The temperature field of the liquid flowing in a tube is described by the following equation [4, 5,

7]:

DT3 = A V2T

c = A)V T3 dt

The Laplacian in a system of cylindrical coordinates is:

Equation (20) in cylindrical coordinates is expressed as

In conformity with Newton’s law, on the contact surface between the tube and the liquid (r= r0) the following equality holds:

= — a(T3 — T2 ), d r

where a is the coefficient of convective heat transfer on the surface, W/m2K.

At the fixed points of the tube y = 0 and y = y1 we will determine the temperature as

T3(г, ф,0) = -3,0 ; Тз(г, ф,y) = Тзд. (23)

Having passed in Eq. (21) and boundary conditions (22), (23) to dimensionless variables (6)-(9), we obtain:

In order to solve the problem (24)-(26), function ©3 (р, ф,п) is divided to the components [6, 10]: ©3 (лФn) = ©3l)(лФn) + ©3l)(лФn) + ©3з)(лФn), (27) each of them satisfying Eq. (24) and the following boundary conditions: P U=1 =- Bi(©(1)-© 2 )p=1; (28) ©з1)(аа0)=0, ©31)(аап )=0; (29) ©32)(t,^,n)=0; (30) ©32)(p,^,n ) = ©3,1, ©32W,0) = 0; (31) ©33)(1,<^n) = 0 . (32) and ©33)(ЛА°) = ©3,0 , ©33)(ААП ) = 0 . (33) By Fourier’s method of separation of variables [6], using boundary conditions (31) one obtains:

By Fourier’s method of separation of variables [8], using boundary conditions (32) and (33) we obtain:

Pe_

®{()(P, P,rl) = e 2 2 C022Jk(imp)shpmn +

m=1

+ e 2 7 ІІ C(l cos к (p — (Pmp)shPmV,

k=1m=1 V 2 J

2. Conclusion

The calculations show that in the function describing the temperature field in the plate section O1ABC (17) the temperature is directly proportional to the solar energy density q, which could be expected since the greater the energy density the greater the temperature in the solar collector. The plate thickness is inversely proportional to the temperature, which means that the smaller the plate thickness the greater the temperature in the plate; however, if we decrease the plate thickness the angle ф will be smaller, which means smaller heat flow on chord BC — the main way through which the heat is flowing from the plate to the tube’s coating (31) and further to the liquid in this tube. Considering the temperature function (17) it can be concluded that the smaller distance b between tubes the greater the temperature in the plate; this, however, reduces the area irradiated by sun rays, which would mean that the collector is more inertial. Even such simple reasoning about the collector’s parameters allows for the conclusion that the mathematical description presented here can help to find the optimal sizes for the absorber, which, taken for the whole collector, would provide its maximum efficiency [2,3].

References

[1] P. Shipkovs, G. Kashkarova, K. Lebedeva, J. Shipkovs, M. Vanags, K. Leitans. “The mathematical description for heat conduction process on the surface of a solar collector’s absorber”. World Renewabe Energy Congress IX, Florence, Italy, 2006, CD Full Proceedings, 5 pp.

[2] P. Shipkovs, T. Esbensen, G. Kashkarova, K. Lebedeva, J. Shipkovs. Solar energy use in Latvian conditions. Journal of applied research official journal of Lithuanian Applied Sciences Academy, Lithuania, 2005,

Nr. 2, 68-73 pp.

[3] P. Shipkovs, G. Kashkarova, K. Lebedeva, J. Shipkovs. Prespectives for solar termal energy in Baltic states. Solar World Congress 2005, Orlando, Florida, USA, August 8-12, 2005, CD proceedings, 6 pp.

[4] Исаченко В. П., Осипова В. А., Сукомел А. С. (1969) Теплопередача Москва: Энергия 724 стр. (In Russian).

[5] Лыков А. В. (1972) Тепломассообмен Москва: Энергия 560 стр. . (In Russian)

[6] Будак Б. М., Самарский А. А., Тихонов А. Н. (1972) Сборник задач по математи-ческой физике Москва: Наука 688 стр. (In Russian).

[7] Лыков А. В. (1967) Теория теплопроводности Москва: Высшая школа 600 стр. (In Russian).

[8] Riekstins E.(1969) Matematiskas fizikas metodes Riga: Zvaigzne 620 lpp. (In Latvian).