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Convective heat dissipation with lattice-frame materials
Mechanics of Materials 36 (2004) 767–780 www.elsevier.com/locate/mechmat
Convective heat dissipation with lattice-frame materials T. Kim, C.Y. Zhao, T.J. Lu *, H.P. Hodson Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 30 January 2003; received in revised form 26 June 2003
Abstract This paper presents experimental results on heat transfer and pressure drop for a compact heat sink made of fully triangulated, lightweight (porosity 0.938), aluminum lattice-frame materials (LFMs). Due to the inherent structural anisotropy of the LFMs, two mutually perpendicular orientations were selected for the measurements. Constant heat flux was applied to the heat sink under steady state conditions, and dissipated by forced air convection. The experimental data were compared with those predicted from an analytical model based on fin analogy. The experimental results revealed that pressure drop is strongly dependent upon the orientation of the structure, due mainly to the flow blockage effect. For heat transfer measurements, typical local temperature distributions on the substrate under constant heat flux conditions were captured with infrared camera. The thermal behavior of LFMs was found to follow closely that of cylinder banks, with early transition Reynolds number (based on strut diameter) equal to about 300. The Nusselt number prediction from the fin-analogy correlates well with experimental measurements, except at low Reynolds numbers where a slightly underestimation is observed. Comparisons with empty channels and commonly used heat exchanger media show that the present LFM heat sink can remove heat approximately seven times more efficient than an empty channel and as efficient as a bank of cylinders at the same porosity level. The aluminum LFMs are extremely stiff and strong, making them ideal candidates for multifunctional structures requiring both heat dissipation and mechanical load carrying capabilities. 2003 Elsevier Ltd. All rights reserved.
1. Introduction A variety of technologies such as pin fins, cylinder banks and wire screens have been developed to increase the heat transfer area density of a heat sink without increasing its overall dimensions in the past century. More recently, cellular metal foams have been processed and used to construct lightweight and compact heat sinks. In such sys-
*
Corresponding author. Fax: +44-1223-332662. E-mail address: [email protected] (T.J. Lu).
tems, the high thermal conductivity of metal, combined with its ability to promote eddies and mix the coolant fluid, has enabled such heat sinks to remove up to five times more heat than that by a traditional pin-fin array, at a third of the weight, although the increase in the pressure drop is relatively high (e.g., Bastawros et al., 1998; Lu et al., 1998; Lu, 1999; Kim and Kim, 1999; Calmidi and Mahajan, 2000; Haack et al., 2001; Kim et al., 2002). In situations where a heat sink is also required to carry a certain amount of structural load, lattice-frame materials (LFMs) appear to be more
0167-6636/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2003.07.001
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
Nomenclature channel cross-section area ( ¼ H W ) specific heat [J/kg K] LFM bar diameter [m] hydraulic diameter of channel [m] friction factor mean heat transfer coefficient ( ¼ q=ðTw Tin Þ) h1 local heat transfer coefficient around LFM struts H , L, W channel height, length and width [m] I heat sink efficiency index ( ¼ j=f ) j j-Colburn factor ( ¼ StPr2=3 ) k thermal conductivity [W/mK] l individual strut length [m] n cell number along flow direction N total number of cells in flow direction NuDh Nusselt number based on hydraulic diameter ( ¼ hDh =kf ) Nud averaged Nusselt number based on bar diameter DP static pressure drop [Pa] P perimeter of wetted cross-section area [m] Pr Prandtl number Q heat released by a heating element [W] q heat flux [W/m2 ] Red Reynolds number based on LFM strut diameter
A Cp d Dh f h
attractive than conventional heat dissipation media. LFMs are a new development made possible by computer-based design and numerically controlled processing. They consist of a 3D network of cylindrical struts (Fig. 1). The important advance is in the reduction in scale. Recent developments in manufacturing techniques allow the diameter of these 3D lattice structures to vary from millimeters to tens of centimeters. For the LFMs studied in this paper, the diameter of the strut is on the order of one millimeter and the relative density of the structure is 6%. (The relative density is defined as the ratio of the LFM structure density to the density of the solid of which the LFM is made.) The stiffness and
ReDh S SR Sx , S y St Tin Tw ðxÞ Um x, y, z
Reynolds number based on channel hydraulic diameter surface area of wetted cross-section [m2 ] strut ratio ( ¼ l=d) longitudinal and transverse unit cell pitches Stanton number ( ¼ NuDh =ðReDh PrÞ) coolant inlet temperature [K] local substrate temperature coolant inlet mean velocity [m/s] global coordinate system
Greek symbols e porosity q density [kg/m3 ] qrel relative density g local coordinate along cell strut h non-dimensional temperature Subscripts f coolant fluid h hydraulic parameter in inlet of heat sink m arithmetic mean out outlet of heat sink s LFM solid cell SA surface area w substrate wall
strength of LFMs are typically an order of magnitude higher than those of cellular foams made of the same material because, unlike most cellular structures, they are fully triangulated and hence does not contain any bending mechanism (Deshpande et al., 2001a,b). Tetrahedral cells, as shown in Figs. 1 and 2, constitute the triangulated frame. When the structure is loaded, the struts support either axial tensile stressing or axial compressive stressing. Thus, the deformation is stretchingdominated and the frame collapses by stretching of the struts (Deshpande et al., 2001a,b). The LFM structures can also be stacked so that they form a multilayered lattice material block while maintaining the same mechanical properties. In
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
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O-B Z d
O-A H
Y b
X
c
S y0
f e
SX
SY
a
(a)
SS x0x0 (b)
Fig. 1. (a) Geometrical configuration of a LFM unit cell; (b) top view of (a), with filled symbols representing projections of the cylindrical struts at the bottom substrate of the channel and empty symbols denoting those at upper substrate.
Fig. 2. Two different frontal faces of the LFM model: (a) orientation A (O–A); (b) orientation B (O–B).
comparison, the deformation and failure of cellular metal foams are governed by cell-wall bending, and are sensitive to inherent geometrical imperfections induced during processing (Chen et al., 1999). Consequently, a precise computer-designed LFM exhibits important morphological advantages over cellular foams (in terms of structural imperfections), in addition to the availability of materials. LFMs are ideal candidates for applications (e.g., supersonic and hypersonic aircrafts) where, in addition to carry structural loads at minimum weight, efficient heat dissipation is also demanded.
The microstructural morphologies of an LFM resemble those of cellular metal foams, and hence the overall heat transfer performance of both materials is expected to be similar if the cells are of similar size. On the other hand, another analogy can be made with banks of cylinders (BOCs) that have been widely used in heat sink configurations (Zukauskas, 1987). By using the configurational similarity that three cylindrical struts comprising a unit LFM cell are inclined relative to the longitudinal and transverse directions (Figs. 1 and 2), the available heat transfer and hydraulic resistance data on BOCs may be used to direct heat sink design with LFMs. A cross-flow analogy of cylinder banks has been used to develop heat transfer models for cellular metal foams (Bastawros et al., 1998; Lu et al., 1998; Lu, 2002; Kim et al., 2002). Lattice-frame structures resemble metal foams in microstructural morphology and banks of cylinders in macrostructural configuration. These three structures all provide a solid–air contact area many times greater than the duct surface area so that the temperature difference between the heated wall surface and the coolant can be drastically reduced as a consequence of convective heat transfer. The successful implementation of LFMs as compact heat sinks demands an integrated approach that combines modeling and testing to quantify their heat removal performance as a function of morphological parameters, relative density and fluid properties. Thus, this work is
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Fig. 3. Test rig with LFM model in the test section.
aimed at investigating general features of thermal and hydraulic behaviors of metal lattice-frame materials as a novel heat sink medium in a forced air convection channel (Fig. 3). The paper is organized as follows. In Section 2, an analytical model is developed, by using fin analogy, to predict temperature distributions (and hence Nusselt number) in the LFM structure as well as along the cover plates. Section 3 presents detailed experimental procedures for heat transfer and pressure drop measurements, with a discussion on measurement uncertainties. The experimental results are presented in Section 4; comparison is made with analytical predictions as well as conventional heat dissipation media.
2. Specifications of LFM heat sink With reference to Fig. 3, the LFM heat sink is a typical sandwich structure consisting of a tetrahedral core and two solid face sheets. Both the LFM core and the substrates are made of an aluminum–silicon casting alloy LM25, of wt.% composition Al–7Si–0.3Mg. The panel is investment cast from injection-molded polystyrene preforms; details of processing procedures can be found in Deshpande et al. (2001a). The overall dimensions of the LFM heat sink tested are 0.13 m (L) · 0.145 m (W ) · 0.012 m (H ). The ratio of the channel height to LFM strut diameter, H =d, is 6 and the slenderness ratio, SR ¼ l=d is 7.35. Detailed specifications of the LFM sample and operating condition of the measurements are listed in Table 1. The solid phase thermal conductivity
Table 1 Specifications of the LFM unit cell and typical operating parameters for measurements Specification of LFM (see Fig. LFM cell bar diameter (d) Longitudinal cell pitch (Sx ) Transverse cell pitch (Sy ) Cell strut length (l) Cell height (H ) Material
1) 0.002 m 0.0127 m 0.0147 m 0.0147 m 0.012 m LM25
Typical operating parameters Inlet coolant mean velocity Bar Reynolds number Input heat flux Inlet coolant temperature Outlet coolant temperature Test section inlet pressure
2.0–26.0 m/s 250 < Red < 3300 4, 8 and 16 kW/m2 300.0 K 305.0–360.0 K 1 bar (ambient conditions)
of LM25 is 150.84 W/mK (from Aluminum Federation, UK). Pressure drop and heat transfer measurements are performed for two orientations of the LFM, as illustrated in Fig. 2; the relevant parameters are listed in Table 2. Here, the relative density of pffiffiffi 2 the LFM is defined as qrel ¼ ð 3=2Þðd=lÞ ðl=H Þ, porosity is e ¼ 1 qrel , and the surface area density of the heat sink, i.e., the ratio of the total wetted heat transfer area including the two face sheets and Table 2 Parameters for the LFM samples Sample Relative density of the LFM (qrel ) Porosity (e) Surface area density (qSA ) Frontal area/free flow area
LFM O–A
LFM O–B 0.062
0.73
0.938 123.68 m1 0.36
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
the LFM core to the volume of the heat sink, is given by qSA ¼ ½3=d þ 2ðSx lÞ 3ðd 2 =4Þ =ðHSx lÞ.
3. Analysis The fluid dynamics and thermal transport in a LFM heat sink are complex, and hence it is difficult to analytically examine the details of flow and heat transfer within the LFM microstructure by using the Navier–Stokes equation and the associated energy equation. The main objective of this section is to construct an approximate analytical solution. Thus, the rules played by a few important parameters such as strut diameter and solid thermal conductivity can be identified and the physical understanding of the problem could be more improved. 3.1. Mathematical formulations
Eq. (1) is solved subjected to the following boundary conditions: dTs ks ¼ q0 dg g¼0 ð2Þ dTs ks ¼0 dg g¼l It is noted here that q0 is the heat flux entering the strut ad, and is generally not identical to the uniform heat flux q imposed on the outer surface of the substrate. This is because the solid LFM strut can drastically reduce the thermal resistance at the three-struts joint, and thus lead to a much higher heat flux at the joint. The value of q0 is not specified, and must be determined. Eq. (1) solved with boundary conditions (2) leads to the following analytical solution of the solid temperature distribution along strut ad: q0 coshðmðl gÞÞ ð3Þ sinhðmlÞ ks m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m ¼ h1 P =ðks AÞ ¼ 2 h1 =ðks dÞ, and d is the strut diameter. From Eq. (3), the wall temperature can be obtained as
Ts ðg; nÞ Tf ðnÞ ¼
The unit cell shown in Fig. 1 can be taken as an analytical domain. As shown in Fig. 1(a), each unit cell comprises three inclined struts. The heat imposed on the bottom wall is taken away either directly by the coolant at the substrate surface or by the struts via conduction and convection. To determine the heat transfer coefficient, h ¼ q= ðTw Tin Þ, the substrate temperature needs to be found. In this analysis, both the fluid and wall temperature are assumed to be constant within a unit cell, but these will increase in subsequent unit cells along the stream direction. Since the heat transfer mechanism for the three inclined struts are the same, here only one strut ad is used for analysis (Fig. 1(a)). The solid temperature variation along the strut ad in the ÔnÕth cell from the entrance is governed by d2 Ts ðg; nÞ h1 P ðTs ðg; nÞ Tf ðnÞÞ ¼ 0 dg2 ks S
771
q0 coshðmlÞ km sinhðmlÞ q0 ¼ tanh1 ðmlÞ ks m
Tw ðnÞ Tf ðnÞ ¼
ð4Þ
It should be pointed out that the mean fluid temperature Tf ðnÞ and heat flux q0 still need to be determined in order to finally obtain the wall temperature Tw ðnÞ. The mean fluid temperature Tf ðN Þ can be obtained by considering energy balance: qf Sy Hum Cp ½Tf ðnÞ Tf ð0Þ ¼ qSy Sx n
ð5Þ
from which ð1Þ
where g is the local coordinate along the ad strut, Tf ðnÞ is the mean fluid temperature in the nth cell, ks is the solid conductivity, h1 is the local heat transfer coefficient around the strut, and P and S are the perimeter and area of the cross-section of strut ad, respectively.
Tf ðnÞ ¼ Tf ð0Þ þ
qSx n qf Hum Cp
ð6Þ
To determine the heat flux q0 , we note that the heat flux at the inner surface of the substrate not connected to the strut is q1 ¼ ½Tw ðN Þ Tf ðN Þ h1 ¼
h1 q0 coshðmlÞ ks m sinhðmlÞ
ð7Þ
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
where h1 is the local heat transfer coefficient near the substrate surface. For simplicity and also because of the strong coolant mixing by the LFM microstructure, it is assumed that h1 takes the same value as the local heat transfer coefficient around the struts, h1 . The total heat imposed on the outer surface of a unit cell can therefore be expressed as Sx Sy q ¼ ðSx Sy pd 2 =4Þq1 þ 3pd 2 q0 =4
ð8Þ
Substitution of (7) into (8) yields, q0 ¼
Sx Sy q h1 coshðmlÞ 3 2 þ pd ðSx Sy pd 2 =4Þ ks m sinhðmlÞ 4
ð9Þ
qSx N 1 coshðmlÞ þ qf Hum Cp km sinhðmlÞ Sx Sy q h1 coshðmlÞ 3 2 þ pd ðSx Sy pd 2 =4Þ ks m sinhðmlÞ 4 ð10Þ
With (10), the unit-cell averaged Nusselt number can be obtained as 0 Dh B B Sx n þ coshðmlÞ NuðnÞ ¼ kf @ qf Hum Cp km sinhðmlÞ
0:36 0:71Re0:5 ðkf =dÞ d Pr 0:6 0:36 0:35Red Pr ðkf =dÞ
Red ¼ 1 500 ¼ 500 1000 ¼ 103 2 105 ð13Þ
The effect of cylinder inclinations with respect to the flow direction on h1 has also been given in Zukauskas (1987), and is found to be relatively small in comparison with its much larger knockdown effect on pressure drop. In the present study, the strut inclination angle with respect to the horizontal surface is a 54:7, so the values of h1 calculated from empirical correlation (13) are multiplied by a factor of 0.9, following the recommendation of Zukauskas (1987).
4. Experimental details 4.1. Experimental apparatus
1 Sx Sy h 3 1 coshðmlÞ þ pd 2 ðSx Sy pd 2 =4Þ ks m sinhðmlÞ 4
C C A
ð11Þ The overall Nusselt number of the LFM heat sink is given by N 1 X NuðnÞ N 1
0:36 ðkf =dÞ h1 ¼ 1:04Re0:4 d Pr
¼
Tw ðN Þ ¼ Tf ð0Þ þ
Nu ¼
The analytical model is not complete until the local heat transfer coefficient h1 in (11) is determined. To find h1 on the surface of each cylindrical strut, the lattice-frame material is taken as comprising of three sets of inclined cylinders arranged in staggered arrays (Figs. 1 and 2). For staggered arrays of cylinders in cross-flow, the following correlations based on extensive experimental measurements can be used to calculate h1 , with an accuracy of ±15% (Zukauskas, 1987): ¼
Finally, the wall temperature Tw ðnÞ can be obtained by substituting Eqs. (6) and (9) into Eq. (4), as
3.2. Local heat transfer coefficient
ð12Þ
where N is the total number of unit cells in the flow direction.
The experimental apparatus consists of four main sections: coolant supply, test section, test model, and data acquisition system. Air in ambient conditions was used as a coolant and drawn through the channel passage by a suction type air blower. One wire screen and one honeycomb were inserted in the channel before the contraction. The coolant then flew through a 9:1 contraction and a parallel section, in which the ratio of section length to channel height was 20, before flow reaches the test section. As shown in Fig. 3, the LFM filled heat channel was encapsulated by perspex sidewalls. A flow rate regulator is located between the exit of the test section and the suction device.
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
The thermal performance of a heat sink channel with a single LFM layer was measured, with the bottom substrate exposed to heating and the top substrate insulated, i.e., asymmetry heating imposed by a heating element (silicone-rubber etched foil from WatlowTM Inc.). A total of four static pressure tapings were placed on the top substrate along the flow direction. In actual heat sink applications, multilayered LFMs is preferable so the asymmetry of the thermal boundary condition was imposed for single layer experiment. The amount of heat released from the heating element was adjusted by changing the supply voltage. To minimize heat loss from direct contact, the external surface of the heating element was covered with a thermal insulation material, Tancast 8 . Because the electrical resistance foil was etched in a zigzag pattern on the silicone-rubber pads, there was a possible gap (unspecified by the manufacturer) between two neighboring columns of the foil. Consequently, a pure copper heat spreader plate, 0.9 mm thick with 12 uniformly spaced transverse slits of 0.5 mm depth and 0.15 mm width, were inserted between the heating element and the LFM substrate to ensure the uniformity of the heat flux entering the LFM heat sink. Five thin foil (0.013 mm thickness) T-type copper–constantan thermocouples (from Rhopoint Inc.) were inserted on the lower substrate along the longitudinal direction (i.e., flow direction). There were two additional T-type bead thermocouples, positioned separately at the inlet and outlet of the test section to measure the coolant temperature at midheight at each location. In addition, thermal images of local temperature distribution on the upper substrate (unheated surface) were captured by an infrared camera, AGEMA ThermoVision 870, to ensure a linear increment of substrate temperature subject to the constant heat flux thermal boundary condition. 4.2. Experimental and data acquisition procedures The experiments were run for several minutes until the flow inside the channel becomes hydraulically and thermally stabilized. All measurements were performed under steady state conditions. A Pitot tube was positioned before the test section to
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measure stagnation pressure and static pressure at the inlet of the test section. Because the blockage ratio, i.e., the ratio of channel height (12 mm) to tube outer diameter (0.51 mm), was 23.5, wall interference effects on the Pitot tube were expected to be small. All measurements were repeated until significant data repetition was ensured, i.e., 5% uncertainty interval (see measurement uncertainties discussed later). A friction factor is used to quantify the flow resistance across the LFM structure, defined as 4f ¼
DP Dh DP ðnDh =LÞð1=ðqU 2 =2ÞÞ ¼ 2 L qU =2 n
where Dh , Um , L, n and DP =n are the hydraulic diameter of the heat sink channel, mean coolant velocity (averaged over a channel height) at the inlet of the test section, LFM sample length, unit cell size, and static pressure drop per unit cell, respectively. The mean heat transfer coefficient h and mean Nusselt number NuDh (and Nud ) for fixed values of heat flux and Reynolds number are obtained as h ¼ q=ðTw Tin Þ and NuDh ¼ hDh =kf
ðand Nud ¼ hd=kf Þ RL where Tw ¼ ð1=LÞ 0 Tw ðxÞDx is the spatially averaged substrate wall temperature. Here, L is longitudinal substrate length, and Tw ðxÞ is measured local substrate temperature. The inlet coolant temperature Tin is used as a reference for all calculations. 4.3. Measurement uncertainties
An uncertainty analysis was performed by following the method of Coleman and Steele (1999). The maximum heat loss through insulation materials was estimated to be less than 2% out of total input heat. Heat loss through side walls was assumed to be negligible due to the small conduction area. The coolant temperature difference between the inlet and outlet can be either measured, ðDT Þmeasured , or calculated directly from energy balance:
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780 1.1
∆ T energy _ balance
∆Tmeasured
1
0.9
Input heat flux, q = 4 KW/m2 Input heat flux, q = 8 KW/m2 Input heat flux, q = 16 KW/m2
0.8
0.7
0
500
1000
1500
2000
2500
3000
Red Fig. 4. Error estimation by comparing measured temperature with that calculated from enthalpy balance equation for three different values of heat input, with DT ¼ Tout Tin .
ðDT Þenergy
balance
_ pÞ ¼ Tf;in Tf;out ¼ Q=ðmC
where Q is the input heat and m_ is mass flow rate. Fig. 4 plots the ratio ðDT Þmeasured =ðDT Þenergy balance as a function of Red for three levels of heat input, where Red is the Reynolds number based on mean inlet velocity and bar diameter. The measured values were within 92% of predicted values for most of the bar Reynolds number regime. The thermal conductivity kf of air varied slightly in the operating temperature range of 300.0–360.0 K. An arithmetic mean value was used for kf , with an uncertainty estimated to be within 6.6%. From these, the uncertainty in the measured heat transfer coefficient and Nusselt number was estimated to be less than 5.0% and 8.3%, whilst the uncertainty in the pressure drop and friction factor measurements was estimated to be less than 5.0% and 9.7%, respectively. A root-mean-square method was used for these calculations. 5. Results and discussion Since heat transfer and pressure drop measurements have been performed on a new type of
heat exchanger medium that has vastly different geometrical features in comparison with other types of heat dissipation medium, a consistency check is necessary. Thus, the LFM strut diameter based Reynolds number is used in presenting the test results and comparing with empirical correlation of banks of cylinder arrays, whilst comparison with other available heat sink media was made by using the hydraulic diameter based Reynolds number. 5.1. Pressure drop Static pressure drop measurements across the LFMs were performed. Results of pressure drop and the corresponding friction factor are depicted in Fig. 5(a) and (b), respectively. Fig. 5 indicates that the flow was typically laminar when Red < 300, and in transition from laminar to turbulent when 300 < Red < 400 for both LFM orientations. The transition Reynolds number is low due to interactions between shedding vortices and wakes from LFM struts. Because of the LFM morphology, flow patterns are three dimensional even in a unit cell. Strong flow interaction with wakes and
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780 18
Orientation A (O-A) Orientation B (O-B)
16 14
O-A
dP/L [KPa/m]
12 10 8 6 4
O-B 2 0
0
5
10
15
25
20
Um [m/s]
(a)
for a smooth pipe from Moody chart Orientation A (O-A), LM25 metallic model Orientation B (O-B), LM25 metallic model 100
f 10-1
O-B
16 / Red (for Laminar flow)
(b)
100
occupied by the structure/free flow area) of O–A and O–B is 0.73 and 0.36, respectively. The resistance encountered by the flow in orientation O–A is twice as large as that in O–B, indicating that pressure drop was mainly contributed by the flow blockage effect due to the inserted structure, i.e., a higher flow resistance occurred in the higher blocked orientation. It appeared to be proportional to the blockage ratio, but difficult to justify due to lack of test data. Systematic parametric studies on the dependence of pressure loss on porosity, blockage ratio, Reynolds number, etc. are currently under investigation. In addition, comparison with a smooth pipe from the Moody chart (Bejan, 1995) was made and plotted together in Fig. 5 to underline the additional pressure loss due to the insertion of the LFM structure for the laminar flow region. About 3.3 times higher pressure drop than the empty channel was recorded for O–B and 7.5 times for O–A. 5.2. Heat transfer characteristics
O-A
10-2
775
1100
2100 3100
Re d
Fig. 5. Flow resistance of the LFM heat sink as a function of Reynolds number for orientation O–A and orientation O–B: (a) pressure drop per unit length, with curve fitting also shown; (b) friction factor.
shedding vortices especially near the vertices is expected. The friction factor, f , approached an asymptotic value for both test models after the flow transition, i.e., turbulent flow prevailed when 500 6 Red 6 3200, with f 0:28 for O–A and f 0:12 for O–B. Flow blockage ratio ( ¼ area
To characterize the capacity of heat removal by the LFMs in forced convection, local heat transfer coefficients measured at selected longitudinal locations, i.e., on centerline (point ÔfÕ in Fig. 1(a)) were spatially averaged. In addition, Fig. 6 plots the local temperature distribution on the upper substrate plate (i.e., without the heating pad) along the mainstream direction captured by the infrared camera. The flow direction was from left to right, and the area of interest was denoted in dotted lines. Note that there is minimal endwall effect, and the substrate surface temperature varies linearly with the x-axis, typical of a constant heat flux boundary. The spatially averaged (mean) Nusselt number Nud of the heat sink system was plotted in Fig. 7 for both orientations as a function of the Reynolds number. Shorter cylinder spacing and narrower flow passage in LFM O–A may lead to reduced wake velocity and increased mainstream velocity, resulting in slightly less efficient heat transfer than that from LFM O–B. Similar observations were reported in Zukauskas (1987) for banks of cylinder arrays.
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
Fig. 6. Local temperature distribution of the upper substrate captured by infrared camera for q ¼ 500 W/m2 and Um ¼ 4 m/s.
0:56
LFM(O-A) LFM(O-B) Correlations by Zukauskas (for Staggered BOCs) 2
10
Empirical correlation for the LFMs
Nu d
Nud=0.428(Red)0.56Pr0.36
101 Correlations by Zukauskas Nud=1.04(Red)0.4Pr0.36 for Red<500 Nud=0.71(Red)0.5Pr0.36 for 500
100 200
1200
2200 3200 4200
Re d Fig. 7. Comparison of heat transfer measurement with empirical correlation of Zukauskas (Deshpande et al., 2001b) for typical staggered banks of cylinder arrays.
To compare with the results of typical staggered banks of cylinders summarized in Zukauskas (1987), the averaged Nusselt number was correlated as a function of Reynolds number and Prandtl number: Nud ¼ 0:428ðRed Þ0:56 Pr0:36 or
ð14aÞ
NuDh ¼ 1:23ðReDh Þ
Pr0:36
ð14bÞ
where Nud and NuDh are mean Nusselt numbers based on the LFM strut diameter and channel hydraulic diameter, respectively. Eqs. (14a) and (14b) are valid for both O–A and O–B orientations in the range of 120 6 Red 6 3200, where the Prandtl number Pr is assumed to be constant ( ¼ 0.71). As seen in Fig. 7, the heat transfer characteristics of LFMs are similar to that of banks of cylinder arrays due to their structural resemblance. The discrepancy between the two orientations was negligible at low Reynolds number but reaches 10% at high Reynolds number. As the coolant velocity increases, heated flow near the substrates may tend to be confined and recirculates especially behind the LFM vertices where the distance between neighboring cylinders is short. Consequently, the heat removed from the substrate surfaces may not be fully convected. This may be one of the limiting cases in heat removal by the use of a low thermal diffusivity fluid (e.g., air) as coolant. 5.3. Fin-analogy estimation The fin-analogy based analytical model presented in Section 3 gave local Nusselt number per unit cell, with local heat transfer coefficient
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
777
350
Measured substrate temperature 340
T [K]
330
Predicted substrate temperature 320
Predicted fluid temperature 310
300
At Red=500 290
1
2
3
4
5
6
7
8
9
10
N, cell number
(a) 0
-0.5
Predicted strut temperature
qD h /ks
Ts − Tw
-1
-1.5
-2
Red = 2000 Red = 500
-2.5
-3 0
0.2
(b)
0.4
0.6
0.8
1
y/H
Fig. 8. (a) Predicted solid and fluid temperature variations along the flow direction; (b) non-dimensional temperature of solid strut.
prescribed from empirical correlations for banks of cylinders (Zukauskas, 1987). The predicted fluid and wall temperature variations are shown in Fig. 8(a) for Red ¼ 500, together with the measured wall temperatures for comparison. The uniform
heat flux imposed on the outer surface of the substrate dictates that the two curves are parallel. Overall, the predictions are in close agreement with the measurements. Fig. 8(b) depicts the nondimensional strut temperature as a function of
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
3
10
j ¼ St Pr2=3
Experimental data of the LFM O-A Experimental data of the LFM O-B Analytical estimation using fin analogy Empty channel correlation from Dittus & Boelter
where St is the Stanton number ( ¼ NuDh =ðReDh PrÞ) and Pr is the Prandtl number. Since heat sink design parameters such as the overall heat transfer coefficient and pressure drop are interdependent, an efficiency index accounting for not only flow resistance but also heat transfer is often required to rank the overall performance of different heat dissipation media. For convenience, such an efficiency index is introduced here as
Nu Dh
O-B O-A 2
10
Estimated by fin analogy
NuD=0.023 (ReD)4/5Pr0.4
101 3 10
I ¼ j=f
by Dittus and Boelter 4
10
5
10
Re Dh Fig. 9. Comparison of predicted overall Nusselt number with experimental data.
y=H within one unit cell for selected values of Red , with Tw and Ts representing solid temperature at the wall and local solid temperature varying along the channel height, respectively. The predicted and measured average Nusselt number (based on the channel hydraulic diameter) is presented in Fig. 9, and compared with that of an empty channel (Dittus and Boelter, 1930). In general, the predicted Nusselt number follows that of the empty channel, with approximately six times higher magnitude, whilst the experimental data show that the magnitude of heat transfer enhancement over the empty channel decreases as the Reynolds number is increased in the range of 3000 6 ReDh 6 35 000. As previously mentioned, the measurement range covers the end of laminar flow as well as the whole transition. Higher heat enhancement was observed in these regimes, accompanied with higher flow friction. Generally speaking, the prediction from the analytical model is in satisfactory agreement with experimental data except within the lower Reynolds number regime.
where j is the Colburn factor defined above and f is the friction factor. It is evident that a high value of I is desirable for optimal heat sink design. Comparisons were made with typical heat dissipation media, e.g., banks of cylinders, pin-fin arrays, cross-rod matrices, and randomly stacked sphere matrices (Kays and London, 1984). To be consistent with the available data, the hydraulic diameter has been used as a characteristic length scale for the LFMs. As shown in Fig. 10, the overall performance of LFM O–B is about twice higher than that of cylinder banks as well as LFM O–A, due mainly to its lower flow resistance. The plain pin-fin arrays offer the highest efficiency
10
LFM: Orientation A LFM: Orientation B Bank of cylinders Wavy-fin plate-fin surface Cross-rod matrices, inline stacking Plain pin-fin Infinitely randomly stacked sphere matrix
0
x
I (=j / f)
778
LFM O-B
10
-1
x
x
x
x x
x
x
LFM O-A
5.4. Thermal efficiency index In order to facilitate quantification of heat transfer performance of the LFM heat sink with other available heat exchanger media, the j-Colburn factor (Colburn, 1933) was used:
10
-2
10
3
10
4
10
5
Re Dh Fig. 10. Efficiency index, I (¼ j=f ), plotted as a function of Reynolds number based on channel hydraulic diameter for LFMs and common heat sink media (Lu, 1999).
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
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Table 3 Configurational data of heat dissipation media Configurations
Porosity
Surface area density [m1 ]
Materials
Lattice-frame material Staggered bank of cylindrical tubes
0.938 N/A
123.68 165.02
LM25 N/A
Wavy-fin plate-fin surface
0.822
1138
Aluminum
Crossed-rod matrices, in-line stacking Infinitely randomly stacked sphere matrix
0.832
N/A
N/A
0.37–0.39
N/A
N/A
although their ability to dissipate heat is relatively low, whereas cross-rods and randomly stacked sphere matrices appear to have the lowest heat removal efficiency. Details of the data compared and the corresponding configurational specifications can be found in Table 3 and in Kays and London (1984). However, it is emphasized that the data shown in Fig. 10 for different heat dissipation media must be used cautiously, as a systematic comparison is at present not possible.
6. Conclusion The thermal and hydraulic properties of latticeframe materials subjected to forced air convection cooling have been evaluated by steady state measurements as well as analytical modeling based on the fin analogy. The model can predict well the behavior of Nusselt number in the higher Reynolds number regime, but gives slightly underestimation in the lower Reynolds number regime including laminar and flow transition. Friction factor, averaged heat transfer coefficient, and averaged Nusselt number were quantified in the range of bar Reynolds number, 120 6 Red 6 3200 (or, 1300 6 ReDh 6 35000 based on channel hydraulic diameter). The performance of a LFM heat sink was found to be somewhat similar to that of conventional cylindrical tube arrays due to morphological similarities. The highly porous LFM structure (porosity 0.94) promotes turbu-
Comments
Longitudinal and transverse pitch are Sx ¼ 2d and Sy ¼ d where d is tube diameter. Fig. 10–9 from Zukauskas, 1987–– Transient test Fig. 10–74 from Zukauskas, 1987 Fig. 10–98 from Zukauskas, 1987 Fig. 7–10 from Zukauskas, 1987
lent flow, which causes thermal dispersion effect, for the whole range of Reynolds number investigated: about six times more heat can be removed in comparison with an empty channel. The multifunctional LFMs can find attractive applications where a structure is required to effectively dissipate heat in addition to carry mechanical loads.
Acknowledgements This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC grant number EJA/U83) and by the US Office of Naval Research (ONR/ONRIFO grant number N000140110271). The authors wish to thank Mr. Victor Yung at the Whittle Laboratory of Cambridge University for helpful discussions, and Dr. Vikram Deshpande and Prof. N.A. Fleck of Cambridge University for the assistance on the preparation of test specimens.
References Bastawros, A.-F., Evans, A.G., Stone, H.A., 1998. Evaluation of cellular metal heat dissipation media. Technical Report MECH 325, DBAS, Harvard University. Bejan, A., 1995. Convection Heat Transfer, second ed. John Wiley & Sons, New York. Calmidi, V.C., Mahajan, R.L., 2000. Forced convection in high porosity metal foams. Trans. ASME J. Heat Transfer 122, 557–565.
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Chen, C., Lu, T.J., Fleck, N.A., 1999. Effect of imperfections on the yielding of two-dimensional foams. J. Mech. Phys. Solids 47, 2235–2272. Colburn, A.P., 1933. A method of correlating forced convection heat transfer data and a comparison with fluid friction. Trans. Am. Inst. Chem. Eng. 29, 174–210. Coleman, H.W., Steele, W.G., 1999. Experimentation and Uncertainty Analysis for Engineers, second ed. John Wiley & Sons, New York. Deshpande, V.S., Fleck, N.A., Ashby, M.F., 2001a. Effective properties of the octet-truss lattice material. J. Mech. Phys. Solids 49, 1747–1769. Deshpande, V.S., Ashby, M.F., Fleck, N.A., 2001b. Foam topology: bending versus stretching dominated architectures. Acta Mater. 49, 1035–1040. Dittus, H.W., Boelter, L.M.K., 1930. Heat transfer in automobile radiators of the tubular type. Univ. California Pub. Eng. 2, 443–461, reprinted in Int. Commun. Heat Mass Transfer 12, 1985, 3–22. Haack, D., Butcher, K., Kirn, T., Hodson, H.P., Lu, T.J., 2001. Novel lightweight metal foam heat exchanger. ASME International Mechanical Engineering Congress & Exposition, Proceedings V.3 (PID-4B: Heat Exchangers Applica-
tions for Change of Phase Media and Fuel Cells Systems, Book No. I00548), New York. Kays, W.M., London, A.L., 1984. Compact Heat Exchangers, third ed. McGraw-Hill. Kim, S.J., Kim, D., 1999. Forced convection in microstructures for electronic equipment cooling. Trans. ASME J. Heat Transfer 121, 639–645. Kim, T., Fuller, A.J., Hodson, H.P., Lu, T.J., 2002. An experimental study on thermal transport in lightweight metal foams at high Reynolds numbers. In: Proceedings of the International Symposium of Compact Heat Exchangers, Grenoble, France, pp. 227– 232. Lu, T.J., 1999. Heat transfer efficiency of metal honeycombs. Int. J. Heat Mass Transfer 42, 2031–2040. Lu, T.J., 2002. Ultralight porous metals: from fundamentals to applications. Acta Mech. Sinica 18, 457–479. Lu, T.J., Stone, H.A., Ashby, M.F., 1998. Heat transfer in open-cell metal foams. Acta Mater. 46, 3619–3635. Zukauskas, A., 1987. Convective heat transfer in cross flow. In: Kakac, S., Shah, R.K., Aung, W. (Eds.), Handbook of Single-Phase Convective Heat Transfer, New York (Chapter 6).
Convective heat dissipation with lattice-frame materials T. Kim, C.Y. Zhao, T.J. Lu *, H.P. Hodson Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UK Received 30 January 2003; received in revised form 26 June 2003
Abstract This paper presents experimental results on heat transfer and pressure drop for a compact heat sink made of fully triangulated, lightweight (porosity 0.938), aluminum lattice-frame materials (LFMs). Due to the inherent structural anisotropy of the LFMs, two mutually perpendicular orientations were selected for the measurements. Constant heat flux was applied to the heat sink under steady state conditions, and dissipated by forced air convection. The experimental data were compared with those predicted from an analytical model based on fin analogy. The experimental results revealed that pressure drop is strongly dependent upon the orientation of the structure, due mainly to the flow blockage effect. For heat transfer measurements, typical local temperature distributions on the substrate under constant heat flux conditions were captured with infrared camera. The thermal behavior of LFMs was found to follow closely that of cylinder banks, with early transition Reynolds number (based on strut diameter) equal to about 300. The Nusselt number prediction from the fin-analogy correlates well with experimental measurements, except at low Reynolds numbers where a slightly underestimation is observed. Comparisons with empty channels and commonly used heat exchanger media show that the present LFM heat sink can remove heat approximately seven times more efficient than an empty channel and as efficient as a bank of cylinders at the same porosity level. The aluminum LFMs are extremely stiff and strong, making them ideal candidates for multifunctional structures requiring both heat dissipation and mechanical load carrying capabilities. 2003 Elsevier Ltd. All rights reserved.
1. Introduction A variety of technologies such as pin fins, cylinder banks and wire screens have been developed to increase the heat transfer area density of a heat sink without increasing its overall dimensions in the past century. More recently, cellular metal foams have been processed and used to construct lightweight and compact heat sinks. In such sys-
*
Corresponding author. Fax: +44-1223-332662. E-mail address: [email protected] (T.J. Lu).
tems, the high thermal conductivity of metal, combined with its ability to promote eddies and mix the coolant fluid, has enabled such heat sinks to remove up to five times more heat than that by a traditional pin-fin array, at a third of the weight, although the increase in the pressure drop is relatively high (e.g., Bastawros et al., 1998; Lu et al., 1998; Lu, 1999; Kim and Kim, 1999; Calmidi and Mahajan, 2000; Haack et al., 2001; Kim et al., 2002). In situations where a heat sink is also required to carry a certain amount of structural load, lattice-frame materials (LFMs) appear to be more
0167-6636/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmat.2003.07.001
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Nomenclature channel cross-section area ( ¼ H W ) specific heat [J/kg K] LFM bar diameter [m] hydraulic diameter of channel [m] friction factor mean heat transfer coefficient ( ¼ q=ðTw Tin Þ) h1 local heat transfer coefficient around LFM struts H , L, W channel height, length and width [m] I heat sink efficiency index ( ¼ j=f ) j j-Colburn factor ( ¼ StPr2=3 ) k thermal conductivity [W/mK] l individual strut length [m] n cell number along flow direction N total number of cells in flow direction NuDh Nusselt number based on hydraulic diameter ( ¼ hDh =kf ) Nud averaged Nusselt number based on bar diameter DP static pressure drop [Pa] P perimeter of wetted cross-section area [m] Pr Prandtl number Q heat released by a heating element [W] q heat flux [W/m2 ] Red Reynolds number based on LFM strut diameter
A Cp d Dh f h
attractive than conventional heat dissipation media. LFMs are a new development made possible by computer-based design and numerically controlled processing. They consist of a 3D network of cylindrical struts (Fig. 1). The important advance is in the reduction in scale. Recent developments in manufacturing techniques allow the diameter of these 3D lattice structures to vary from millimeters to tens of centimeters. For the LFMs studied in this paper, the diameter of the strut is on the order of one millimeter and the relative density of the structure is 6%. (The relative density is defined as the ratio of the LFM structure density to the density of the solid of which the LFM is made.) The stiffness and
ReDh S SR Sx , S y St Tin Tw ðxÞ Um x, y, z
Reynolds number based on channel hydraulic diameter surface area of wetted cross-section [m2 ] strut ratio ( ¼ l=d) longitudinal and transverse unit cell pitches Stanton number ( ¼ NuDh =ðReDh PrÞ) coolant inlet temperature [K] local substrate temperature coolant inlet mean velocity [m/s] global coordinate system
Greek symbols e porosity q density [kg/m3 ] qrel relative density g local coordinate along cell strut h non-dimensional temperature Subscripts f coolant fluid h hydraulic parameter in inlet of heat sink m arithmetic mean out outlet of heat sink s LFM solid cell SA surface area w substrate wall
strength of LFMs are typically an order of magnitude higher than those of cellular foams made of the same material because, unlike most cellular structures, they are fully triangulated and hence does not contain any bending mechanism (Deshpande et al., 2001a,b). Tetrahedral cells, as shown in Figs. 1 and 2, constitute the triangulated frame. When the structure is loaded, the struts support either axial tensile stressing or axial compressive stressing. Thus, the deformation is stretchingdominated and the frame collapses by stretching of the struts (Deshpande et al., 2001a,b). The LFM structures can also be stacked so that they form a multilayered lattice material block while maintaining the same mechanical properties. In
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
769
O-B Z d
O-A H
Y b
X
c
S y0
f e
SX
SY
a
(a)
SS x0x0 (b)
Fig. 1. (a) Geometrical configuration of a LFM unit cell; (b) top view of (a), with filled symbols representing projections of the cylindrical struts at the bottom substrate of the channel and empty symbols denoting those at upper substrate.
Fig. 2. Two different frontal faces of the LFM model: (a) orientation A (O–A); (b) orientation B (O–B).
comparison, the deformation and failure of cellular metal foams are governed by cell-wall bending, and are sensitive to inherent geometrical imperfections induced during processing (Chen et al., 1999). Consequently, a precise computer-designed LFM exhibits important morphological advantages over cellular foams (in terms of structural imperfections), in addition to the availability of materials. LFMs are ideal candidates for applications (e.g., supersonic and hypersonic aircrafts) where, in addition to carry structural loads at minimum weight, efficient heat dissipation is also demanded.
The microstructural morphologies of an LFM resemble those of cellular metal foams, and hence the overall heat transfer performance of both materials is expected to be similar if the cells are of similar size. On the other hand, another analogy can be made with banks of cylinders (BOCs) that have been widely used in heat sink configurations (Zukauskas, 1987). By using the configurational similarity that three cylindrical struts comprising a unit LFM cell are inclined relative to the longitudinal and transverse directions (Figs. 1 and 2), the available heat transfer and hydraulic resistance data on BOCs may be used to direct heat sink design with LFMs. A cross-flow analogy of cylinder banks has been used to develop heat transfer models for cellular metal foams (Bastawros et al., 1998; Lu et al., 1998; Lu, 2002; Kim et al., 2002). Lattice-frame structures resemble metal foams in microstructural morphology and banks of cylinders in macrostructural configuration. These three structures all provide a solid–air contact area many times greater than the duct surface area so that the temperature difference between the heated wall surface and the coolant can be drastically reduced as a consequence of convective heat transfer. The successful implementation of LFMs as compact heat sinks demands an integrated approach that combines modeling and testing to quantify their heat removal performance as a function of morphological parameters, relative density and fluid properties. Thus, this work is
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Fig. 3. Test rig with LFM model in the test section.
aimed at investigating general features of thermal and hydraulic behaviors of metal lattice-frame materials as a novel heat sink medium in a forced air convection channel (Fig. 3). The paper is organized as follows. In Section 2, an analytical model is developed, by using fin analogy, to predict temperature distributions (and hence Nusselt number) in the LFM structure as well as along the cover plates. Section 3 presents detailed experimental procedures for heat transfer and pressure drop measurements, with a discussion on measurement uncertainties. The experimental results are presented in Section 4; comparison is made with analytical predictions as well as conventional heat dissipation media.
2. Specifications of LFM heat sink With reference to Fig. 3, the LFM heat sink is a typical sandwich structure consisting of a tetrahedral core and two solid face sheets. Both the LFM core and the substrates are made of an aluminum–silicon casting alloy LM25, of wt.% composition Al–7Si–0.3Mg. The panel is investment cast from injection-molded polystyrene preforms; details of processing procedures can be found in Deshpande et al. (2001a). The overall dimensions of the LFM heat sink tested are 0.13 m (L) · 0.145 m (W ) · 0.012 m (H ). The ratio of the channel height to LFM strut diameter, H =d, is 6 and the slenderness ratio, SR ¼ l=d is 7.35. Detailed specifications of the LFM sample and operating condition of the measurements are listed in Table 1. The solid phase thermal conductivity
Table 1 Specifications of the LFM unit cell and typical operating parameters for measurements Specification of LFM (see Fig. LFM cell bar diameter (d) Longitudinal cell pitch (Sx ) Transverse cell pitch (Sy ) Cell strut length (l) Cell height (H ) Material
1) 0.002 m 0.0127 m 0.0147 m 0.0147 m 0.012 m LM25
Typical operating parameters Inlet coolant mean velocity Bar Reynolds number Input heat flux Inlet coolant temperature Outlet coolant temperature Test section inlet pressure
2.0–26.0 m/s 250 < Red < 3300 4, 8 and 16 kW/m2 300.0 K 305.0–360.0 K 1 bar (ambient conditions)
of LM25 is 150.84 W/mK (from Aluminum Federation, UK). Pressure drop and heat transfer measurements are performed for two orientations of the LFM, as illustrated in Fig. 2; the relevant parameters are listed in Table 2. Here, the relative density of pffiffiffi 2 the LFM is defined as qrel ¼ ð 3=2Þðd=lÞ ðl=H Þ, porosity is e ¼ 1 qrel , and the surface area density of the heat sink, i.e., the ratio of the total wetted heat transfer area including the two face sheets and Table 2 Parameters for the LFM samples Sample Relative density of the LFM (qrel ) Porosity (e) Surface area density (qSA ) Frontal area/free flow area
LFM O–A
LFM O–B 0.062
0.73
0.938 123.68 m1 0.36
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
the LFM core to the volume of the heat sink, is given by qSA ¼ ½3=d þ 2ðSx lÞ 3ðd 2 =4Þ =ðHSx lÞ.
3. Analysis The fluid dynamics and thermal transport in a LFM heat sink are complex, and hence it is difficult to analytically examine the details of flow and heat transfer within the LFM microstructure by using the Navier–Stokes equation and the associated energy equation. The main objective of this section is to construct an approximate analytical solution. Thus, the rules played by a few important parameters such as strut diameter and solid thermal conductivity can be identified and the physical understanding of the problem could be more improved. 3.1. Mathematical formulations
Eq. (1) is solved subjected to the following boundary conditions: dTs ks ¼ q0 dg g¼0 ð2Þ dTs ks ¼0 dg g¼l It is noted here that q0 is the heat flux entering the strut ad, and is generally not identical to the uniform heat flux q imposed on the outer surface of the substrate. This is because the solid LFM strut can drastically reduce the thermal resistance at the three-struts joint, and thus lead to a much higher heat flux at the joint. The value of q0 is not specified, and must be determined. Eq. (1) solved with boundary conditions (2) leads to the following analytical solution of the solid temperature distribution along strut ad: q0 coshðmðl gÞÞ ð3Þ sinhðmlÞ ks m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m ¼ h1 P =ðks AÞ ¼ 2 h1 =ðks dÞ, and d is the strut diameter. From Eq. (3), the wall temperature can be obtained as
Ts ðg; nÞ Tf ðnÞ ¼
The unit cell shown in Fig. 1 can be taken as an analytical domain. As shown in Fig. 1(a), each unit cell comprises three inclined struts. The heat imposed on the bottom wall is taken away either directly by the coolant at the substrate surface or by the struts via conduction and convection. To determine the heat transfer coefficient, h ¼ q= ðTw Tin Þ, the substrate temperature needs to be found. In this analysis, both the fluid and wall temperature are assumed to be constant within a unit cell, but these will increase in subsequent unit cells along the stream direction. Since the heat transfer mechanism for the three inclined struts are the same, here only one strut ad is used for analysis (Fig. 1(a)). The solid temperature variation along the strut ad in the ÔnÕth cell from the entrance is governed by d2 Ts ðg; nÞ h1 P ðTs ðg; nÞ Tf ðnÞÞ ¼ 0 dg2 ks S
771
q0 coshðmlÞ km sinhðmlÞ q0 ¼ tanh1 ðmlÞ ks m
Tw ðnÞ Tf ðnÞ ¼
ð4Þ
It should be pointed out that the mean fluid temperature Tf ðnÞ and heat flux q0 still need to be determined in order to finally obtain the wall temperature Tw ðnÞ. The mean fluid temperature Tf ðN Þ can be obtained by considering energy balance: qf Sy Hum Cp ½Tf ðnÞ Tf ð0Þ ¼ qSy Sx n
ð5Þ
from which ð1Þ
where g is the local coordinate along the ad strut, Tf ðnÞ is the mean fluid temperature in the nth cell, ks is the solid conductivity, h1 is the local heat transfer coefficient around the strut, and P and S are the perimeter and area of the cross-section of strut ad, respectively.
Tf ðnÞ ¼ Tf ð0Þ þ
qSx n qf Hum Cp
ð6Þ
To determine the heat flux q0 , we note that the heat flux at the inner surface of the substrate not connected to the strut is q1 ¼ ½Tw ðN Þ Tf ðN Þ h1 ¼
h1 q0 coshðmlÞ ks m sinhðmlÞ
ð7Þ
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
where h1 is the local heat transfer coefficient near the substrate surface. For simplicity and also because of the strong coolant mixing by the LFM microstructure, it is assumed that h1 takes the same value as the local heat transfer coefficient around the struts, h1 . The total heat imposed on the outer surface of a unit cell can therefore be expressed as Sx Sy q ¼ ðSx Sy pd 2 =4Þq1 þ 3pd 2 q0 =4
ð8Þ
Substitution of (7) into (8) yields, q0 ¼
Sx Sy q h1 coshðmlÞ 3 2 þ pd ðSx Sy pd 2 =4Þ ks m sinhðmlÞ 4
ð9Þ
qSx N 1 coshðmlÞ þ qf Hum Cp km sinhðmlÞ Sx Sy q h1 coshðmlÞ 3 2 þ pd ðSx Sy pd 2 =4Þ ks m sinhðmlÞ 4 ð10Þ
With (10), the unit-cell averaged Nusselt number can be obtained as 0 Dh B B Sx n þ coshðmlÞ NuðnÞ ¼ kf @ qf Hum Cp km sinhðmlÞ
0:36 0:71Re0:5 ðkf =dÞ d Pr 0:6 0:36 0:35Red Pr ðkf =dÞ
Red ¼ 1 500 ¼ 500 1000 ¼ 103 2 105 ð13Þ
The effect of cylinder inclinations with respect to the flow direction on h1 has also been given in Zukauskas (1987), and is found to be relatively small in comparison with its much larger knockdown effect on pressure drop. In the present study, the strut inclination angle with respect to the horizontal surface is a 54:7, so the values of h1 calculated from empirical correlation (13) are multiplied by a factor of 0.9, following the recommendation of Zukauskas (1987).
4. Experimental details 4.1. Experimental apparatus
1 Sx Sy h 3 1 coshðmlÞ þ pd 2 ðSx Sy pd 2 =4Þ ks m sinhðmlÞ 4
C C A
ð11Þ The overall Nusselt number of the LFM heat sink is given by N 1 X NuðnÞ N 1
0:36 ðkf =dÞ h1 ¼ 1:04Re0:4 d Pr
¼
Tw ðN Þ ¼ Tf ð0Þ þ
Nu ¼
The analytical model is not complete until the local heat transfer coefficient h1 in (11) is determined. To find h1 on the surface of each cylindrical strut, the lattice-frame material is taken as comprising of three sets of inclined cylinders arranged in staggered arrays (Figs. 1 and 2). For staggered arrays of cylinders in cross-flow, the following correlations based on extensive experimental measurements can be used to calculate h1 , with an accuracy of ±15% (Zukauskas, 1987): ¼
Finally, the wall temperature Tw ðnÞ can be obtained by substituting Eqs. (6) and (9) into Eq. (4), as
3.2. Local heat transfer coefficient
ð12Þ
where N is the total number of unit cells in the flow direction.
The experimental apparatus consists of four main sections: coolant supply, test section, test model, and data acquisition system. Air in ambient conditions was used as a coolant and drawn through the channel passage by a suction type air blower. One wire screen and one honeycomb were inserted in the channel before the contraction. The coolant then flew through a 9:1 contraction and a parallel section, in which the ratio of section length to channel height was 20, before flow reaches the test section. As shown in Fig. 3, the LFM filled heat channel was encapsulated by perspex sidewalls. A flow rate regulator is located between the exit of the test section and the suction device.
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
The thermal performance of a heat sink channel with a single LFM layer was measured, with the bottom substrate exposed to heating and the top substrate insulated, i.e., asymmetry heating imposed by a heating element (silicone-rubber etched foil from WatlowTM Inc.). A total of four static pressure tapings were placed on the top substrate along the flow direction. In actual heat sink applications, multilayered LFMs is preferable so the asymmetry of the thermal boundary condition was imposed for single layer experiment. The amount of heat released from the heating element was adjusted by changing the supply voltage. To minimize heat loss from direct contact, the external surface of the heating element was covered with a thermal insulation material, Tancast 8 . Because the electrical resistance foil was etched in a zigzag pattern on the silicone-rubber pads, there was a possible gap (unspecified by the manufacturer) between two neighboring columns of the foil. Consequently, a pure copper heat spreader plate, 0.9 mm thick with 12 uniformly spaced transverse slits of 0.5 mm depth and 0.15 mm width, were inserted between the heating element and the LFM substrate to ensure the uniformity of the heat flux entering the LFM heat sink. Five thin foil (0.013 mm thickness) T-type copper–constantan thermocouples (from Rhopoint Inc.) were inserted on the lower substrate along the longitudinal direction (i.e., flow direction). There were two additional T-type bead thermocouples, positioned separately at the inlet and outlet of the test section to measure the coolant temperature at midheight at each location. In addition, thermal images of local temperature distribution on the upper substrate (unheated surface) were captured by an infrared camera, AGEMA ThermoVision 870, to ensure a linear increment of substrate temperature subject to the constant heat flux thermal boundary condition. 4.2. Experimental and data acquisition procedures The experiments were run for several minutes until the flow inside the channel becomes hydraulically and thermally stabilized. All measurements were performed under steady state conditions. A Pitot tube was positioned before the test section to
773
measure stagnation pressure and static pressure at the inlet of the test section. Because the blockage ratio, i.e., the ratio of channel height (12 mm) to tube outer diameter (0.51 mm), was 23.5, wall interference effects on the Pitot tube were expected to be small. All measurements were repeated until significant data repetition was ensured, i.e., 5% uncertainty interval (see measurement uncertainties discussed later). A friction factor is used to quantify the flow resistance across the LFM structure, defined as 4f ¼
DP Dh DP ðnDh =LÞð1=ðqU 2 =2ÞÞ ¼ 2 L qU =2 n
where Dh , Um , L, n and DP =n are the hydraulic diameter of the heat sink channel, mean coolant velocity (averaged over a channel height) at the inlet of the test section, LFM sample length, unit cell size, and static pressure drop per unit cell, respectively. The mean heat transfer coefficient h and mean Nusselt number NuDh (and Nud ) for fixed values of heat flux and Reynolds number are obtained as h ¼ q=ðTw Tin Þ and NuDh ¼ hDh =kf
ðand Nud ¼ hd=kf Þ RL where Tw ¼ ð1=LÞ 0 Tw ðxÞDx is the spatially averaged substrate wall temperature. Here, L is longitudinal substrate length, and Tw ðxÞ is measured local substrate temperature. The inlet coolant temperature Tin is used as a reference for all calculations. 4.3. Measurement uncertainties
An uncertainty analysis was performed by following the method of Coleman and Steele (1999). The maximum heat loss through insulation materials was estimated to be less than 2% out of total input heat. Heat loss through side walls was assumed to be negligible due to the small conduction area. The coolant temperature difference between the inlet and outlet can be either measured, ðDT Þmeasured , or calculated directly from energy balance:
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780 1.1
∆ T energy _ balance
∆Tmeasured
1
0.9
Input heat flux, q = 4 KW/m2 Input heat flux, q = 8 KW/m2 Input heat flux, q = 16 KW/m2
0.8
0.7
0
500
1000
1500
2000
2500
3000
Red Fig. 4. Error estimation by comparing measured temperature with that calculated from enthalpy balance equation for three different values of heat input, with DT ¼ Tout Tin .
ðDT Þenergy
balance
_ pÞ ¼ Tf;in Tf;out ¼ Q=ðmC
where Q is the input heat and m_ is mass flow rate. Fig. 4 plots the ratio ðDT Þmeasured =ðDT Þenergy balance as a function of Red for three levels of heat input, where Red is the Reynolds number based on mean inlet velocity and bar diameter. The measured values were within 92% of predicted values for most of the bar Reynolds number regime. The thermal conductivity kf of air varied slightly in the operating temperature range of 300.0–360.0 K. An arithmetic mean value was used for kf , with an uncertainty estimated to be within 6.6%. From these, the uncertainty in the measured heat transfer coefficient and Nusselt number was estimated to be less than 5.0% and 8.3%, whilst the uncertainty in the pressure drop and friction factor measurements was estimated to be less than 5.0% and 9.7%, respectively. A root-mean-square method was used for these calculations. 5. Results and discussion Since heat transfer and pressure drop measurements have been performed on a new type of
heat exchanger medium that has vastly different geometrical features in comparison with other types of heat dissipation medium, a consistency check is necessary. Thus, the LFM strut diameter based Reynolds number is used in presenting the test results and comparing with empirical correlation of banks of cylinder arrays, whilst comparison with other available heat sink media was made by using the hydraulic diameter based Reynolds number. 5.1. Pressure drop Static pressure drop measurements across the LFMs were performed. Results of pressure drop and the corresponding friction factor are depicted in Fig. 5(a) and (b), respectively. Fig. 5 indicates that the flow was typically laminar when Red < 300, and in transition from laminar to turbulent when 300 < Red < 400 for both LFM orientations. The transition Reynolds number is low due to interactions between shedding vortices and wakes from LFM struts. Because of the LFM morphology, flow patterns are three dimensional even in a unit cell. Strong flow interaction with wakes and
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780 18
Orientation A (O-A) Orientation B (O-B)
16 14
O-A
dP/L [KPa/m]
12 10 8 6 4
O-B 2 0
0
5
10
15
25
20
Um [m/s]
(a)
for a smooth pipe from Moody chart Orientation A (O-A), LM25 metallic model Orientation B (O-B), LM25 metallic model 100
f 10-1
O-B
16 / Red (for Laminar flow)
(b)
100
occupied by the structure/free flow area) of O–A and O–B is 0.73 and 0.36, respectively. The resistance encountered by the flow in orientation O–A is twice as large as that in O–B, indicating that pressure drop was mainly contributed by the flow blockage effect due to the inserted structure, i.e., a higher flow resistance occurred in the higher blocked orientation. It appeared to be proportional to the blockage ratio, but difficult to justify due to lack of test data. Systematic parametric studies on the dependence of pressure loss on porosity, blockage ratio, Reynolds number, etc. are currently under investigation. In addition, comparison with a smooth pipe from the Moody chart (Bejan, 1995) was made and plotted together in Fig. 5 to underline the additional pressure loss due to the insertion of the LFM structure for the laminar flow region. About 3.3 times higher pressure drop than the empty channel was recorded for O–B and 7.5 times for O–A. 5.2. Heat transfer characteristics
O-A
10-2
775
1100
2100 3100
Re d
Fig. 5. Flow resistance of the LFM heat sink as a function of Reynolds number for orientation O–A and orientation O–B: (a) pressure drop per unit length, with curve fitting also shown; (b) friction factor.
shedding vortices especially near the vertices is expected. The friction factor, f , approached an asymptotic value for both test models after the flow transition, i.e., turbulent flow prevailed when 500 6 Red 6 3200, with f 0:28 for O–A and f 0:12 for O–B. Flow blockage ratio ( ¼ area
To characterize the capacity of heat removal by the LFMs in forced convection, local heat transfer coefficients measured at selected longitudinal locations, i.e., on centerline (point ÔfÕ in Fig. 1(a)) were spatially averaged. In addition, Fig. 6 plots the local temperature distribution on the upper substrate plate (i.e., without the heating pad) along the mainstream direction captured by the infrared camera. The flow direction was from left to right, and the area of interest was denoted in dotted lines. Note that there is minimal endwall effect, and the substrate surface temperature varies linearly with the x-axis, typical of a constant heat flux boundary. The spatially averaged (mean) Nusselt number Nud of the heat sink system was plotted in Fig. 7 for both orientations as a function of the Reynolds number. Shorter cylinder spacing and narrower flow passage in LFM O–A may lead to reduced wake velocity and increased mainstream velocity, resulting in slightly less efficient heat transfer than that from LFM O–B. Similar observations were reported in Zukauskas (1987) for banks of cylinder arrays.
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T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
Fig. 6. Local temperature distribution of the upper substrate captured by infrared camera for q ¼ 500 W/m2 and Um ¼ 4 m/s.
0:56
LFM(O-A) LFM(O-B) Correlations by Zukauskas (for Staggered BOCs) 2
10
Empirical correlation for the LFMs
Nu d
Nud=0.428(Red)0.56Pr0.36
101 Correlations by Zukauskas Nud=1.04(Red)0.4Pr0.36 for Red<500 Nud=0.71(Red)0.5Pr0.36 for 500
100 200
1200
2200 3200 4200
Re d Fig. 7. Comparison of heat transfer measurement with empirical correlation of Zukauskas (Deshpande et al., 2001b) for typical staggered banks of cylinder arrays.
To compare with the results of typical staggered banks of cylinders summarized in Zukauskas (1987), the averaged Nusselt number was correlated as a function of Reynolds number and Prandtl number: Nud ¼ 0:428ðRed Þ0:56 Pr0:36 or
ð14aÞ
NuDh ¼ 1:23ðReDh Þ
Pr0:36
ð14bÞ
where Nud and NuDh are mean Nusselt numbers based on the LFM strut diameter and channel hydraulic diameter, respectively. Eqs. (14a) and (14b) are valid for both O–A and O–B orientations in the range of 120 6 Red 6 3200, where the Prandtl number Pr is assumed to be constant ( ¼ 0.71). As seen in Fig. 7, the heat transfer characteristics of LFMs are similar to that of banks of cylinder arrays due to their structural resemblance. The discrepancy between the two orientations was negligible at low Reynolds number but reaches 10% at high Reynolds number. As the coolant velocity increases, heated flow near the substrates may tend to be confined and recirculates especially behind the LFM vertices where the distance between neighboring cylinders is short. Consequently, the heat removed from the substrate surfaces may not be fully convected. This may be one of the limiting cases in heat removal by the use of a low thermal diffusivity fluid (e.g., air) as coolant. 5.3. Fin-analogy estimation The fin-analogy based analytical model presented in Section 3 gave local Nusselt number per unit cell, with local heat transfer coefficient
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
777
350
Measured substrate temperature 340
T [K]
330
Predicted substrate temperature 320
Predicted fluid temperature 310
300
At Red=500 290
1
2
3
4
5
6
7
8
9
10
N, cell number
(a) 0
-0.5
Predicted strut temperature
qD h /ks
Ts − Tw
-1
-1.5
-2
Red = 2000 Red = 500
-2.5
-3 0
0.2
(b)
0.4
0.6
0.8
1
y/H
Fig. 8. (a) Predicted solid and fluid temperature variations along the flow direction; (b) non-dimensional temperature of solid strut.
prescribed from empirical correlations for banks of cylinders (Zukauskas, 1987). The predicted fluid and wall temperature variations are shown in Fig. 8(a) for Red ¼ 500, together with the measured wall temperatures for comparison. The uniform
heat flux imposed on the outer surface of the substrate dictates that the two curves are parallel. Overall, the predictions are in close agreement with the measurements. Fig. 8(b) depicts the nondimensional strut temperature as a function of
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
3
10
j ¼ St Pr2=3
Experimental data of the LFM O-A Experimental data of the LFM O-B Analytical estimation using fin analogy Empty channel correlation from Dittus & Boelter
where St is the Stanton number ( ¼ NuDh =ðReDh PrÞ) and Pr is the Prandtl number. Since heat sink design parameters such as the overall heat transfer coefficient and pressure drop are interdependent, an efficiency index accounting for not only flow resistance but also heat transfer is often required to rank the overall performance of different heat dissipation media. For convenience, such an efficiency index is introduced here as
Nu Dh
O-B O-A 2
10
Estimated by fin analogy
NuD=0.023 (ReD)4/5Pr0.4
101 3 10
I ¼ j=f
by Dittus and Boelter 4
10
5
10
Re Dh Fig. 9. Comparison of predicted overall Nusselt number with experimental data.
y=H within one unit cell for selected values of Red , with Tw and Ts representing solid temperature at the wall and local solid temperature varying along the channel height, respectively. The predicted and measured average Nusselt number (based on the channel hydraulic diameter) is presented in Fig. 9, and compared with that of an empty channel (Dittus and Boelter, 1930). In general, the predicted Nusselt number follows that of the empty channel, with approximately six times higher magnitude, whilst the experimental data show that the magnitude of heat transfer enhancement over the empty channel decreases as the Reynolds number is increased in the range of 3000 6 ReDh 6 35 000. As previously mentioned, the measurement range covers the end of laminar flow as well as the whole transition. Higher heat enhancement was observed in these regimes, accompanied with higher flow friction. Generally speaking, the prediction from the analytical model is in satisfactory agreement with experimental data except within the lower Reynolds number regime.
where j is the Colburn factor defined above and f is the friction factor. It is evident that a high value of I is desirable for optimal heat sink design. Comparisons were made with typical heat dissipation media, e.g., banks of cylinders, pin-fin arrays, cross-rod matrices, and randomly stacked sphere matrices (Kays and London, 1984). To be consistent with the available data, the hydraulic diameter has been used as a characteristic length scale for the LFMs. As shown in Fig. 10, the overall performance of LFM O–B is about twice higher than that of cylinder banks as well as LFM O–A, due mainly to its lower flow resistance. The plain pin-fin arrays offer the highest efficiency
10
LFM: Orientation A LFM: Orientation B Bank of cylinders Wavy-fin plate-fin surface Cross-rod matrices, inline stacking Plain pin-fin Infinitely randomly stacked sphere matrix
0
x
I (=j / f)
778
LFM O-B
10
-1
x
x
x
x x
x
x
LFM O-A
5.4. Thermal efficiency index In order to facilitate quantification of heat transfer performance of the LFM heat sink with other available heat exchanger media, the j-Colburn factor (Colburn, 1933) was used:
10
-2
10
3
10
4
10
5
Re Dh Fig. 10. Efficiency index, I (¼ j=f ), plotted as a function of Reynolds number based on channel hydraulic diameter for LFMs and common heat sink media (Lu, 1999).
T. Kim et al. / Mechanics of Materials 36 (2004) 767–780
779
Table 3 Configurational data of heat dissipation media Configurations
Porosity
Surface area density [m1 ]
Materials
Lattice-frame material Staggered bank of cylindrical tubes
0.938 N/A
123.68 165.02
LM25 N/A
Wavy-fin plate-fin surface
0.822
1138
Aluminum
Crossed-rod matrices, in-line stacking Infinitely randomly stacked sphere matrix
0.832
N/A
N/A
0.37–0.39
N/A
N/A
although their ability to dissipate heat is relatively low, whereas cross-rods and randomly stacked sphere matrices appear to have the lowest heat removal efficiency. Details of the data compared and the corresponding configurational specifications can be found in Table 3 and in Kays and London (1984). However, it is emphasized that the data shown in Fig. 10 for different heat dissipation media must be used cautiously, as a systematic comparison is at present not possible.
6. Conclusion The thermal and hydraulic properties of latticeframe materials subjected to forced air convection cooling have been evaluated by steady state measurements as well as analytical modeling based on the fin analogy. The model can predict well the behavior of Nusselt number in the higher Reynolds number regime, but gives slightly underestimation in the lower Reynolds number regime including laminar and flow transition. Friction factor, averaged heat transfer coefficient, and averaged Nusselt number were quantified in the range of bar Reynolds number, 120 6 Red 6 3200 (or, 1300 6 ReDh 6 35000 based on channel hydraulic diameter). The performance of a LFM heat sink was found to be somewhat similar to that of conventional cylindrical tube arrays due to morphological similarities. The highly porous LFM structure (porosity 0.94) promotes turbu-
Comments
Longitudinal and transverse pitch are Sx ¼ 2d and Sy ¼ d where d is tube diameter. Fig. 10–9 from Zukauskas, 1987–– Transient test Fig. 10–74 from Zukauskas, 1987 Fig. 10–98 from Zukauskas, 1987 Fig. 7–10 from Zukauskas, 1987
lent flow, which causes thermal dispersion effect, for the whole range of Reynolds number investigated: about six times more heat can be removed in comparison with an empty channel. The multifunctional LFMs can find attractive applications where a structure is required to effectively dissipate heat in addition to carry mechanical loads.
Acknowledgements This work was supported by the UK Engineering and Physical Sciences Research Council (EPSRC grant number EJA/U83) and by the US Office of Naval Research (ONR/ONRIFO grant number N000140110271). The authors wish to thank Mr. Victor Yung at the Whittle Laboratory of Cambridge University for helpful discussions, and Dr. Vikram Deshpande and Prof. N.A. Fleck of Cambridge University for the assistance on the preparation of test specimens.
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