So far it has been ascertained that lumped reaction networks can be over-parametrized, a fact that gives space for reducing the number of reactions present in the model. Eventually, one important question arises, namely, what do we gain with the model reduction? Kinetic models constructed using discrete lumping, in general, are already simple. The main advantage of kinetic model reduction is the higher confidence in the identified parameters. I applied bootstrapping on the lumped kinetic models to estimate the confidence intervals of the identified kinetic parameters. The procedure involves the following steps:
1. Identify the parameters of the kinetic model (full or reduced network) using the experimental data available.
2. Generate a set of experimental conditions using a normal distribution with the experimental data as expected values and a ±5% variation normally distributed (2σ confidence). I have chosen the reactor temperature (T) and the amount of raw material used in a batch (m0) in case of thermo-catalytic pyrolysis, and T and reactor feed liquid hourly space velocity (LHSV) in case of VGO hydrocracking.
3. Using the identified kinetic parameters in Step 1, generate a set of simulated experimental data using these experimental conditions, adding no further measurement error.
4. Identify a kinetic parameter set for every generated data set.
5. Because of Step 3, the value of the objective function can reach zero, and the location of this global minimum is known from Step 1. I have also established a threshold for the objective function and deemed the identification step successful if the value of the objective function was below this value (𝑓1(𝑥𝑛) ≤ 10 (Eq. (4.1)) for thermo-catalytic pyrolysis and 𝑓2(𝑥𝑛) ≤ 0.05 (Eq. (5.1)) for VGO hydrocracking).
6. After reaching 100 successful identification steps, the algorithm was terminated.
With this algorithm the confidence in the identified model parameters can be assessed separately since there is no measurement error (because of Step 3) and the error of the optimization algorithm was also eliminated (in Step 5). If the confidence of a model parameter is high, the identified values will be close to each other (low standard deviation) and vice versa. The 100 identified values for each kinetic parameter were used to fit a probability distribution function. I tried out different solutions; here I use a Weibull-distribution fitted to each parameter and give 95% confidence intervals relative to the expected values to characterize parameter uncertainty. Lakshmanan and White used Weibull-distribution to model the distribution of activation energy [244]. Sánchez et al. applied several distributions to approximate distillation curves of the products of VGO hydrocracking, reaching the conclusion that only distributions of at least three parameters could be fitted appropriately, highlighting the Weibull and γ distributions among them [245]. It should be noted that based on my review of the literature, there is no direct antecedent of the method presented in this Section;
nevertheless, the application of the Weibull distribution definitely looks promising.
Table 6.6. Confidence intervals for the identified parameters for VGO hydrocracking.
VGO-N0-R15 VGO-N0-R7
Expected 95% confidence Expected 95% confidence
k0,1 8.73·106 35% 9.14·106 25%
The corresponding probability distribution functions of these 100 successful identification steps are shown in from Figure S1 to Figure S22 in the Appendix.
For the sake of the length of this section, only the parameters of the more complex VGO hydrocracking model are considered here. The expected values of the identified kinetic parameters and the respective confidence intervals are denoted in Table 6.6. The 95% confidence intervals are given relative to the expected values. Except for k0,6, the parameter confidence is retained, or, in many cases, become narrower for the reduced reaction network. Since the latter already has fewer parameters, this indicates higher confidence in the model itself. In order to demonstrate that, I have generated a set of 10,000 parameters using the fitted probability distributions and calculated the value of the objective function at each point. Results are plotted in histograms (Figure 6.7).
Figure 6.7. Occurrence of objective function values calculated using the PDFs of the kinetic parameters for VGO hydrocracking a) complete network, b) reduced
network
For seven reactions, the objective function mostly returned the minimum value, in contrast to the results of the full reaction network. The reason behind this might be the correlation of the kinetic parameters (via the contributions of the related reactions to the same pseudocomponent mass concentrations); hence the probability distributions might not be the most accurate, yet the tendencies are
clear in comparison to the reduced model. In any case, the identification of 30 kinetic parameters with high correlations could be problematic, and this can be clearly avoided if we eliminate as many correlated variables as possible.
6.6 Chapter summary
Global Sensitivity Analysis is an easy to use and powerful method to distinct between the more and the less important model parameters. In this Chapter, I have applied this technique to reduce the number of reactions present in the two lumped reaction networks introduced in Chapter 3. It is essential to note that the reduction of these networks would be not useful just because the model fit remains more or less the same. The usefulness of this development would be at least questionable in the light of the vast computing capacities available nowadays.
Instead, based on the results discussed in this paper we can conclude that the reduction of these reaction networks greatly contributes to lessening the uncertainty in kinetic parameters, leading to more unbiased parameter estimation.
The lower uncertainty of the underlying kinetic model is critical to achieve proper reactor design or operation and the proposed methods can contribute significantly to realize this objective. I showed that global sensitivity analysis is an effective tool in carrying out the model reduction step as it requires minimal information about the kinetic parameters; hence, it can be implemented before the actual parameter identification step. This is a major advantage compared to the methods described in Chapter 5 although I did not reach a reduced reaction network with all its states observable here.
Accordingly, if we define a number of pseudocomponents, we can screen an arbitrary large set of reactions and only identify the relevant reaction pathways, thus automating the building up of the network. This will be a key step in automating the lumping process itself, i.e. determining which lumps have decisive roles in describing the behavior of the chemical system investigated.
7 Structure of lumped reaction networks with correlating parameters
The previous two Chapters dealt with the reduction of lumped reaction networks and the related advantages. The results indicate that the application of sparse reaction networks (i.e., where the ratio of reactions to components is relatively low) is desired. On the other hand, the absolute number of reactions was manageable even before the reduction of the kinetic model. This leads to the idea to increase the number of pseudocomponents present. In the case of the P-N0-R10 reaction network, this can be achieved with relative ease as the liquid product composition is available in a higher level of detail than just the introduced heavy and light liquid (L+ and L–) lumps.
Furthermore; in a chemical reaction network, whether it is a lumped or detailed one, the kinetic parameters might correlate to some extent simply through the amount of products to be formed. If the formation of two or more products has a strong correlation, we might even combine the corresponding reactions to reduce the size of the reaction network. This modeling step instinctively occurs when we have thousands of reactions, but it is usually not carried out in case of a lumped model because of its elementary nature.
In this Chapter I utilize both approaches. I show that the correlations between the amounts of liquid products can be utilized to increase both the number of correlated and uncorrelated lumps considered by applying some not so complicated structural modifications to the original model. This way, we can optimize the structure of the lumped reaction network so that we can capture the characteristics of the measurement rather efficiently without using an overcomplicated lumped reaction model full of uncertain parameters.