Model singularly perturbed problems of the soil heat selection

Abstract

Побудовано математичну модель двовимірного нестаціонарного процесу конвективно-дифузійного поширення тепла в трубці ґрунтового теплообмінника за умови превалювання конвективних його складових над дифузійними, яка дозволяє отримати розподіл температури теплоносія вздовж трубки теплообмінника та всередині самої трубки. Розвинуто числово-асимптотичний метод для розв’язання відповідної сингулярно збуреної задачі з умовою третього роду на бічній границі. Наведено результати комп’ютерних розрахунків, здійснено їх аналіз.In this paper we considered mathematical modeling of groundwater heat selection using the ground heat exchanger tubes, which has important applications for the solution of design problems of modern heating systems using renewable energy sources. We built a mathematical model of the two-dimensional unsteady convection - diffusion process of heat in a heat exchanger tube soil, under conditions of the prevalence of convective its components over diffusive ones. This let us get the temperature distribution of the coolant along the heat exchanger tubes and within the tube. Thus we assumed homogeneity of the soil structure, the lack of thermal resistance of the wall and ground heat exchanger pressure loss in the tube bends. This model allows to describe processes of heat transfer in both horizontal and vertical heat exchanger. We built asymptotic expansion of the solution of the corresponding singularly perturbed problem with the condition of the third kind on the side of the boundary (surface of the tube), in which the tube was describes the interaction of the heat exchanger with an array of the ground, which laid up. Application of the asymptotic method allowed to split a complex process into its component parts and stand-alone supplement to the convective component solution amendments on the exit from the tube and side adjustments that take into account the influence of ambient temperature. We performed a series of computer experiments on the results of which we can conclude that the effectiveness of screening low potential ground heat using heat exchanger through the soil for its further use for heating. The fact that the water content of the soil yield a greater increase in heat energy compared with dry sand and clay using horizontal heat exchangers was confirmed. Besides, it was confirmed that vertical heat exchanger is effective for all types of soil, but significant problem is the loss of heat accumulated in depth during the rise of the coolant in the heat exchanger tube through the upper cool layers of the soil