Determination of the optimal cross-sectional size of a flat plate solar collector

. The article is devoted to the determination of thermal stresses in a flat plate solar collector and the calculation of its optimal size. The solution is found when moving to an elliptical coordinate system. The obtained result is useful because by varying the geometric shape of the cross-section, it is possible to reduce the level of thermal stresses that occur, and, consequently, to increase the reliability and durability of the solar collector design.


Introduction
Solar energy is one of the most suitable options for generating electricity, because it is inexhaustible, free (in terms of its availability) and environmentally friendly. In this connection, there are many works in the literature devoted to optimizing and increasing the efficiency of solar panels [1][2][3][4].
The main and most expensive element of the solar heating system (SST) is the solar collector (SC). Therefore, the task of increasing its thermal efficiency and optimizing the mass-dimensional characteristics and parameters of thermal engineering improvement is in the constant field of view of many researchers.
In 2021, the total area of installed SCS as part of various solar installations in the world amounted to 715 million m2 [1]. At the same time, from 2000 to 2018, the total area of installed SCS in the world increased by 7.6 times [5], of which more than 71% are tubular vacuum collectors SCS [6], most of which use double-layer tubes of Chinese manufacture [7]. The remaining 29% is accounted for by flat solar collectors (FSC), which are widely used in European countries.
It should be noted that the pace of SC application is currently declining, and the production of SC in the world is constantly falling.
Over the past 15 years, the weight and size characteristics and parameters of thermal engineering perfection of flat SCS have practically not changed. They are quite well developed in world practice and have reached parameters close to their limit values. In other words, there is no need for a significant increase in the efficiency of using solar thermal energy in SST by improving the weight and size characteristics and thermal parameters of individual FSC structures in the near future. Therefore, it seems promising to increase the efficiency of using existing flat solar collectors in heat supply systems by optimizing their operating parameters.
The main purpose of the work is to reduce the probability of deformation of the solar collector shell.

Main part
To date, a flat collector is the most common. It consists mainly of an absorber, a transparent coating and a collector housing. The absorber located inside the collector converts solar radiation into heat and gives it to the heat carrier.
A flat solar collector is a product designed to heat water under the influence of sunlight. In other words, it converts the energy of the sun into the thermal energy of a liquid coolant, and the heating can reach very significant temperatures. That is why flat collectors are used in hot water and heating systems, especially in private homes.
The basis of such a collector is an absorber, in which the coolant is heated (water is used in household models, a liquid with antifreeze properties can be used in factory models). The absorber is a thin-sheet metal plate (necessarily black in color for better radiation absorption), to which a small-section serpentine tube fits snugly (welded) from below. Water circulates through this tube, heating up during passage under the metal plate. The absorber is placed inside the housing (wooden or metal), and its walls and bottom are necessarily insulated. This is done to prevent heat loss, which will inevitably occur if the absorber meets the housing. The front part of the case is covered with tempered glass with special optical properties (plexiglass can be used in homemade models).
Moreover, two variants of the device are possible: the water-coolant enters directly into the tank, mixing with cold (single-circuit system) or it passes through the liquid in the tank through a thin tube, warming its volume like a boiler. As it passes, the coolant cools down and returns to the collector without mixing with the water in the tank (two-circuit system).
When operating flat collectors, several factors must be taken into account. Firstly, trees or other houses) should in no case shade their surface, this will lead to a critical drop in productivity. Secondly, after heavy precipitation (for example, with hail) or wind, it is necessary to inspect the front glass. The fact is that hail or wind-blown debris can damage its surface. This mainly concerns cellular polycarbonate, which is most often used in household models. Such scratches lead to a sharp deterioration in the optical properties, which means that the passage of sunlight worsens. In other words, to reduce productivity. It is more expedient to use flat collectors in summer, in conditions of warm and sunny weather. In this case, they will fully justify all the costs of purchasing, installing and integrating the system. Moreover, an excellent option for minimizing costs is the independent manufacture of a flat collector.
There is an opinion that flat solar collectors are much stronger in their design than vacuum ones, and there is an explanation for this, a flat collector consists of a single panel, and unlike a vacuum collector, it looks stronger in appearance and physical data. But even imported flat solar collectors can be damaged. On the other hand, cases of severe weather, large hail, hurricanes or falling aircraft that can damage solar collectors do not occur so often in the usual average area.
Unlike traditional fuel and electric heat generators, the efficiency of PSC depends very much on the average temperature of the absorbing heat exchange panel, which usually has a construction of sheet pipes ( fig.1). Determining the average temperature of the absorbing heat exchange panel is a rather difficult task [8], since this requires a detailed study of the temperature distribution in the collector plane along the x and y axes ( fig.2, a). Under the influence of heat transferred to the liquid, it heats up, and a temperature gradient occurs in it in the direction of flow (along the y axis). Since the total temperature level in any part of the tank is determined by the level of the local temperature of the liquid, the spatial picture of the temperature field will look similar to that shown in fig. 2, b. Temperature distributions in the x-axis direction at any value of y and in the y-axis direction at any value of x are shown in fig. 2, c and d. To find the temperature distribution, it is necessary to solve the Poisson equation, which is an elliptic partial differential equation [8] The determination of thermal stresses in a heat-generating element is reduced to solving a planar thermoelasticity problem. The stress function F is found from the solution of the equations where λ is the coefficient of thermal conductivity of concrete, α is the coefficient of linear expansion, E is the Young's modulus, v is the Poisson's ratio.
We will conduct a comparative analysis of the corresponding components of the thermal stress tensor for circular or elliptical sections. In a circular cylinder, the components of the stress tensor have the form In an elliptical cylinder For further analysis, let us consider the behavior of one of the tensor components. The ratio of the considered thermal stresses is equal to [9]   to the mechanics of a deformable solid is carried out. It is established that the transition to the elliptical cross-section of the collector reduces the level of thermal stresses arising. Thus, by varying the geometric shape of the cross-section, it is possible to reduce the level of thermal stresses that occur, and, consequently, to increase the reliability and durability of solar collector structures.