A Procedural Approach for Finding Kinetic Parameters of Polypropylene Gasification in Super Critical Water Using Genetic Algorithm

doi:10.21203/rs.3.rs-2635577/v1

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A Procedural Approach for Finding Kinetic Parameters of Polypropylene Gasification in Super Critical Water Using Genetic Algorithm

https://doi.org/10.21203/rs.3.rs-2635577/v1

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Plastic gasification under super critical conditions could yield a wide range of hydrocarbons depending on operating conditions. Phenomenological models describing such process involve multiple competing reaction routes. Different reaction mechanisms and kinetic parameters could be used to model the process. Introduced in this study, is a robust/procedural approach for finding the kinetic parameters. In this approach, genetic algorithm is utilized to find the rate constants at specific temperatures prior to finding the kinetic parameters (pre-exponential factor and activation energy). Rate constants are obtained sequentially, starting with the highest temperature and ending with the lowest temperature. Then, linear regression is used to calculate the kinetic parameters for all reactions. Values of kinetic parameters, obtained via linear regression, are used to set the limits for the genetic algorithm to find more accurate values of the kinetic parameters. The relative/normalized deviation, from the original data (main species concentration versus residence time), using this approach is 1.2%. The respective normalized deviation using linear regression was 2.7%.

A procedural approach for using GA in finding kinetic parameters has been established in this work. In this approach, no prior assumptions were made on the search range for each parameters. The approach is meant to set a standard method of obtaining kinetic parameters using an optimization tool.

The objective of this study is to develop a structured approach to finding the kinetic parameters for a global reaction mechanism describing a gasification process under supercritical conditions. The end goal is to be able to predict the concentrations of main species as a function of residence time and temperature, using a phenomenological model. Such models can be simple as few reactions [1]–[9] or as complicated as 10 ~ 20 reactions [10]–[16] depending on the process. This is specifically true in the case of plastics gasification under supercritical conditions [10]–[16], where competing reaction routs are observed. Given the complexity of such process, operating conditions can be optimized to maximize either liquid yield or gaseous hydrocarbons or hydrogen yield. Hence the need for a robust approach for finding kinetic parameters that can be obtained from bench scale experiments and utilized in predicting the outcome of a large scale/industrial reactors.

Although there could be a general correlation between the number of tracked species and the size of the phenomenological model, the same process can be described using different number of reactions and different combinations of reactions [17]–[20]. As mentioned before, the end goal is to use bench scale experimental data to obtain the kinetic parameters of a suggested model/mechanism that can simulate the performance of a full scale reactor as a function of time and temperature. Given the size of reaction mechanisms usually used to model gasification processes, optimization techniques are popular tools for finding the kinetic parameters [3], [4], [21]–[27]. However, researchers either have to go through a process of trial and error or make some assumptions on the range or the values of some of the kinetic parameters in order to find a suitable set of kinetic parameters for a given model. Assumptions on range or certain values, such as the pre-exponential factors, are usually made to avoid numerical instabilities or to reach results using reasonable computational resources or a reasonable time. The former problem is specifically true for the model used in this study; when a wide range of search, for all kinetic parameters, numerical instability problems where encountered due to unrealistic/negative reaction rate values. Hence the motivation to construct a procedural/robust approach to obtain the kinetic parameters. This approach makes no assumptions on the values of any parameter or the search range for the optimization tool. The approach is described in the next section.

In this approach, an assumed reaction mechanism is used without using any prior knowledge of its kinetic parameters or the range of each parameter to find another set of kinetic parameters that result in the best fit of main species concentration versus time and temperature. Although the approach is intended to obtain the kinetic parameters from experimental data, the authors believe an ideal phenomenological model is best suited to test the robustness of the procedural approach. For that reason, a reaction mechanism and kinetic parameters from the literature are used to generate concentration versus time data points at different temperatures. An ideal plug flow reactor was simulated using an open source code, Cantera [28], to generate the concretion profiles of nine species. The plug flow reactor (PFR) operating conditions used to generate the original concentration versus time data is shown in Table 1. The reaction mechanism and original kinetic parameters are shown in Table 2.

Table 1

PFR operating conditions used for data generation
Reactor type	Temperatures (°C)	Pressure (MPa)	Lengths (m)	Input concentration	Inlet velocity (m/s)
Plug flow reactor (PFR)	600, 650, 700, 750 and 800	23	0.45, 0.40, 0.35, 0.3 and 0.25	5% PP (weight)	0.001

Table 2

Reaction mechanism and kinetic parameters used for data generation
Reaction ID (r)	Reaction	Pre-exponential factor (\({A}, {{s}{e}{c}}^{-1}\))	Activation energy (\({E}, {J}/{k}{m}{o}{l}\))
1	C₃H₆➡ C₂H₂ + CH₄	7.37E-02	1.26E + 07
2	C₃H₆ + H₂O ➡ 2 CH₄ + CO	6.45E + 12	3.11E + 08
3	C₃H₆ + 6 H₂O ➡ 3 CO₂ + 9 H₂	3.28E-01	4.36E + 07
4	C₃H₆ + 3 H₂O ➡ 3 CO + 6 H₂	3.82E + 02	1.38E + 08
5	C₃H₆ + 3 H₂➡ 3 CH₄	1.13E + 00	2.14E + 07
6	C₂H₂ + H₂O ➡ CH₄ + CO	3.08E + 03	1.23E + 08
7	C₂H₂ + 4 H₂O ➡ 2 CO₂ + 5 H₂	8.60E + 08	2.58E + 08
8	C₃H₆ + 9 C₂H₂➡ 3 C₇H₈	2.02E + 00	4.06E + 07
9	2 C₇H₈➡ 1.4 C₁₀H₈ + 2.4 H₂	1.22E-01	1.73E + 07
10	5 C₂H₂➡ C₁₀H₈ + H₂	8.75E + 02	1.19E + 08
11	1.6 C₁₀H₈➡ C₁₆H₈ + 2.4 H₂	4.13E + 11	2.83E + 08
12	CO + H₂O ➡⬅ CO₂ + H₂	3.15E + 10	2.41E + 08
13	CO + 3 H₂➡⬅ CH₄ + H₂O	1.31E + 19	4.31E + 08

Genetic Algorithm (GA) was used as the optimization tool. The objective function minimized by the GA is expressed by Eq. 1. The objective function is the absolute deviation from original concertation values for all tracked species at all time steps at a given temperature. The procedures for finding the kinetic parameters are explained through the following steps:

Search for reaction rate constants, for each reaction at the highest temperature, using an optimization tool. GA was used in this study.
Find the reaction rate constants at the next lower temperature using rate constant values obtained from previous step: set the upper limit for each rate constant (k_r,750) equal k_r,800 found in previous step. Set the lower limit, for each reaction, 100 times smaller than the upper limit.
Repeat the previous step until the rate constants at the lowest temperature are obtained
Using linear regression, plot ln(k_r) versus 1/T. Find the kinetic parameters; the pre-exponential factor “A” and the activation energy “E” for each reaction.
Use the kinetic parameters obtained from linear regression to set a narrow and specific search limits for each reaction. Use an optimization tool to find more accurate values of “A” and “E” using data from the entire temperature range. GA is used in this study for optimization.
Compare relative errors of concentrations versus time use the kinetic parameters obtained from linear regression and the kinetic parameters obtained using GA to confirm a better fit is obtained using the GA.

Results of each step are discussed in more details in the next section: “Results and discussion”. Summary of the GA parameters is shown in the appendix (Figure A 1).

Shown in Table 1 are the operating conditions for the reactor model used to generate and find the kinetic parameters. The reactor is an ideal plug flow reactor with a constant operating temperature and pressure. Reactor temperature varied from 600°C to 800°C in 50 degrees increments. The reactor pressure was kept constant at 23 MPa, similar to the study where the reaction mechanism was obtained [10]. Inlet velocity was kept constant as well as the inlet concentration of PP at 5%, weight basis. The reactor length varied from 0.24m to 0.25m in 0.05m increments in a descending order, opposite to the reactor temperature. The logic behind varying the reactor length while maintaining the same flow velocity, is to obtain similar concentration profiles versus time from all temperatures. That help generate good quality data (concentrations versus residence time) and useful objective function values while changing the reactor temperature.

Shown in Table 2 is the reaction mechanism and kinetic parameter used to generate concentration versus residence time data from an ideal PFR. Time dependant concentrations are chosen since that’s the common output that can be obtain experimentally. Time dependency data can be obtained, experimentally, by sampling along the length of the PFR or changing its feeding rate (i.e. altering the residence time of reactants).

Shown in Table 3 are the parameters used to find the reaction rate constant values at 800°C. As mentioned in the approach section, these procedures make no assumptions about the range of reaction rate constants or the order of magnitude of the kinetic parameters. Any attempt to obtain the kinetic parameters (A and E) without having a preconceived idea about the possible range for each reaction resulted in negative reaction rates and numerical instabilities in the modeling tool, Cantera. Hence the approach start with looking for rate constants at the upper temperature limit (800°C in this study). As seen in Table 3, the search limit, for the rate constants, is four order of magnitudes and is the same for all reactions. Hence the population size for the GA model is relatively large; four order of magnitudes per variable. Given the large population size and the fact that the rate constants obtained from this step are not the final values that will be used to obtain the kinetic parameters, a relatively low number of iterations is acceptable.

Table 3

First iteration GA parameters at highest temperature
Rate constant (\({{s}{e}{c}}^{-1}\))	Search limit (all reactions)	Population size	Number of iterations
\({{k}}_{{r},{800}^{^\circ }{C}}\)	1E-5–1E-1	13 X 1E4	500

The objective function minimized by the GA model in this simulation (and all subsequent GA simulations for rate constants) is the singe value obtain from Eq. 1. The objective function is the sum of absolute differences between concentrations values obtain by the GA model and original data for all modeled species (9 species), at all time steps/intervals (150 time steps).

Equation 1. \(Objective\:function \left(OF\right)=\sum _{all\:time\:steps}\sum _{all\:species}\left|{\left[C\right]}_{GA}-{\left[C\right]}_{orig.}\right| (mol/L)\)

Shown in Table 4 are the GA results for the rate constant values at 800°C. Progression of the OF values is shown in the Appendix (Figure A 2). The final OF value (absolute deviation) was 0.38 mol/L. For convenience, a relative deviation from original data is calculated using Eq. 2. Relative deviation from original data, at 800°C, is ~ 1%. Relative deviations at all temperatures are shown in Table 10.

Table 4

Initial GA results at highest temperature (800°C)
Rate constants (\({{s}{e}{c}}^{-1}\))	GA search results at 800°C
\({{k}}_{1,{800}^{^\circ }{C}}\)	2.03E-02
\({{k}}_{2,{800}^{^\circ }{C}}\)	4.84E-03
\({{k}}_{3,{800}^{^\circ }{C}}\)	1.95E-03
\({{k}}_{4,{800}^{^\circ }{C}}\)	2.96E-04
\({{k}}_{5,{800}^{^\circ }{C}}\)	8.29E-02
\({{k}}_{6,{800}^{^\circ }{C}}\)	4.43E-03
\({{k}}_{7,{800}^{^\circ }{C}}\)	5.40E-04
\({{k}}_{8,{800}^{^\circ }{C}}\)	1.81E-02
\({{k}}_{9,{800}^{^\circ }{C}}\)	1.28E-02
\({{k}}_{10,{800}^{^\circ }{C}}\)	1.83E-03
\({{k}}_{11,{800}^{^\circ }{C}}\)	7.11E-03
\({{k}}_{12, {800}^{^\circ }{C}}\)	8.87E-02
\({{k}}_{13,{800}^{^\circ }{C}}\)	1.33E-02

Given a successful / low relative deviation from original data for the rate constants obtained at the highest temperature, they are used to set the limits for subsequent GA search for rate constants at lower temperatures (see Table 5). At this point, a narrower and specific range is assigned for each reaction; for the subsequent temperature, 750°C, the rate constants upper search limits is the rate constant at 800°C and the lower limit is 2 order of magnitudes lower than the upper limit for each reaction. Same limits are applied for subsequent reactions. Because of the narrower and specific range for each rate constant, population size is smaller (2000 X number of variables) and number of iterations is low as well (500 iterations). Following this approached, the GA reached even lower OF/NOF in shorter time duration at subsequent temperatures. Results of the GA search for the rate constants at 750°C (as well as lower temperatures) are listed in Table 6. OF values versus iterations at each temperature are respectively shown in the appendix (Figure A 3, Figure A 4, Figure A 5 and Figure A 6).

Table 5

GA parameters for subsequent iterations at lower temperatures
Rate constants (\({{s}{e}{c}}^{-1}\))	GA search limits
\({{k}}_{{r},{750}^{^\circ }{C}}\)	\([{k}_{r,{800}^{^\circ }C}/100-{k}_{r,{800}^{^\circ }C}], r= 1-13\)
\({{k}}_{{r},{700}^{^\circ }{C}}\)	\([{k}_{r,{750}^{^\circ }C}/100-{k}_{r,{750}^{^\circ }C}], r= 1-13\)
\({{k}}_{{r},{650}^{^\circ }{C}}\)	\([{k}_{r,{700}^{^\circ }C}/100-{k}_{r,{700}^{^\circ }C}], r= 1-13\)
\({{k}}_{{r},{600}^{^\circ }{C}}\)	\([{k}_{r,{650}^{^\circ }C}/100-{k}_{r,{650}^{^\circ }C}], r= 1-13\)
* Population size: 13 X 2000. Iterations: 500

Table 6

GA results for reaction rate constants at subsequent temperatures
Rate constant (\({{s}{e}{c}}^{-1}\))	Temperature
Rate constant (\({{s}{e}{c}}^{-1}\))	750°C	700°C	650°C	600°C
\({{k}}_{1}\)	1.68E-02	1.54E-02	1.43E-02	1.30E-02
\({{k}}_{2}\)	1.19E-03	2.18E-04	3.78E-05	1.31E-05
\({{k}}_{3}\)	1.90E-03	1.50E-03	1.12E-03	8.12E-04
\({{k}}_{4}\)	1.31E-05	5.83E-07	1.56E-08	4.08E-10
\({{k}}_{5}\)	8.18E-02	7.85E-02	6.92E-02	5.94E-02
\({{k}}_{6}\)	1.70E-03	7.72E-04	3.45E-04	1.34E-04
\({{k}}_{7}\)	1.54E-05	3.42E-06	1.21E-06	7.07E-07
\({{k}}_{8}\)	1.63E-02	1.21E-02	9.93E-03	7.27E-03
\({{k}}_{9}\)	1.28E-02	1.27E-02	1.26E-02	1.11E-02
\({{k}}_{10}\)	7.69E-04	3.90E-04	1.70E-04	7.28E-05
\({{k}}_{11}\)	1.59E-03	2.94E-04	4.33E-05	7.53E-06
\({{k}}_{12}\)	1.75E-02	3.88E-03	7.72E-04	8.00E-05
\({{k}}_{13}\)	1.60E-03	9.74E-04	4.31E-04	1.07E-04

Equation 2. \(Normalized\:objective\:function \left(NOF\right)=\frac{OF}{\sum _{all time steps}\sum _{all species}{⌊C⌋}_{orig.}}\times 100 \left(\%\right)\)

Given the rate constants values at each temperature, linear regression is used to obtain the kinetic parameters for each reaction, by plotting ln(k) versus 1/T (see Fig. 1). Results of the linear regression are shown in Table 7.

Table 7

Kinetic parameters results using linear regression
Reaction ID (r)	Pre-exponential factor (\({{A}}_{{r}, {L}{R}} {{s}{e}{c}}^{-1}\))	Activation energy (\({{E}}_{{r}, {L}{R}} {J}/{k}{m}{o}{l}\))
1	1.19E-01	1.63E + 07
2	1.39E + 09	2.37E + 08
3	1.19E-01	3.59E + 07
4	9.95E + 21	5.26E + 08
5	3.85E-01	1.33E + 07
6	1.24E + 04	1.34E + 08
7	7.49E + 07	2.41E + 08
8	1.10E + 00	3.63E + 07
9	2.25E-02	4.80E + 06
10	1.76E + 03	1.24E + 08
11	9.08E + 10	2.70E + 08
12	9.10E + 11	2.68E + 08
13	1.40E + 06	1.70E + 08

Kinetic parameters obtained by linear regression (Table 7) resulted in normalized objective function (relative deviation from original data) in the range of ~ 0.5% − 6.5%, with an average / normalized relative deviation of ~ 2.7%. Although acceptable, a better fit is obtained by re-running the GA search for all kinetic parameters for all reactions, using the values obtained by linear regression to help set the search limits for the GA. Narrower and specific search limits for each parameter are set as shown in Table 8. The pre-exponential factor (A) spans a range of 2 order of magnitude. However, given the exponential effect of the activation energy (E), a range of ± 20% was used as limits for the GA search. Population size was 400 X number of variables (26). Results of GA search for the kinetic parameters are shown in Table 9. The objective function minimized for finding the kinetic parameters is similar to that used to find the rate constants, except concentration values at all temperatures are added to the summation (see Eq. 3 and Eq. 4). Final values of the objective functions (absolute difference in concentrations of all species at all time intervals for all temperatures) are shown in Table 10) for convenience, relative difference (as defined by Eq. 4) is also shown in Table 10. NOF (relative difference between original concentration values and concentrations obtained by the kinetic parameters found via the GA) ranged from ~ 0.82–2.1% with an average deviation of ~ 1.2%.

Table 8

GA search parameters using kinetic parameters from linear regression
Reaction ID (r)	Pre-exponential factor search range \(\left({{s}{e}{c}}^{-1}\right)\)		Activation energy search range \(({J}/{k}{m}{o}{l})\)
\({r}= 1-13\)	\({A}_{r,LR}/10\)	\({A}_{r, LR}\times 10\)	\({E}_{r, LR}-20\%\)	\({E}_{r,LR}+20\%]\)
* Population size: 26 X 400. Iterations: 1000

Table 9

GA search results for the kinetic parameters
Reaction	Pre-exponential factor (\({{A}}_{{r},{G}{A}} {{s}{e}{c}}^{-1}\))	Activation energy (\({{E}}_{{r},{G}{A}} {J}/{k}{m}{o}{l}\))
r₁	1.24E-01	1.66E + 07
r₂	9.29E + 09	2.53E + 08
r₃	2.38E-01	4.12E + 07
r₄	6.26E + 22	5.56E + 08
r₅	2.77E-01	1.08E + 07
r₆	4.59E + 04	1.44E + 08
r₇	2.83E + 07	2.87E + 08
r₈	2.63E + 00	4.15E + 07
r₉	2.98E-02	5.54E + 06
r₁₀	1.34E + 04	1.42E + 08
r₁₁	8.20E + 11	2.89E + 08
r₁₂	3.83E + 12	2.79E + 08
r₁₃	6.70E + 06	1.78E + 08

Equation 3. \(Objective\:function \left({OF}_{A, E}\right)=\sum _{all\:temp.}\sum _{all\:time\:steps}\sum _{all species}\left|{\left[C\right]}_{GA}-{\left[C\right]}_{orig.}\right| (mol/L)\)

Equation 4. \(Normalized\:objective\:function \left({NOF}_{A,E}\right)=\frac{{OF}_{A,E}}{\sum _{all temp.}\sum _{all time steps}\sum _{all species}{⌊C⌋}_{orig.}}\times 100 \left(\%\right)\)

For the purpose of visual comparison between calculated concentrations using kinetic parameters obtained by the suggested approach and original data, Fig. 2, Fig. 3, Fig. 4 and Fig. 5 are utilized. These figures show concentrations of major species versus time at all temperatures. Figure 2 and Fig. 3 represent the behaviour of the main reactant (PP) and the main product (H₂) respectively. Figure 4 and Fig. 5 represent the behaviour of two intermediates; C₂H₂ and CH₄ respectively. Solid lines represent the original data. Square markers represent concentrations obtained from the procedural approach followed above.

Figure 6, Fig. 7, Fig. 8 and Fig. 9 demonstrate the relative importance of the suggested reaction mechanism in the literature as well as the kinetic parameters. Shown in Fig. 6 are three sets of rate constants at 600°C, k_600,r; original data rate constants, rate constants calculated using kinetic parameters obtained by linear regression as well as the rate constants calculated using kinetic parameters obtained from the GA search. Shown in Fig. 7 are the rate constants at 800°C, k_800,r. Shown in Fig. 8 and Fig. 9 are the respective sets of kinetic parameters; original A_orig, E_orig, A_LR and E_LR obtained by linear regression and A_GA and E_GA obtained by the GA search. The purpose of these figures is show the discrepancy in rate constants and kinetic parameters between the original data and the kinetic parameters obtained either by linear regression or GA, however, the excellent agreement in the final results of concentrations versus time shown in Fig. 2 to Fig. 5. Noting the logarithmic scale in Fig. 6 and Fig. 7, it’s obvious the rate constants for reactions 2, 4, 7, 12 and 13 are much different from the original data. This indicates that the process can be expressed via other mechanisms or some of these reaction routes are parallel and, effectively combined, can yield the same concentration profiles.

Also, given the log scale in Fig. 8 and Fig. 9, a more pronounced discrepancy in the values of calculated kinetic parameters (pre-exponential factor, Fig. 8) and (activation energy, Fig. 9) is noticed. That confirms the conclusion that the process can be modeled using different mechanisms. Also, it is possible that the compensation effect plays a role and could result in obtaining the similar values of rate constants using various combinations of kinetic parameters. The important conclusion in this study is, given the end goal represented in the objective function (reactor output versus time and temperature), there is no one answer (mechanism or set of kinetic parameters) can describe the process. Hence, a procedural / systematic approach should be followed while obtaining the kinetic parameters using optimization techniques. This procedural approach is presented in this investigation. The relative/normalized deviation in concentrations (9 species) versus time for at all temperatures (5 temperatures), at all time steps (150 increments) was 1.2%.

Table 10

Absolute differences and relative difference from original data for all methods
Method	600°C		650°C		700°C		750°C		800°C		All Ts
Method	OF (mol/L)	NOF (%)	OF (mol/L)	NOF (%)	OF (mol/L)	NOF (%)	OF (mol/L)	NOF (%)	OF (mol/L)	NOF (%)	OF (mol/L)	NOF (%)
\({{k}}_{{G}{A}}\)	0.02	0.06	0.01	0.03	0.04	0.13	0.07	0.22	0.38	1.07	-	-
\({{A}}_{{r},{L}{R}} , {{E}}_{{r},{L}{R}}\)	0.72	2.3	0.15	0.49	0.54	1.74	0.59	1.86	2.31	6.51	4.32	2.69
\({{A}}_{{r},{G}{A}} , {{E}}_{{r},{G}{A}}\)	0.36	1.15	0.35	1.13	0.26	0.84	0.67	2.11	0.29	0.82	1.93	1.2

A robust approach for finding kinetic parameters for a phenomenological model of plastic gasification under super critical conditions is introduced in this study. In this approach, rate constants are obtained at each temperature prior to finding the kinetic parameters. First, the rate constants at the highest temperature, only, is obtained using genetic algorithm. The values of rate constants at this temperature is used to set a narrow and specific range of search for the rate constants of each reaction at the next lower temperature. This step is repeated until the rate constants at the lowest temperature is obtained. After finding the rate constants for all reactions at all temperatures, linear regression is used to calculate the kinetic parameters; pre-exponential factors add activation energies for all reactions. Then these values of kinetic parameters are used to set the limits for the genetic algorithm to find more accurate values of the kinetic parameters. The relative/normalized deviation, from original data, using this approach, for all concentrations (9 species) versus time, for at all temperatures (5 temperatures), at all time steps (150 increments) was 1.2%. The respective normalized deviation using linear regression was 2.7%.

Acknowledgment

The authors would like to thank the Advanced Clean Energy program for sponsoring this work.

Declarations

Competing Interests

The authors have no relevant financial or non-financial interests to disclose

Author Contributions

All Authors made substantial contributions to enhance the procedural approach. All authors revised the initial draft critically. All Authors have revised the final version for submission.

Data Availability

The data generated during the current study is reported in the main body of the manuscript. Extra data related to the genetic algorithm performance is reported in the supplementary data / appendix.

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A Procedural Approach for Finding Kinetic Parameters of Polypropylene Gasification in Super Critical Water Using Genetic Algorithm

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