saturs - Latvijas Lauksaimniecības universitāte

U. Iljins, I. Ziemelis The Optimization of Some Parameters of a Flat Plate Solar CollectorI⎧ ∂UI⎪λi= αrU⎪ ∂yy=0y=0⎪III⎪ ∂U− λg= αU⎪ ∂yy=δ1+ δ 2 + δ 3⎪⎪IIIU = U⎪y=δ1y=δ1⎨III∂U∂U⎪λ− λi=⎪ ∂y∂yy=δ1y=δ1⎪II⎪ ∂UIIλ+ αg( U⎪ ∂yy=δ1y=δ1+ δ 2⎪⎪III∂UII⎪−λg+ αg( U⎪ ∂y⎩y=δ1+ δ 2IIIn∑i=1+ δ 2y=δ1+ δ 2 + δ 3q δ ( x − xi−Uy = δ1+ δ 2III−U)y=δ1+ δ 2IIIoi) = 0y=δ1+ δ 2)(19.1)(19.2)(19.3)(19.4)(19.5)(19.6)For further development of the solution, expressions (15) in turn have to be inserted into the boundaryconditions (19) (see the point where function U is expanded as Y k⋅X k). For instance, inserting the formula for U Iinto condition (19.1) the following is obtained:λ ( AI BI ) X ( x)( AI BIi∑ µk k − k k= αr ∑ k + k ) Xk( x). (20)kIn the formula (20), by regrouping all members to one side and then grouping them at the particular functionsX k(x), it is obtained that this equality is equal to zero then and only then, if coefficients are equal to zero at allX k(x). It means thatwhereλ µ ( AI I) ( AIBIk − B k = α k + k ). (21)i krFrom expression (21) the coherence between coefficients A I and k BI can be found:kBIk= AIkλiµk/ αr−1= Aλ µ / α + 1ikλiµk=λ µThen the function U I can be written in the form:ikr/ α/ αrk−1+ 1Ik⋅ϕIk(22)rϕ Ik. (23)I IIU = ∑ A k (exp( µ ky)+ ϕ k exp( −µ ky)) cos µ kx.(24)kInserting U III into equation (19.2) and making an analogous procedure as shown before, the function U III canbe obtained as following:IIIUIII III= ∑ B k ( ϕ k exp( µ k (y − δ1− δ2)) + exp( −µ k (y − δ1− δ2)))cosµk x . (25)kwhereLLU Raksti 12 (307), 2004; 67-75 1-1871