Temperature mode of a room at proportional-integrated regulation of climate systems

The complex including a room serviced by automated microclimate control systems, heat and mass exchange equipment of such systems and their technical means of automation is very complicated for mathematical description of transient processes under conditions of thermal disturbances when solving problems of ensuring comfort of internal meteorological parameters and synthesis of automatic control systems. The nonlinearity of the process of temperature wave propagation in the arrays of enclosing structures leads to a significant error in the case of representation of the room as a linear inertial link. Therefore, the correct mathematical description of transient processes in this case leads to nonlinear differential equations of the Emden–Fowler type, the solution of which in the considered conditions is not expressed in elementary functions and requires numerical methods. One of the most complicated variants in this case is the use of combined proportional-integral law of air conditioning system control in the absence of local heating-cooling systems. Then the solution can be obtained in a parametric form containing a generalized dimensionless parameter of the automated climate control system, including the proportional and integral components transfer factors and the air throughput of the system. The paper shows that in this case, with the growth of the integral component, the dynamic error, i.e., the largest deviation of the room temperature from the setpoint, and the control time are reduced; at the same time, for small moments of time, all the calculated curves describing the temperature behavior asymptotically coincide with the initial heating curve. After reducing the solution to a dimensionless form, the universal dependencies for the indoor temperature behavior depending on the characteristics of the room and regulator parameters, suitable for use in the engineering practice, are obtained. The calculations performed for a typical representative room, in comparison with the relevant experimental data, confirm the reliability of the obtained results and their applicability for mass design.