Estimation of Natural Convection Heat Transfer from Plate-Fin Heat Sinks in a Closed Enclosure

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1 World Academy o Science, Engineering Technology International Journal o Mechanical Mechatronics Engineering Vol:8, o:8, 014 Estimation o atural Convection Heat Transer rom Plate-Fin Heat Sins in a Closed Enclosure Han-Taw Chen, Chung-Hou Lai, Tzu-Hsiang Lin, Ge-Jang He International Science Index, Mechanical Mechatronics Engineering Vol:8, o:8, 014 waset.org/publication/ Abstract This study applies the inverse method threedimensional CFD commercial sotware in conunction with the experimental temperature data to investigate the heat transer luid low characteristics o the plate-in heat sin in a closed rectangular enclosure or various values o in height. The inverse method with the inite dierence method the experimental temperature data is applied to determine the heat transer coeicient. The -ε turbulence model is used to obtain the heat transer luid low characteristics within the ins. To validate the accuracy o the results obtained, the comparn o the average heat transer coeicient is made. The calculated temperature at selected measurement locations on the plate-in is also compared with experimental data. Keywords Inverse method, FLUET, -ε model, Heat transer characteristics, Plate-in heat sin. I. ITRODUCTIO ATURAL convection heat transer with heat sin extended suraces has been widely used in engineering applications. In recent years, with the rapid development o electronic technology, promoting the heat transer rate under the woring process at the desired operating temperature may play an important role to ensure reliable operation o the electronic components. The appropriate design o the heat sin has gradually become attractive or these applications because they provide a more economical solution to the above problem. A great amount o research has been devoted to predict natural convection heat transer between two plate-ins. Elenbass [1] studied the natural convection heat transer or dierent values o the in size, temperature inclination angle. Sparrow Bahrami [] showed the heat transer coeicient between the two vertical plate-ins with dierent in spacing through experiments conclusions rom [1]. Bodoia Osterle [3] applied numerical method to study the heat transer low characteristic between the two vertical plate-ins. In order to increase the heat transer rate o the heat sin under natural convection heat transer, numerous investigations have been conducted to study heat transer empirical correlation o rectangular ins. Vollaro et al. [4] ound the optimum in spacing or the plate-ins on a vertical plate. Basaya et al. [5] applied numerical method to investigate the optimum in spacing or the plate ins on a horizontal plate. Harahap Setio [6] obtained empirical correlation or the plate ins on the horizontal vertical plate by the experiment. Han-Taw Chen is with Department o Mechanical Engineering, ational Cheng Kung University, Tainan 701, Taiwan (Phone: ; htchen@mail.ncu.edu.tw). For many engineering applications, the heat sin was oten placed in a closed enclosure. This has orced many researchers to study the natural convection heat transer luid low in closed enclosure. ada [7] obtained empirical relationship o natural convection heat transer in the horizontal vertical closed narrow enclosure with the ins on a heated rectangular plate by the experiment. Yalcin et al. [8] applied the inite volume method to investigate the optimum in spacing in height in limited enclosure. Tari Mehrtash [9] applied ASYS FLUET solver with zero-equation turbulence model to investigate the natural convection heat transer rom the inclined plate-in heat sins. Their suggested correlations were in good agreement with literature data. However, it can be ound rom [7]-[9] that the calculated temperature did not compare with the experimental data at the selected measurement locations on the in. The present study urther applies the inverse method the commercial sotware FLUET [10] in conunction with the -ε turbulence model the experimental temperature data to determine the heat transer characteristics within the ins on a heated horizontal plate or various values o the air velocity the in spacing. The results also show that the number o the total grid points may slightly increase with the in spacing. The comparn o the average heat transer coeicient will be made in order to validate the accuracy o the results obtained. The calculated temperature at selected measurement locations on the middle in is also compared with experimental data. II. MATHEMATICAL FORMULATIO OF IVERSE SCHEME The two-dimensional inverse heat conduction problem is irst introduced to estimate the unnown heat transer coeicient on the ins o the plate-in heat sins or various values o the in height. The temperature o the in at the selected measurement locations the ambient air temperature are measured rom the experimental apparatus constructed in a closed rectangular enclosure. The inverse method in conunction with the inite dierence method, the experimental temperature data least squares method is used to predict the heat transer coeicient or the three vertical ins attached to a heated horizontal plate. Due to the assumption o non-uniorm distribution o heat transer coeicient, the entire in is divided into several sub-in regions beore perorming an inverse method. The heat transer coeicient in each sub-in region is assumed to be an unnown constant. The measurement locations sub-in regions is shown in Fig. 1. 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2 World Academy o Science, Engineering Technology International Journal o Mechanical Mechatronics Engineering Vol:8, o:8, 014 T(x, 0) = T 0 at y = 0 (3) T = 0 at y = H (4) y International Science Index, Mechanical Mechatronics Engineering Vol:8, o:8, 014 waset.org/publication/ Fig. 1 Physical geometry o the measurement locations sub-in regions Fig. Three parallel rectangular ins mounted on the top surace o a horizontal plate in a rectangular enclosure Under the assumption o the thin in, the temperature gradient in the z-direction (the in thicness) can be neglected the in temperature varies only in the x- y-directions. Moreover, the total surace area o the edge o the in relative to the total surace area o the in is small enough. This implies that the actual heat transer rate dissipated through the in tip is much smaller than the total heat transer rate drawn rom the in base. Thus, the boundary conditions at the edge surace o the ins may be assumed to be insulated [11], [1]. Under the assumption o steady state constant thermal properties, the heat conduction equation or a thin in can be expressed as T T h ( x, y ) + = ( T T ) y t, 0<x<L, 0<y<H (1) Its corresponding boundary conditions are T = 0 at x = 0 x = L () where T denotes the in temperature. x y are the Cartesian coordinates. L, H t denote the length, height thicness o the in, respectively. h(x,y) is the unnown heat transer coeicient. is the thermal conductivity o the in. T 0 T respectively denote the in base temperature the ambient air temperature. The unnown heat transer coeicient in each sub-in region is assumed to a constant. Thus the inite dierence orm o (1) in the -th sub-in region can be expressed as T i+ 1, T i, + T i 1, T i, + 1 T i, + T i, 1 h + = T i, l x l y t or i = 1,,, x, = 1,,, y, = 1,,,. x y are, respectively, the number o nodes in the x- y-directions. l x l y are deined as l x = L/( x -1) l y = L/( y -1). is the number o sub-in region. The inite dierence orm o the boundary conditions ()-(4) can be written as (5) T, = T 0, T x-1, = T x+1, or = 1,,, y (6) Ti, 1 = T0 Ti, y-1 = Ti, y+1 or i = 1,,, x (7) The dierence equations or the nodes at the interace between two neighboring sub-in regions the intersection o our neighboring sub-in regions are similar to those shown in [1]. In order to avoid repetition, they will not be shown in this manuscript. Rearrangement o (5) in conunction with their corresponding dierence equations can produce the ollowing matrix equation as [ K][T] = [F] (8) where [K] is the global conduction matrix. [T] is the matrix representing the nodal temperatures. [F] is the orce matrix. The in temperatures at the selected measurement locations can be obtained rom (8) using the Gauss elimination algorithm. In order to estimate the unnown heat transer coeicient h in the -th sub-in region, additional inormation o the measured in temperatures at measurement locations can be required. The more the number o the analysis sub-in region are, the more accurate the estimate o the unnown heat transer coeicient may be. However, it might be diicult to measure the temperature distribution on the middle in o the present problem using the inrared thermography. Excessive thermocouples in the in may also signiicantly interrupt the International Scholarly Scientiic Research & Innovation 8(8) scholar.waset.org/ /999901

3 World Academy o Science, Engineering Technology International Journal o Mechanical Mechatronics Engineering Vol:8, o:8, 014 International Science Index, Mechanical Mechatronics Engineering Vol:8, o:8, 014 waset.org/publication/ low heat transer behaviors within the ins. Thus, the thermocouple may be applied to measure the in temperatures at the selected measurement locations. The heat transer rate dissipated rom the -th sub-in region q can be written as q = h A ( T T ) da or = 1,,, (9) The average heat transer coeicient on the in h can be deined as [1] A ( T T ) h( x, y) da h = A ( T T ) da (10) h = A hda / A (11) Due to limitations o the number o thermocouples, (10) (11) may not be suitable or the present study. Thus, the average heat transer coeicient may be approximated as [1] h h A A / (1) =1 t h h / =1 t (13) where h denotes the heat transer coeicient at the -th grid point. t represents the total number o grid points on the lateral surace. A A are represented, respectively, lateral surace area o the in area o the -th sub-in region. The average heat transer coeicient h on the in is obtained rom (1) or the inverse method rom (13) or the commercial sotware FLUET. The actual total heat transer rate dissipated rom the in to the ambient Q can be expressed as Q = q = 1 (14) The heat transer coeicient based on the in base temperature can be deined as h Q = A ( T o T ) (15) The least-squares minimization technique can be applied to minimize the sum o the squares o the deviations between the calculated measured in temperatures at the selected measurement locations. The error in the estimates E(h 1, h,, h ) is minimized is deined as cal mea [ ] E ( h 1, h,..., h ) = T, inv T (16) = 1 where T mea T cal, inv are, respectively, the experimental calculated in temperatures at the -th measurement location. T cal, inv is determined rom (8). The estimated value o h or = 1,,, can be determined i the value o E(h 1, h,, h ) is minimum. The details o estimating the unnown value h can be ound in [11]. In order to avoid repetition, they are not shown in this manuscript. The computational procedure o this study is repeated until the value o (T mea - T cal, inv)/t mea or = 1,,, is less than Once the value o h or = 1,,, is determined, h h can be obtained rom (1) (15). III. THREE-DIMESIOAL UMERICAL AALYSIS A hal section o a in is selected or the computation o the commercial sotware. Under the assumptions o the steady-state constant thermal properties, the three-dimensional heat conduction equation or the thin in can be expressed as T T T + + = 0 x y, 0<x<L, 0<y<H, 0<z<t/ (17) The boundary conditions at z = 0 z = t/ can be expressed as T = h(t T ) at z = t/ (18) T = 0 at z = 0 (19) where x, y z are the Cartesian coordinates. Many researchers used various numerical methods to investigate the heat transer characteristics o plate-in tube heat exchangers. However, little inormation is available in the open literature to investigate the eect o the grid points on the numerical results. Thereore, the present study applies computational luid dynamics commercial sotware FLUET [10] with the -ε model to determine the heat transer luid low characteristics within the ins o a plate-in heat sin on heated horizontal plate or various values o the in spacing S. Understing the details o the local heat transer luid low distributions within the ins can be very important in the design o the heat sin. At the same time, in order to validate the accuracy o the present results urther, the computational luid dynamics commercial sotware FLUET [10] in conunction with the experimental temperature data is also applied to determine the heat transer coeicient in temperature. The ambient air with constant properties can be assumed to be incompressible. The low within the ins o the heat sin may be assumed to be three-dimensional, symmetric, turbulent, steady no viscous dissipation. Due to the in base International Scholarly Scientiic Research & Innovation 8(8) scholar.waset.org/ /999901

4 World Academy o Science, Engineering Technology International Journal o Mechanical Mechatronics Engineering Vol:8, o:8, 014 International Science Index, Mechanical Mechatronics Engineering Vol:8, o:8, 014 waset.org/publication/ temperature higher than the ambient temperature, the buoyancy eect is considered. The laminar low the ε turbulence model are applied to simulate the low within the ins. It is ound that the ε model can be the best one even i the Rayleigh number is smaller than Thus, the low within the ins is modeled by the ε model. The continuity, momentum energy equations within the ins are expressed in tensor orm as u i = 0 i (0) ' ' u u u u i u i p u i i ρ u = + µ ( + δi ) ρ (1) i i 3 i T a T ρu a = a ' T ' u a ρc p () where x i, i = 1,, 3, are respectively x, y z. u i, p, T a g are a component o the velocity, pressure, air temperature gravitational acceleration in the x direction, respectively. β is the volumetric thermal expansion coeicient. δ i is the Kronecer delta. ρ, ν, c p a are respectively the density, laminar inematic viscosity, speciic heat thermal conductivity o the air. They are all assumed to be constant. The Reynolds stress tensor turbulent heat luxes in conunction with the Boussinesq approximation are given as ' u i i u ' ρ u = µ t S i ( ρ + µ t ) δ (3) i 3 i ' ' T ρ c a p u i T a = (4) a where the mean strain rate tensor S i is deined as 1 u u i Si = ( + ). Pr t is the turbulent Prtl number. The i turbulent viscosity µ t is deined as µ t = ρc µ / ε. The ε equations o this model with the buoyancy eect can be expressed as: µ ρu = µ + t [( ) ] + G σ + G b ρε ε µ t ε ε ε ρu = [( µ + ) ] + c G c x x x 1ε ε ρ σ ε (5) (6) due to the velocity gradient. G b is the production o turbulent inetic energy due to buoyancy. They are deined as: G u u i u = i t ( + ) i G b µ (7) µ t T = βg a Prt (8) where Pr t is the turbulent Prtl number or energy. The values o Pr t, c µ, c 1ε, c ε, σ σ ε are given as Pr t = 0.85, c µ = 0.09, c 1ε = 1.44, c ε = 1.9, σ = 1.0 σ ε = 1.3. The terms c 1ε (ε/)g c ε ρ(ε /) represented, respectively, the shear generation viscous dissipation o ε. The matching condition o the temperature heat lux at the in-luid interace can be written as: T = T a T T = a a (9) At the solid surace, no-slip conditions are speciied. The in base temperature T 0 is constant. The in temperature at selected measurement locations, T T 0 are obtained rom the present experiment. The boundary condition o top wall is constant temperature T. Other walls are adiabatic. IV. EXPERIMETAL APPARATUS The present experiment is conducted in a closed rectangular enclosure. The length, width height o the rectangular enclosure are L e = 0.39 m, W e = 0.16 m, H e = 0.39 m, respectively. The test in with 0.1 m in length m in thicness is made o stainless material AISI 304. Two dierent values o in height, H = 0.04 m H = 0.06 m, are used. The ambient air temperature the in temperature are measured using the T-type thermocouple. The diameter o the T-type thermocouple is about 0.13 mm. The limit o its error is ±0.4% or 0 C T 350 C. The schematic diagram o three parallel rectangular ins mounted on the top surace o a heated horizontal plate is shown in Fig.. In order to heat three parallel ins, a square heater with 0.08 m in length is ixed on the bottom o this plate using the adhesive tapes (itto Deno Co., Ltd). The test ins horizontal plate enclosed with an insulation material are heated to about 7600 seconds using 40W heater. Four thermocouples placed in a gap between the ins the horizontal plate are ixed to (L/5, 0), (L/5, 0), (3L/5, 0) (4L/5, 0). The average o the our measured temperatures is taen as the in base temperature T 0. Contact thermal resistance between the in the horizontal plate is assumed to be negligible. The ambient air temperature T is measured at 0.3 m over the in. where σ σ ε are turbulent Prtl numbers or ε. respectively. G is the production o turbulent inetic energy International Scholarly Scientiic Research & Innovation 8(8) scholar.waset.org/ /999901

5 World Academy o Science, Engineering Technology International Journal o Mechanical Mechatronics Engineering Vol:8, o:8, 014 International Science Index, Mechanical Mechatronics Engineering Vol:8, o:8, 014 waset.org/publication/ TABLE I COMPARISO OF T, h AD h FOR S = 0.0 m AD VARIOUS VALUES OF H H = 0.04 m H = 0.06 m T 0 = K, T 0 = K T = K T = K h (W/m -K) h (W/m -K) T 1 T T 3 T 4 T 5 T 6 T 7 T 8 T mea T num T mea Inverse FLUET Inverse FLUET Re. [13] Ra s t V. RESULTS AD DISCUSSIO T num All physical properties are evaluated at the average temperature o the in base ambient air temperatures. All computations are perormed with t/s 0., = 14.9 W/m-K. x = 1 y = 17 are perormed or the inverse method commercial sotware FLUET. The commercial sotware FLUET [10] is applied to determine the heat transer characteristics within the ins o the plate-in heat sin. An unstructured grid system containing a non-uniorm distribution o the grid points is used to determine all o the numerical results. High density grid points between the ins the coarse grid points in the remainder o the closed enclosure are used. The initial guess o the unnown heat transer coeicients h is taen as unity or the inverse method. The rectangular in is divided into eight regions, i.e., = 8. The eight thermocouples are ixed at the selected positions on the in, as shown in Fig. 1. T num denotes the in temperature obtained using the commercial sotware FLUET [10] at the -th measurement location. Various heat transer correlations have been proposed or the present problem or the natural convection heat transer rom the plate-in heat sins. Among these heat transer correlations, correlations rom [13] may be selected to be compared with the results obtained. An empirical correlation can be obtained rom [13] as ollows 0.5 Ra s 0.39 u = ( ) + (0.081Ra s ) 1500 (30) where 00 Ra s S/L 0.0. The Rayleigh number Ra s the usselt number u are deined as ollows: 3 gβ(t0 T )S Ra s = (31) αν h S u = air (3) (a) Fig. 3 Temperature distribution between plate-ins or S = 0.0 m (a) H = 0.04 m, (b) H = 0.06 m Comparn o T, h h is shown in Table I or two dierent values o H S = 0.0 m. It is ound that the values o h obtained rom the inverse method slightly deviate rom those rom (30). In order to determine a more accurate numerical result, the appropriate grid points may be selected such that the numerical results o h h are as close as possible to those obtained rom the inverse method. In addition, the dierence o the experimental data numerical in temperature is also small at the selected measurement locations. Table I shows that the numerical results o h h are in good agreement with those obtained rom the inverse method. The calculated temperature o the in at selected measurement locations also agrees with the experimental data. This implies that the present results have good accuracy good reliability. The obtained results o h h decrease with increasing in height. An interesting inding is that the total number o grid points t increases with the in height. The value o t varies with the in height Fig. 3 shows the temperature distribution o the air between the ins or z = 0.5 mm, S = 0.0 m H = 0.04 m 0.06 m. The ins have a higher temperature the temperature gradient close to the in base, as shown in Table I. The temperature variation in the vicinity o the in base seems to be enormous. VI. COCLUSIO The present study proposes the inverse method the computational luid dynamics commercial sotware FLUET in conunction with the experimental temperature data to determine the heat transer characteristics within the ins o plate-in heat sin. The results show that the present numerical results o h h using the -ε model are in good agreement with the present inverse results. The present inverse numerical results o h also agree with those obtained rom [13]. This implies that the present results are reliable. This implies that the application o the -ε model to solve the present problem is appropriate. The total number o grid points may vary with the in height in order to obtain more accurate results. The commercial sotware FLUET in conunction with inverse results or experimental data can be applied to obtain a more accurate heat transer characteristics o plate-in heat sin with an appropriate low model grid points. This study is useul in electronics cooling applications. (b) International Scholarly Scientiic Research & Innovation 8(8) scholar.waset.org/ /999901

6 World Academy o Science, Engineering Technology International Journal o Mechanical Mechatronics Engineering Vol:8, o:8, 014 ACKOWLEDGMET The authors grateully acnowledge the inancial support provided by the ational Science Council o the Republic o China under Grant o. SC 10-1-E MY3. We would also lie to than Proessor Chin-Hsiang Cheng at ational Cheng Kung University to give us the commercial sotware FLUET. International Science Index, Mechanical Mechatronics Engineering Vol:8, o:8, 014 waset.org/publication/ REFERECES [1] W. Elenbass, Heat dissipation o parallel plates by ree convection, Physical, Vol. 9, pp. -8, 194. [] E. M. Sparrow, P. A. Bahrami, Experiments on natural convection between heated vertical plates with either open or closed edges, ASME J. Heat Transer, vol. 10, pp. 1-7, [3] J. R. Bodoia, J. F. Osterle, The development o ree convection between heat vertical plates, ASME J. Heat Transer, vol. 84, pp , 196. [4] A. de Lieto Vollaro, S. Grignai, F. Gugliermetti, Optimum design o vertical rectangular in arrays, Int. J. Therm. Sci., vol. 38, pp , [5] S. Basaya, M. Sivirioglu, M. Oze, Parametric study o natural convection heat transer orm horizontal rectangular in arrays, Int. J. Therm. Sci., vol. 39, pp , 000. [6] F. Harahap, D. Setio, Correlations or heat dissipation natural convection heat transer rom horizontally-based, vertically-inneed arrays, Appl. Energy, vol. 69, pp. 9-38, 001. [7] S. A. ada, atural convection heat transer in horizontal vertical closed narrow enclosure with heated rectangular inned base plate, Int. J. Heat Mass transer, vol. 50, pp , 007. [8] H. G. Yalcin, S. Basaya, M. Sivrioglu, umerical analysis o natural convection heat transer rom rectangular shrouded in arrays on a horizontal surace, Int. Comm. Heat Mass Transer, vol. 45, pp , 00. [9] I. Tari, M. Mehrtash, atural convection heat transer rom inclined plate-in heat sins, Int. J. Heat Mass transer, vol. 56, pp , 013. [10] FLUET Dynamics Sotware, FLUET, Lehanon, H, 010. [11] H.T. Chen, L.S. Liu, S.K. Lee, Estimation o heat-transer characteristics rom ins mounted on a horizontal plate in natural convection, CMES: Computer Modeling in Engineering & Sciences 65, , 010. [1] H.T. Chen, S.T. Lai, L.Y. Haung, Investigation o heat-transer characteristics in plate-in heat sin, Appl. Thermal Eng. 50, , 013. [13] G. D. Raithby, K. G. T. Holls, atural Convection, Hboo o Heat Transer Fundamentals, nd ed., W. M. Rohsenow, J. P. Hartnett E.. Ganic, eds, McGraw-Hill, ew Yor, International Scholarly Scientiic Research & Innovation 8(8) scholar.waset.org/ /999901