Thermal calculation of a flat plate solar collector

. The article is devoted to the thermal calculation of a flat plate solar collector. In this case, the boundary conditions are boundary conditions of the third kind. The solution is found in the transition to an elliptical coordinate system. The author has obtained an analytical solution for the distribution of the temperature field of a collector with an elliptical cross-section at zero ambient temperature with partial adiabatic isolation in the form of a functional series using hypergeometric functions


Introduction
The increase in demand for electricity leads to the rapid depletion of traditional fossil fuels and exacerbates the problem of environmental pollution.Therefore, there is a need to develop alternative (renewable) energy sources to ensure sustainable energy supplies to consumers, as well as to reduce local and global environmental pollution [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 [5][6][7].
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.

Main part
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 which describes a stationary temperature field with boundary conditions on the body surface: the first half is the heat transfer the second heat transfer is not To obtain the formula describing the temperature field, we use the elliptic coordinate system α, β, 0≤α<∞, -π≤β≤π.If α=α0 is the equation of the body surface, then The Poisson equation in elliptic coordinates has the form ) cos ( 1 (5) and the boundary conditions are given by the dependence for α=α0 0≤β≤π the substitution of the form ( cos where: Bithe number of Bio, which characterizes the intensity of heat exchange between the surface of the body and the environment, h-the coefficient of heat transfer between the medium and the surface of the body. And the boundary condition (7) will take the form The solution of equation ( 8) is given by the dependence We find the constants Bn from the boundary condition ( and integrating from - to 0, we get We find the constants An from the boundary condition ( 9) then all with odd n vanish.Therefore and integrating by β from 0 to π, we get from where elliptic integrals are used to calculate the integrals in the last formula where E is a complete elliptic integral of the second kind where F is a complete elliptic integral of the first kind.
The coefficients a2n are determined by the formula where B(y,z) is the beta function, F1(y,z,m;k) is a hypergeometric function, which is a special function represented by a hypergeometric series that includes many other special functions as specific or limiting cases in which A2n is found from the ratio (28) and B2n+1 from the ratio (15).
We will analyse the resulting solution.In the case of a surface without adiabatic isolation, for reasons of symmetry, we should put Bn =0.Then we get the solution given in [9,10].

Conclusion
In this paper, a thermal calculation of a flat plate solar collector with an elliptical crosssection was obtained under boundary conditions of the third kind.The solution is obtained for the case of adiabatic isolation of half of the wall in the system of elliptic coordinates in the form of a trigonometric series containing hypergeometric functions.The obtained result is compared with the case of the absence of adiabatic isolation and compared with other results of the author's research.

Fig. 1 .
Fig. 1.Flat solar collector of the sheet-tube type: 1 -upper hydraulic collector; 2-lower hydraulic collector; 3 -parallel pipes in the amount of n pieces, located at a distance W from each other.

Fig. 2 .
Fig. 2. Temperature distribution of the absorbing heat exchange panel of a flat solar collector of the sheet-tube type.