Kanat dizilerinde ışınım ve iletimle ısı geçişi

Abstract

ÖZET Bu çalışmada dikdörtgen kesitli boyuna kanatlarda ısı geçişi ve sıcaklık dağılımının ışınım ve üretimin etkisinde çeşitli fiziksel büyüklüklerle nasıl değiştiği bir takım kabuller yapılarak incelenmiştir. Bölüm 2 de açıklandığı gibi kanat üzerinde alınan diferansiyel bir parçada enerji dengesi yazılarak bir integrodiferansiyel denklem elde edilmiştir. Bu denklem bilgisayarda programlanabilecek hale getirildikten sonra FTN77 programlama dili ile Newton Raphson iterasyon yöntemi esas alınarak bir program yapılmıştır. Bu programdan elde edilen sonuçlarla bu modelde ışınım iletim ve toplam ısı geçişleri hesaplanmıştır. Program sonuçlan grafiksel olarak Bölüm 3 'de gösterilmiştir.

SUMMARY HEAT TRANSFER BY CONDUCTION AND RADIATİON IN FIN ARRAYS Physical situations that involve only conduction and radiation are fairly common. Some examples are heat losses through the walls of a vacuum Dewar, heat transfer through `superinsulation` made up of seperated layers of highly reflective material, and heat losses and temperature distributions in satellite and spacecraft structures. This study theoretically investigates the radiative and conduction heat transfer characteristics of the fin array shown in Fig.l to achive the following primary objectives : (1) tiie evaluation of the total heat transfer from the fin ensemble (2) the evaluation of the local fin temperature and (3) the influence of the important parameters on the thermal performance of the fin array. An infinite array of thin fins of thickness b, width W and infinite length are attached black bases that is held at a constant temperatures Tbl and Tb2 as pictured in Fig.l. The fin surface radiates in a difluse gray manner and the fins are in vacuum. Set up the equation necessary for describing the local fin temperature. Because the fins are thin, it will be assumed that the local temperature of the fin is constant across the thickness b. An energy balance will now be derived for the circled differential element of one fin shown in the inset of Fig. 1. Hence from symmetry, only half the fin thickness need be considered. Also the problem is simplified because the temperature distribution Tf(f) of the adjacent fin is the same as Tflx). Thus the energy balance need be considered for only one fin. The conduction terms for the energy into and out of the element dx per unit time and per unit length of fin in the z direction are:w 1ı*Ö QvM dx 1iio(x) ete Qc,o(s) l _, d. Fıg 1. Geometry for the determination of local temperatures on parallel fins The radiation terms are formulated by using Poljakfs net radiation method. The incoming radiation to the element originates from the acent fin and from the base surfaces, The irradition The irradition The irradition from the adjacent from the base 1 from the base 2 fin (3) q^i(x)dx = dx j qR,0(OdFto-df + dx^dF^ + dxaT^dF^ <T=o (4) The outgoing radiation is composed of emission plus reflected incident radiation qR,o(x)dx = eg Tf4(x)+ (1 - e ) qw(x)dx (5) The energy balance on the element is composed of the conduction and radiation quantities q^dx + Qc,0(x)= qwdx + Q^x) (6) By substituting the conduction terms and assuming constant thermal conductivity, the energy balance becomes IXbd2Tf 2 dx2 <kidx = q^dx - k-- f-dx (7) Equation (7) along with (4) and (5) for qKi(x) and q^Cx) give three equations in the unknows q^Cx), q^0(x) and Tf(x). Eliminating the two energy rates q^jCx) and q^Cx) from the three equations result in f (^(Z)- Qz*l£*y& +& +fef& = tfQC) LİÎSS (8) Z{0V w 2«NC dZ2' ^'^ ``` VT`/ *-B w 2*NC dX2 w In the above relation, the following dimonsionless quantities are used. 9(X)=Il t B = ±, Nc =^S-, X=^ and Z=^- (9) Tbl w kb w w Elemental configuration factors appearing in the above are all obtained for energy exchange between differantial elements. F* b = -0 - i * ) = -(1 -, * ) (10) <#W <e = -7 TSTT^ = -7 ~ ^Z (1 1) ^ 2[a2+(^_x)2]3/2 2[Ba+(Z_X)a]3/2 The non-linear nature of the above equations necessitates an iterative solution. These equations were discretized by finite differences and solved explicitly. When finite difference techniques are used in the solution of combined conduction radition problems, the energy equation is replaced by a set of simultaneous nonlinear algebraic equations. To solve a set of nonlinear equations, presented a rapid convergence iteration method for the digital computer based on the Newton Raphson technique. In the Newton Raphson procedure an approximate value for each temperature is assumed. Let tj_t be this approximation for the j 'th temperature and 8j_t is temperature rate for this point. Then a correction factor Axj will be computed so that Gj^S^-Axj. This corrected temperature rate is used to compute a new Axj and the process continued until the Axj becomes smaller than a specified value.The FTN77 program structure for the solution of this problem is shown in fig.2. The program for temperature distribution listed in Appendix A. Main Program that provides input data and cells Subroutine EQNS that provides the simultaneous equations Subroutine SIMUL that performs the Newton Raphson solution of the equations in Subroutine EQNS with appropriate aplication of Subrotunes PARDIF and GAUSS Subroutine PARDIF that extracts partial derivatives Subroutine GAUSS that solves a set of linear simultaneous equations Fig.2 Program Structure The Newton Raphson method, while it is a powerful iteration technique, should be used carefully because if the initial trial is too far off from the correct result, the solution may not converge. Some insight into the nature of the function being solved is therefore always helpful. Then the temperature distributions are found from the computer program, we may calculate the heat flux for all surfaces. Outgoing and incoming radiation for the fin surfaces are found by (4) and (5) equations. The net rate of the radiative heat transfer from the fin surfaces is equal to the difference between the outgoing and incoming radiation. The conduction heat transfer for the element dx calculate from (1). The radiative heat transfer for basel and base2 calculated from the enclosure theory for diffuse gray surfaces. a N 1 N bi j=i fcj j=4 (12) The Total heat transfer from the base 1 equal to the sum of the conduction and radiative heat transfer. We made a new program for the total heat transfer that is listed in Appendix B.Result: There are essentially five independent parameters e, k, a/w, Tb2/Tbl, b and a representative number of values were selected as listed below. The fin temperature distributions are given in fig 3. For small a/w, the radiative heat transfer has a large influence on the temperature distribution. k=50w/mK E=0,1 Tb2/Tb1=0,5 a/w=0.01 - a/w=0.1 a/w=1 Fig 3. The fin temperature distribution The total heat transfer from the base 1 are presented fig. 4. In this figure we may see the augmentation of the total heat transfer by e. So it is clearly appear mat ; for bigger values of a/w, q will take bigger values. When the thermal radiation is significant, the radiant interactionof the extended surfaces with adjacent fins and bases has an important effect on the heat exchange.k=10w/mK 10000 t QT(W) 1000 - ? -E=0,1 E=0,9 MN Fig.4 Total heat transfer