Explicit analytical solution of a differential equation model for solar heating systems

Abstract

In the issue of the environmental protection, the role of the widespread application of solar heating systems is very important. Well-usable mathematical models or their solutions can help to promote this aim. In the literature, analytical solutions for the models of solar heating systems are rather rare, despite of their advantages (fast and easy usability, low computational demand, low cost).In the present paper, we intend to (partly) make up for this shortage with respect to a two-dimensional ordinary differential equation model of complete collector-storage systems. The worked out new analytical system solution provides easy- and fast-to-use explicit formulas (for direct algebraic substitutions) to determine both the temperature of the collector and the temperature of the storage.The analytical solution is validated with the measured data of a real solar heating system. As a part of the validation of the (two-dimensional) system solution, the analytical solution of the (one-dimensional) collector sub-model (which is already known and successfully used in the literature) is also validated separately with the measured data of a real solar collector. Based on the validation containing a relatively long time period of 19 days, the error of the new system solution is 3.4% in case of the collector and 2.0% in case of the storage, which means an outstanding precision in the field.