The lumping method most generally can be explained as a form of clustering, during which we take a complex chemical reaction network and replace two or more (or even hundreds) of chemical species with one pseudocomponent (aka.
lump), thus greatly simplifying the chemical reaction network in question. We can do this multiple times, forming a lumped reaction network. There are two common methods for reaching this target [65]: a priori lumping, which is carried out based on empirical rules such as constraining the total number of species and/or reactions, and a posteriori lumping, where the detailed reaction network is generated first (although its parameters are not identified) and the component grouping is carried out based on the properties of the reaction network. The application of lumping methods in modeling complex reaction systems dates back to almost 50 years [66,67]. At first, it was classified as a problem related to the petroleum industry and was applied as an accessory tool, pointing out that some systems are only approximately lumpable.
By and large, there are four main approaches of modeling processes involving complex reaction networks, from simple to sophisticated: single reaction models, discrete lumping, continuous lumping and detailed kinetic mechanisms.
Single reaction models involve one power-law type equation in that the rate of conversion is expressed by a reaction rate coefficient and a function of conversion describing the type of the reaction [68,69]. Chandrasekaran et al. developed a kinetic model based on thermogravimetric analysis (TGA) results to determine and optimize pyrolysis process parameters (including the catalyst used in the reactor) that predicted the overall activation energy of the process on a given catalyst [70]. The apparent activation energy also indicates the difficulty of decomposing the polymer feedstock based on its composition [71]. A closely
related and more advanced approach is the Distributed Activation Energy Model (DAEM) that is also used for modeling the conversion of the feedstock, which consists of several different species, each having a contribution to the decomposition process described by a pseudo-nth-order rate equation, resulting in an activation energy distribution function [72]. Reaction enthalpy contributions can be taken into account as well, leading to detailed reactor simulations [73].
However, the application of single reaction models requires measuring the amount of remaining polymer in the reactor, and that were not available in the case investigated.
During discrete lumping, components are grouped together in such a way that one compound can only be part of one pseudocomponent. Using this method, we can form multiple component groups that in turn will form a lumped reaction network where each predefined group (lump) is treated as a single component.
These models can handle multiple products in an explicit form with relative ease, while both parallel and consecutive reactions steps can be identified with different rates. The lumping approach is commonly used to give researchers a better understanding of experimental data [74,75] or to model industrial-scale processes [76,77]. A great deal of reported applications comes from the oil industry. A five-lump kinetic model for the hydrocracking of heavy oils under moderate conditions was proposed by Sánchez et al., which was capable of predicting component concentrations with an average absolute error of <5% at temperatures of 380−420 °C and liquid hourly space velocity (LHSV) values of 0.33-1.5 h-1 [78].
The effect of pressure on the kinetics of hydrocracking can also be studied with hydrotreatment [82]. While the majority of publications involve quasi-homogeneous phase models, the lumped kinetic modeling approach is applicable for describing multiphase systems in detail as well [83,84]. A detailed,
two-dimensional, non-isothermal, heterogeneous model was established by Forghani et al., by applying two different reaction kinetic networks between four lumps that is also applicable for the scale-up of green diesel production [85]. Lumped kinetic models can also be used in the case of treating various oils from renewable sources, such as biomass tar cracking [86], catalytic cracking of vegetable oils [87,88], or waste cooking oil [89].
Moreover, this approach is not limited to the modeling of hydrocracking.
Csukás et al. developed a dynamic simulation model for plastic waste pyrolysis in tubular reactors at laboratory and pilot-scale with 14 lumps collapsed into four measured groups. The vapor/liquid ratio along the reactor length was also determined [90]. An attempt was also made to incorporate a priori information into the determination of the reaction network, though there are several reaction mechanisms proposed in the literature that are not always based on the same considerations. One approach considers that products are mainly formed from the polymer feedstock directly, with interactions between the products to a degree [91,92]. Al-Salem et al. used such a mechanism that included the primary conversion of the feedstock into five different lumps with the further conversion of waxes to liquids and aromatics (also formed from the polymer) [93]. Another method is to use a more consecutive reaction scheme where lighter products can be formed from each heavier lump (e.g., gases can be formed from both polymer feedstock and liquid intermediates) [94,95]. In addition to that, Westerhout et al.
suggested that PE and PP degrade randomly, producing a range of intermediates considered as a separate lump and the actual products are formed by further cracking in secondary (and ternary) steps [96]. In Section 3.1, I propose a similar mechanism that involves a cracked polymer intermediate.
There are some remarks that kinetic models using discrete lumps are so elementary that their results cannot be reproduced because the feedstock and product compositions are not recognized in-depth sufficiently [97,98]. On the other hand, with appropriate selection of the pseudocomponents considered in the model, it is possible to describe the behavior of the system in detail, e.g., to model catalyst deactivation [99]. Furthermore, there are some cases, e.g., interim measurements or preliminary experimental design procedures, where more
complex methods are not applicable simply because there are no detailed measurement data available.
In the case of continuous lumping, the reaction mixture is represented by a continuous function (such as a function of true boiling point (TBP)) that is then discretized in order to recover the concentration of the sought pseudo-components (defined by TBP range) [100,101]. The advantage of this approach is that any number of lumps can be defined and the reaction rate coefficient can be correlated to the normalized TBP, thus reducing the number parameters to be identified. The disadvantage is that some underlying ideas come directly from the field of hydrocracking (e.g., the form of the so-called yield distribution function);
nevertheless, this method has found its way into the field of modeling other processes as well [102].
Lastly, it should be noted that methods based on detailed kinetic mechanisms are also applicable to model complex reaction systems. These involve a significantly higher number of species and reactions present that makes the identification of model parameters increasingly difficult. One of the possible solutions to this problem is to decompose the problem into smaller subtasks that can be solved sequentially [103]. This approach has been successfully applied to pyrolysis reactors previously [104]. Population balance models can also be used to determine the molecular weight distribution of a polymer during thermal degradation [105,106]. Still, more complex methods generally require more comprehensive measurement data and the acquisition of that is not always feasible.