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Heat transfers from pin-fin arrays experiencing forced convection
Applied Energy 67 (2000) 419±442
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Heat transfers from pin-®n arrays experiencing forced convection M. Tahat a,*, Z.H. Kodah a, B.A. Jarrah a, S.D. Probert b a
Department of Mechanical Engineering, Jordan University of Science and Technology, PO Box 3030, Irbid, Jordan b Department of Applied Energy, Cran®eld University, Bedford MK43 OAL, UK
Abstract Steady-state heat-transfers from pin-®n arrays have been investigated experimentally for staggered and in-line arrangements of the pin ®ns, which were orthogonal to the mean air¯ow. For the applied conditions, the optimal spacings of the ®ns in the span-wise and streamwise directions have been determined. The dependences of the Nusselt number upon the Reynolds number and pin-®n pitch (in both directions) have been deduced. # 2000 Elsevier Science Ltd. All rights reserved.
1. The challenge and solution In industrial processes, internal heat-generation may cause overheating and hence sometimes even system failure. So an eective means of removing this heat is often required. Cooling for electronic systems is usually needed to maintain the component temperatures lower than 40 C [1±4] in order to achieve a prolonged mean-life between failures, and so avoid having to make frequent replacements. Fins are used primarily to increase a body's heat-emitting surface area and consequently to enhance its heat-loss rate capability. During operation, the ®ns are at lower temperatures than those of their base plate. For cooling a turbine blade, the circular pin-®ns, spanning the assembly passage between the suction and pressure surfaces of the aerofoil [3,5,6], also serve as structural supports. The rate of heat transfer from a pin-®n assembly to its surrounding environment depends on: (i) the temperature distributions over the pin ®ns as well as the assembly's base; (ii) the pin-®n geometry; (iii) the thermal conductivities of the materials employed; (iv) the
* Corresponding author. Fax: +962-2-709-5018. E-mail address: [email protected] (M. Tahat). 0306-2619/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(00)00032-5
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Nomenclature area, (m2) uniform clearance between the extreme span-wise pin ®ns of the array and the wind-tunnel's walls (see Fig. 1), (m) C vertical clearance between the vertical-®n tips and the upper horizontal shroud (see Fig. 2), (m) speci®c heat of air at atmospheric pressure, (J/kg K) Cp d diameter of each pin ®n (see Fig. 1), (m) D hydraulic diameter (=4Aff =Wb ), (m) f friction factor F grey-body shape-factor : G air-mass ¯ow rate per unit area ( m=Aff ), (kg/m2 s) h heat-transfer coecient Ð see Eq. (8), (W/m2 K) H height of each pin ®n of the considered array (see Fig. 1), (m) thermal conductivity of air, (W/m K) kair L length of the pin-®n assembly (see Fig. 1), (= 0.3 m) : m air's mass ¯ow rate, (kg/s) N number of pin ®ns Nu Nusselt number ( hav d=k) p: static-air pressure, (N/m2) Q steady-state rate of heat loss, (W) Re Reynolds number (=Gd/m) S ®n pitch, i.e center-to-center separation between adjacent pin ®ns (m) T temperature, (K) Tf z ®n's surface temperature at height z above the ®n base, (K) V mean speed of the air in the wind-tunnel duct, (m/s) W width, (m) x, y, z set of Cartesian coordinates (as de®ned in Fig. 3) p overall air-pressure drop along the array, (N/m2) dynamic viscosity of air, (kg/ms) density of air, (kg/m3) Stefan±Boltzmann constant, (W/m2 K4) A B
Subscripts a of the ambient environment av average b of the base plate for the pin-®n assembly conv convective d based on the diameter of the pin-®n f of the ®n of the ®ns in the x and y directions, respectively fx, fy
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in loss max out rad s sf t total x,y z 1,4 2,3
421
corresponding to free ¯ow at the inlet loss maximum at the outlet radiative of the whole surface (in contact with the air) of the pin-®n assembly surface of the pin ®n for an unrestricted cross-section of the wind tunnel (see Fig. 1) total in the span-and stream-wise directions, respectively (see Fig. 1) in the vertical direction entrance and exhaust of the pin-®n assembly, respectively after the ®rst row and prior the last row of the pin-®n assembly,
air-¯ow rate; and (v) especially for low air-¯ow rates, the orientation of the heat exchanger [3,4]. The eects of some of these factors are interrelated. For a speci®ed temperature of the base plate of the pin-®n assembly, the rate of heat transmission can be enhanced by increasing the mean heat-transfer coecient, the emitting surface area or both. An increase in the heat-transfer coecient can be achieved by forced convection or changing the geometrical con®guration of the heat exchanger. However, in practice, this means is limited by the maximum-permitted pressure drop through the assembly [3,4]. 2. Steady-state heat-transfers from a horizontal base-plate, with cylindrical pin ®ns protruding vertically upwards The heat-transfer modes of interest for this system are conduction, convection and radiation through the air. The magnitude of each mode depends on the temperature of the pin-®n array's base, the assembly material(s) and geometry as well as the neighbouring air-¯ow rate. So the heat balance equation from the whole system is: : : : : Q total Qconv Qrad Qloss
1
where : : Q conv Cp m Tout ÿ Tin Also, to a ®rst approximation,
2
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Fig. 1. Schematic representation of the shrouded aligned pin-®n assembly, within the wind tunnel.
: Tin Tout Q conv hav As Tb ÿ 2
3
where d As Wb L dNfx Nfy H ÿ 4
4
The mean surface temperature, Tav , in Eq. (3) is de®ned by: Tav
Tb A Tf Af As
5
According to steady-state heat transfer analysis, Tf Z Ta C1 enx C2 eÿnx
6
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423
Fig. 2. Array of in-line pin ®ns.
and n2 dh= kAf
7
C1 and C2 are constants which can be determined from the boundary conditions at the ®n tips and base, e.g. at z 0, T: 0 Tb . It is apparent, from Eq. (3), that Qconv can be enhanced by increasing (i) the heattransfer surface area As ; (ii) the heat-transfer coecient hav : or (iii) the temperature of the base plate, Tb . In this investigation, the base-plate temperature remains constant throughout the experimental test, so that the rate of heat transfer can only be increased by either increasing the heat-transfer coecient or the surface area or both. From Eqs. (2) and (3), the average convective plus conductive heat-transfer coecient : 8 hav Cp m Tout ÿ Tin = As Tb ÿ Tout ÿ Tin =2 It should : be realized, from this equation, that, as As rises, hav may decrease even though Qconv increases. The radiative heat-transfer rate from a pin ®n depends on: (i) the temperature of its radiating surface; (ii) the temperature distribution of its surroundings; (iii) the
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emissivities of the ®n, base and environment: and (iv) the shielding eect of adjacent ®ns. The radiating surface temperature and the shielding eect determine the greybody shape-factor F. The total steady-state rate of radiative heat-transfer can be evaluated from ÿ : Q rad FAs T4av ÿ T4a
9
From previous steady-state heat-transfer investigations using a similar system [3,4], with (Tb 40 0:5 C), it has been deduced that the total steady-state rate of radiative heat-transfer from polished duralumin pin-®n assembly was less than 0.5% of the total steady-state rate of heat losses from the ®n arrays of length of 250 mm and height of 32 mm. The pin-®ns and base plate were of duralumin; their surfaces being highly polished and so of low emissivity. For the present investigation, the magnitude of the total wild-heat losses plus radiation arrays. The exact value of : is less than 5% of the total input to the pin-®n : : determined in each instance so that Q could then be found from Qrad Qloss was conv : the measured Qtotal as shown in Eq. (1). Hence, the average convective heat-transfer coecient hav could be deduced via : Qconv 10 hav Tin ÿ Tout As Tb ÿ 2 The free-¯ow sectional area Aff is calculated as Aff Wb H ÿ Nx Hd
11
: The mass ¯ow rate, m based on the mean velocity, V, of air in the wind tunnel duct is de®ned as : 12 m At V where At HWb For 250 K4
13
Tin Tout 4400 K; 2
expressions for the speci®c heat, dynamic viscosity and thermal conductivity of air at atmospheric pressure [7] are, respectively: Cp 9:8185 7:7 10ÿ4 Tin Tout =2 10ÿ2 J=kg K
14
air 4:9934 4:483 10ÿ2 Tin Tout =2 10ÿ6 kg=ms
15
Fig. 3. Schematic representation of the experimental rig and associated instruments.
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kair 3:7415 7:495 10ÿ2 Tin Tout 10ÿ3 W=m K
16
2.1. Pressure-drop and friction-factor characteristics The main aim of the heat exchanger is to transfer heat rapidly. In considering the pressure drop for air passing through the heat exchanger, it was essential to take into consideration the entrance pressure-drop, exhaust pressure-rise due to change in the area, and the energy losses occurring due to the ¯ow around and through the ®n array, i.e. frictional losses. Pressure-drop measurements of the air immediately adjacent to the pin-®n assembly, at the vertical plane sections 1±4 orthogonal to the direction of the mean air-¯ow, as shown in Figs. 3 and 4, were obtained by positioning four static-pressure probes located through the horizontal roof of the wind-tunnel duct, as well as along pin-®n array were recorded. The average reading at each section for each ®n geometry, as well as each mass-¯ow rate, were used to deduce the overall pressure drop, via P P1 ÿ P2 P2 ÿ P3 P3 ÿ P4 P1ÿ2 P2ÿ3 P3ÿ4 P1ÿ4
Fig. 4. Schematic diagram of the locations of the pressure taps in the wind-tunnel test section.
17
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427
The pressure drop associated with the ¯ow through the pin-®n array can be represented by the non-dimensional friction-factor, f, which is de®ned by the D'Arcy relationship [8]: f
2P2ÿ3 : L m 2 D Aff
18
Fig. 5. Steady-state rate of heat loss from the heat exchanger to the air via the in-line arrangement of pin ®ns.
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3. Experimental appararatus 3.1. Pin-®n assembly Each regular array assessed consisted of circular-sectioned pin-®ns (each with d 8 mm and H 90 mm) uniformly separated and protruding vertically upwards from a 300 by 250 mm horizontal rectangular base (see Fig. 1). Each array is characterised by its pin-®n spacings in the mean air-¯ow and orthogonal directions. For their least separation, the maximum number of pin ®ns is 341, whereas for the largest spacing, the array consists of only 12 pin ®ns. The spacing can be varied from 1.09 to 83.92 mm in the span-wise direction and from 9.86 to 63.44 mm in the stream-wise direction (see Fig. 2). Each pin ®n can be removed and replaced with a stud made from
Fig. 6. Eect of ®n spacing in span wise direction on the steady-state rate of heat loss for in-line con®gurations.
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429
the same batch of material as used for the assembly base: when screwed in, the studs' heads are ¯ush with the upper surface of the horizontal base-plate. The rectangular base, as well as the pin ®ns, were manufactured from duralumin, i.e. a light aluminum-alloy (No. 2024). For each test, there was zero clearance between the tips of the pin ®ns and the thermally well-insulated horizontal shroud (i.e C 0) [1,2,4,6].
Fig. 7. Eect of ®n spacing in the streamwise direction on the steady-state rate of heat loss for in-line con®gurations.
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3.2. Heating system The base of the heat exchanger was heated uniformly by four 500W electric-resistor strips as the main heater. The assembly was ®rmly bolted together to the bottom surface of the rectangular base (see Fig. 3). The presence of thin layers of high thermal-conductivity heat-sink putty ensured that good thermal contact existed between the main heater and the rectangular base, as well as between the pin-®n roots and the rectangular base [3].
Fig. 8. Eect of Reynolds number on the heat transfer coecient for in-line con®gurations.
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431
The lower horizontal surface and the sides of the main heater block (when operational) were insulated thermally with 50 mm thick mineral-wool blankets. A horizontal guard heater, rated at 50 W, was positioned parallel to the main heater, below the mineral-wool blanket, with yet another 50 mm layer of mineral-wool placed below it (see Fig. 2). The whole system of heat-exchanger base, main and guard heaters, with associated thermal insulation, was located in a well-®tting opentopped wooden box. The horizontal upper edges of this box and the top surfaces of
Fig. 9. Eect of ®n spacing in spanwise direction on the heat transfer coecient for in-line con®gurations.
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the laterally placed thermal insulant, during each experiment, were ¯ush with the upper surface of the base of the pin-®n array [3,4]. The power supplied to the main heater could be adjusted by altering the Variac setting and was measured by an in-line electronic Wattmeter. The dissipation in the guard heater was adjusted until the steady-state temperature dierence, across the layer of insulant, sandwiched between the two heaters, was zero. Then, under all the test conditions employed, more than 98% of the heat generated in the main heater passed, to the air of the surrounding environment, through the ®nned heat-exchanger. The steady-state temperature at the base of the pin-®n array was measured by
Fig. 10. Eect of ®n spacing in streamwise direction on the heat transfer coecient for in-line arrangements.
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an appropriately distributed set of four copper-constantan thermo-junctions, embedded within the rectangular base. Each thermo-junction was bonded in position with a thin layer of epoxy resin, so as to ensure good thermal-contact ensued. The average value obtained from these thermo-junctions was regarded as the mean overall base temperature. This was maintained constant during each experiment at 40 (0.5) C [3,4].
Fig. 11. Eect of Reynolds number on the steady-state rate of heat loss for staggered pin-®n con®gurations.
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The inlet and outlet air-stream temperatures in the wind-tunnel duct were measured using eight thermojunctions: four were located immediately prior to the entrance to the pin-®n assembly and another four just downstream of the array. Each of these thermojunctions could be traversed across either the whole inlet or the outlet cross-sections of the wind-tunnel duct (see Fig. 3). All the thermocouples, as well as those indicating the ambient air temperature were connected, through ribbon cables, to a data-logger, which was used to interpret the temperatures, at half-hourly intervals: when consecutive values were identical, it was assumed that steady-state conditions had been attained [3,4].
Fig. 12. Eect of ®n spacing in spanwise direction on the steady-state rate of heat loss for staggered con®gurations.
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3.3. Wind tunnel The main body of the rectangular cross-section wind-tunnel duct (see Fig. 2) was manufactured from wood and was 3 m long, with a constant internal width of 405 mm. However, the uniform vertical height of the duct, and hence the duct's crosssectional area, could be varied. Dierent duct heights were obtained by means of an adjustable horizontal roof (or shroud). Approximately half-way along the length of the wind-tunnel duct was the test section. The roof and side walls of this test section
Fig. 13. Eect of ®n-spacing in streamwise direction on the steady-state rate of heat loss for staggered con®gurations.
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were made of 5 mm Perspex, so enabling the pin-®n array (and the air, via smoke¯ow visualization around it) to be observed [3,4]. A bell-mouth section was ®tted as the entrance to the wind-tunnel duct, followed by some ¯ow straighteners, i.e. sieves that allowed the air-¯ow to pass to and through the shrouded length without encountering any sudden change of crosssection of the passage. The exhaust air from the pin-®n assembly was passed through an insulated chamber, where mixing was accomplished by two sets of sieves, one being of relatively low porosity and the other of higher porosity. The latter was situated upstream of the former. The two sets of sieves were mounted orthogonal to
Fig. 14. Eect of Reynolds number on the heat-transfer coecient for staggered con®gurations.
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437
the undisturbed ¯ow-stream. At the exhaust end of the duct, a gradual crosssectional area contraction duct was connected via ¯exible tubing to a single-speed, single-stage fan, which was capable of providing a maximum suction rate of 0.33 kg/ s. Pitot-static tubes were employed to measure the dynamic pressure before and after the pin-®n array, so as to be able to deduce the mean air-¯ow speed. The wind tunnel was operated in the suction mode, (i.e. the fan sucked air through the pin-®n array and the test section) with the fan and motor assembly on the exhaust side of the test section. This avoided the air-stream being heated by the fan's motor prior to its passage through the heat exchanger.
Fig. 15. Eect of ®n spacing in the spanwise direction on the heat-transfer coecient for staggered con®gurations.
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The velocity pro®le of the inlet air-stream to the pin-®n assembly was identi®ed via two traversable pitot-static tubes. An electronic analogue micro-manometer was employed to measure the pressure drops [3,4]. 4. Experimental results and discussion Various ®n arrays were employed together with a range of inlet air speeds. The resulting convection correlations are valid for 3:138 103 4Red 46:683 103 0:004074Sx =Wb 40:3316 and 0:032874Sy =L40:2115
4.1. In-line arrangement The steady-state rate of heat loss rose as the Reynolds number was increased, but decreased with increasing pin-®n spacing for both the stream-wise and the span-wise directions (see Fig. 5). The optimal value of the spacing in the span-wise direction divided by the base at 0.01340.0033 (see Fig. 6). That for the stream-wise width, Sx =Wbÿopt , occurred direction, i.e. Sy =L opt , was slightly beyond the range employed in this investigation, but, to a ®rst approximation, is likely to be somewhat similar (see Fig. 7). The average heat-transfer coecient (hav ) increased with Reynolds number for each pin-®n spacing (see Fig. 8). The dimensionless optimal pin-®n pitch in the spanwise direction ÿSx =Wb opt occurred at 0.1350.055 (see Fig. 9) and for the streamwise direction, Sy =L opt at 0.1730.027 (Fig. 10). 4.2. Staggered arrangement The steady-state rate of heat loss was enhanced as the Reynolds number increased. Also, it decreased with increasing pin-®n spacing in both the stream-wise and spanwise directions (see Figs. 11±13). The optimal pin-®n spacings in both the span-wise and stream-wise directions were not withinÿ the range of the current experimental investigation. However, Sx =Wb opt and Sy =L opt were likely to be less than 0.05 and 0.02 respectively (see Figs. 12 and 13). The heat-transfer coecient increased with Reynolds number (see Fig. 14). The optimal pin-®n spacing Sx opt divided by the base width Wb for the maximum
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439
steady-state ÿ heat-transfer coecient occurred at 0.190.03 (see Fig. 15), whereas that for Sy opt =L ensued at 0.1000.025 (see Fig. 16). 4.3. Correlations It was assumed [5] that the steady-state convective behaviour of the pin-®n array could be described by n2 n3 Sy Sx 19 Nud aRen1 Wb L
Fig. 16. Eect of ®n spacing in streamwise direction on the heat-transfer coecient for staggered pin-®n con®gurations.
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where a, n1 , n2 and n3 are arbitrary constants, which were determined by a regression analysis. For the range of experimental variables tested, steady-state heattransfer correlations were obtained by least-squares ®ts for both the in-line and staggered arrangements of pin ®ns (see Figs. 17 and 18, respectively). 4.3.1. In-line arrangement Nud 9:02 10ÿ3 Re1:011
Sx Wb
0:285 0:212 Sy L
where
Sx 40:332 0:0044 Wb Sy 40:212 0:0334 L and
Fig. 17. Generalised heat-transfer correlation for pin-®n in-line con®gurations.
20
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441
Fig. 18. Generalised heat-transfer correlation for pin-®n staggered con®gurations.
3:3414 103 4Red 46:683 103 4.3.2. Staggered arrangement
Nud 7:04 10ÿ3 Re0:953
Sx Wb
where 0:0044 Sx =Wb 40:332 ÿ 0:0334 Sy =L 40:152 and 3:138 103 4Re44:98 103
0:091 0:053 Sy L
21
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5. Conclusion These steady-state design data correlations facilitate predicting the performances of aligned and staggered pin ®ns, when used as arrays in heat exchangers and hence optimal designs to be chosen. References [1] Yao Peng. Heat-transfer and friction-loss characteristics of pin ®n cooling con®gurations. Trans ASME, Journal of Engineering for Gas Turbines and Power 1984;106:246±51. [2] Chung BT, Iyer JR. Optimal design of longitudinal rectangular ®ns and cylindrical spines with variable heat-transfer coecient. Heat Transfer Engineering 1993;14(1):31±42. [3] Razelos P, Imre K. The optimal dimensions of circular ®ns. Trans ASME, Journal of Heat Transfer 1980;102:420±4. [4] Chyu MK. Heat transfer and pressure drop for a short pin-®n with pin-endwall ®llet. Trans ASME, Journal of Heat Transfer 1990;112:926±32. [5] Tahat MA, Babus'hag RF, Probert SD. Forced steady-state convections from pin-®n arrays. Applied Energy 1994;48:335±51. [6] Babus'hag RF, Akintude K, Probert SD. Thermal performance of a pin-®n assembly. International Journal of Heat and Fluid Flow 1995;16(1):50±5. [7] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. 2nd ed. New York: John Wiley, 1985. [8] Gerencser DS, Razani A. Optimal of radiative±convective arrays of pin-®ns including mutual irradiation between ®ns. Journal of Heat Transfer 1995;38(5):899±907.
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Heat transfers from pin-®n arrays experiencing forced convection M. Tahat a,*, Z.H. Kodah a, B.A. Jarrah a, S.D. Probert b a
Department of Mechanical Engineering, Jordan University of Science and Technology, PO Box 3030, Irbid, Jordan b Department of Applied Energy, Cran®eld University, Bedford MK43 OAL, UK
Abstract Steady-state heat-transfers from pin-®n arrays have been investigated experimentally for staggered and in-line arrangements of the pin ®ns, which were orthogonal to the mean air¯ow. For the applied conditions, the optimal spacings of the ®ns in the span-wise and streamwise directions have been determined. The dependences of the Nusselt number upon the Reynolds number and pin-®n pitch (in both directions) have been deduced. # 2000 Elsevier Science Ltd. All rights reserved.
1. The challenge and solution In industrial processes, internal heat-generation may cause overheating and hence sometimes even system failure. So an eective means of removing this heat is often required. Cooling for electronic systems is usually needed to maintain the component temperatures lower than 40 C [1±4] in order to achieve a prolonged mean-life between failures, and so avoid having to make frequent replacements. Fins are used primarily to increase a body's heat-emitting surface area and consequently to enhance its heat-loss rate capability. During operation, the ®ns are at lower temperatures than those of their base plate. For cooling a turbine blade, the circular pin-®ns, spanning the assembly passage between the suction and pressure surfaces of the aerofoil [3,5,6], also serve as structural supports. The rate of heat transfer from a pin-®n assembly to its surrounding environment depends on: (i) the temperature distributions over the pin ®ns as well as the assembly's base; (ii) the pin-®n geometry; (iii) the thermal conductivities of the materials employed; (iv) the
* Corresponding author. Fax: +962-2-709-5018. E-mail address: [email protected] (M. Tahat). 0306-2619/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S0306-2619(00)00032-5
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M. Tahat et al. / Applied Energy 67 (2000) 419±442
Nomenclature area, (m2) uniform clearance between the extreme span-wise pin ®ns of the array and the wind-tunnel's walls (see Fig. 1), (m) C vertical clearance between the vertical-®n tips and the upper horizontal shroud (see Fig. 2), (m) speci®c heat of air at atmospheric pressure, (J/kg K) Cp d diameter of each pin ®n (see Fig. 1), (m) D hydraulic diameter (=4Aff =Wb ), (m) f friction factor F grey-body shape-factor : G air-mass ¯ow rate per unit area ( m=Aff ), (kg/m2 s) h heat-transfer coecient Ð see Eq. (8), (W/m2 K) H height of each pin ®n of the considered array (see Fig. 1), (m) thermal conductivity of air, (W/m K) kair L length of the pin-®n assembly (see Fig. 1), (= 0.3 m) : m air's mass ¯ow rate, (kg/s) N number of pin ®ns Nu Nusselt number ( hav d=k) p: static-air pressure, (N/m2) Q steady-state rate of heat loss, (W) Re Reynolds number (=Gd/m) S ®n pitch, i.e center-to-center separation between adjacent pin ®ns (m) T temperature, (K) Tf z ®n's surface temperature at height z above the ®n base, (K) V mean speed of the air in the wind-tunnel duct, (m/s) W width, (m) x, y, z set of Cartesian coordinates (as de®ned in Fig. 3) p overall air-pressure drop along the array, (N/m2) dynamic viscosity of air, (kg/ms) density of air, (kg/m3) Stefan±Boltzmann constant, (W/m2 K4) A B
Subscripts a of the ambient environment av average b of the base plate for the pin-®n assembly conv convective d based on the diameter of the pin-®n f of the ®n of the ®ns in the x and y directions, respectively fx, fy
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in loss max out rad s sf t total x,y z 1,4 2,3
421
corresponding to free ¯ow at the inlet loss maximum at the outlet radiative of the whole surface (in contact with the air) of the pin-®n assembly surface of the pin ®n for an unrestricted cross-section of the wind tunnel (see Fig. 1) total in the span-and stream-wise directions, respectively (see Fig. 1) in the vertical direction entrance and exhaust of the pin-®n assembly, respectively after the ®rst row and prior the last row of the pin-®n assembly,
air-¯ow rate; and (v) especially for low air-¯ow rates, the orientation of the heat exchanger [3,4]. The eects of some of these factors are interrelated. For a speci®ed temperature of the base plate of the pin-®n assembly, the rate of heat transmission can be enhanced by increasing the mean heat-transfer coecient, the emitting surface area or both. An increase in the heat-transfer coecient can be achieved by forced convection or changing the geometrical con®guration of the heat exchanger. However, in practice, this means is limited by the maximum-permitted pressure drop through the assembly [3,4]. 2. Steady-state heat-transfers from a horizontal base-plate, with cylindrical pin ®ns protruding vertically upwards The heat-transfer modes of interest for this system are conduction, convection and radiation through the air. The magnitude of each mode depends on the temperature of the pin-®n array's base, the assembly material(s) and geometry as well as the neighbouring air-¯ow rate. So the heat balance equation from the whole system is: : : : : Q total Qconv Qrad Qloss
1
where : : Q conv Cp m Tout ÿ Tin Also, to a ®rst approximation,
2
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Fig. 1. Schematic representation of the shrouded aligned pin-®n assembly, within the wind tunnel.
: Tin Tout Q conv hav As Tb ÿ 2
3
where d As Wb L dNfx Nfy H ÿ 4
4
The mean surface temperature, Tav , in Eq. (3) is de®ned by: Tav
Tb A Tf Af As
5
According to steady-state heat transfer analysis, Tf Z Ta C1 enx C2 eÿnx
6
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423
Fig. 2. Array of in-line pin ®ns.
and n2 dh= kAf
7
C1 and C2 are constants which can be determined from the boundary conditions at the ®n tips and base, e.g. at z 0, T: 0 Tb . It is apparent, from Eq. (3), that Qconv can be enhanced by increasing (i) the heattransfer surface area As ; (ii) the heat-transfer coecient hav : or (iii) the temperature of the base plate, Tb . In this investigation, the base-plate temperature remains constant throughout the experimental test, so that the rate of heat transfer can only be increased by either increasing the heat-transfer coecient or the surface area or both. From Eqs. (2) and (3), the average convective plus conductive heat-transfer coecient : 8 hav Cp m Tout ÿ Tin = As Tb ÿ Tout ÿ Tin =2 It should : be realized, from this equation, that, as As rises, hav may decrease even though Qconv increases. The radiative heat-transfer rate from a pin ®n depends on: (i) the temperature of its radiating surface; (ii) the temperature distribution of its surroundings; (iii) the
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emissivities of the ®n, base and environment: and (iv) the shielding eect of adjacent ®ns. The radiating surface temperature and the shielding eect determine the greybody shape-factor F. The total steady-state rate of radiative heat-transfer can be evaluated from ÿ : Q rad FAs T4av ÿ T4a
9
From previous steady-state heat-transfer investigations using a similar system [3,4], with (Tb 40 0:5 C), it has been deduced that the total steady-state rate of radiative heat-transfer from polished duralumin pin-®n assembly was less than 0.5% of the total steady-state rate of heat losses from the ®n arrays of length of 250 mm and height of 32 mm. The pin-®ns and base plate were of duralumin; their surfaces being highly polished and so of low emissivity. For the present investigation, the magnitude of the total wild-heat losses plus radiation arrays. The exact value of : is less than 5% of the total input to the pin-®n : : determined in each instance so that Q could then be found from Qrad Qloss was conv : the measured Qtotal as shown in Eq. (1). Hence, the average convective heat-transfer coecient hav could be deduced via : Qconv 10 hav Tin ÿ Tout As Tb ÿ 2 The free-¯ow sectional area Aff is calculated as Aff Wb H ÿ Nx Hd
11
: The mass ¯ow rate, m based on the mean velocity, V, of air in the wind tunnel duct is de®ned as : 12 m At V where At HWb For 250 K4
13
Tin Tout 4400 K; 2
expressions for the speci®c heat, dynamic viscosity and thermal conductivity of air at atmospheric pressure [7] are, respectively: Cp 9:8185 7:7 10ÿ4 Tin Tout =2 10ÿ2 J=kg K
14
air 4:9934 4:483 10ÿ2 Tin Tout =2 10ÿ6 kg=ms
15
Fig. 3. Schematic representation of the experimental rig and associated instruments.
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kair 3:7415 7:495 10ÿ2 Tin Tout 10ÿ3 W=m K
16
2.1. Pressure-drop and friction-factor characteristics The main aim of the heat exchanger is to transfer heat rapidly. In considering the pressure drop for air passing through the heat exchanger, it was essential to take into consideration the entrance pressure-drop, exhaust pressure-rise due to change in the area, and the energy losses occurring due to the ¯ow around and through the ®n array, i.e. frictional losses. Pressure-drop measurements of the air immediately adjacent to the pin-®n assembly, at the vertical plane sections 1±4 orthogonal to the direction of the mean air-¯ow, as shown in Figs. 3 and 4, were obtained by positioning four static-pressure probes located through the horizontal roof of the wind-tunnel duct, as well as along pin-®n array were recorded. The average reading at each section for each ®n geometry, as well as each mass-¯ow rate, were used to deduce the overall pressure drop, via P P1 ÿ P2 P2 ÿ P3 P3 ÿ P4 P1ÿ2 P2ÿ3 P3ÿ4 P1ÿ4
Fig. 4. Schematic diagram of the locations of the pressure taps in the wind-tunnel test section.
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427
The pressure drop associated with the ¯ow through the pin-®n array can be represented by the non-dimensional friction-factor, f, which is de®ned by the D'Arcy relationship [8]: f
2P2ÿ3 : L m 2 D Aff
18
Fig. 5. Steady-state rate of heat loss from the heat exchanger to the air via the in-line arrangement of pin ®ns.
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3. Experimental appararatus 3.1. Pin-®n assembly Each regular array assessed consisted of circular-sectioned pin-®ns (each with d 8 mm and H 90 mm) uniformly separated and protruding vertically upwards from a 300 by 250 mm horizontal rectangular base (see Fig. 1). Each array is characterised by its pin-®n spacings in the mean air-¯ow and orthogonal directions. For their least separation, the maximum number of pin ®ns is 341, whereas for the largest spacing, the array consists of only 12 pin ®ns. The spacing can be varied from 1.09 to 83.92 mm in the span-wise direction and from 9.86 to 63.44 mm in the stream-wise direction (see Fig. 2). Each pin ®n can be removed and replaced with a stud made from
Fig. 6. Eect of ®n spacing in span wise direction on the steady-state rate of heat loss for in-line con®gurations.
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the same batch of material as used for the assembly base: when screwed in, the studs' heads are ¯ush with the upper surface of the horizontal base-plate. The rectangular base, as well as the pin ®ns, were manufactured from duralumin, i.e. a light aluminum-alloy (No. 2024). For each test, there was zero clearance between the tips of the pin ®ns and the thermally well-insulated horizontal shroud (i.e C 0) [1,2,4,6].
Fig. 7. Eect of ®n spacing in the streamwise direction on the steady-state rate of heat loss for in-line con®gurations.
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3.2. Heating system The base of the heat exchanger was heated uniformly by four 500W electric-resistor strips as the main heater. The assembly was ®rmly bolted together to the bottom surface of the rectangular base (see Fig. 3). The presence of thin layers of high thermal-conductivity heat-sink putty ensured that good thermal contact existed between the main heater and the rectangular base, as well as between the pin-®n roots and the rectangular base [3].
Fig. 8. Eect of Reynolds number on the heat transfer coecient for in-line con®gurations.
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431
The lower horizontal surface and the sides of the main heater block (when operational) were insulated thermally with 50 mm thick mineral-wool blankets. A horizontal guard heater, rated at 50 W, was positioned parallel to the main heater, below the mineral-wool blanket, with yet another 50 mm layer of mineral-wool placed below it (see Fig. 2). The whole system of heat-exchanger base, main and guard heaters, with associated thermal insulation, was located in a well-®tting opentopped wooden box. The horizontal upper edges of this box and the top surfaces of
Fig. 9. Eect of ®n spacing in spanwise direction on the heat transfer coecient for in-line con®gurations.
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the laterally placed thermal insulant, during each experiment, were ¯ush with the upper surface of the base of the pin-®n array [3,4]. The power supplied to the main heater could be adjusted by altering the Variac setting and was measured by an in-line electronic Wattmeter. The dissipation in the guard heater was adjusted until the steady-state temperature dierence, across the layer of insulant, sandwiched between the two heaters, was zero. Then, under all the test conditions employed, more than 98% of the heat generated in the main heater passed, to the air of the surrounding environment, through the ®nned heat-exchanger. The steady-state temperature at the base of the pin-®n array was measured by
Fig. 10. Eect of ®n spacing in streamwise direction on the heat transfer coecient for in-line arrangements.
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an appropriately distributed set of four copper-constantan thermo-junctions, embedded within the rectangular base. Each thermo-junction was bonded in position with a thin layer of epoxy resin, so as to ensure good thermal-contact ensued. The average value obtained from these thermo-junctions was regarded as the mean overall base temperature. This was maintained constant during each experiment at 40 (0.5) C [3,4].
Fig. 11. Eect of Reynolds number on the steady-state rate of heat loss for staggered pin-®n con®gurations.
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The inlet and outlet air-stream temperatures in the wind-tunnel duct were measured using eight thermojunctions: four were located immediately prior to the entrance to the pin-®n assembly and another four just downstream of the array. Each of these thermojunctions could be traversed across either the whole inlet or the outlet cross-sections of the wind-tunnel duct (see Fig. 3). All the thermocouples, as well as those indicating the ambient air temperature were connected, through ribbon cables, to a data-logger, which was used to interpret the temperatures, at half-hourly intervals: when consecutive values were identical, it was assumed that steady-state conditions had been attained [3,4].
Fig. 12. Eect of ®n spacing in spanwise direction on the steady-state rate of heat loss for staggered con®gurations.
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3.3. Wind tunnel The main body of the rectangular cross-section wind-tunnel duct (see Fig. 2) was manufactured from wood and was 3 m long, with a constant internal width of 405 mm. However, the uniform vertical height of the duct, and hence the duct's crosssectional area, could be varied. Dierent duct heights were obtained by means of an adjustable horizontal roof (or shroud). Approximately half-way along the length of the wind-tunnel duct was the test section. The roof and side walls of this test section
Fig. 13. Eect of ®n-spacing in streamwise direction on the steady-state rate of heat loss for staggered con®gurations.
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were made of 5 mm Perspex, so enabling the pin-®n array (and the air, via smoke¯ow visualization around it) to be observed [3,4]. A bell-mouth section was ®tted as the entrance to the wind-tunnel duct, followed by some ¯ow straighteners, i.e. sieves that allowed the air-¯ow to pass to and through the shrouded length without encountering any sudden change of crosssection of the passage. The exhaust air from the pin-®n assembly was passed through an insulated chamber, where mixing was accomplished by two sets of sieves, one being of relatively low porosity and the other of higher porosity. The latter was situated upstream of the former. The two sets of sieves were mounted orthogonal to
Fig. 14. Eect of Reynolds number on the heat-transfer coecient for staggered con®gurations.
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the undisturbed ¯ow-stream. At the exhaust end of the duct, a gradual crosssectional area contraction duct was connected via ¯exible tubing to a single-speed, single-stage fan, which was capable of providing a maximum suction rate of 0.33 kg/ s. Pitot-static tubes were employed to measure the dynamic pressure before and after the pin-®n array, so as to be able to deduce the mean air-¯ow speed. The wind tunnel was operated in the suction mode, (i.e. the fan sucked air through the pin-®n array and the test section) with the fan and motor assembly on the exhaust side of the test section. This avoided the air-stream being heated by the fan's motor prior to its passage through the heat exchanger.
Fig. 15. Eect of ®n spacing in the spanwise direction on the heat-transfer coecient for staggered con®gurations.
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The velocity pro®le of the inlet air-stream to the pin-®n assembly was identi®ed via two traversable pitot-static tubes. An electronic analogue micro-manometer was employed to measure the pressure drops [3,4]. 4. Experimental results and discussion Various ®n arrays were employed together with a range of inlet air speeds. The resulting convection correlations are valid for 3:138 103 4Red 46:683 103 0:004074Sx =Wb 40:3316 and 0:032874Sy =L40:2115
4.1. In-line arrangement The steady-state rate of heat loss rose as the Reynolds number was increased, but decreased with increasing pin-®n spacing for both the stream-wise and the span-wise directions (see Fig. 5). The optimal value of the spacing in the span-wise direction divided by the base at 0.01340.0033 (see Fig. 6). That for the stream-wise width, Sx =Wbÿopt , occurred direction, i.e. Sy =L opt , was slightly beyond the range employed in this investigation, but, to a ®rst approximation, is likely to be somewhat similar (see Fig. 7). The average heat-transfer coecient (hav ) increased with Reynolds number for each pin-®n spacing (see Fig. 8). The dimensionless optimal pin-®n pitch in the spanwise direction ÿSx =Wb opt occurred at 0.1350.055 (see Fig. 9) and for the streamwise direction, Sy =L opt at 0.1730.027 (Fig. 10). 4.2. Staggered arrangement The steady-state rate of heat loss was enhanced as the Reynolds number increased. Also, it decreased with increasing pin-®n spacing in both the stream-wise and spanwise directions (see Figs. 11±13). The optimal pin-®n spacings in both the span-wise and stream-wise directions were not withinÿ the range of the current experimental investigation. However, Sx =Wb opt and Sy =L opt were likely to be less than 0.05 and 0.02 respectively (see Figs. 12 and 13). The heat-transfer coecient increased with Reynolds number (see Fig. 14). The optimal pin-®n spacing Sx opt divided by the base width Wb for the maximum
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steady-state ÿ heat-transfer coecient occurred at 0.190.03 (see Fig. 15), whereas that for Sy opt =L ensued at 0.1000.025 (see Fig. 16). 4.3. Correlations It was assumed [5] that the steady-state convective behaviour of the pin-®n array could be described by n2 n3 Sy Sx 19 Nud aRen1 Wb L
Fig. 16. Eect of ®n spacing in streamwise direction on the heat-transfer coecient for staggered pin-®n con®gurations.
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where a, n1 , n2 and n3 are arbitrary constants, which were determined by a regression analysis. For the range of experimental variables tested, steady-state heattransfer correlations were obtained by least-squares ®ts for both the in-line and staggered arrangements of pin ®ns (see Figs. 17 and 18, respectively). 4.3.1. In-line arrangement Nud 9:02 10ÿ3 Re1:011
Sx Wb
0:285 0:212 Sy L
where
Sx 40:332 0:0044 Wb Sy 40:212 0:0334 L and
Fig. 17. Generalised heat-transfer correlation for pin-®n in-line con®gurations.
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441
Fig. 18. Generalised heat-transfer correlation for pin-®n staggered con®gurations.
3:3414 103 4Red 46:683 103 4.3.2. Staggered arrangement
Nud 7:04 10ÿ3 Re0:953
Sx Wb
where 0:0044 Sx =Wb 40:332 ÿ 0:0334 Sy =L 40:152 and 3:138 103 4Re44:98 103
0:091 0:053 Sy L
21
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5. Conclusion These steady-state design data correlations facilitate predicting the performances of aligned and staggered pin ®ns, when used as arrays in heat exchangers and hence optimal designs to be chosen. References [1] Yao Peng. Heat-transfer and friction-loss characteristics of pin ®n cooling con®gurations. Trans ASME, Journal of Engineering for Gas Turbines and Power 1984;106:246±51. [2] Chung BT, Iyer JR. Optimal design of longitudinal rectangular ®ns and cylindrical spines with variable heat-transfer coecient. Heat Transfer Engineering 1993;14(1):31±42. [3] Razelos P, Imre K. The optimal dimensions of circular ®ns. Trans ASME, Journal of Heat Transfer 1980;102:420±4. [4] Chyu MK. Heat transfer and pressure drop for a short pin-®n with pin-endwall ®llet. Trans ASME, Journal of Heat Transfer 1990;112:926±32. [5] Tahat MA, Babus'hag RF, Probert SD. Forced steady-state convections from pin-®n arrays. Applied Energy 1994;48:335±51. [6] Babus'hag RF, Akintude K, Probert SD. Thermal performance of a pin-®n assembly. International Journal of Heat and Fluid Flow 1995;16(1):50±5. [7] Incropera FP, DeWitt DP. Fundamentals of heat and mass transfer. 2nd ed. New York: John Wiley, 1985. [8] Gerencser DS, Razani A. Optimal of radiative±convective arrays of pin-®ns including mutual irradiation between ®ns. Journal of Heat Transfer 1995;38(5):899±907.