**1. Introduction**

Effective transformation of carbon dioxide, which arises when burning fuels in industrial installations on substitution natural gas (SNG), is one of the challenges of the twenty-first century implemented in new research and implementation projects, especially that the pressure of the European Union (EU) to reduce CO2 emissions increases with every decade. In case of facing problems with carbon capture and storage (CCS) technologies, which are expensive, and at the same time due to the lack of social acceptance for the transport of compressed CO2 and its underground storage, the European Commission has recently started to suggest the implementation of carbon capture and utilization (CCU) technology or transformation of carbon dioxide into socially acceptable products [1–3].

1970s and 1980s are years, which we were looking for a solution that would turn sulfur into a product like gypsum. Now, we face a similar problem of what to do with carbon dioxide. Industrial methanation processes, which are based on the transformation of carbon dioxide, e.g., from coal-fired power plants and chemical and metallurgical plants or from cement plants to methane as a result of hydrogen-derived water electrolysis fed with surplus cheap energy or from coal gasification [4].

The purpose of the use of surplus energy from wind farms for hydrogen production is to address the problems related to renewable energy sources. A serious drawback of such installations is the lack of human influence on the current volume of production, which depends directly on the forces of nature, i.e., in the case of a wind farm depends on the instantaneous strength of the wind. Hydrogen produced in the process of electrolysis of water, the additional product of which is oxygen (it can be sold or, for example, used in power coal gasification or for oxygen combustion—in a method that simplifies CO2 capture). Although, carbon dioxide will come from pilot installations for capturing CO2 from exhaust gases. CO2 processing as part of previous research projects in their power plants (mobile installation of amine CO2 capture in Łaziska Power Plant in a project shared with Tauron PE and variable pressure installation in Łagisza Power Plant) is one of the examples of industrial methanation [5].

A research and development organization CEA deals with CO2 methanation in modular structural reactors. The first test reactor was started with a capacity of 3–4 m<sup>3</sup> CO2/h at the inlet. One of the companies dealing in the production, purchase of equipment, and their integration with the entire installation will be prepared by West Technology & Trading (deals in the design and construction of industrial installations, among others) [6].

Industrial research and process optimization play an important role in the development and implementation of industrial installations converting CO2 to methane, which enables effective conversion of CO2 into methane. CO2 management options using surplus energy from renewable sources, as well as the possibility of energy storage by converting surplus electricity into a natural gas substitute. All this also increases the stability of the entire energy system, and above all, it is socially acceptable [7].

Among the advantages of the proposed solution, project participants also mention the ease of the use of captured CO2 without its high compression, transport by pipeline or underground storage, as well as reduction of dependence on natural gas and its imports from outside Europe through the use of gas networks. The risks associated with the project are related to technological issues. This is, for example, the problem of long-term stabilization of the system's efficiency (e.g., catalyst aging) and its durability, as well as the necessity of successful scaling from a demonstration to a commercial installation, for example, on a scale corresponding to the capacity of converting CO2 from a power unit. However, the key aspect in technology is the type of used catalysts [8]. The chapter presents a model of fixedbed methanation reactors for one-, two-, and three-dimensional models with a pseudohomogeneous or heterogeneous phase with the application of various kinetic models. The main kinetic models include Kopyscinski, Xu-Froment, van Herwijnen, and van Doesburg. When creating mathematical models, various types of software were used, including MATLAB, OpenFOAM, FEMLAB, Fortran, Modelica, as well as Aspen Plus and ACM. Modeling the methanation reaction allows to effectively predict the rate of methanation reaction depending on the temperature, pressure, or initial concentrations of reagents. Therefore, model research focuses on the modeling of reactions and reactors; however, the reaction mechanism or the degree of catalyst deactivation factors significantly influences the development of the entire process.
