Heat Transfer Analysis of a Space Radiating Fin with Variable Thermal Conductivity

Transcription

1 Heat Transfer Analysis of a Space Radiating Fin ith Variale Thermal Conductivity Farzad Bazdidi-Tehrani, azdid@iust.ac.ir Department of Mechanical ngineering, Iran University of Science and Technology, Tehran, Iran Mohammad Hadi Kamrava, kamrava@mecheng.iust.ac.ir Department of Mechanical ngineering, Iran University of Science and Technology, Tehran, Iran ABSTRACT fficient heat removal systems are a prerequisite for the safe and satisfactory operation of a satellite. Application of fins such as in space radiating fins represents an important part of the satellite thermal control system. Consideration of constant thermo-physical properties, such as thermal conductivity, may e helpful for the evaluation of fin s temperature distriution. Hoever, a temperature dependent thermal conductivity is considered presently for hen there is a large temperature difference. The finite volume method is employed to simulate numerically the temperature distriution in a space radiating fin. The present results are compared ith those of to other analytical methods and good agreement is shon. Keyords: Finite volume method, Heat transfer, Space radiating fin, Variale thermal conductivity 1. INTRODUCTION Satellites are playing an increasingly important role in the quality of human s life. very Satellite has different susystems hich control its performance. One of these susystems is the thermal control unit devised to lose the generating heat that is produced in other sections, particularly, in the electronic susystem. Considering the space medium, the heat transfer modes are solely conduction and thermal radiation. The main role in the satellite heat transfer is carried out y radiation. Usually, space radiating fins are used for performing the heat loss y radiation. One of the main ojectives of a thermal design is to maintain the temperature of a heat dissipating component at or elo a specified value. The majority of orks reported on space radiating fins, including references [1] - [4], is on the asis of the constant thermal conductivity of the fin material. Bartas and Sellers [1] have studied a heat rejecting system consisting of parallel tues joined y e plates that serve or served (ecause of

2 joined) as extended surfaces. Stockman et al. [2] have compared one-dimensional and todimensional analyses in the central fin-tue radiators and have shon good agreement eteen onedimensional and to-dimensional analyses. Cockfield [3] has discussed the role of the radiator as a structural element in the spacecraft applications. Naumann [4] has investigated analytically/numerically the thermal design of heat pipe/fin type space radiators for the case of uniformly tapered fins as ell as for flat fins ith constant thermal conductivity assumption. hereas, the temperature difference eteen the fin ase and its tip is very high in the actual situation. Hence, the variation of material s thermal conductivity should e taken into consideration. Arslanturk [5] has evaluated the temperature distriution along a radiating fin y the analytical ADM 1 method. He has assumed a temperature dependent thermal conductivity of fin material and compared his results ith those of Naumann [4] and the agreement is shon to e satisfactory. The Adomian decomposition method provides an analytical solution in terms of an infinite poer series. Also, Hosseini and Ghanarpour [6] have applied the HM 2 method for a variale thermal conductivity, displaying their results to e in good agreement ith those of reference [5]. The HM is a ne method to solve a non-linear differential equation analytically and it is on the asis of the perturation technique. The present heat transfer analysis of a space radiating fin is carried out numerically y developing a computer code hich is ased on the finite volume method [7]. The governing equation for the onedimensional configuration is the energy equation encompassing the conduction and thermal radiation modes, under the steady-state condition. The thermal conductivity of fin material is assumed as a linear function of temperature. Also, all radiating surfaces are considered as diffuse and gray. The contact resistance is regarded as negligile. The results on the temperature distriution are otained for different geometric and thermal variales. 2. ROBLM DFINITION The configuration of present radiating fin is shon in Fig. 1. It is ased on an optimized shape provided y Naumann [4] and Arslanturk [5]. The fin material is considered as Aluminum (Al) It is assumed that the fin has a length, and thickness, D connected to a tue at its ase hich acts as a heat pipe. All of the geometrical parameters are measured in meter. The fin ase at x 0 is held at constant temperature, T and oth side surfaces of fin can radiate to outer vacuum space, hich is considered at asolute zero temperature. There are no gradients across the thickness of fin and no significant radiation from the edges, ecause the thickness is assumed to e thin enough. 1 Adomian Decomposition Method 2 Homotopy erturation Method

3 Fig.1. Fin configuration Also, it is presumed that the fin has diffuse and gray surfaces ith a constant emissivity. Radiation exchange eteen surfaces of pipe and fin is negligile. Assuming D, the prolem is solved as one-dimensional heat flo (in the x direction). The governing energy equation is as follos: D d dx dt k T 2 T x 4 (1) dx here, T K is temperature, is Stefan-Boltzmann constant and m K conductivity of fin material hich is assumed to depend on the temperature linearly [5]: k T k T T k is thermal conductivity at fin ase and 1 K here, k. is the thermal 1 (2) is slope of temperature-conductivity curve. The oundary conditions for the present fin geometry are defined as in equation (3): dt 0 dx T T at at x x 0 (3) 3. NUMRICAL DTAILS The present prolem is aimed at the prediction of temperature distriution in an extended surface hile comined conduction-radiation heat transfer is taking place. One-dimensional conduction across the fin and radiation heat loss from side surfaces are taken into account. The governing equation for the present computation is considered as follos [7]: d dt k S 0 (4) dx dx here, S is radiation heat loss that has a negative sign. The discretization procedure, ased on the finite volume technique [7], is carried out on the aove equation to solve the prolem numerically.

4 To derive the discretization equation, the grid point cluster shon in Fig. 2 is employed. The main point is, here the temperature is to e determined. This grid point has the grid points and as its east and est neighors, respectively. The narro line shos the face of the control volume hile letters e and denote these faces. For the one-dimensional prolem under consideration, the thickness in the y and z direction is assumed as unity. The grid points are distriuted in x-direction uniformly. Fig.2 Grid point cluster for the one-dimensional prolem Hence, equation (5) is resulted from the integration of equation (4): dt dt k k e Sdx 0 (5) dx dx e The derivatives of dt dx in equation (5) are evaluated from pieceise linear profile. Then, the resulting equation is as follos: k e T T x e k T T x Sx 0 (6) here, S is the average value of S over the control volume. It is useful to cast the discretization equation (6) into the folloing form: a T a T a T (7) here, a a a k x a Sx k x a (8)

5 4. TRATMNT OF RADIATION HAT LOSS TRM The treatment of radiation heat loss term is considered in equation (4). As shon in equation (1), this term is a function of the dependent variale T itself and it is then desirale to acknoledge this dependence in constructing the discretization equation. The discretization equation is solved y the technique for linear algeraic equations and only a linear dependence can e accountale. The average value S is expressed as: S S S T (9) C here, S stands for the constant part of S, hile C S is the coefficient of T. ith the linearized heat loss expression, the discretization equation ould still look like q. (7), ut the to coefficient definitions in equation (8) ould change. The ne definitions for those to are expressed as: a a a S x (10) S x C 4 The comparison eteen equations (1) and (4) shos that S is the function oft. Thus, the heat loss term is linearized y replacing 4 * 4 * 3 * T y T 4T T T herever it appears. Here, the superscript, *, refers to a previous iteration value. This technique of linearization is on the asis of the Taylor series [7]. To solve the set of matrices from equations (7) to (10), TDMA 3 technique [7] is used. 5. RSULTS AND DISCUSSION In the present ork, the temperature distriution on a set of heat pipe for the temperature dependent thermal conductivity of fin material is evaluated. The present results are compared ith those of Naumann [4] and Arslanturk [5]. For more precise and also simpler analysis, some dimensionless parameters are defined as follos: T T 2 2 T kd 3 x T here, is dimensionless temperature, is a thermo-geometric parameter, is dimensionless length, and is a dimensionless coefficient for thermal conductivity function. The dimensionless thermo-geometric parameter is a function of some thermal, optical and geometrical properties. In this paper is varied y length and other parameters are constant. (11) 3 Tri Diagonal-Matrix Algorithm

6 The present predictions on the dimensionless fin temperature distriution in the axial direction (i.e., fin length, x) are depicted y Fig. 3, for constant thermal conductivity 0. The fin ase temperature and the other variales are assumed as: T 700K, k 257 m. K, 0.85, m It can e seen that the temperature decreases as the fin length increases. Also, the present numerical results are directly compared ith those availale y Naumann [4] and good agreement is shon. Fig.3 Dimensionless fin temperature distriution for constant thermal conductivity 0 Fig. 4 represents the dimensionless temperature distriution at a fixed thermo-geometric parameter,, and for different in the range It is oserved that as is increased from 0.0 toard 0.6 the temperature difference eteen the fin ase and its tip is decreased and consequently the fin can radiate to outer space ith an overally higher temperature. This may lead to an increase in the fin overall efficiency. The values of conductivity of Al at T 100K and T 1000K can e taken from the literature [8] as k 300 m. K and k 200 m. K, respectively. Assuming that the minimum temperature ithin the fin, i.e., tip temperature, is100 K, and employing equations (2) and (11) can e found that the thermal conductivity parameter can e found as 0. 55for the case of variale conductivity. The average conductivity at the same temperature range is k 250 m. K.

7 Fig.4 Dimensionless temperature distriution for different at fixed 1 Fig. 5 demonstrates the distriution of fin temperature for different negative values of in the range to 0.0, at fixed 1. The results for negative values of are in line ith those for positive in Fig. 4. That is, as is increased from -0.4 toard 0.0 the temperature difference eteen the ase and tip is decreased. Fig.5 Dimensionless temperature distriution for different negative at fixed 1

8 The present dimensionless fin tip temperature predictions (i.e., at x=) for different, at constant 1, are depicted in Tale 1. They are compared ith the to recently availale analytical solutions of Arslanturk [5] and Hosseini and Ghanarpour [6]. The percentage of deviation is less than 5% in all cases. Hence, it can e concluded that the present numerical computations are in good agreement ith the orks of others. Tale1. Dimensionless tip temperature at 1 Method resent ork Reference [5] Reference [6] As mentioned efore, the thermo-geometric parameter,, is an important parameter and its influence on the fin axial temperature distriution, at fixed 1, is displayed in Fig.6. It is oserved that as is increased from 1 to 10, the fin tip temperature and hence overall temperature along the fin length are decreased significantly. Fig.6 Dimensionless temperature distriution for different, at fixed 1

9 6. CONCLUSIONS In the present ork, the heat pipe-fin configuration concerning a space radiating fin ith the temperature dependent thermal conductivity of the fin material is investigated numerically. The nonlinear governing equation is solved y the finite volume method. Also, the linearization technique is used for the radiation heat loss term. The temperature distriution along a radiating fin is predicted for different geometric and thermal parameters. The main conclusions may e dran as follos: For a constant thermo-geometrical parameter, the fin tip temperature is increased hile the dimensionless coefficient for thermal conductivity is increased in the range -0.4 to 0.6. Fin tip temperature decreases if increases from 1 to 10, at constant. Dimensionless tip temperature predictions are compared ith to other analytical solutions displaying good agreement ith one another. 7. RFRNCS 1. Bartas, J.G., Sellers,.H., (1960), Radiation fin effectiveness, J. Heat Transf. 82C, Stockman, Norert O., Bittner, dard C., Sprague, Sprague, arl L., (1966), Comparison of one and to dimensional heat transfer calculations in central fin-tue radiators, NASA TN D Cockfield, R.D., (1968), Structural optimization of a space radiator, J. Spacecraft Rockets 5 (10), Naumann, R.J., (2004), Optimizing the design of space radiators, Int. J. Thermophys. 25, Arslanturk, Cihat, (2006), Optimum design of space radiators ith temperature-dependent thermal conductivity, Applied Thermal ngineering 26, Hosseini, M. J., Ghanarpour, M., (2009), Solution of Temperature Distriution in a Radiating Fin Using Homotopy erturation Method, Mathematical rolems in ngineering Volume 2009, Article ID atankar, S.V., (1980), Numerical Heat transfer and fluid flo, Hemisphere ulication Corporation, Taylor and Francis 8. Incropera, Frank., Deitt, David., (2002), Introduction to heat transfer, John iley and sons, 4 th edition