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In accordance with the operation principle of the rotary kiln of limestone and its shortcomings, the authors developed the foundations of a mathematical model. The ultimate goal of building algorithms for the proposed method is to reduce fuel consumption when using a rotary kiln. The paper proposes to use a mathematical model based on the method of energy balances. The scientific novelty is that for a rotary kiln for the first time it is proposed to use the method of energy balances by heating zones, depending on the type of fuel burned. The procedure of inputting initial data for mathematical modeling of heat transfer processes allows us to identify the largest heat fluxes in various zones of the furnace. After that, an optimization procedure is carried out using, among other things, objective functions, as well as nonlinear programming. In the end, such calculation algorithms allow us to make recommendations to reduce heat losses in a rotary kiln and a refrigerator, as well as to reduce specific fuel consumption.

It is proposed to put the linear programming method into the basis of the optimization problem of a mathematical economic model of fuel type selection in the boiler house. The linear programming method is used in modern computer programs for calculating various economic models. It is proposed to use graphical or vector methods for solving optimization problems when constructing a mathematical model after analyzing the boiler room operation data. Scientific novelty consists in the fact that for the first time it is proposed to use linear programming methods for an energy technology complex when choosing coal-water fuel based on various coals. Such calculation algorithms allow us to make recommendations to reduce fuel losses during transportation and storage. The solution of the optimization problem of a mathematical economic model can lead to a significant reduction in fuel consumption and an increase in economic efficiency when using energy technology complexes based on coal-water fuel.

An energy balancing scheme has been developed for building a mathematical model of a tubular furnace for heating oil in accordance with the principle of its operation. This will allow taking into account all the heat fluxes, and the result of the work will be a reduction in fuel consumption for the tube furnace. The use of a particular calculation scheme depends on the nature of the task and the features of the boundary conditions of heat transfer. The main factors when choosing a calculation scheme are the accuracy and duration of the calculation. The scientific novelty is that for a tubular heating furnace for the first time it is proposed to use the method of energy balances in the chambers of radiation and convection. The practical significance of the work is to reduce fuel consumption for a tube furnace, as well as the possibility of utilizing the heat of oil refining products when building a mathematical model of a distillation column. The calculation algorithm using the energy balances method for mathematical modeling of heat transfer processes allows balance equations for all furnace sections, including the radiation chamber and the convection chamber. After which several successive approximations are carried out using linear programming. In addition, there are technological solutions to optimize fuel use.