# A Robust Method for the Estimation of Kinetic Parameters for Systems Including Slow and Rapid Reactions—From Differential-Algebraic Model to Differential Model

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Development of a Robust Method

- T1.
- Mass balances of all components in the batch reactor are written down (ODEs)
- T2.
- A quasi-steady-state hypothesis is applied to the intermediates—the ODEs become a system of DAEs is created
- T3.
- The DAE system is converted to a set of implicit ODEs by differentiation
- T4.
- The system of implicit ODEs is converted to a set of explicit ODEs
- T5.
- The system of explicit ODEs is implemented in a combined stiff ODE solver—parameter estimation software Modest.

#### 2.1. Example: Consecutive Reactions with Slow and Rapid Steps

_{2}of the rapid reaction step 2 to find the value of k

_{2}by which the solution of the general case approaches that of the special case.

_{1}= 1 and assuming arbitrary values for the equilibrium constants K

_{1}= 10 and K

_{2}= 10, the concentration profiles depicted in Figure 2 were obtained for the special case.

_{2}of the rapid reaction step 2. Other parameters k

_{1}= 1, K

_{1}= 10 and K

_{2}= 10 were kept the same as in the special case. The sum of squared residuals was calculated for the difference between simulated data points of the special case and the general case (number of simulated data points was 101 in the time frame of 0–10 in each case). Concentration profiles for the general case of consecutive reaction system corresponding to k

_{2}values of 1, 5, 10, 100 and 300 are depicted in Figure 3.

_{1}= 1, k

_{2}= 1, 5, 10, 100 and 300, K

_{1}= 10 and K

_{2}= 10 compared to the special case are presented in Table 1.

_{2}is displayed in Figure 4. The general case approaches the special case as the rate constant k

_{2}of the rapid reaction step 2 exceeds 100 i.e., 100 times the value of the rate parameter k

_{1}of the slow reaction step 1.

#### 2.2. Parallel Reactions with Slow and Rapid Steps

_{A}(0) and C

_{S}(0), C

_{R}(0) = 0 as a function of equilibrium constant K

_{2}in the special case of a parallel reaction system are depicted in Figure 6.

_{1}= 1, K

_{1}= 10 and K

_{2}= 2 are displayed in Figure 7.

_{2}of the rapid reaction step 2 to find the value of k

_{2}by which the solution of the general case approaches that of the special case. Concentration profiles for the general case of parallel reactions corresponding to k

_{2}values of 2, 10 and 50 are depicted in Figure 8.

_{2}while k

_{1}= 1, K

_{1}= 10 and K

_{2}= 2, (the number of simulated data points was 301 in the time interval of 0–15) are shown in Table 2. The behavior of the sum of squared residuals as a function is displayed in Figure 9. The general case approaches the special case in this example when k

_{2}is >50.

#### 2.3. Example: Synthesis of Dimethyl Carbonate from Methanol and Carbon Dioxide

_{2}-MgO) in an isothermal and isobaric laboratory-scale batch reactor [5]. The thermodynamics for this reaction is extremely unfavorable, so a way to shift the equilibrium is to include an additive to the reaction mixture; the role of the additive was to act as a chemical dehydration agent, i.e., to capture the water formed in the reaction. Butylene oxide (BO) was selected as the additive [5]. In this way, the process gains a more irreversible character and can be forced to the side of the products. Methylene butylate (MB) appears as an intermediate species, forming butylene glycol (BG) and thus preventing the water formation. The reaction scheme is displayed below in Figure 10, where * denotes a vacant site on the surface of the catalyst, and MeOH* denotes adsorbed methanol on the catalyst surface.

_{2}was constantly added to the system by keeping the pressure constant. Thus, the saturation concentration of CO

_{2}was presumed, and Henry’s law was applied to relate the partial pressure of CO

_{2}and the concentration of dissolved CO

_{2}. A large excess of methanol was used. Based on the reaction mechanism displayed above, the rate equations for the rate determining steps were derived. The details of the derivation of the rate equations are given as supplementary material in Appendix A: Derivation of the Rate Equations.

#### 2.3.1. Basic Mass Balances

_{2}concentration due to controlled pressure can be written as follows (ρ

_{B}= mass of catalyst-to-liquid volume, i.e., the catalyst bulk density)

#### 2.3.2. Differential-Algebraic Problem

_{1}K

_{5}/K

_{3}),

_{2}O], [MB], [MeOH]).

#### 2.3.3. Transformation to ODEs

_{2}and r

_{4}) were obtained from the mechanism—these rates include only the concentrations of CO

_{2}, MeOH, DMC, MB, BO, BG and H

_{2}O, respectively (Appendix A: Derivation of the Rate Equations). The rate equations are given below. The ODEs were solved with the backward difference method during the parameter estimation, which was performed with the Levenberg–Marquardt algorithm [7].

#### 2.3.4. Parameter Estimation Results

_{2}and k

_{4}were estimated from isothermal experiments shown as Equations (79) and (80). The experimental temperature was 150 °C, and the pressure was 45 bar of CO

_{2}initially. Parameters K

_{1}and 1/K

_{3}(Table 3, Equations (47) and (48)) turned out not to be significant and were approximated to zero in the rate expression r

_{2}. Parameter α was determined separately from the plot according to Equation (47); α = 1.3 × 10

^{−3}. The thermodynamic equilibrium constant was estimated from theoretical calculations: K = 0.08 × 10

^{−5}, at 150 °C.

## 3. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Notation

c | concentration |

c* | concentration of an intermediate |

f | function |

k | reaction rate constant |

r | Rate |

t | Time |

y | concentration variable |

α | parameter in rate equation |

β | parameter in rate equation |

γ | merged concentration |

ρ_{B} | catalyst bulk density (mass of catalyst-to-liquid volume) |

ω | merged parameter |

[] | concentration |

## Appendix A. Derivation of the Rate Equations

## References

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**2010**, 49, 9609–9617. [Google Scholar] [CrossRef] - Eta, V. Catalytic Synthesis of Dimethyl Carbonate from Carbon Dioxide and Methanol. Ph.D. Thesis, Åbo Akademi, Turku, Finland, 2011. [Google Scholar]
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**Figure 2.**Concentration profiles of the special case of consecutive reactions with parameters: k

_{1}= 1, K

_{1}= 10 and K

_{2}= 10.

**Figure 3.**Concentration profiles of the general case of consecutive reactions with parameters: k

_{1}= 1, k

_{2}= 1, 5, 10, 100 and 300, K

_{1}= 10 and K

_{2}= 10.

**Figure 4.**Sum of squared residuals as a function of the rate parameter k

_{2}of the rapid reaction step 2 in the general case of consecutive reaction system compared to the special case (k

_{1}= 1, K

_{1}= 10 and K

_{2}= 10).

**Figure 6.**Initial values C

_{A}(0) and C

_{S}(0), C

_{R}(0) = 0 as a function of equilibrium constant K

_{2}in the special case of a parallel reaction system, where step 1 is slow and step 2 is rapid.

**Figure 7.**Concentration profiles of the special case of a parallel reaction system with parameters: k

_{1}= 1, K

_{1}= 10 and K

_{2}= 2.

**Figure 8.**Concentration profiles of the general case of a parallel reaction system with parameters: k

_{1}= 1, k

_{2}= 2, 10 and 50, K

_{1}= 10 and K

_{2}= 2.

**Figure 9.**The sum of the squared residuals as a function of the rate parameter k

_{2}of the fast reaction step 2 in the general case of a parallel reaction system compared to the special case (k

_{1}= 1, K

_{1}= 10 and K

_{2}= 2).

**Figure 11.**Performance of the solution of ordinary differential equations (ODEs) and parameter estimation method: synthesis of dimethyl carbonate (DMC) (150 °C, initial CO

_{2}pressure of 45 bar). The smaller concentrations in the left figure (butylene oxide (BO), DMC, butylene glycol (BG), methylene butylate (MB)) are magnified in the right figure.

**Table 1.**The sum of squared residuals of the general case simulations as compared to the special case (k

_{1}= 1, K

_{1}= 10 and K

_{2}= 10).

k_{2} | S |
---|---|

1 | 3.6113 |

5 | 0.2373 |

10 | 0.0644 |

100 | 0.006522 |

300 | 0.00071645 |

**Table 2.**The sum of squared residuals of the general case of parallel reaction system compared to the special case with different values of k

_{2}(k

_{1}= 1, K

_{1}= 10 and K

_{2}= 2).

k_{2} | S |
---|---|

2 | 3.31901 |

5 | 1.68380 |

10 | 1.14954 |

50 | 0.89085 |

100 | 0.88929 |

1000 | 0.888893 |

10,000 | 0.888889 |

Parameter | Value | Error/% | |
---|---|---|---|

k_{2} | 1.60 × 10^{−5} | 6.8 | |

k_{4} | 1.12 × 10^{−2} | 6.7 | |

K_{1} = 0, 1/K_{3} = 0, α = 1.3 × 10^{−3}, K = 0.08 × 10^{−5}, at 150 °C |

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**MDPI and ACS Style**

Salmi, T.; Tirronen, E.; Wärnå, J.; Mikkola, J.-P.; Murzin, D.; Eta, V.
A Robust Method for the Estimation of Kinetic Parameters for Systems Including Slow and Rapid Reactions—From Differential-Algebraic Model to Differential Model. *Processes* **2020**, *8*, 1552.
https://doi.org/10.3390/pr8121552

**AMA Style**

Salmi T, Tirronen E, Wärnå J, Mikkola J-P, Murzin D, Eta V.
A Robust Method for the Estimation of Kinetic Parameters for Systems Including Slow and Rapid Reactions—From Differential-Algebraic Model to Differential Model. *Processes*. 2020; 8(12):1552.
https://doi.org/10.3390/pr8121552

**Chicago/Turabian Style**

Salmi, Tapio, Esko Tirronen, Johan Wärnå, Jyri-Pekka Mikkola, Dmitry Murzin, and Valerie Eta.
2020. "A Robust Method for the Estimation of Kinetic Parameters for Systems Including Slow and Rapid Reactions—From Differential-Algebraic Model to Differential Model" *Processes* 8, no. 12: 1552.
https://doi.org/10.3390/pr8121552