Mathematical modeling of thermal processes in building enclosures

In the development of automatic control systems (ACS) for heat supply of buildings, mathematical models of their control objects play a significant role, one of the most important elements of which being the enclosing structures. The existing models of thermal processes indoors either do not take into account the dynamic effect of the enclosing structures, or are complex analytical expressions, the use of which in the engineering practice of the synthesis of ACS for the indoor thermal regime is difficult. The purpose of the study was to create a linear mathematical model of the thermal interaction of the external and internal air environment through the enclosing structures. To achieve this goal, a functional diagram of a heated room as a control object was developed, which describes the mechanism of thermal interaction of its elements. On its basis, design schemes of thermal processes for external and internal enclosing structures were developed. They were used to create systems of equations linking temperatures on the surfaces of layers of external enclosing structures and air temperatures outside and inside the premises. At the same time, the assumption was introduced that the dynamic temperature distribution over the thickness of the enclosing structure is linear. To find the temperatures at the boundaries of the layers, the average temperature value over the thickness of the structure was used. As a result of the study, equations for the dynamics of heat transfer processes were obtained, and a generalized structure of a model of multilayer external enclosing structures of a room as an element of a control object was developed. The resulting model can be used to develop a generalized mathematical model of the thermal regime of both individual rooms and the building as a whole.