- Email: [email protected]
Effect of pore structure on heat transfer performance of lotus-type porous copper heat sink
International Journal of Heat and Mass Transfer 144 (2019) 118641
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of pore structure on heat transfer performance of lotus-type porous copper heat sink Xiaobang Liu a, Yanxiang Li a,b,⇑, Huawei Zhang a,b, Yuan Liu a,b, Xiang Chen a,b a b
School of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, PR China
a r t i c l e
i n f o
Article history: Received 19 June 2019 Received in revised form 16 August 2019 Accepted 25 August 2019 Available online 4 September 2019 Keywords: Lotus-type porous copper Heat sink Pore structure Open pores
a b s t r a c t A special kind of microchannel heat sink is fabricated by a lotus-type porous copper in which long cylindrical pores are arranged in one direction. By applying the straight fin model to the lotus-type porous copper, the effect of the pore structure including porosity, pore diameter and open porosity on its heat transfer performance can be predicted. Given an available pump pressure, the structural parameter window of lotus-type porous copper which meets the heat dissipation requirement can be given. By improving the process control pore structure can be optimized and 80% pores in optimized lotus-type porous copper samples are open when the sample length along the pore growth direction is 16 mm. The heat transfer performance of optimized lotus-type porous copper heat sinks was tested. At a pressure drop of 100 kPa, a heat transfer coefficient of 7.86 (W/cm2 K) could be achieved. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Lotus-type porous metals [1] are also called as Gasar metals [2– 4]. They are fabricated by a directional solidification process in pressurized hydrogen or nitrogen. During the solidification process, the pores and the metal grow together, and finally a structure in which cylindrical pores are regularly arranged is formed. This technology was developed in the USSR, and then was patented in the US [5]. Since oriented cylindrical pores can serve as smooth transmission channels, lotus-type porous metals are very promising in the fields of heat sinks [6,7], filters, and biomaterials [8], among others. When fluid flows through the cylindrical pores, a large amount of heat can be quickly carried away by convective heat transfer of the fluid and heat conduction of the substrate. Thus, lotus-type porous copper is very suitable for heat dissipation, which combines a large specific surface area of the porous structure and the high thermal conductivity of copper. In general, the porosity of lotustype porous copper is 0.2–0.6 and pore diameter is 0.1–1 mm. Therefore, the lotus-type porous copper can be regarded as a special microchannel heat sink. To date, some researchers have investigated the heat transfer performance of lotus-type porous copper (hereafter referred to as
⇑ Corresponding author at: School of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China. E-mail address: [email protected] (Y. Li). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118641 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
lotus copper). Chiba and Ogushi et al. [6,9] compared the heat transfer performance of lotus copper heat sink with conventional groove fin and microchannel heat sink for water cooling. They spliced 3 sections of lotus copper with a length of 3 mm (along pore growth direction) into a heat sink. Under the same pumping power, the heat transfer performance of the lotus copper heat sink was 4 times greater than the conventional groove fins and 1.3 times greater than the microchannel heat sink. The lotus copper sample had pores with an average diameter of 0.3 mm and a porosity of 0.39 and a heat transfer coefficient of 8 W/cm2 K was obtained. They also found that the fin efficiency of lotus copper fin could be predicted by the straight fin model by using the effective thermal conductivity perpendicular to the pore axis and the surface area ratio between the lotus copper fin and the straight fin. Then, Chiba et al. [10] investigated the heat transfer performance of lotus copper heat sink for air cooling. They considered the distribution of pore diameter and found that the heat transfer coefficient and pressure drop per unit fin length of the nonuniform pore model showed a good agreement with that of the uniform pore model. Muramatsu et al. [11] compared the heat transfer performance of two types of porous copper heat sink for air cooling. One was lotus copper and the other was porous copper machined with a deterministic pattern of holes. Muramatsu pointed out that in lotus copper the open porosity needed to be considered. Zhang et al. [12] studied the heat transfer performance of lotus copper heat sinks and found that open porosity had a great
2
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Nomenclature A Ab Aeff b CP d d di f h0 h1 hexp hcal H kf ks keff? L n m Nup P DP Pr q Q Rcond Rconv Rconv;b Rheat Rtotal
area of heater cross-sectional area of a lotus copper fin total convective heat transfer area of lotus copper width of lotus copper fin specific heat capacity of fluid pore diameter average pore diameter diameter of ith pore volumetric flow rate convective heat transfer coefficient in the pores heat transfer coefficient of the equivalent fin experimental heat transfer coefficient theoretical heat transfer coefficient height of lotus copper sample thermal conductivity of fluid thermal conductivity of copper effective thermal conductivity of lotus copper perpendicular to the pore growth direction length of lotus copper sample number of pores fin factor Nusselt number peripheral length of the equivalent fin pressure drop Prandtl number heat flux density heat flow quantity conductive thermal resistance of lotus copper convective thermal resistance of lotus copper convective thermal resistance of a lotus copper fin caloric thermal resistance of lotus copper theoretical total thermal resistance
influence on the heat transfer performance. The lotus copper had pores with an average diameter of 0.39 mm and a porosity of 0.29. When the length of lotus copper sample was 20 mm, the corresponding open porosity was only 0.11 and a heat transfer coefficient of 5(W/cm2 K) could be obtained. By cutting the sample into several sections, the length of a single segment along the pore growth direction was reduced. Thus, the open porosity improved, and the flow rate increased, thereby enhancing the heat transfer performance. When the pumping power was 5 W, decreasing the length of the single lotus copper segment from 20 mm to 10 mm, 5 mm, and 2.5 mm resulted in a gradual increase in the heat transfer coefficient from 5(W/cm2 K) to 6(W/cm2 K), 7.5(W/cm2 K), and 8(W/cm2 K). Moreover, Liu et al. [13] used Fluent-3D numerical simulation to study the heat transfer performance of lotus copper heat sinks when liquid GaInSn was taken as the coolant. As pointed out by Muramatsu and Zhang, some pores do not penetrate through the sample, which are called ‘‘closed pores” here, while those pores which penetrate through the sample are called ‘‘open pores” here. There is a great difference in the number of open pores between different samples, even though the porosity remains unchanged. Therefore, in order to quantify the contribution of open pores, the open porosity must be introduced and is defined as the ratio of the area of open pores to the crosssectional area of the whole sample (Porosity is the ratio of the area of pores to the cross-sectional area of the whole sample). Only open pores can play a role in convective heat transfer while closed pores will decrease the effective thermal conductivity, resulting in a discounted heat transfer performance. Thus, the open porosity is critically important to the heat transfer performance of lotus copper heat sink.
Rexp total Re s t T w;max T f ;in DT max
v
W
experimental value of total thermal resistance Reynolds number interpore spacing thickness of solid copper plate maximum surface temperature of heat sink inlet water temperature maximum temperature difference velocity of fluid in pores width of lotus copper sample
Greek symbols a ratio of total convective heat transfer area and area of copper substrate e porosity e average porosity e1 porosity measured on the bottom cross-section e2 porosity measured on the top cross-section u open porosity b open ratio c aspect ratio of pores q density of fluid g fin efficiency t dynamic viscosity of fluid 1 pressure loss coefficient d difference x ratio of thermal conductivity of pore to thermal conductivity of copper Subscripts f fluid s solid
However, the open porosity is always smaller than the porosity and is dependent on the sample length along the pore growth direction [12]. As the sample length of lotus copper increases, the open porosity will decrease. In the research of Chiba [9], Ogushi [6] and Muramatsu [11], in order to obtain a high open porosity, they spliced 3 sections of lotus copper with a length of 3 mm (along pore growth direction) into a heat sink. However, those lotus copper samples used in their testing were too short along pore growth direction to make practical heat sink products. Only the heat sink made by lotus copper with a greater length can cool the high heat flux density devices with larger area and, at the same time, has sufficient structural strength, which is important for practical application. Much effort has been made to optimize the fabrication process to increase the open porosity on a longer lotus copper sample [14,15]. It is widely recognized that a flat solid/liquid interface can improve the pore straightness and parallelism, which contributes to increasing the open porosity. Yun He et al. [14] introduced the Bridgman method to fabricate lotus copper ingots. They obtained a planar solid/liquid interface by optimizing the withdrawing speed, thereby increasing the open porosity. As mentioned above, the heat transfer performance of lotus copper heat sink mainly depends on three influencing factors: porosity, pore diameter and open porosity. However, there are few studies on the effect of open porosity on heat transfer performance. In addition, similar to metal honeycombs [16] and cellular metals [17], the heat transfer performance of lotus copper heat sinks can be maximized by optimizing the porosity and pore diameter. However, there is no systematic study on the effects of pore diameter and porosity on the overall heat transfer performance of lotus copper heat sink.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Thus, the main objectives of the present paper are as follows: (1) to optimize the fabrication process to increase the open porosity and hence enhance the heat transfer performance of lotus copper heat sink; (2) to study the effects of porosity, pore diameter and open porosity on overall heat transfer performance, and to obtain the optimal pore structure for maximum heat dissipation. In the present work, we first observed the pore structure of optimized lotus copper and the porosity, open porosity and pore diameter were measured. Then, we predicted the heat transfer performance by applying the straight fin model into lotus copper heat sink. The effects of pore diameter, porosity and open porosity on the heat transfer performance of lotus copper heat sink have been investigated. The structural parameter window of lotus copper which meets the heat dissipation requirement under an available pump pressure is given. Finally, we examined experimentally the heat transfer performance of lotus copper heat sinks which were fabricated under optimized conditions.
the top and bottom surfaces of each sample and average value of two surface was taken. The average porosity was calculated using the following formula:
e¼
e1 þ e2 2
The experimental procedure is as follows: (1) Fabrication of lotus copper ingots. (2) Characterization of the pore structure of lotus copper. (3) Preparation of lotus copper heat sinks. (4) Test of the heat transfer performance of lotus copper heat sinks. 2.1. Fabrication of lotus copper ingots Cylindrical lotus copper ingots were fabricated using a Bridgman directional solidification apparatus. The porosity and pore diameter could be modulated through the application of different hydrogen and argon pressures [18,19]. The open porosity could be improved by optimizing the withdrawing speed. Under optimum conditions, the solid/liquid interfaces were almost planar and straight pores were obtained. The optimum withdrawing speed increased as the ingot size decreased. For the lotus copper ingot with 150 mm in diameter and 200 mm in height (weighing about 20 kg), the optimum withdrawing speed was 0.5 mm/s. For the small ingot with 100 mm in diameter and 130 mm in height (weighing about 6 kg), the optimum withdrawing speed was 1 mm/s. Detailed fabrication procedures have been reported previously [14]. Cuboid samples were cut from the central part of two optimized lotus copper ingots by a wire cutting machine. 2.2. Characterization of pore structure The sample dimension of lotus copper was 16 mm (length along the pore growth direction) * 26 mm (width) * 5 mm (height). Cuboid samples were first polished with a series of sand papers from 800 to 2000 grits. Then, they were degreased with acetone and cleaned with diluted HCl and ethanol successively in an ultrasonic cleaner. The pore structure of lotus copper samples is shown in Fig. 1. Paint was used to enhance the contrast for distinguishing open pores, closed pores and the substrate. Open pores are white and closed pores are black. As shown in Fig. 1, open pores were observed with the help of a flashlight. A beam of light with enough intensity was radiated perpendicularly from one side of the sample. Then, the light could pass through open pores. Thus, the open pores could be picked out easily from the other side of the sample. All open pores and closed pores were marked on the scanned picture. The open porosity, porosity, and pore diameter could be obtained by an image analysis software. In order to avoid the influence of incomplete pores on the edge, only the 25 mm * 4mm area in the center was counted. Statistical analysis was performed on
ð1Þ
e1 and e2 are porosities measured on the bottom and top crosssection, respectively. The porosity in cross-section of lotus copper almost keeps constant along the solidification height [19]. The pore diameter will vary not much on the cross-section. Chiba et al. [10] considered the distribution of pore diameter and found that the heat transfer coefficient and pressure drop per unit fin length of the non-uniform pore model showed a good agreement with that of the uniform pore model. The average pore diameter reflected the combined result of all pores, and it was calculated using the following formula [13]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 di d¼ n
2. Experimental procedure
3
ð2Þ
where di is the diameter of ith pore on two cross-sections. 2.3. Preparation of lotus copper heat sinks Thereafter, the cuboid lotus copper sample was bonded to a solid copper plate with a thickness of 1 mm, width of 29 mm and length of 20 mm by vacuum diffusion welding to form the heat sink, as shown in Fig. 2. The diffusion welding process was as follows: vacuum pumping (vacuum degree: 10-3 Pa, and compressive pre-stress: 1 MPa), followed by heating (setting temperature: 1100 K), forcing (compressive stress: 5 MPa and duration time: 10 min; a stop block was used to limit over deformation), and cooling (in the vacuum furnace). 2.4. Test of heat transfer performance Fig. 3a shows a schematic diagram of the apparatus used for evaluating the heat transfer performance of lotus copper heat sinks. Water was selected as liquid coolant and circulated in a loop by a water pump. The flow rate could be controlled by adjusting valves and a flow divider. The inlet pressure and outlet pressure of the heat sink were measured by two high-precision pressure sensors with a range of 0–300 kPa. The relative accuracy was ±0.3 kPa. Thus, the pressure drop between the inlet and outlet could be obtained. The heat flux load was applied to the heat sink by a high-power heating unit. The heat flux was generated by eight pieces of heat generation resistance sheet and transferred to the heat sink from a copper conduction block. The heat flux input section was 16 * 25 mm and the heating power could be regulated by voltage and current. To reduce the interfacial thermal resistance, a layer of thermal conductive silicone was applied between the heating unit and heat sink. T-type thermocouples with an accuracy of ±0.1 °C were used to measure the inlet water temperature, outlet water temperature and the hot surface temperature of heat sinks. The hot surface temperature of heat sinks was measured at three different positions on the copper base plate (near the inlet, in the middle, near the outlet). Three parallel slots with a depth of 0.5 mm and width of 0.5 mm located in the surface of copper base plate of the heat sink were made by a wire cutting machine to embed the thermocouples. One is 1 mm from the inlet of lotus copper. One is in the middle of lotus copper. One is 1 mm from the outlet of lotus copper. The three parallel slots were perpendicular to the flow direction
4
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Fig. 1. Pore structure of lotus copper samples.
Fig. 2. (a) A schematic drawing of diffusion welding of lotus copper heat sinks; (b) a lotus copper heat sink sample.
Fig. 3. (a) A schematic drawing of the apparatus used for evaluating the heat transfer performance. The outer view (b) and profile drawing (c) of the test section.
of fluid. Thermocouples with a diameter of 0.2 mm were pressed into and sealed inside these slots by solder wire for achieving good contact between thermocouples and the copper base plate. T-type thermocouples were connected to an Agilent 34970A data acquisition module to obtain temperature data. In order to reduce heat loss, an organic glass with a very low thermal conductivity was used to fabricate the fluid flow channel with a cross-sectional dimension that was 25 mm wide by 4 mm high. Polyurethane foam was set at the bottom and two side gaps
of heat sink for heat isolation. Experimental results on thermal equilibrium and heat loss showed that the heat quantity generated by the heating element was almost the same as that carried away by running water and the difference was less than 5%. Thus, the electric power of heating unit can be approximately regarded as the heating power. Epoxy adhesive and Kafuter liquid sealant 609 were used for water sealing under pressurized environment. The outer view and profile drawing of test section are shown in Fig. 3b and c.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
5
When the heat transfer reached a steady state, the inlet water temperature, outlet water temperature, and heat sink surface temperature were constant. All the data were then recorded. The experimental value of total thermal resistance (Rexp total ) of the heat sink could be described as follows:
Rexp total ¼ DT max =Q
ð3Þ
DT max ¼ T w;max T f ;in is the difference between the maximum surface temperature of heat sink T w;max (normally equal to the surface temperature of copper plate near outlet position), and the inlet water temperature T f ;in . DT max is the maximum temperature difference and it corresponds to the maximum thermal resistance and minimum heat transfer coefficient. Q is heat flow quantity into heat sink and approximately equal to the electric power of heater. Correspondingly, the experimental heat transfer coefficient (hexp ) of the heat sink is:
hexp ¼ 1= Rexp total A
ð4Þ
The heat transfer performance was almost invariable with the heat flux density [13]. Thus, in the experiment, the heat flow quantity was set as 200 W and the corresponding heat flux density qðq ¼ Q =AÞ is 50 W/cm2.A is the area of heater (16 mm * 25 mm). 2.5. Uncertainty analysis An error analysis following the method described by Coleman and Steele [20] was performed to determine the uncertainty in the experimental data. The uncertainties for volumetric flow rate measurement and pressure drop measurement were estimated to be less than 5% and 3%, respectively. The total heat loss in the present study was estimated less than 5%. For the present measurements, the temperature difference was more than 6 °C. The uncertainty in temperature difference was 4%. The uncertainty in thermal resistance was due to errors in the measurements of heat flow and temperature difference. The total uncertainty in heat transfer coefficient was due to errors in the measurements of the heat flow, temperature difference and heating area, and it could be obtained as
dhexp ¼ hexp
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 dQ dA dDT max þ þ Q A DT max
Fig. 4. (a) An ideal lotus copper heat sink model; (b) equivalent fin model of lotus copper.
(4) Axial heat conduction in the substrate and the fluid is negligible. (5) A constant convective heat transfer coefficient h0 between the fluid and pore wall in the pores under laminar flow is assumed. The size of the lotus copper sample is L * W * H. t is the thickness of solid copper plate. The total thermal resistance of lotus copper heat sink is the sum of three components: conductive thermal resistance Rcond , convective thermal resistance Rconv , and caloric thermal resistance Rheat [21]. The conductive thermal resistance of lotus copper Rcond is due to conduction from the heater through the substrate. The convective thermal resistance of lotus copper Rconv is due to convection from the lotus copper heat sink to the coolant fluid. The caloric thermal resistance of lotus copper Rheat is due to heating of the fluid as it absorbs energy passing through the heat exchanger.
Rtotal ¼ Rcond þ Rconv þ Rheat ð5Þ
The total uncertainty of heat transfer coefficient was estimated to be 6.5%. 3. Theoretical calculation 3.1. Model description The fin approach is widely used for analysis of microchannel heat sinks [21–24]. It also can be used to obtain analytical solutions of the heat transfer performance of honeycombs, open-celled metal foams and other cellular metals [16,17,25]. Here, the fin model is introduced to lotus copper heat sink to predict its heat transfer performance. During the derivation of analytical model, assumptions are made as follows: (1) As shown in Fig. 4a, the lotus copper has an ideal structure: all pores have the same pore diameter and are infinitely long; pores possess an ideal hexagonal distribution in the matrix. (2) The top surface of lotus copper heat sink is kept at constant heat flux and the bottom surface and side surfaces are thermally insulated. (3) The usual assumptions of steady state, and constant thermal/ physical properties of both fluid and solid are also made.
ð6Þ
The theoretical heat transfer coefficient hcal of the lotus copper heat sink can be expressed as:
hcal ¼
1 Rtotal A
ð7Þ
Conductive thermal resistance of lotus copper Rcond can be expressed as:
Rcond ¼
t ks A
ð8Þ
ks is the thermal conductivity of copper. A is the contact area between the heat sink and heater. Caloric thermal resistance of lotus copper Rheat can be expressed as:
Rheat ¼
1
qC P f
ð9Þ
q and C P are the density and specific heat capacity of fluid. f is the volumetric flow rate. Rcond and Rheat can be reduced by decreasing the thickness of solid copper plate and increasing the volumetric flow rate. Next, the convective thermal resistance of lotus copper Rconv will be discussed. The use of lotus copper, rather than coolant flowing over the solid copper plate surface, extends convective heat transfer area. Thus, we multiply the substrate surface area A by a factor a. a is the ratio of the total convective heat transfer area of
6
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
lotus copper Aeff and the area of copper substrate A and it can be expressed as:
a¼
Aeff A
ð10Þ
Aeff depends on the pore structure, including the porosity e, pore diameter d, and height H of the lotus copper. It can be expressed as
Aeff ¼ npdL ¼
HW e p d2
pdL ¼
4
4He 4He WL ¼ A d d
1 h0 Aeff
keff? ¼
ð12Þ
h0 is the convective heat transfer coefficient in the pores. Next, we consider the convective thermal resistance of a lotus copper fin Rconv;b . Combining Eqs. (10)–(12), the convective thermal resistance of a lotus copper fin can be further expressed as
Rconv ;b ¼
d 4h0 HeAb
ð13Þ
Ab is the cross-sectional area of a lotus copper fin ðAb ¼ bLÞ. b is the pffiffiffi width of lotus copper fin ðb ¼ 3sÞ. s is the interpore spacing s ¼ 0:95 pdffie . Increasing the height H of lotus copper can reduce Rconv ;b . However, we have assumed infinitely conductive channel walls. For a substrate with finite thermal conductivity, there is limited benefit in increasing H beyond the point at which thermal resistance due to conduction along the height of lotus copper becomes comparable to convective thermal resistance. To account for a finite wall conductivity ks which implies a nonuniform temperature along the height, we can multiply by a correction factor 1/g, where g is known as the ‘‘fin efficiency.” Considering the symmetry boundary, the lotus copper fin is modeled as a straight fin as shown in Fig. 4b. Thus, Rconv;b can be expressed as
d 1 Rconv ;b ¼ 4h0 HeAb g
ð14Þ
In the equivalent fin, the fin efficiency g is expressed as the following equations:
8 < g ¼ thðmHÞ mH ffiffiffiffiffiffiffiffiffiffiffiffi q h1 P :m ¼ k A eff?
ð15Þ
b
P is the peripheral length of the equivalent fin P ¼ 2ðb þ LÞ. h1 is the heat transfer coefficient of the equivalent fin and can be expressed as [6]:
h1 ¼
h0 PH
4He Ab d
ð16Þ
ð17Þ
where x is the ratio of thermal conductivity of pore kp to thermal conductivity of copper ks ðx ¼ kp =ks Þ. Because the thermal conductivity of the gas or fluid in the pores is negligible compared with that of copper, keff\ is derived as the following equation by setting x ¼ 0 [26].
ð11Þ
n is the number of pores in the lotus copper. At each cross-section along the length of the channel, we initially assume that the walls are infinitely thermally conductive so that the temperature is uniform. The convective thermal resistance of lotus copper Rconv can be expressed as
Rconv ¼
keff? ðx þ 1Þ þ eðx 1Þ ¼ ks ðx þ 1Þ eðx 1Þ
1e ks 1þe
ð18Þ
Combining Eqs. (15) and (16), it gives
sffiffiffiffiffiffiffiffiffiffiffiffi 4h0 e m¼ keff? d
ð19Þ
Thus, the fin efficiency g is independent of the width of equivalent fin b. The flow in pores of the lotus copper fin resembles that in a cylindrical pipe. It is assumed that the convective heat transfer coefficient h0 between the fluid and pore wall in the pores under a laminar flow ðRe < 3000Þ is constant and it can be expressed as follows [29]:
10=9 3=10 Nup ¼ 5:364 1 þ ð220=pÞXþ 8 " #5=3 93=10 < = p=ð115:2X þ Þ 1þ 1 3=5 2=3 1=2 þ 10=9 : ; ½1þðPr=0:0207Þ ½1þfð220=pÞX g X þ ¼ ðL=dÞ=ðRePrÞ
ð20Þ
ð21Þ
where Re is the Reynolds number ðRe ¼ v d=tÞ, Nup is the Nusselt numberðNup ¼ h0 d=kf Þ, and Pr is the Prandtl number. v is the flow velocity of fluid in pores. t and kf are the dynamic viscosity and thermal conductivity of fluid, respectively. The total volumetric flow rate through lotus copper is:
f ¼ v HW e
ð22Þ
Substituting Eq. (22) into Eq. (9), the caloric thermal resistance Rheat can be expressed as:
Rheat ¼
1
qC P f
¼
1
qC P v HW e
ð23Þ
For a given coolant-fluid, clearly the only way to significantly increase h0 is to reduce the pore diameter d. However, with a decrease of pore diameter d, pressure drop due to that fluid flows through pores of the lotus copper increases and can be expressed as follows:
DP ¼
64 L 1 2 1 qv þ 1 qv 2 Re d 2 2
ð24Þ
1 is the total of pressure loss coefficient when fluid flows into and out of the pores: here, it is assumed to be 1 ¼ 1:4 [10].
4HeA
d b where PH is the ratio between the convective heat transfer area of lotus copper fin and that of the equivalent straight fin. keff? is the effective thermal conductivity of lotus copper perpendicular to the pore growth direction. Due to anisotropy of lotus copper, the effective thermal conductivities of lotus copper parallel and perpendicular to the pore axis are different. Ogushi et al. investigated the effective thermal conductivities of lotus copper parallel and perpendicular to the pore axis both experimentally and analytically [26,27]. They applied Behrens’ effective thermal conductivity model of composite materials into lotus copper. keff\ can be expressed by the following equation [28]
3.2. The effect of pump pressure on heat transfer performance Pump pressure drives the fluid to flow through pores in the lotus copper, and the driving force can be quantified by the pressure drop DP. Fig. 5 shows the influence of pressure drop on the thermal resistance and theoretical heat transfer coefficient hcal of the lotus copper heat sink. With increasing pump pressure, flow velocity increases, which enhances the heat transfer between water and the pore wall. As a result, Rconv and Rheat decrease and the theoretical heat transfer coefficient increases.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
7
Fig. 5. Effect of pressure drop on the theoretical heat transfer coefficient and thermal resistance of lotus copper heat sink.
3.3. The effect of pore structure on the heat transfer performance As discussed, the heat transfer performance of lotus copper depends on the pore structure, including the porosity e, pore diameter d, open porosity u and height H of the lotus copper. Fig. 6 shows the influence of pore diameter and porosity on the thermal resistance and heat transfer performance of the lotus copper heat sink.
the convective heat transfer coefficient in the pores h0 increases and the convective heat transfer area Aeff increases. Thus, the convective thermal resistance Rconv decreases. Therefore, by assuming a practical limit on the available pressure, we can calculate an optimum pore diameter d which minimizes Rconv þ Rheat (see Fig. 6a). It corresponds to the maximum value of hcal, as shown in Fig. 6b. With the increase of pump pressure, the optimum pore diameter gradually decreases to 0.1 mm or less.
3.3.1. The effect of pore diameter on heat transfer performance There will probably be some maximum pressure available to pump the coolant. For a given pump pressure, as pore diameter d decreases, the volumetric flow rate f decreases, resulting in an increase of Rheat . On the other hand, as pore diameter d decreases,
3.3.2. The effect of porosity on heat transfer performance As shown in Fig. 6c. caloric thermal resistance Rheat decreases as the porosity e increases. As for Rconv , on the one hand, as the porosity e increases, the convective heat transfer area Aeff increases. On the other hand, as the porosity e increases, the effective thermal
Fig. 6. Effect of pore diameter and porosity on the thermal resistance and theoretical heat transfer coefficient of lotus copper heat sink.
8
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Fig. 7. The effect of pore structure on theoretical heat transfer coefficient of lotus copper heat sink.
conductivity of the lotus copper perpendicular to the pore growth direction keff? decreases and the fin efficiency g decreases. Thus, the porosity e also has an optimum value. With the decrease of pore diameter, the optimum porosity slightly increases (see Fig. 6d). At a given pump pressure, the heat transfer performance could be effectively improved by selecting suitable structural parameters. Fig. 7 shows the effect of pore structure on the theoretical heat transfer coefficient hcal of lotus copper heat sink. As shown in Fig. 7, the shape of the function resembles an inverted funnel and the apex of funnel corresponds to the maximum heat transfer coefficient. Under a pressure drop of 100 kPa, the maximum heat transfer coefficient can reach 14(W/cm2K) at an optimized pore structural parameter (e ¼ 0:6; d ¼ 0:07 mm). With increasing pump pressure, the theoretical heat transfer coefficient hcal increases and the funnel moves upward and expands outward. 3.3.3. The effect of open porosity on heat transfer performance However, not all pores are open in actual lotus copper, which will result in a discounted heat transfer performance. Here, we want to consider the actual pore structure and modify the ideal model. Only open pores can be used for convective heat transfer while closed pores will decrease the effective thermal conductivity of lotus copper perpendicular to the pore axis, resulting in a
discounted heat transfer performance. Assuming that open pores are uniformly distributed, the porosity e is replaced by the open porosity u where involves convective heat transfer area such as Eqs. (11), (13), (14), (16), (19), (22), (23). In addition, in consideration of the effect of closed pores on the effective thermal conductivity of lotus copper perpendicular to the pore axis, porosity e is used in the calculation of keff? in Eq. (18), which includes both open pores and closed pores. 3.3.4. Structural parameter window meeting a designed heat transfer performance Given an available pump pressure, the structural parameter window of lotus copper heat sink to acquire a designed heat transfer performance could be established. Fig. 8 show the structural parameter window of lotus copper heat sink under different pressure drops and heat transfer performances. With the increase of designed heat transfer performance, the structural parameter window meeting the target value becomes narrow. With the increase of available pump pressure, the structural parameter window meeting the target heat transfer performance expands. When the pressure drop increases up to 100 kPa, the desired structural parameter window covers the common structural parameter zone of lotus copper (eð0:2 0:6Þ; dð0:1 1 mmÞ). By optimizing the pore structure, the heat transfer coefficient can reach 9(W/cm2K) and even higher.
Fig. 8. Structural parameter window of the lotus copper heat sink under different pressure drops and heat transfer performances.
9
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641 Table 1 Structural parameters of lotus copper samples. Heat sink
Porosity
A B C D E
0.35 0.34 0.35 0.36 0.37
e
Pore diameter d (mm)
Open porosity u
Open ratio b (%)
0.40 0.44 0.42 0.59 0.62
0.28 0.29 0.28 0.29 0.31
80 85 80 81 84
Red circle dots represent the selected lotus copper samples in present experiment. They are plotted using pore diameter and open porosity in Fig. 8. Structural parameters of selected lotus copper samples are listed in Table 1. Blue and green circle dots represent lotus copper samples reported in previous studies. Chiba and Ogushi et al both used a lotus copper sample with a porosity of 0.39 and pore diameter of 0.3 mm, and obtained a heat transfer coefficient of 8 W/(cm2K). However, the open porosity has not been given in the research of Chiba et al. [6,9]. Since the sample length was only 3 mm, we assumed that all pores were open and the open porosity was equal to the porosity. Zhang et al. [12] used a lotus copper sample with a porosity of 0.29 and pore diameter of 0.39 mm. When the sample length was 20 mm, the open porosity was given and it was 0.11. When the length of a single segment along the pore growth direction was reduced to 10 mm, 5 mm, and 2.5 mm, corresponding open porosities were estimated to be 0.20, 0.25, 0.27 respectively. They are calculated according to the Eq. (26), which will be described in Section.4. Decreasing the length of the single lotus copper segment from 20 mm to 10 mm, 5 mm, and 2.5 mm resulted in a gradual increase in the heat transfer coefficient from 5(W/cm2 K) to 6(W/cm2 K), 7.5(W/cm2 K), and 8(W/cm2 K) [12].
The predicted heat transfer coefficients are basically consistent with the experimental results in previous studies. Open ratio b can be expressed by
b¼
u 100% e
ð25Þ
80% pores are open in selected lotus copper samples. Under a pressure drop of 100 kPa, the theoretical heat transfer coefficient of selected lotus copper samples ranges from 6(W/cm2 K) to 8 (W/cm2 K). 4. Experimental results and discussion The heat transfer performance of lotus copper heat sinks was tested. The experimental results show that the heat transfer coefficient can reach up to 7.86(W/cm2 K), under a pressure drop of 100 kPa. The experimental results are basically consistent with the theoretical values. Fig. 9 show the influence of hydrodynamic parameters including volumetric flow rate, pressure drop, and pumping power on the heat transfer performance of lotus copper heat sinks. With the increase of volumetric flow rate, the convective heat transfer
Fig. 9. (a) Volumetric flow rate as a function of pressure drop; (b) the effect of volumetric flow rate on heat transfer coefficient; (c) the effect of pressure drop on heat transfer coefficient; (d) the effect of pumping power on heat transfer coefficient.
10
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
between fluid and pore wall enhances, so the heat transfer performance of lotus copper heat sink improves, as shown in Fig. 9b. With increasing volumetric flow rate of cooling water, the pressure drop between inlet and outlet water pressures becomes greater (see Fig. 9a). Taking pressure drop as an evaluation index, the heat transfer coefficient increases as the pressure drop increases. Pumping power is the consumption power for driving the fluid and approximately equals to the product of volumetric flow rate and pressure drop. It is also chosen as an evaluation index. When the pumping power is less than 5 W, the heat transfer coefficient increases rapidly with an increase of the pumping power. When the pumping power is greater than 5 W, the increase rate of the heat transfer coefficient reduces. The way to enhance the heat transfer performance by increasing the volumetric flow rate will increase the pumping power rapidly, so there is limited benefit in further increasing the pumping power. Experimental curves exhibit wavy trends, which have also been observed in the research of Zhang [12]. In our opinion, in addition to experimental error, it is probably related to the non-uniform diameter distribution in actual lotus copper. The distribution of volumetric flow rate will vary with the change of pressure drop due to the non-uniform diameter distribution. However, it will not affect the change trend of the heat transfer performance of lotus copper heat sinks. Here it needs to be emphasized that lotus copper is different from the machined porous copper, metal honeycombs and other periodic cellular metals. The open porosity is dependent on the sample length along the pore growth direction. As the sample length of lotus copper increases, the open porosity will decrease. In order to ensure a high open porosity, a short sample length was usually adopted by previous researchers [6,9,11]. However, only the heat sink made by lotus copper with a greater length can cool the high heat flux density devices with larger area and, at the same time, has sufficient structural strength, which is important for practical application. In the present work, the length of lotus copper sample is 16 mm which is 5 times of that in previously reported works [6,9] and a similar heat transfer coefficient still can be obtained. It proves that by optimizing the preparation process the open porosity is increased. Below we will explain the relationship between the open porosity and the sample length, porosity, pore diameter, the aspect ratio of pores. Fig. 10a shows the pore growth process during the solidification of lotus copper [30]. During the solidification process, the interruption of pores and new nucleation will result in closed pores. Assuming that pore coalescence does not occur and the aspect ratio of pores c is constant, the open porosity for the sample with a length of L along pore growth direction can be expressed as
u¼e 1
L dc
ð26Þ
Fig. 10b shows the relationship between the open porosity and sample length. As the sample length decreases, the open porosity increases linearly and gradually becomes close to the porosity. As the aspect ratio c increases, the absolute value of the slope gradually decreases. When the aspect ratio tends to infinity, the open porosity is equal to the porosity and independent of the sample length L. Fig. 10c shows the relationship between the open porosity and pore diameter. As the pore diameter decreases, the open porosity decreases. When the pore diameter decreases to a certain value at which the pore length is equal to the sample length, the open porosity is zero. Two ways can be used to increase the open porosity of lotus copper: one way is to artificially shorten the sample length by cutting [6,12], and the other way is to increase the aspect ratio of pores by optimizing the fabrication process. However, it is difficult to make practical heat sink products with short lotus copper samples since too many kerfs will split the lotus copper matrix and damage structural strength and heat transfer performance of the heat sink. Therefore, optimizing the pore structure of lotus copper is the most effective way to enhance its heat transfer performance. This method simplifies the subsequent processing and can cool high heat flux density devices with a larger surface area. From the perspective of fabrication, achieving a higher heat transfer performance over a larger heat dissipation area puts higher demands on the pore structure of lotus copper. Pores with a higher aspect ratio are desired and a narrower parameter window of pore structure is allowed.
5. Conclusions Based on the equivalent fin model, the heat transfer performance of lotus copper heat sink can be predicted. The effects of porosity, pore diameter and open porosity on heat transfer performance have been investigated and the optimal pore structure for maximum heat dissipation has been obtained. Given an available pump pressure, the structural parameter window of the lotus copper to acquire a designed heat transfer performance can be established. Pore structures of optimized lotus copper samples are observed. 80% of pores are open in optimized lotus copper samples with a length of 16 mm. The experimental results of the heat transfer performance of lotus copper heat sinks are basically consistent with the theoretical values. Under a pressure drop of 100 kPa, a heat transfer coefficient of 7.86(W/cm2 K) can be obtained. The corresponding open porosity and pore diameter of the lotus copper sample are 0.28 and 0.4 mm respectively. When pore structure is optimized further, a heat transfer performance of 9(W/cm2 K) and even higher can be achieved, which desires pores with a higher aspect ratio and allows a narrower parameter window of pore structure.
Fig. 10. (a) A schematic diagram of pore growth during the solidification of lotus copper. (b) The relationship between the open porosity and sample length. (c) The relationship between the open porosity and pore diameter.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Declaration of Competing Interest This manuscript has not been published or presented elsewhere in part or in entirety, and is not under consideration by another journal. All authors have approved the manuscript and agreed with submission to your esteemed journal. Acknowledgments The authors gratefully acknowledge the financial support by National Natural Science Foundation of China (grant number 51371104). The authors wish to thank Dr. Yun He for the fabrication of lotus copper ingots and thank Dr. Liutao Chen for insightful discussions. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2019.118641. References [1] H. Nakajima, Fabrication, properties and application of porous metals with directional pores, Prog. Mater Sci. 52 (7) (2007) 1091–1173. [2] V. Shapovalov, Porous metals, MRS Bull. 19 (04) (1994) 24–28. [3] L. Drenchev, J. Sobczak, N. Sobczak, et al., A comprehensive model of ordered porosity formation, Acta Mater. 55 (19) (2007) 6459–6471. [4] L. Drenchev, J. Sobczak, S. Malinov, et al., Modelling of structural formation in ordered porosity metal materials, Modell. Simul. Mater. Sci. Eng. 14 (4) (2006) 663–675. [5] Shapovalov V I. Method for manufacturing porous articles. US,5181549,199301-26.[P]. 1993-01-26. [6] T. Ogushi, H. Chiba, H. Nakajima, Development of lotus-type porous copper heat sink, Mater. Trans. 47 (9) (2006) 2240–2247. [7] H. Du, D. Lu, J. Qi, et al., Heat dissipation performance of porous copper with elongated cylindrical pores, J. Mater. Sci. Technol. 30 (9) (2014) 934–938. [8] K. Alvarez, S. Hyun, T. Nakano, et al., In vivo osteocompatibility of lotus-type porous nickel-free stainless steel in rats, Mater. Sci. Eng., C 29 (4) (2009) 1182– 1190. [9] H. Chiba, T. Ogushi, H. Nakajima, et al., Heat transfer capacity of lotus-type porous copper heat sink, JSME. Int. J. B. 47 (3) (2004) 516–521. [10] H. Chiba, T. Ogushi, H. Nakajima, Heat transfer capacity of lotus-type porous copper heat sink for air cooling, J. Therm. Sci. Tech-Jpn. 5 (2) (2010) 222–237. [11] K. Muramatsu, T. Ide, H. Nakajima, et al., Heat transfer and pressure drop of lotus-type porous metals, J. Heat Transfer 135 (2013) 0726017.
11
[12] H. Zhang, L. Chen, Y. Liu, et al., Experimental study on heat transfer performance of lotus-type porous copper heat sink, Int. J. Heat Mass Transf. 56 (1–2) (2013) 172–180. [13] Y. Liu, H.F. Chen, H.W. Zhang, et al., Heat transfer performance of lotus-type porous copper heat sink with liquid GaInSn coolant, Int. J. Heat Mass Transf. 80 (2015) 605–613. [14] Y. He, Y. Li, H. Zhang, et al., Influence of withdrawing speed on the porous structures of Gasar ingots fabricated by Bridgman method, J. Mater. Process. Technol. 245 (2017) 106–114. [15] Y. He, Y. Li, H. Zhang, et al., Fabrication of Gasar ingots with straight parallel pores by a Bridgman method, J. Mater. Process. Technol. 249 (2017) 128–134. [16] T.J. Lu, Heat transfer efficiency of metal honeycombs, Int. J. Heat Mass Transf. 42 (11) (1999) 2031–2040. [17] S. Gu, T.J. Lu, A.G. Evans, On the design of two-dimensional cellular metals for combined heat dissipation and structural load capacity, Int. J. Heat Mass Transf. 44 (11) (2001) 2163–2175. [18] S. Yamamura, H. Shiota, K. Murakami, et al., Evaluation of porosity in porous copper fabricated by unidirectional solidification under pressurized hydrogen, Mater. Sci. Eng., A 318 (1–2) (2001) 137–143. [19] Y. Liu, Y. Li, J. Wan, et al., Metal-gas eutectic growth during unidirectional solidification, Metall. Mater. Trans. A 37A (9) (2006) 2871–2878. [20] Hugh W, Coleman, W. Glenn Steele - Experimentation, Validation, and Uncertainty Analysis for Engineers, third ed., Wiley. [21] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE. Electr. Device. L. EDL-2 (5) (1981) 126–129. [22] Roy W. Knight, Donald J. Hall, John S. Goodling, et al. Heat sink optimization with application to microchannels, IEEE Trans. Compon., Hybrids, Manuf. Technol., 1992. [23] D. Liu, S.V. Garimella, Analysis and optimization of the thermal performance of microchannel heat sinks, Int. J. Numer. Methods Heat Fluid Flow 15 (1) (2005) 7–26. [24] C.Y. Zhao, T.J. Lu, Analysis of microchannel heat sinks for electronics cooling, Int. J. Heat Mass Transf. 45 (24) (2002) 4857–4869. [25] T.J. Lu, H.A. Stone, M.F. Ashby, Heat transfer in open-cell metal foams, Acta Mater. 46 (1998) 3619–3635. [26] T. Ogushi, H. Chiba, H. Nakajima, Measurement and analysis of effective thermal conductivities of lotus-type porous copper, J. Appl. Phys. 95 (10) (2004) 5843. [27] H. Chiba, T. Ogushi, H. Nakajima, et al., Steady state comparative-longitudinal heat flow method using specimen of different thicknesses for measuring thermal conductivity of lotus-type porous metals, J. Appl. Phys. 103 (1) (2008) 13515. [28] E. Behrens, Thermal conductivities of composite materials, J. Compos. Mater. 1 (2) (1968) 2–17. [29] R.K. Shah, A.L. London, Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data, Academic Press, New York, 1978, p. 148. [30] X. Liu, Y. Li, J. Wang, et al., The pore growth process and pore coalescence process in Gasar copper, Mater. Charact. 137 (2018) 231–243.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Effect of pore structure on heat transfer performance of lotus-type porous copper heat sink Xiaobang Liu a, Yanxiang Li a,b,⇑, Huawei Zhang a,b, Yuan Liu a,b, Xiang Chen a,b a b
School of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China Key Laboratory for Advanced Materials Processing Technology, Ministry of Education, PR China
a r t i c l e
i n f o
Article history: Received 19 June 2019 Received in revised form 16 August 2019 Accepted 25 August 2019 Available online 4 September 2019 Keywords: Lotus-type porous copper Heat sink Pore structure Open pores
a b s t r a c t A special kind of microchannel heat sink is fabricated by a lotus-type porous copper in which long cylindrical pores are arranged in one direction. By applying the straight fin model to the lotus-type porous copper, the effect of the pore structure including porosity, pore diameter and open porosity on its heat transfer performance can be predicted. Given an available pump pressure, the structural parameter window of lotus-type porous copper which meets the heat dissipation requirement can be given. By improving the process control pore structure can be optimized and 80% pores in optimized lotus-type porous copper samples are open when the sample length along the pore growth direction is 16 mm. The heat transfer performance of optimized lotus-type porous copper heat sinks was tested. At a pressure drop of 100 kPa, a heat transfer coefficient of 7.86 (W/cm2 K) could be achieved. Ó 2019 Elsevier Ltd. All rights reserved.
1. Introduction Lotus-type porous metals [1] are also called as Gasar metals [2– 4]. They are fabricated by a directional solidification process in pressurized hydrogen or nitrogen. During the solidification process, the pores and the metal grow together, and finally a structure in which cylindrical pores are regularly arranged is formed. This technology was developed in the USSR, and then was patented in the US [5]. Since oriented cylindrical pores can serve as smooth transmission channels, lotus-type porous metals are very promising in the fields of heat sinks [6,7], filters, and biomaterials [8], among others. When fluid flows through the cylindrical pores, a large amount of heat can be quickly carried away by convective heat transfer of the fluid and heat conduction of the substrate. Thus, lotus-type porous copper is very suitable for heat dissipation, which combines a large specific surface area of the porous structure and the high thermal conductivity of copper. In general, the porosity of lotustype porous copper is 0.2–0.6 and pore diameter is 0.1–1 mm. Therefore, the lotus-type porous copper can be regarded as a special microchannel heat sink. To date, some researchers have investigated the heat transfer performance of lotus-type porous copper (hereafter referred to as
⇑ Corresponding author at: School of Materials Science and Engineering, Tsinghua University, Beijing 100084, PR China. E-mail address: [email protected] (Y. Li). https://doi.org/10.1016/j.ijheatmasstransfer.2019.118641 0017-9310/Ó 2019 Elsevier Ltd. All rights reserved.
lotus copper). Chiba and Ogushi et al. [6,9] compared the heat transfer performance of lotus copper heat sink with conventional groove fin and microchannel heat sink for water cooling. They spliced 3 sections of lotus copper with a length of 3 mm (along pore growth direction) into a heat sink. Under the same pumping power, the heat transfer performance of the lotus copper heat sink was 4 times greater than the conventional groove fins and 1.3 times greater than the microchannel heat sink. The lotus copper sample had pores with an average diameter of 0.3 mm and a porosity of 0.39 and a heat transfer coefficient of 8 W/cm2 K was obtained. They also found that the fin efficiency of lotus copper fin could be predicted by the straight fin model by using the effective thermal conductivity perpendicular to the pore axis and the surface area ratio between the lotus copper fin and the straight fin. Then, Chiba et al. [10] investigated the heat transfer performance of lotus copper heat sink for air cooling. They considered the distribution of pore diameter and found that the heat transfer coefficient and pressure drop per unit fin length of the nonuniform pore model showed a good agreement with that of the uniform pore model. Muramatsu et al. [11] compared the heat transfer performance of two types of porous copper heat sink for air cooling. One was lotus copper and the other was porous copper machined with a deterministic pattern of holes. Muramatsu pointed out that in lotus copper the open porosity needed to be considered. Zhang et al. [12] studied the heat transfer performance of lotus copper heat sinks and found that open porosity had a great
2
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Nomenclature A Ab Aeff b CP d d di f h0 h1 hexp hcal H kf ks keff? L n m Nup P DP Pr q Q Rcond Rconv Rconv;b Rheat Rtotal
area of heater cross-sectional area of a lotus copper fin total convective heat transfer area of lotus copper width of lotus copper fin specific heat capacity of fluid pore diameter average pore diameter diameter of ith pore volumetric flow rate convective heat transfer coefficient in the pores heat transfer coefficient of the equivalent fin experimental heat transfer coefficient theoretical heat transfer coefficient height of lotus copper sample thermal conductivity of fluid thermal conductivity of copper effective thermal conductivity of lotus copper perpendicular to the pore growth direction length of lotus copper sample number of pores fin factor Nusselt number peripheral length of the equivalent fin pressure drop Prandtl number heat flux density heat flow quantity conductive thermal resistance of lotus copper convective thermal resistance of lotus copper convective thermal resistance of a lotus copper fin caloric thermal resistance of lotus copper theoretical total thermal resistance
influence on the heat transfer performance. The lotus copper had pores with an average diameter of 0.39 mm and a porosity of 0.29. When the length of lotus copper sample was 20 mm, the corresponding open porosity was only 0.11 and a heat transfer coefficient of 5(W/cm2 K) could be obtained. By cutting the sample into several sections, the length of a single segment along the pore growth direction was reduced. Thus, the open porosity improved, and the flow rate increased, thereby enhancing the heat transfer performance. When the pumping power was 5 W, decreasing the length of the single lotus copper segment from 20 mm to 10 mm, 5 mm, and 2.5 mm resulted in a gradual increase in the heat transfer coefficient from 5(W/cm2 K) to 6(W/cm2 K), 7.5(W/cm2 K), and 8(W/cm2 K). Moreover, Liu et al. [13] used Fluent-3D numerical simulation to study the heat transfer performance of lotus copper heat sinks when liquid GaInSn was taken as the coolant. As pointed out by Muramatsu and Zhang, some pores do not penetrate through the sample, which are called ‘‘closed pores” here, while those pores which penetrate through the sample are called ‘‘open pores” here. There is a great difference in the number of open pores between different samples, even though the porosity remains unchanged. Therefore, in order to quantify the contribution of open pores, the open porosity must be introduced and is defined as the ratio of the area of open pores to the crosssectional area of the whole sample (Porosity is the ratio of the area of pores to the cross-sectional area of the whole sample). Only open pores can play a role in convective heat transfer while closed pores will decrease the effective thermal conductivity, resulting in a discounted heat transfer performance. Thus, the open porosity is critically important to the heat transfer performance of lotus copper heat sink.
Rexp total Re s t T w;max T f ;in DT max
v
W
experimental value of total thermal resistance Reynolds number interpore spacing thickness of solid copper plate maximum surface temperature of heat sink inlet water temperature maximum temperature difference velocity of fluid in pores width of lotus copper sample
Greek symbols a ratio of total convective heat transfer area and area of copper substrate e porosity e average porosity e1 porosity measured on the bottom cross-section e2 porosity measured on the top cross-section u open porosity b open ratio c aspect ratio of pores q density of fluid g fin efficiency t dynamic viscosity of fluid 1 pressure loss coefficient d difference x ratio of thermal conductivity of pore to thermal conductivity of copper Subscripts f fluid s solid
However, the open porosity is always smaller than the porosity and is dependent on the sample length along the pore growth direction [12]. As the sample length of lotus copper increases, the open porosity will decrease. In the research of Chiba [9], Ogushi [6] and Muramatsu [11], in order to obtain a high open porosity, they spliced 3 sections of lotus copper with a length of 3 mm (along pore growth direction) into a heat sink. However, those lotus copper samples used in their testing were too short along pore growth direction to make practical heat sink products. Only the heat sink made by lotus copper with a greater length can cool the high heat flux density devices with larger area and, at the same time, has sufficient structural strength, which is important for practical application. Much effort has been made to optimize the fabrication process to increase the open porosity on a longer lotus copper sample [14,15]. It is widely recognized that a flat solid/liquid interface can improve the pore straightness and parallelism, which contributes to increasing the open porosity. Yun He et al. [14] introduced the Bridgman method to fabricate lotus copper ingots. They obtained a planar solid/liquid interface by optimizing the withdrawing speed, thereby increasing the open porosity. As mentioned above, the heat transfer performance of lotus copper heat sink mainly depends on three influencing factors: porosity, pore diameter and open porosity. However, there are few studies on the effect of open porosity on heat transfer performance. In addition, similar to metal honeycombs [16] and cellular metals [17], the heat transfer performance of lotus copper heat sinks can be maximized by optimizing the porosity and pore diameter. However, there is no systematic study on the effects of pore diameter and porosity on the overall heat transfer performance of lotus copper heat sink.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Thus, the main objectives of the present paper are as follows: (1) to optimize the fabrication process to increase the open porosity and hence enhance the heat transfer performance of lotus copper heat sink; (2) to study the effects of porosity, pore diameter and open porosity on overall heat transfer performance, and to obtain the optimal pore structure for maximum heat dissipation. In the present work, we first observed the pore structure of optimized lotus copper and the porosity, open porosity and pore diameter were measured. Then, we predicted the heat transfer performance by applying the straight fin model into lotus copper heat sink. The effects of pore diameter, porosity and open porosity on the heat transfer performance of lotus copper heat sink have been investigated. The structural parameter window of lotus copper which meets the heat dissipation requirement under an available pump pressure is given. Finally, we examined experimentally the heat transfer performance of lotus copper heat sinks which were fabricated under optimized conditions.
the top and bottom surfaces of each sample and average value of two surface was taken. The average porosity was calculated using the following formula:
e¼
e1 þ e2 2
The experimental procedure is as follows: (1) Fabrication of lotus copper ingots. (2) Characterization of the pore structure of lotus copper. (3) Preparation of lotus copper heat sinks. (4) Test of the heat transfer performance of lotus copper heat sinks. 2.1. Fabrication of lotus copper ingots Cylindrical lotus copper ingots were fabricated using a Bridgman directional solidification apparatus. The porosity and pore diameter could be modulated through the application of different hydrogen and argon pressures [18,19]. The open porosity could be improved by optimizing the withdrawing speed. Under optimum conditions, the solid/liquid interfaces were almost planar and straight pores were obtained. The optimum withdrawing speed increased as the ingot size decreased. For the lotus copper ingot with 150 mm in diameter and 200 mm in height (weighing about 20 kg), the optimum withdrawing speed was 0.5 mm/s. For the small ingot with 100 mm in diameter and 130 mm in height (weighing about 6 kg), the optimum withdrawing speed was 1 mm/s. Detailed fabrication procedures have been reported previously [14]. Cuboid samples were cut from the central part of two optimized lotus copper ingots by a wire cutting machine. 2.2. Characterization of pore structure The sample dimension of lotus copper was 16 mm (length along the pore growth direction) * 26 mm (width) * 5 mm (height). Cuboid samples were first polished with a series of sand papers from 800 to 2000 grits. Then, they were degreased with acetone and cleaned with diluted HCl and ethanol successively in an ultrasonic cleaner. The pore structure of lotus copper samples is shown in Fig. 1. Paint was used to enhance the contrast for distinguishing open pores, closed pores and the substrate. Open pores are white and closed pores are black. As shown in Fig. 1, open pores were observed with the help of a flashlight. A beam of light with enough intensity was radiated perpendicularly from one side of the sample. Then, the light could pass through open pores. Thus, the open pores could be picked out easily from the other side of the sample. All open pores and closed pores were marked on the scanned picture. The open porosity, porosity, and pore diameter could be obtained by an image analysis software. In order to avoid the influence of incomplete pores on the edge, only the 25 mm * 4mm area in the center was counted. Statistical analysis was performed on
ð1Þ
e1 and e2 are porosities measured on the bottom and top crosssection, respectively. The porosity in cross-section of lotus copper almost keeps constant along the solidification height [19]. The pore diameter will vary not much on the cross-section. Chiba et al. [10] considered the distribution of pore diameter and found that the heat transfer coefficient and pressure drop per unit fin length of the non-uniform pore model showed a good agreement with that of the uniform pore model. The average pore diameter reflected the combined result of all pores, and it was calculated using the following formula [13]: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Pn 2 i¼1 di d¼ n
2. Experimental procedure
3
ð2Þ
where di is the diameter of ith pore on two cross-sections. 2.3. Preparation of lotus copper heat sinks Thereafter, the cuboid lotus copper sample was bonded to a solid copper plate with a thickness of 1 mm, width of 29 mm and length of 20 mm by vacuum diffusion welding to form the heat sink, as shown in Fig. 2. The diffusion welding process was as follows: vacuum pumping (vacuum degree: 10-3 Pa, and compressive pre-stress: 1 MPa), followed by heating (setting temperature: 1100 K), forcing (compressive stress: 5 MPa and duration time: 10 min; a stop block was used to limit over deformation), and cooling (in the vacuum furnace). 2.4. Test of heat transfer performance Fig. 3a shows a schematic diagram of the apparatus used for evaluating the heat transfer performance of lotus copper heat sinks. Water was selected as liquid coolant and circulated in a loop by a water pump. The flow rate could be controlled by adjusting valves and a flow divider. The inlet pressure and outlet pressure of the heat sink were measured by two high-precision pressure sensors with a range of 0–300 kPa. The relative accuracy was ±0.3 kPa. Thus, the pressure drop between the inlet and outlet could be obtained. The heat flux load was applied to the heat sink by a high-power heating unit. The heat flux was generated by eight pieces of heat generation resistance sheet and transferred to the heat sink from a copper conduction block. The heat flux input section was 16 * 25 mm and the heating power could be regulated by voltage and current. To reduce the interfacial thermal resistance, a layer of thermal conductive silicone was applied between the heating unit and heat sink. T-type thermocouples with an accuracy of ±0.1 °C were used to measure the inlet water temperature, outlet water temperature and the hot surface temperature of heat sinks. The hot surface temperature of heat sinks was measured at three different positions on the copper base plate (near the inlet, in the middle, near the outlet). Three parallel slots with a depth of 0.5 mm and width of 0.5 mm located in the surface of copper base plate of the heat sink were made by a wire cutting machine to embed the thermocouples. One is 1 mm from the inlet of lotus copper. One is in the middle of lotus copper. One is 1 mm from the outlet of lotus copper. The three parallel slots were perpendicular to the flow direction
4
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Fig. 1. Pore structure of lotus copper samples.
Fig. 2. (a) A schematic drawing of diffusion welding of lotus copper heat sinks; (b) a lotus copper heat sink sample.
Fig. 3. (a) A schematic drawing of the apparatus used for evaluating the heat transfer performance. The outer view (b) and profile drawing (c) of the test section.
of fluid. Thermocouples with a diameter of 0.2 mm were pressed into and sealed inside these slots by solder wire for achieving good contact between thermocouples and the copper base plate. T-type thermocouples were connected to an Agilent 34970A data acquisition module to obtain temperature data. In order to reduce heat loss, an organic glass with a very low thermal conductivity was used to fabricate the fluid flow channel with a cross-sectional dimension that was 25 mm wide by 4 mm high. Polyurethane foam was set at the bottom and two side gaps
of heat sink for heat isolation. Experimental results on thermal equilibrium and heat loss showed that the heat quantity generated by the heating element was almost the same as that carried away by running water and the difference was less than 5%. Thus, the electric power of heating unit can be approximately regarded as the heating power. Epoxy adhesive and Kafuter liquid sealant 609 were used for water sealing under pressurized environment. The outer view and profile drawing of test section are shown in Fig. 3b and c.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
5
When the heat transfer reached a steady state, the inlet water temperature, outlet water temperature, and heat sink surface temperature were constant. All the data were then recorded. The experimental value of total thermal resistance (Rexp total ) of the heat sink could be described as follows:
Rexp total ¼ DT max =Q
ð3Þ
DT max ¼ T w;max T f ;in is the difference between the maximum surface temperature of heat sink T w;max (normally equal to the surface temperature of copper plate near outlet position), and the inlet water temperature T f ;in . DT max is the maximum temperature difference and it corresponds to the maximum thermal resistance and minimum heat transfer coefficient. Q is heat flow quantity into heat sink and approximately equal to the electric power of heater. Correspondingly, the experimental heat transfer coefficient (hexp ) of the heat sink is:
hexp ¼ 1= Rexp total A
ð4Þ
The heat transfer performance was almost invariable with the heat flux density [13]. Thus, in the experiment, the heat flow quantity was set as 200 W and the corresponding heat flux density qðq ¼ Q =AÞ is 50 W/cm2.A is the area of heater (16 mm * 25 mm). 2.5. Uncertainty analysis An error analysis following the method described by Coleman and Steele [20] was performed to determine the uncertainty in the experimental data. The uncertainties for volumetric flow rate measurement and pressure drop measurement were estimated to be less than 5% and 3%, respectively. The total heat loss in the present study was estimated less than 5%. For the present measurements, the temperature difference was more than 6 °C. The uncertainty in temperature difference was 4%. The uncertainty in thermal resistance was due to errors in the measurements of heat flow and temperature difference. The total uncertainty in heat transfer coefficient was due to errors in the measurements of the heat flow, temperature difference and heating area, and it could be obtained as
dhexp ¼ hexp
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 dQ dA dDT max þ þ Q A DT max
Fig. 4. (a) An ideal lotus copper heat sink model; (b) equivalent fin model of lotus copper.
(4) Axial heat conduction in the substrate and the fluid is negligible. (5) A constant convective heat transfer coefficient h0 between the fluid and pore wall in the pores under laminar flow is assumed. The size of the lotus copper sample is L * W * H. t is the thickness of solid copper plate. The total thermal resistance of lotus copper heat sink is the sum of three components: conductive thermal resistance Rcond , convective thermal resistance Rconv , and caloric thermal resistance Rheat [21]. The conductive thermal resistance of lotus copper Rcond is due to conduction from the heater through the substrate. The convective thermal resistance of lotus copper Rconv is due to convection from the lotus copper heat sink to the coolant fluid. The caloric thermal resistance of lotus copper Rheat is due to heating of the fluid as it absorbs energy passing through the heat exchanger.
Rtotal ¼ Rcond þ Rconv þ Rheat ð5Þ
The total uncertainty of heat transfer coefficient was estimated to be 6.5%. 3. Theoretical calculation 3.1. Model description The fin approach is widely used for analysis of microchannel heat sinks [21–24]. It also can be used to obtain analytical solutions of the heat transfer performance of honeycombs, open-celled metal foams and other cellular metals [16,17,25]. Here, the fin model is introduced to lotus copper heat sink to predict its heat transfer performance. During the derivation of analytical model, assumptions are made as follows: (1) As shown in Fig. 4a, the lotus copper has an ideal structure: all pores have the same pore diameter and are infinitely long; pores possess an ideal hexagonal distribution in the matrix. (2) The top surface of lotus copper heat sink is kept at constant heat flux and the bottom surface and side surfaces are thermally insulated. (3) The usual assumptions of steady state, and constant thermal/ physical properties of both fluid and solid are also made.
ð6Þ
The theoretical heat transfer coefficient hcal of the lotus copper heat sink can be expressed as:
hcal ¼
1 Rtotal A
ð7Þ
Conductive thermal resistance of lotus copper Rcond can be expressed as:
Rcond ¼
t ks A
ð8Þ
ks is the thermal conductivity of copper. A is the contact area between the heat sink and heater. Caloric thermal resistance of lotus copper Rheat can be expressed as:
Rheat ¼
1
qC P f
ð9Þ
q and C P are the density and specific heat capacity of fluid. f is the volumetric flow rate. Rcond and Rheat can be reduced by decreasing the thickness of solid copper plate and increasing the volumetric flow rate. Next, the convective thermal resistance of lotus copper Rconv will be discussed. The use of lotus copper, rather than coolant flowing over the solid copper plate surface, extends convective heat transfer area. Thus, we multiply the substrate surface area A by a factor a. a is the ratio of the total convective heat transfer area of
6
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
lotus copper Aeff and the area of copper substrate A and it can be expressed as:
a¼
Aeff A
ð10Þ
Aeff depends on the pore structure, including the porosity e, pore diameter d, and height H of the lotus copper. It can be expressed as
Aeff ¼ npdL ¼
HW e p d2
pdL ¼
4
4He 4He WL ¼ A d d
1 h0 Aeff
keff? ¼
ð12Þ
h0 is the convective heat transfer coefficient in the pores. Next, we consider the convective thermal resistance of a lotus copper fin Rconv;b . Combining Eqs. (10)–(12), the convective thermal resistance of a lotus copper fin can be further expressed as
Rconv ;b ¼
d 4h0 HeAb
ð13Þ
Ab is the cross-sectional area of a lotus copper fin ðAb ¼ bLÞ. b is the pffiffiffi width of lotus copper fin ðb ¼ 3sÞ. s is the interpore spacing s ¼ 0:95 pdffie . Increasing the height H of lotus copper can reduce Rconv ;b . However, we have assumed infinitely conductive channel walls. For a substrate with finite thermal conductivity, there is limited benefit in increasing H beyond the point at which thermal resistance due to conduction along the height of lotus copper becomes comparable to convective thermal resistance. To account for a finite wall conductivity ks which implies a nonuniform temperature along the height, we can multiply by a correction factor 1/g, where g is known as the ‘‘fin efficiency.” Considering the symmetry boundary, the lotus copper fin is modeled as a straight fin as shown in Fig. 4b. Thus, Rconv;b can be expressed as
d 1 Rconv ;b ¼ 4h0 HeAb g
ð14Þ
In the equivalent fin, the fin efficiency g is expressed as the following equations:
8 < g ¼ thðmHÞ mH ffiffiffiffiffiffiffiffiffiffiffiffi q h1 P :m ¼ k A eff?
ð15Þ
b
P is the peripheral length of the equivalent fin P ¼ 2ðb þ LÞ. h1 is the heat transfer coefficient of the equivalent fin and can be expressed as [6]:
h1 ¼
h0 PH
4He Ab d
ð16Þ
ð17Þ
where x is the ratio of thermal conductivity of pore kp to thermal conductivity of copper ks ðx ¼ kp =ks Þ. Because the thermal conductivity of the gas or fluid in the pores is negligible compared with that of copper, keff\ is derived as the following equation by setting x ¼ 0 [26].
ð11Þ
n is the number of pores in the lotus copper. At each cross-section along the length of the channel, we initially assume that the walls are infinitely thermally conductive so that the temperature is uniform. The convective thermal resistance of lotus copper Rconv can be expressed as
Rconv ¼
keff? ðx þ 1Þ þ eðx 1Þ ¼ ks ðx þ 1Þ eðx 1Þ
1e ks 1þe
ð18Þ
Combining Eqs. (15) and (16), it gives
sffiffiffiffiffiffiffiffiffiffiffiffi 4h0 e m¼ keff? d
ð19Þ
Thus, the fin efficiency g is independent of the width of equivalent fin b. The flow in pores of the lotus copper fin resembles that in a cylindrical pipe. It is assumed that the convective heat transfer coefficient h0 between the fluid and pore wall in the pores under a laminar flow ðRe < 3000Þ is constant and it can be expressed as follows [29]:
10=9 3=10 Nup ¼ 5:364 1 þ ð220=pÞXþ 8 " #5=3 93=10 < = p=ð115:2X þ Þ 1þ 1 3=5 2=3 1=2 þ 10=9 : ; ½1þðPr=0:0207Þ ½1þfð220=pÞX g X þ ¼ ðL=dÞ=ðRePrÞ
ð20Þ
ð21Þ
where Re is the Reynolds number ðRe ¼ v d=tÞ, Nup is the Nusselt numberðNup ¼ h0 d=kf Þ, and Pr is the Prandtl number. v is the flow velocity of fluid in pores. t and kf are the dynamic viscosity and thermal conductivity of fluid, respectively. The total volumetric flow rate through lotus copper is:
f ¼ v HW e
ð22Þ
Substituting Eq. (22) into Eq. (9), the caloric thermal resistance Rheat can be expressed as:
Rheat ¼
1
qC P f
¼
1
qC P v HW e
ð23Þ
For a given coolant-fluid, clearly the only way to significantly increase h0 is to reduce the pore diameter d. However, with a decrease of pore diameter d, pressure drop due to that fluid flows through pores of the lotus copper increases and can be expressed as follows:
DP ¼
64 L 1 2 1 qv þ 1 qv 2 Re d 2 2
ð24Þ
1 is the total of pressure loss coefficient when fluid flows into and out of the pores: here, it is assumed to be 1 ¼ 1:4 [10].
4HeA
d b where PH is the ratio between the convective heat transfer area of lotus copper fin and that of the equivalent straight fin. keff? is the effective thermal conductivity of lotus copper perpendicular to the pore growth direction. Due to anisotropy of lotus copper, the effective thermal conductivities of lotus copper parallel and perpendicular to the pore axis are different. Ogushi et al. investigated the effective thermal conductivities of lotus copper parallel and perpendicular to the pore axis both experimentally and analytically [26,27]. They applied Behrens’ effective thermal conductivity model of composite materials into lotus copper. keff\ can be expressed by the following equation [28]
3.2. The effect of pump pressure on heat transfer performance Pump pressure drives the fluid to flow through pores in the lotus copper, and the driving force can be quantified by the pressure drop DP. Fig. 5 shows the influence of pressure drop on the thermal resistance and theoretical heat transfer coefficient hcal of the lotus copper heat sink. With increasing pump pressure, flow velocity increases, which enhances the heat transfer between water and the pore wall. As a result, Rconv and Rheat decrease and the theoretical heat transfer coefficient increases.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
7
Fig. 5. Effect of pressure drop on the theoretical heat transfer coefficient and thermal resistance of lotus copper heat sink.
3.3. The effect of pore structure on the heat transfer performance As discussed, the heat transfer performance of lotus copper depends on the pore structure, including the porosity e, pore diameter d, open porosity u and height H of the lotus copper. Fig. 6 shows the influence of pore diameter and porosity on the thermal resistance and heat transfer performance of the lotus copper heat sink.
the convective heat transfer coefficient in the pores h0 increases and the convective heat transfer area Aeff increases. Thus, the convective thermal resistance Rconv decreases. Therefore, by assuming a practical limit on the available pressure, we can calculate an optimum pore diameter d which minimizes Rconv þ Rheat (see Fig. 6a). It corresponds to the maximum value of hcal, as shown in Fig. 6b. With the increase of pump pressure, the optimum pore diameter gradually decreases to 0.1 mm or less.
3.3.1. The effect of pore diameter on heat transfer performance There will probably be some maximum pressure available to pump the coolant. For a given pump pressure, as pore diameter d decreases, the volumetric flow rate f decreases, resulting in an increase of Rheat . On the other hand, as pore diameter d decreases,
3.3.2. The effect of porosity on heat transfer performance As shown in Fig. 6c. caloric thermal resistance Rheat decreases as the porosity e increases. As for Rconv , on the one hand, as the porosity e increases, the convective heat transfer area Aeff increases. On the other hand, as the porosity e increases, the effective thermal
Fig. 6. Effect of pore diameter and porosity on the thermal resistance and theoretical heat transfer coefficient of lotus copper heat sink.
8
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Fig. 7. The effect of pore structure on theoretical heat transfer coefficient of lotus copper heat sink.
conductivity of the lotus copper perpendicular to the pore growth direction keff? decreases and the fin efficiency g decreases. Thus, the porosity e also has an optimum value. With the decrease of pore diameter, the optimum porosity slightly increases (see Fig. 6d). At a given pump pressure, the heat transfer performance could be effectively improved by selecting suitable structural parameters. Fig. 7 shows the effect of pore structure on the theoretical heat transfer coefficient hcal of lotus copper heat sink. As shown in Fig. 7, the shape of the function resembles an inverted funnel and the apex of funnel corresponds to the maximum heat transfer coefficient. Under a pressure drop of 100 kPa, the maximum heat transfer coefficient can reach 14(W/cm2K) at an optimized pore structural parameter (e ¼ 0:6; d ¼ 0:07 mm). With increasing pump pressure, the theoretical heat transfer coefficient hcal increases and the funnel moves upward and expands outward. 3.3.3. The effect of open porosity on heat transfer performance However, not all pores are open in actual lotus copper, which will result in a discounted heat transfer performance. Here, we want to consider the actual pore structure and modify the ideal model. Only open pores can be used for convective heat transfer while closed pores will decrease the effective thermal conductivity of lotus copper perpendicular to the pore axis, resulting in a
discounted heat transfer performance. Assuming that open pores are uniformly distributed, the porosity e is replaced by the open porosity u where involves convective heat transfer area such as Eqs. (11), (13), (14), (16), (19), (22), (23). In addition, in consideration of the effect of closed pores on the effective thermal conductivity of lotus copper perpendicular to the pore axis, porosity e is used in the calculation of keff? in Eq. (18), which includes both open pores and closed pores. 3.3.4. Structural parameter window meeting a designed heat transfer performance Given an available pump pressure, the structural parameter window of lotus copper heat sink to acquire a designed heat transfer performance could be established. Fig. 8 show the structural parameter window of lotus copper heat sink under different pressure drops and heat transfer performances. With the increase of designed heat transfer performance, the structural parameter window meeting the target value becomes narrow. With the increase of available pump pressure, the structural parameter window meeting the target heat transfer performance expands. When the pressure drop increases up to 100 kPa, the desired structural parameter window covers the common structural parameter zone of lotus copper (eð0:2 0:6Þ; dð0:1 1 mmÞ). By optimizing the pore structure, the heat transfer coefficient can reach 9(W/cm2K) and even higher.
Fig. 8. Structural parameter window of the lotus copper heat sink under different pressure drops and heat transfer performances.
9
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641 Table 1 Structural parameters of lotus copper samples. Heat sink
Porosity
A B C D E
0.35 0.34 0.35 0.36 0.37
e
Pore diameter d (mm)
Open porosity u
Open ratio b (%)
0.40 0.44 0.42 0.59 0.62
0.28 0.29 0.28 0.29 0.31
80 85 80 81 84
Red circle dots represent the selected lotus copper samples in present experiment. They are plotted using pore diameter and open porosity in Fig. 8. Structural parameters of selected lotus copper samples are listed in Table 1. Blue and green circle dots represent lotus copper samples reported in previous studies. Chiba and Ogushi et al both used a lotus copper sample with a porosity of 0.39 and pore diameter of 0.3 mm, and obtained a heat transfer coefficient of 8 W/(cm2K). However, the open porosity has not been given in the research of Chiba et al. [6,9]. Since the sample length was only 3 mm, we assumed that all pores were open and the open porosity was equal to the porosity. Zhang et al. [12] used a lotus copper sample with a porosity of 0.29 and pore diameter of 0.39 mm. When the sample length was 20 mm, the open porosity was given and it was 0.11. When the length of a single segment along the pore growth direction was reduced to 10 mm, 5 mm, and 2.5 mm, corresponding open porosities were estimated to be 0.20, 0.25, 0.27 respectively. They are calculated according to the Eq. (26), which will be described in Section.4. Decreasing the length of the single lotus copper segment from 20 mm to 10 mm, 5 mm, and 2.5 mm resulted in a gradual increase in the heat transfer coefficient from 5(W/cm2 K) to 6(W/cm2 K), 7.5(W/cm2 K), and 8(W/cm2 K) [12].
The predicted heat transfer coefficients are basically consistent with the experimental results in previous studies. Open ratio b can be expressed by
b¼
u 100% e
ð25Þ
80% pores are open in selected lotus copper samples. Under a pressure drop of 100 kPa, the theoretical heat transfer coefficient of selected lotus copper samples ranges from 6(W/cm2 K) to 8 (W/cm2 K). 4. Experimental results and discussion The heat transfer performance of lotus copper heat sinks was tested. The experimental results show that the heat transfer coefficient can reach up to 7.86(W/cm2 K), under a pressure drop of 100 kPa. The experimental results are basically consistent with the theoretical values. Fig. 9 show the influence of hydrodynamic parameters including volumetric flow rate, pressure drop, and pumping power on the heat transfer performance of lotus copper heat sinks. With the increase of volumetric flow rate, the convective heat transfer
Fig. 9. (a) Volumetric flow rate as a function of pressure drop; (b) the effect of volumetric flow rate on heat transfer coefficient; (c) the effect of pressure drop on heat transfer coefficient; (d) the effect of pumping power on heat transfer coefficient.
10
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
between fluid and pore wall enhances, so the heat transfer performance of lotus copper heat sink improves, as shown in Fig. 9b. With increasing volumetric flow rate of cooling water, the pressure drop between inlet and outlet water pressures becomes greater (see Fig. 9a). Taking pressure drop as an evaluation index, the heat transfer coefficient increases as the pressure drop increases. Pumping power is the consumption power for driving the fluid and approximately equals to the product of volumetric flow rate and pressure drop. It is also chosen as an evaluation index. When the pumping power is less than 5 W, the heat transfer coefficient increases rapidly with an increase of the pumping power. When the pumping power is greater than 5 W, the increase rate of the heat transfer coefficient reduces. The way to enhance the heat transfer performance by increasing the volumetric flow rate will increase the pumping power rapidly, so there is limited benefit in further increasing the pumping power. Experimental curves exhibit wavy trends, which have also been observed in the research of Zhang [12]. In our opinion, in addition to experimental error, it is probably related to the non-uniform diameter distribution in actual lotus copper. The distribution of volumetric flow rate will vary with the change of pressure drop due to the non-uniform diameter distribution. However, it will not affect the change trend of the heat transfer performance of lotus copper heat sinks. Here it needs to be emphasized that lotus copper is different from the machined porous copper, metal honeycombs and other periodic cellular metals. The open porosity is dependent on the sample length along the pore growth direction. As the sample length of lotus copper increases, the open porosity will decrease. In order to ensure a high open porosity, a short sample length was usually adopted by previous researchers [6,9,11]. However, only the heat sink made by lotus copper with a greater length can cool the high heat flux density devices with larger area and, at the same time, has sufficient structural strength, which is important for practical application. In the present work, the length of lotus copper sample is 16 mm which is 5 times of that in previously reported works [6,9] and a similar heat transfer coefficient still can be obtained. It proves that by optimizing the preparation process the open porosity is increased. Below we will explain the relationship between the open porosity and the sample length, porosity, pore diameter, the aspect ratio of pores. Fig. 10a shows the pore growth process during the solidification of lotus copper [30]. During the solidification process, the interruption of pores and new nucleation will result in closed pores. Assuming that pore coalescence does not occur and the aspect ratio of pores c is constant, the open porosity for the sample with a length of L along pore growth direction can be expressed as
u¼e 1
L dc
ð26Þ
Fig. 10b shows the relationship between the open porosity and sample length. As the sample length decreases, the open porosity increases linearly and gradually becomes close to the porosity. As the aspect ratio c increases, the absolute value of the slope gradually decreases. When the aspect ratio tends to infinity, the open porosity is equal to the porosity and independent of the sample length L. Fig. 10c shows the relationship between the open porosity and pore diameter. As the pore diameter decreases, the open porosity decreases. When the pore diameter decreases to a certain value at which the pore length is equal to the sample length, the open porosity is zero. Two ways can be used to increase the open porosity of lotus copper: one way is to artificially shorten the sample length by cutting [6,12], and the other way is to increase the aspect ratio of pores by optimizing the fabrication process. However, it is difficult to make practical heat sink products with short lotus copper samples since too many kerfs will split the lotus copper matrix and damage structural strength and heat transfer performance of the heat sink. Therefore, optimizing the pore structure of lotus copper is the most effective way to enhance its heat transfer performance. This method simplifies the subsequent processing and can cool high heat flux density devices with a larger surface area. From the perspective of fabrication, achieving a higher heat transfer performance over a larger heat dissipation area puts higher demands on the pore structure of lotus copper. Pores with a higher aspect ratio are desired and a narrower parameter window of pore structure is allowed.
5. Conclusions Based on the equivalent fin model, the heat transfer performance of lotus copper heat sink can be predicted. The effects of porosity, pore diameter and open porosity on heat transfer performance have been investigated and the optimal pore structure for maximum heat dissipation has been obtained. Given an available pump pressure, the structural parameter window of the lotus copper to acquire a designed heat transfer performance can be established. Pore structures of optimized lotus copper samples are observed. 80% of pores are open in optimized lotus copper samples with a length of 16 mm. The experimental results of the heat transfer performance of lotus copper heat sinks are basically consistent with the theoretical values. Under a pressure drop of 100 kPa, a heat transfer coefficient of 7.86(W/cm2 K) can be obtained. The corresponding open porosity and pore diameter of the lotus copper sample are 0.28 and 0.4 mm respectively. When pore structure is optimized further, a heat transfer performance of 9(W/cm2 K) and even higher can be achieved, which desires pores with a higher aspect ratio and allows a narrower parameter window of pore structure.
Fig. 10. (a) A schematic diagram of pore growth during the solidification of lotus copper. (b) The relationship between the open porosity and sample length. (c) The relationship between the open porosity and pore diameter.
X. Liu et al. / International Journal of Heat and Mass Transfer 144 (2019) 118641
Declaration of Competing Interest This manuscript has not been published or presented elsewhere in part or in entirety, and is not under consideration by another journal. All authors have approved the manuscript and agreed with submission to your esteemed journal. Acknowledgments The authors gratefully acknowledge the financial support by National Natural Science Foundation of China (grant number 51371104). The authors wish to thank Dr. Yun He for the fabrication of lotus copper ingots and thank Dr. Liutao Chen for insightful discussions. Appendix A. Supplementary material Supplementary data to this article can be found online at https://doi.org/10.1016/j.ijheatmasstransfer.2019.118641. References [1] H. Nakajima, Fabrication, properties and application of porous metals with directional pores, Prog. Mater Sci. 52 (7) (2007) 1091–1173. [2] V. Shapovalov, Porous metals, MRS Bull. 19 (04) (1994) 24–28. [3] L. Drenchev, J. Sobczak, N. Sobczak, et al., A comprehensive model of ordered porosity formation, Acta Mater. 55 (19) (2007) 6459–6471. [4] L. Drenchev, J. Sobczak, S. Malinov, et al., Modelling of structural formation in ordered porosity metal materials, Modell. Simul. Mater. Sci. Eng. 14 (4) (2006) 663–675. [5] Shapovalov V I. Method for manufacturing porous articles. US,5181549,199301-26.[P]. 1993-01-26. [6] T. Ogushi, H. Chiba, H. Nakajima, Development of lotus-type porous copper heat sink, Mater. Trans. 47 (9) (2006) 2240–2247. [7] H. Du, D. Lu, J. Qi, et al., Heat dissipation performance of porous copper with elongated cylindrical pores, J. Mater. Sci. Technol. 30 (9) (2014) 934–938. [8] K. Alvarez, S. Hyun, T. Nakano, et al., In vivo osteocompatibility of lotus-type porous nickel-free stainless steel in rats, Mater. Sci. Eng., C 29 (4) (2009) 1182– 1190. [9] H. Chiba, T. Ogushi, H. Nakajima, et al., Heat transfer capacity of lotus-type porous copper heat sink, JSME. Int. J. B. 47 (3) (2004) 516–521. [10] H. Chiba, T. Ogushi, H. Nakajima, Heat transfer capacity of lotus-type porous copper heat sink for air cooling, J. Therm. Sci. Tech-Jpn. 5 (2) (2010) 222–237. [11] K. Muramatsu, T. Ide, H. Nakajima, et al., Heat transfer and pressure drop of lotus-type porous metals, J. Heat Transfer 135 (2013) 0726017.
11
[12] H. Zhang, L. Chen, Y. Liu, et al., Experimental study on heat transfer performance of lotus-type porous copper heat sink, Int. J. Heat Mass Transf. 56 (1–2) (2013) 172–180. [13] Y. Liu, H.F. Chen, H.W. Zhang, et al., Heat transfer performance of lotus-type porous copper heat sink with liquid GaInSn coolant, Int. J. Heat Mass Transf. 80 (2015) 605–613. [14] Y. He, Y. Li, H. Zhang, et al., Influence of withdrawing speed on the porous structures of Gasar ingots fabricated by Bridgman method, J. Mater. Process. Technol. 245 (2017) 106–114. [15] Y. He, Y. Li, H. Zhang, et al., Fabrication of Gasar ingots with straight parallel pores by a Bridgman method, J. Mater. Process. Technol. 249 (2017) 128–134. [16] T.J. Lu, Heat transfer efficiency of metal honeycombs, Int. J. Heat Mass Transf. 42 (11) (1999) 2031–2040. [17] S. Gu, T.J. Lu, A.G. Evans, On the design of two-dimensional cellular metals for combined heat dissipation and structural load capacity, Int. J. Heat Mass Transf. 44 (11) (2001) 2163–2175. [18] S. Yamamura, H. Shiota, K. Murakami, et al., Evaluation of porosity in porous copper fabricated by unidirectional solidification under pressurized hydrogen, Mater. Sci. Eng., A 318 (1–2) (2001) 137–143. [19] Y. Liu, Y. Li, J. Wan, et al., Metal-gas eutectic growth during unidirectional solidification, Metall. Mater. Trans. A 37A (9) (2006) 2871–2878. [20] Hugh W, Coleman, W. Glenn Steele - Experimentation, Validation, and Uncertainty Analysis for Engineers, third ed., Wiley. [21] D.B. Tuckerman, R.F.W. Pease, High-performance heat sinking for VLSI, IEEE. Electr. Device. L. EDL-2 (5) (1981) 126–129. [22] Roy W. Knight, Donald J. Hall, John S. Goodling, et al. Heat sink optimization with application to microchannels, IEEE Trans. Compon., Hybrids, Manuf. Technol., 1992. [23] D. Liu, S.V. Garimella, Analysis and optimization of the thermal performance of microchannel heat sinks, Int. J. Numer. Methods Heat Fluid Flow 15 (1) (2005) 7–26. [24] C.Y. Zhao, T.J. Lu, Analysis of microchannel heat sinks for electronics cooling, Int. J. Heat Mass Transf. 45 (24) (2002) 4857–4869. [25] T.J. Lu, H.A. Stone, M.F. Ashby, Heat transfer in open-cell metal foams, Acta Mater. 46 (1998) 3619–3635. [26] T. Ogushi, H. Chiba, H. Nakajima, Measurement and analysis of effective thermal conductivities of lotus-type porous copper, J. Appl. Phys. 95 (10) (2004) 5843. [27] H. Chiba, T. Ogushi, H. Nakajima, et al., Steady state comparative-longitudinal heat flow method using specimen of different thicknesses for measuring thermal conductivity of lotus-type porous metals, J. Appl. Phys. 103 (1) (2008) 13515. [28] E. Behrens, Thermal conductivities of composite materials, J. Compos. Mater. 1 (2) (1968) 2–17. [29] R.K. Shah, A.L. London, Laminar flow forced convection in ducts: a source book for compact heat exchanger analytical data, Academic Press, New York, 1978, p. 148. [30] X. Liu, Y. Li, J. Wang, et al., The pore growth process and pore coalescence process in Gasar copper, Mater. Charact. 137 (2018) 231–243.