 13.854 | (5) | 0.0166 |

Test run 8 | |||

${k}_{i}^{\prime}$ϵ [0,100], K _{i} ϵ [0,100] | k_{1} = 19.272 k_{2} = 0.0392 k_{3} = 1.364·10^{−10} k_{5} = 3.808 × 10^{−9} k_{7} = 0.332 K_{1} = 13.782 | (5) | 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 |

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**MDPI and ACS Style**

Borovinskaya, E.
Redundancy-Free Models for Mathematical Descriptions of Three-Phase Catalytic Hydrogenation of Cinnamaldehyde. *Catalysts* **2021**, *11*, 207.
https://doi.org/10.3390/catal11020207

**AMA Style**

Borovinskaya E.
Redundancy-Free Models for Mathematical Descriptions of Three-Phase Catalytic Hydrogenation of Cinnamaldehyde. *Catalysts*. 2021; 11(2):207.
https://doi.org/10.3390/catal11020207

**Chicago/Turabian Style**

Borovinskaya, Ekaterina.
2021. "Redundancy-Free Models for Mathematical Descriptions of Three-Phase Catalytic Hydrogenation of Cinnamaldehyde" *Catalysts* 11, no. 2: 207.
https://doi.org/10.3390/catal11020207