# A Mathematical Model of Industrial Waste-Derived Fuel Droplet Combustion in High-Temperature Air
## 1 Introduction  
The current hazards of fossil fuel depletion in the medium term and of irreversible climate change due to global warming have prompted the design and implementation of sustainable development policy in the main areas of the economy. Quickly growing population, rapid industrialization and global economic changes have increased the demand for electric and thermal energy [1] even as there is a projected deficit of energy resources and fossil fuels. According to the industry estimates [2], even though there was a decrease in the energy demands during the COVID-19 pandemic, humanity is expected to consume 10% more primary energy in 2030 (and 35% more in 2050) compared to the annual consumption in 2019. Coal energy production is a globally widespread technology that generated about seven billion tons of industrial waste in 2020 alone [3], and this figure is projected to rise to 25 billion tons by 2026 [4]. In addition to industrial waste, municipal waste (including combustible waste) pollutes the environment. According to estimates [5], if humanity annually produced 2.1 billion tons of municipal solid waste (MSW) in 2019, then by 2050 people will generate about 3.4 billion tons. Stored at open-air disposal sites, industrial and municipal solid wastes pollute the environment (air, soil and water) as a result of their slow natural decay [5,6]. Overall, this leads to deteriorating conditions for both people and wildlife living in the vicinity [7, 8, 9, 10]. One of the most economically sparing ways to utilize accumulated and unclaimed waste is the method of burning composite liquid fuels (CLF). The results of a cycle ofexperimental research [11; 12; 13] into the combustion of CLF, consisting of wet coal enrichment waste (high-moisture low-rank coal) and fine MSW, have proved the efficiency of using the waste-to-energy technology to generate heat and power with a relatively low level of anthropogenic emissions. The use of such CLFs in practice is limited by the need to develop a number of compositions with predictable energy characteristics. The scientific justification of a high-potential fuel component composition is time-consuming, costly and challenging. The numerical simulation of physical and chemical processes is an alternative approach to the efficient prediction of fuel combustion characteristics under different conditions. In recent years, many mathematical models have been developed to study the influence of environmental conditions and fuel properties on the thermal decomposition characteristics of different grades of coal [14; 15] and a group of typical MSW components taken separately [16; 17; 18; 19]. Studies on the modeling of MSW thermal decomposition for subsequent practical application generally state the problem with significant assumptions and simplifications, which increases the scale of the problem to be solved and lowers the requirements of the numerical simulation algorithm for the PC's computational resources. Yet this approach does not make it possible to explore the physics and chemistry of solid fuel conversion. Studies like that may use empirical models of the conversion of solid fuel in a densified layer divided into several zones (e.g., drying, pyrolysis, combustion, gasification), to estimate the qualitative and quantitative characteristics of gaseous pyrolysis and gasification products [18]. In a number of works [17; 19], the Eulerian-Lagrangian model is used. It describes gas (liquid) flows with a low content of the dispersed phase [20; 21]. Solid fuel particles are considered as a discrete phase (Lagrangian), and gas (liquid) flows are considered as a continuous medium (Eulerian). However, the Eulerian-Lagrangian approach is not suitable for modeling the processes in suspensions, fluidized bed layers and any other mixtures in which the volume ratio of the solid phase cannot be neglected. Fuel conversion in the fluidized bed is described using the Eulerian-Eulerian model (continuum model) [16; 22], which describes different phases as continuous media.