Temperature mode of a room at integrated regulation of split systems

The article considers simplified mathematical formulation and problem solution of internal air temperature changing. The room is equipped with automated local heating and cooling systems under variable thermal influences. At the same time, the study does not pay attention to the influence of the background general exchange system of supply and exhaust mechanical ventilation. It is shown that the main differential equation connecting the most important components of the heat flow in the room for the case under consideration belongs to the class of Emden-Fowler equations. The article provides an analysis of this equation, obtains the structure and variants of its asymptotic solutions. They describe the time dependence of the room air temperature deviation from the setpoint and the expression for the time moment at which the maximum temperature deviation is observed, with an abrupt change in the heat flow and regulation of the equipment of local heating and cooling system according to the integral law. Calculations were carried out to confirm the obtained dependencies using a numerical solution of the original differential equation by the Runge-Kutta method, as well as by comparing them with the results of field measurements in one residential building in Moscow. It is noted that the structure of the analytical solution and the type of dimensionless complexes constructed by reducing the equation to a dimensionless form directly follow from the properties of the Emden-Fowler differential equations. The obtained ratios are proposed to be used for an approximate assessment of the non-stationary thermal regime of an air-conditioned room served by local heating-cooling systems controlled by the integral law. Moreover, these ratios can be used for determination of the necessary characteristics of the regulator, including on the basis of multivariate calculations with a change in the parameters of the problem.