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Performance of a polymeric heat sink with circular microchannels
Applied Thermal Engineering 26 (2006) 787–794 www.elsevier.com/locate/apthermeng
Performance of a polymeric heat sink with circular microchannels Alessandro Barba, Barbara Musi, Marco Spiga
*
Department of Industrial Engineering, University of Parma, Parco Area delle Scienze 181, 43100 Parma, Italy Received 10 May 2005; accepted 13 October 2005 Available online 28 November 2005
Abstract The object of this work is the thermal investigation of a polymeric microchannel heat sink designed for the active cooling of small flat surfaces. Its performance, pressure drop, temperature distribution, and thermal resistance are evaluated. A three-dimensional procedure is developed and applied to a geometrical configuration consisting of a circular microduct (with a gas running through it), embedded in a solid substrate with rectangular cross-section. The conjugate heat transfer problem is solved assuming fully developed laminar flow in forced convection. The bottom side of the heat sink receives a uniform heat flux, while the top side is adiabatic. Considering a gas flow with low Prandtl and Reynolds numbers, the temperature distribution is given by the sum of a linear function (in the stream direction) and a numerical solution obtained in 2-D coordinates resorting to a finite element software, based on the Rayleigh–Ritz–Galerkin method, with user-defined error tolerance. Rarefaction, compressibility and viscous dissipation are neglected, i.e., the Knudsen, Mach and Brinkman numbers are low. The theoretical results are shown in some graphs and compared with experimental data concerning helium and nitrogen flows in Nylon circular microducts. The agreement is quite satisfactory. 2005 Elsevier Ltd. All rights reserved. Keywords: Micro heat sinks; Gas flow; Polymeric materials; Thermal resistance; Conjugate
1. Introduction The heat rejection from a constrained, small space is a hard challenge for conventional cooling techniques and new methods must be settled and improved, in order to enhance heat transfer coefficients, to compensate the small heat transfer surface. This problem is well known in electronic cooling, where microchips can work at allowable temperatures [1–4] thanks to the cooling of a fluid running in trapezoidal or rectangular microchannels; microchannels are defined as ducts having hydraulic diameters in the range of 10–200 lm [5]. The recent developments of micro and nano devices lead to assert that new applications will be found in the near future [6,7] and new researches will be carried out. *
Corresponding author. Tel.: +39 521 905855; fax: +39 521 905705. E-mail address: [email protected] (M. Spiga).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.10.015
Microscale analysis of multi-layered structures and interfaces between different materials is needed in several applications of microelectronics and protective coating of jet and space devices. For space missions small dimensions and lightness of thermal equipments are essential and new effective and compact cooling techniques are necessary [8,9], for thermal control of the new generation satellites (with mass less than 10 kg) and cloth conditioning in severe environmental conditions. One of these promising techniques introduces the cooling obtained by micro heat sinks. The heat sink is made from a conductive solid with microchannels machined in it and used as flow passages for the cooling fluid. The main advantage obtained by micro heat sinks is the enhanced heat transfer coefficient, much higher than in conventional macroscale heat exchangers, for this reason micro heat sinks technology will have a great impact on the future of micro and nano devices.
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Notation c cv Br D De f h H K K1 Kn L Lh m Ma N Nu p P Pr q Q
fluid specific heat at constant pressure [J/ kg K] fluid specific heat at constant volume [J/kg K] Brinkman number, lW2/q hydraulic diameter of the circular channel [m] outer diameter of the tube [m] Darcy Weisbach friction factor heat transfer coefficient (averaged on the cross-section) [W/m2 K] half thickness of the heat sink [m] collectors pressure drop number incremental pressure drop number Knudsen number, kfp/D length of the circular channels [m] heated length of the circular channels [m] coolant mass flow rate in the N channels [kg/s] Mach number, W/Ws number of parallel channels in the heat sink Nusselt number, hD/k pressure [Pa] perimeter of the unit cross-section for the fluid flow [m] Prandtl number, lc/k average heat flux at the fluid–solid wall [W/m2] total power received by heat sink and fluid [W]
In the last years microscale heat exchangers or microscale heat sinks have been proposed and analysed in several papers [10–17]. Basically, a cooling fluid is forced through the microchannels attached to a power source to carry away the heat generated under the heat sink base. To design effective micro heat sinks, it is necessary to know the relationship among key design parameters, such as pressure drop, mass flow rate, heat transfer rate, temperatures of the solid substrate. Therefore, fundamental understanding of fluid flow and heat transfer is required. A classical issue is whether the classical continuum model (Navier–Stokes and energy equations) is valid at the microscale, because rarefaction, compressibility, roughness, electrostatic effects can produce deviations from the classical theory. Due to fabrication technology in electronic materials, most of the papers in the open literature deal with microchannels having rectangular or trapezoidal crosssection. Many authors [18–21] presented detailed pressure and temperature distribution in micro heat sinks in rectangular channels with sides ranging from 57 to 713 lm, using the continuum model. They proved that
R gas constant [J/kg K] Re Reynolds number qWD/l thermal resistance defined in Eq. (10) [C/W] Rth T(x, y, z) temperature [C] TK absolute temperature [K] v(x, y) fluid velocity distribution [m/s] W fluid average velocity [m/s] Ws sound velocity, [c(c 1)TK]0.5 [m/s] x, y, z Cartesian coordinates Greek letters k fluid thermal conductivity [W/m K] kfp fluid mean free path, m (p/2RTK)0.5 [m] c ratio c/cv l fluid dynamic viscosity [Pa s] m fluid kinematic viscosity [m2/s] h azimuthal angle in cylindrical coordinates [rad] q fluid density [kg/m3] Subscripts b fluid bulk in fluid inlet section max maximum in the exit section p polymer s substrate sc single channel w fluid–solid wall
it is valid by showing good agreement with the available experimental results (even if measurements errors can be very influential in microscale and novel phenomena can occur). Microchannels can be used not only in electronic cooling, where rectangular or trapezoidal geometry is usually imposed. In other emerging application (such as cooling of equipments for space missions) microchannels with circular cross-section could be preferred. They can be fabricated in thin polymeric materials, with thermal conductivity as high as possible. Polymeric tubes, assembled in a strip in contact with the hot surface, offer good modelability, durability and lightness. In a recent paper, a micro heat sink with a gas flow in Nylon channels has been fabricated and tested [22]. In literature only two papers, at least to our knowledge, are concerned with the analysis of heat sinks with circular microchannels [23,24]. The first deals with copper or silicon heat sinks with a water flow (Pr 4) in a circular microchannel, where the hydrodynamical and thermal entrance regions play a fundamental role in the behaviour of the component. The authors solve the continuum model equations, presenting numerical results in good
A. Barba et al. / Applied Thermal Engineering 26 (2006) 787–794
agreement with some single-phase data experimentally obtained by Bowers and Mudawar [25] in mini-channel and microchannel heat sinks. The paper by Tso and Mahulikar [24] presents experimental results performed using water flow in circular microchannels in the laminar regime; the test section consists of two aluminium plates containing the circular ducts, with constant wall heat flux boundary condition. The authors correlated the Nusselt number in the fully developed region with the Brinkman number. For gases in laminar flow, the thermal entry length and the viscous dissipation are quite negligible and the theoretical approach must be different. This paper is aimed at presenting a three-dimensional numerical analysis concerning a laminar gas flow in the circular microchannels of a heat sink. As a fundamental hypothesis the axial distribution of the heat flux is considered constant and, consequently, the temperature between the inlet and the outlet is linear. The thermal conductivity of the walls (in polymeric material) is not so high, hence the axial distribution can be reasonably assumed constant. Temperature distribution, thermal resistance and pressure drop are presented as functions of mass flow rate and dissipated power, they are compared with experimental data available in literature.
2. Analysis The heat sink under consideration is schematically depicted in Fig. 1, where the Cartesian coordinates x, y, z and the relevant dimensions are reported. The gas passes through a number of adjacent microchannels and takes heat away from an electric heater attached below, simulated as a constant heat flux at the bottom wall of the sink. The circular tubes are in polymeric materials, embedded in a conductive glue. The horizontal surface at the coordinate y = H receives the uniform heat flux, while the surface at y = H is adiabatic. The analysis is conjugate since convection between fluid and walls is simultaneously considered together with conduction in the solid substrate. The flow is forced, laminar, fully developed and steady. Viscous dissipation, natural convection, rarefaction effects, interfacial electrokinetic and rough effects
near the fluid–solid interface are negligible. Thermal sources are absent. The fluid is Newtonian and incompressible, the temperature changes are moderate, hence the thermophysical properties can be assumed constant for all the materials. The channel walls are rigid and nonporous. Under these hypotheses, the temperature distribution in the micro heat sink can be determined by solving the conjugate problem concerning heat conduction in the solid and convection in the gas region. The suitable energy balance equations, for each material, must be solved, and continuity (for temperature and heat flux) must be imposed on all the interfaces between different materials. Because of geometrical symmetry, a unit cell consisting of a microtube and the surrounding solid is considered. The relevant geometry parameters are the height 2H, the unit cell width De (representing the outer diameter of the tube) and the tube inner diameter D. The 3-D domain is divided in three regions, respectively the gas region inside the tube, the polymeric circular tube, the glue substrate. In the gas region, the Navier–Stokes equation for continuum flow can be applied to the whole length of the microtube. The velocity profile is then represented by the classical second order equation for hydrodynamically developed flow x2 þ y 2 vðx; yÞ ¼ 2W 1 4 ð1Þ D2 In the heated region, the hypothesis of H boundary conditions leads to the usual result dT w dT b oT 4q ¼ ¼ ¼ oz qcWD dz dz
ð2Þ
where the last passage is obtained considering the heat transfer balance in a control volume of the duct of length dz. Assuming constant heat flux on the bottom side of the sink, and o2T/oz2 = 0 (as suggested by the fully developed condition and by the negligible fluid axial conduction), the energy balance equation reads as 2 4qvðx; yÞ o T o2 T ¼k þ ð3Þ DW ox2 oy 2 The average wall temperature Tw is then a linear function of the longitudinal coordinate z; in any point of the wetted perimeter the local temperature is a linear function of z. Be Tw,0 the local wall temperature in an arbitrary point 0 of the wetted perimeter. Following Eq. (2), this temperature reads as T w;0 ¼
Fig. 1. Heat sink and cross-section of its unit cell.
789
4q z þ T w;0;in qcWD
ð4Þ
This result suggests to represent the unknown function T(x, y, z) as the sum of two functions Tw,0 and Txy, the former depending on the longitudinal coordinate z only,
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T max T in Q
the latter depending on the transversal coordinates x, y, and solution to the energy equation
Rth ¼
4q vðx; yÞ ¼ kr2 T xy D W
where Tmax is evaluated in the bottom corner of the rectangular exit cross-section and represents the temperature in the hot spot of the whole heat sink. The pressure drop is ! qW 2 L X Dp ¼ f þ Ki þ K1 ð11Þ D 2 i
ð5Þ
Under these assumptions, the temperature profile Txy in the cross-section is the same throughout all the channel length, i.e., for any value of the longitudinal coordinate z. Analogous considerations are valid for the two solid regions (polymeric material and substrate) where the temperature profiles Txy is the solution to the conduction equation r2 T xy ¼ 0
ð6Þ
The classical energy equations are solved by assigning no slip at the boundary. For any region, the longitudinal contribution is a linear function of the coordinate z, while the transverse profile Txy is deduced by the finite element code FlexPDETM. This software package performs the operation necessary to turn a description of a partial differential equations system into a finite element model and solve the system, resorting to the Rayleigh–Ritz–Galerkin method. The cross-section of the micro heat sink is divided into triangular elements, and an iterative refinement of the grid is performed, until the user-defined error tolerance, correlated with the maximum local residual value, is achieved. The thermal flux at the bottom wall of the heat sink is known, as well as the bulk fluid temperature in the inlet section. Continuity of temperature and heat flux are imposed at all the fluid–solid and solid–solid interfaces. The entrance effects are neglected because the gas flow is assumed hydraulically and thermally fully developed; as a consequence the heat transfer coefficient is constant. The knowledge of the fluid velocity and temperature allows to determine all the characteristics of the thermal problem. Then, in the gas region, the average wall temperature is valuated as I 1 Tw ¼ T xy dP ð7Þ pD w The gas bulk temperature is the surface integral ZZ 4 Tb ¼ vðx; yÞT xy dx dy ð8Þ pD2 W x;y Hence the Nusselt number is Nu ¼
hD qD ¼ k kðT w T b Þ
ð9Þ
The thermal resistance of the whole micro heat sink is defined as
ð10Þ
where f represents the classical Darcy Weisbach friction P factor (f = 64/Re) and K i represents the minor losses due to the inlet and exit, while K1 represent the pressure coefficient in the hydrodynamically developing region, accounting for momentum change and accumulated increment in wall shear between developing flow and developed flow [26]. The rectangular cross-section of Fig. 1 is chosen as the computation domain, numerical solutions of the above model are obtained by the finite element approach; solutions are considered converged when the sum of the absolute dimensionless residuals of the balance equations are less than 105.
3. Model validation and results A first validation of the numerical results is given by the Nusselt number deduced by Eq. (9); for any numerical simulation, the Nusselt number is 4.3636, the typical value corresponding to the classical H boundary condition with fully developed flow in circular channels [26]. An useful benchmark for the results obtained in this paper is given by the experimental data presented in [22] (provided the assumptions on which the model is based are satisfied), where the authors investigated heat transfer characteristics of helium and nitrogen flow in circular Nylon channels, with the following geometrical and physical data: D = 200 lm, De = 380 lm, N = 74, kp = 0.250 W/m K. Experimental uncertainties of the data are not indicated in [22]. The tube length is L = 55 mm, the heated length is Lh = 25 mm (at the centre of the test section), the thickness of the Nylon heat sink is 2H = 1.38 mm. The length of the hydrodynamical entrance region is valuated as [26] Lhy 0:60 þ 0:056Re ¼ 0:035Re þ 1 D
ð12Þ
It results less than about 12 mm, hence in the heated region the velocity profile is fully developed and well represented by Eq. (1). The tubes are glued with a silicon paste having ks = 3 W/m K, that constitutes the substrate of the heat sink. Among the large amount of experimental data presented in [22], only those concerning fully developed
A. Barba et al. / Applied Thermal Engineering 26 (2006) 787–794
laminar flow are considered, where rarefaction and compressibility effects are negligible. The gas is considered to behave as a perfect gas over the variations in temperature attainable in subsonic flow, pressure is always less than 2.7 bar. For gas flows in circular microtubes, the laminar—to turbulent transition occurs for Reynolds numbers from 1700 to 6000 [15]. In all the tests performed and described in this paper, the gas flows with Reynolds numbers less than 1200, the Prandtl numbers for helium and nitrogen are respectively 0.68 and 0.71. The gas flow is laminar and the hydrodynamical and thermal entrance regions are very short (being the Reynolds and Prandtl numbers very low, and the ratio of length to hydraulic diameter very large), hence the flow can be considered fully developed. The flow is forced by a pressure gradient. The effects of viscous dissipation are absent because the Brinkman number is always less than 103 [27]. The mean free paths kfp = m (p/2RTK)0.5 are 0.2 and 0.07 lm, respectively for helium and nitrogen. As a consequence, the Knudsen numbers are respectively 0.001 and 0.0003 and the effects of rarefaction can be neglected [27]. The flow of the gases in a tube may be considered incompressible if the Mach number is less than about 0.3 and if the pressure drop is also less than about 5% of the of the gas pressure in the inlet section [28,29]. The numerical results have been obtained by the finite element model, assuming constant physical properties, evaluated for the gas at the average temperature between inlet and outlet sections. The physical gas properties are indicated in Table 1 as a function of temperature (expressed in Celsius degrees); the gas constant is R = c cv and the density is deduced as q = p/RTK. The experimental results obtained in [22] are referred to helium flows with mass flow rates in the range 3 and 77 mg/s, while the dissipated power by the heat sink is 0.6 < Q < 9 W. For nitrogen flows the test were performed with 39 < m < 312 mg/s and 0.66 < Q < 7.5 W. The numerical solutions requires the knowledge of inlet fluid pressure and temperature, the power Q given to the fluid in the heated length, the mass flow rate. The average heat flux at the fluid–solid wall is then
Hence the longitudinal bulk temperature profile is given by the equation T b ðzÞ ¼
4q z þ T in qcWD
ð13Þ
ð14Þ
and the temperature distribution in the 2-D cross-section is given by the finite element solution of the conjugate problem. The pressure drop deduced by Eq. (11) is shown in Fig. 2, for nitrogen flow, as a function of the mass flow rate in the single channel of the heat sink; the experimental results are indicated by small squares. The gas density has been determined by the state equation q = p/RTK, assuming pressure equal to the average pressure between inlet and outlet sections; the temperature is the average gas temperature. The minor losses due to the inlet and P exit (through two collectors) are valuated assuming K i ¼ 22 in Eq. (11). This high loss coefficient (suggested by the authors of the experimental tests) accounts for the local and distributed pressure drop in two collectors (each 30 mm long), having approximately the same length of the heat sink. The losses in the hydrodynamic entrance region are determined as [26] K 1 ¼ 1:20 þ
1.8 bar 1.6
38 Re
ð15Þ
p
Nitrogen
1.4 1.2 1 0.8 0.6 0.4 0.2
msc (mg/s)
0 0.5
Q q¼ NPLh
791
1
1.5
2
2.5
3
3.5
4
Fig. 2. Pressure drop in the heat sink for nitrogen flow.
Table 1 Physical properties of helium and nitrogen Helium
Nitrogen
c = 5234 J/kg K cv = 3155 J/kg K k = 7 · 107T2 + 0.0003T + 0.1459 W/m K l = 2 · 1011T2 + 4.5 · 108T + 1.86 · 105 Pa s
c = 0.029T + 1039.2 J/kg K cv = 0.03 T + 742.3 J/kg K k = 2 · 107T2 + 8 · 105T + 0.0239 W/m K l = 4 · 1011T2 + 4.6 · 108T + 1.66 · 105 Pa s
4.5
792
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The Reynolds number in Eq. (15) has been valuated at an average temperature of 43 C, based on the experimental data in [22], where pressure was measured by a high-accuracy transducer, between the inlet section of the first collector and the outlet section of the second collector. The measured pressure drop accounts for the losses in the two collectors (60 mm long) and in the test section (55 mm long). The comparison between numerical results and experimental data is satisfactory, even if a deeper insight in the collector losses should be appreciated. The agreement is satisfactory, confirming the reliability of the continuum model, when compressibility, rarefaction, viscous dissipation and electrostatic effects are absent. The relative difference between the numerical results and the measured pressure values is less than 10%. The 2-D temperature distribution, given by the numerical solution of Eq. (3), is depicted in Fig. 3, for the gas region, the polymer annular section and the substrate, assuming nitrogen flow with m = 210.3 mg/s, Q = 6.46 W, Tin = 33 C. The average heat flux at the fluid-wall interface (at the inner tube surface) is q = 2.134 kW/m2, the exit gas bulk temperature is 65.2 C. In the exit section of the heated length, the bottom substrate surface is the hottest region of the heat sink, the hot spot is located in the left (or right) bottom corner of the unit cell; the maximum temperature is Tmax = 78.7 C, while the minimum temperature (in the top surface) is 57.6 C (Fig. 3). The temperature in the hot spot of the heat sink is determined by the numerical solution to Eq. (3); it
Fig. 3. Temperature distribution in the cross-section of the unit cell.
250
T max-T in (˚C)
A: B: C: D: E: F: G: H:
Helium
H
200
G
150 100
F
Q=0.58 W Q=1.26 W Q=1.73 W Q =2.29 W Q =3.05 W Q=6.39 W Q=7.89 W Q=8.79 W
E D
50
C B
0
A
5
F
D A
10
B
15 20
G
C
25
30
35
H E
40 45
50
55
60
65 70
75
80
m (mg/s)
Fig. 4. Maximum temperature of the heat sink and substrate temperature (helium flow).
100 90
T max-T in (˚C)
A: B: C: D: E F: G: H:
Nitrogen
H
80
E
70 60 50
G
D F
C
40
Q=0.67 W Q=1.44 W Q=2.09 W Q=2.32 W Q=3.09 W Q=3.95 W Q=4.04 W Q=6.46 W
H
B
30
E F
20
C
A A
10
G
D B
0 30
60
90
120
150
180
210
240
270
300
330
m (mg/s)
Fig. 5. Maximum temperature of the heat sink and substrate temperature (nitrogen flow).
depends on the gas mass flow rate and the dissipated power. The difference Tmax Tin is an almost linear function of the dissipated power and decreases for increasing mass flow rates. The maximum heat sink temperatures are shown in Figs. 4 and 5 (for helium and nitrogen, respectively) as a function of the mass flow rate, and compared with some experimental data [22], related to a point in the bottom surface of the substrate. Each experimental result is referred to a different dissipated power; it is indicated by small squares in Fig. 4; the eight letters, from A to H, refer the experimental point and the theoretical line to the corresponding dissipated power. For helium flow the agreement between experimental data and numerical results is quite satisfactory. For nitrogen flow a small discrepancy is apparent (with a maximum relative difference of 12%), it can be reasonably justified by some uncertainty in the exact positioning of microthermocouples. The inner wall temperature is not uniformly distributed on the wetted perimeter, it presents a minimum near the top of the sink and a maximum near the bot-
A. Barba et al. / Applied Thermal Engineering 26 (2006) 787–794
tom, where heating occurs. This temperature on the wetted perimeter is shown in Fig. 6, where for symmetry conditions the range of the azimuthal angle is 0–p (h = 0 at the top, h = p at the bottom of the perimeter). A nitrogen flow is considered, with m = 7.8 mg/s, Q = 2.48 W, Tin = 34.6 C. The knowledge of the temperature distribution allows to determine the thermal resistance. The numerical results highlight that the resistance decreases significantly with increasing mass flow rate, while it depends very slightly on the dissipated power; for helium flows the thermal resistance is practically independent of the power. The large amount of results obtained by the numerical solutions can be correlated obtaining a simple and compact representation of the thermal resistance for helium and nitrogen flows in the heat sink Rth ¼ 0:001m0:8654 ðheliumÞ Rth ¼ 0:0163m0:7208 Q0:0224 ðnitrogenÞ
ð16Þ
Fig. 7 presents the thermal resistance for both gas flows in the Nylon microtubes. For nitrogen flow two lines are indicated, for different values of Q, with the intention of proving the negligible effect of the dissipated power on the thermal resistance of the heat sink.
70.2
local T W
Nitrogen
70
793
Considering a mean gas temperature of 43 C, the thermal conductivities are 0.158 and 0.027 W/m K, for helium and nitrogen respectively; the convective heat transfer coefficients at the inner wall of the microducts are then 3436 and 588 W/m2 K. This justifies the higher thermal resistance for nitrogen flows.
4. Conclusions The paper has presented a 3-D analysis of a heat sink with a gas running in parallel circular microchannels. Pressure drops and temperatures are numerically determined and thermal resistance are obtained. The classical theory of continuum flow gives numerical results in very good agreement with the experimental data, obtained in a polymeric heat sink with helium and nitrogen flows at low Reynolds, Prandtl, Knudsen, Mach, and Brinkman numbers. For same mass flow rates and dissipated powers, helium flows allow to obtain lower temperatures in the heat sink, with a lower thermal resistance. The development of strips of polymeric micro heat sinks will require technological efforts, in order to improve the thermal conductivity of materials (polymers and substrate).
Acknowledgements
69.8 69.6
The financial support of the Italian Space Agency (Grant I/R/266/02) and the Ministry of University and Scientific Research (Grant COFIN 2003) is gratefully acknowledged. The authors would like to thank Prof. Marengo and Dr. Chignoli, of the University of Bergamo, for the profitable discussions and suggestions.
69.4 69.2 69 68.8 68.6 68.4
θ degrees
68.2 0
30
60
90
120
150
180
Fig. 6. Wall temperature distribution in the wetted perimeter of the exit section.
30 Rth
25
˚C /W
20 Q=1 W
15 N2
Q=8 W
10 He
5 m (mg/s)
0 0
30
60
90
120
150
180
210
240
270
300
Fig. 7. Thermal resistance for helium and nitrogen flows.
330
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Performance of a polymeric heat sink with circular microchannels Alessandro Barba, Barbara Musi, Marco Spiga
*
Department of Industrial Engineering, University of Parma, Parco Area delle Scienze 181, 43100 Parma, Italy Received 10 May 2005; accepted 13 October 2005 Available online 28 November 2005
Abstract The object of this work is the thermal investigation of a polymeric microchannel heat sink designed for the active cooling of small flat surfaces. Its performance, pressure drop, temperature distribution, and thermal resistance are evaluated. A three-dimensional procedure is developed and applied to a geometrical configuration consisting of a circular microduct (with a gas running through it), embedded in a solid substrate with rectangular cross-section. The conjugate heat transfer problem is solved assuming fully developed laminar flow in forced convection. The bottom side of the heat sink receives a uniform heat flux, while the top side is adiabatic. Considering a gas flow with low Prandtl and Reynolds numbers, the temperature distribution is given by the sum of a linear function (in the stream direction) and a numerical solution obtained in 2-D coordinates resorting to a finite element software, based on the Rayleigh–Ritz–Galerkin method, with user-defined error tolerance. Rarefaction, compressibility and viscous dissipation are neglected, i.e., the Knudsen, Mach and Brinkman numbers are low. The theoretical results are shown in some graphs and compared with experimental data concerning helium and nitrogen flows in Nylon circular microducts. The agreement is quite satisfactory. 2005 Elsevier Ltd. All rights reserved. Keywords: Micro heat sinks; Gas flow; Polymeric materials; Thermal resistance; Conjugate
1. Introduction The heat rejection from a constrained, small space is a hard challenge for conventional cooling techniques and new methods must be settled and improved, in order to enhance heat transfer coefficients, to compensate the small heat transfer surface. This problem is well known in electronic cooling, where microchips can work at allowable temperatures [1–4] thanks to the cooling of a fluid running in trapezoidal or rectangular microchannels; microchannels are defined as ducts having hydraulic diameters in the range of 10–200 lm [5]. The recent developments of micro and nano devices lead to assert that new applications will be found in the near future [6,7] and new researches will be carried out. *
Corresponding author. Tel.: +39 521 905855; fax: +39 521 905705. E-mail address: [email protected] (M. Spiga).
1359-4311/$ - see front matter 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.applthermaleng.2005.10.015
Microscale analysis of multi-layered structures and interfaces between different materials is needed in several applications of microelectronics and protective coating of jet and space devices. For space missions small dimensions and lightness of thermal equipments are essential and new effective and compact cooling techniques are necessary [8,9], for thermal control of the new generation satellites (with mass less than 10 kg) and cloth conditioning in severe environmental conditions. One of these promising techniques introduces the cooling obtained by micro heat sinks. The heat sink is made from a conductive solid with microchannels machined in it and used as flow passages for the cooling fluid. The main advantage obtained by micro heat sinks is the enhanced heat transfer coefficient, much higher than in conventional macroscale heat exchangers, for this reason micro heat sinks technology will have a great impact on the future of micro and nano devices.
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Notation c cv Br D De f h H K K1 Kn L Lh m Ma N Nu p P Pr q Q
fluid specific heat at constant pressure [J/ kg K] fluid specific heat at constant volume [J/kg K] Brinkman number, lW2/q hydraulic diameter of the circular channel [m] outer diameter of the tube [m] Darcy Weisbach friction factor heat transfer coefficient (averaged on the cross-section) [W/m2 K] half thickness of the heat sink [m] collectors pressure drop number incremental pressure drop number Knudsen number, kfp/D length of the circular channels [m] heated length of the circular channels [m] coolant mass flow rate in the N channels [kg/s] Mach number, W/Ws number of parallel channels in the heat sink Nusselt number, hD/k pressure [Pa] perimeter of the unit cross-section for the fluid flow [m] Prandtl number, lc/k average heat flux at the fluid–solid wall [W/m2] total power received by heat sink and fluid [W]
In the last years microscale heat exchangers or microscale heat sinks have been proposed and analysed in several papers [10–17]. Basically, a cooling fluid is forced through the microchannels attached to a power source to carry away the heat generated under the heat sink base. To design effective micro heat sinks, it is necessary to know the relationship among key design parameters, such as pressure drop, mass flow rate, heat transfer rate, temperatures of the solid substrate. Therefore, fundamental understanding of fluid flow and heat transfer is required. A classical issue is whether the classical continuum model (Navier–Stokes and energy equations) is valid at the microscale, because rarefaction, compressibility, roughness, electrostatic effects can produce deviations from the classical theory. Due to fabrication technology in electronic materials, most of the papers in the open literature deal with microchannels having rectangular or trapezoidal crosssection. Many authors [18–21] presented detailed pressure and temperature distribution in micro heat sinks in rectangular channels with sides ranging from 57 to 713 lm, using the continuum model. They proved that
R gas constant [J/kg K] Re Reynolds number qWD/l thermal resistance defined in Eq. (10) [C/W] Rth T(x, y, z) temperature [C] TK absolute temperature [K] v(x, y) fluid velocity distribution [m/s] W fluid average velocity [m/s] Ws sound velocity, [c(c 1)TK]0.5 [m/s] x, y, z Cartesian coordinates Greek letters k fluid thermal conductivity [W/m K] kfp fluid mean free path, m (p/2RTK)0.5 [m] c ratio c/cv l fluid dynamic viscosity [Pa s] m fluid kinematic viscosity [m2/s] h azimuthal angle in cylindrical coordinates [rad] q fluid density [kg/m3] Subscripts b fluid bulk in fluid inlet section max maximum in the exit section p polymer s substrate sc single channel w fluid–solid wall
it is valid by showing good agreement with the available experimental results (even if measurements errors can be very influential in microscale and novel phenomena can occur). Microchannels can be used not only in electronic cooling, where rectangular or trapezoidal geometry is usually imposed. In other emerging application (such as cooling of equipments for space missions) microchannels with circular cross-section could be preferred. They can be fabricated in thin polymeric materials, with thermal conductivity as high as possible. Polymeric tubes, assembled in a strip in contact with the hot surface, offer good modelability, durability and lightness. In a recent paper, a micro heat sink with a gas flow in Nylon channels has been fabricated and tested [22]. In literature only two papers, at least to our knowledge, are concerned with the analysis of heat sinks with circular microchannels [23,24]. The first deals with copper or silicon heat sinks with a water flow (Pr 4) in a circular microchannel, where the hydrodynamical and thermal entrance regions play a fundamental role in the behaviour of the component. The authors solve the continuum model equations, presenting numerical results in good
A. Barba et al. / Applied Thermal Engineering 26 (2006) 787–794
agreement with some single-phase data experimentally obtained by Bowers and Mudawar [25] in mini-channel and microchannel heat sinks. The paper by Tso and Mahulikar [24] presents experimental results performed using water flow in circular microchannels in the laminar regime; the test section consists of two aluminium plates containing the circular ducts, with constant wall heat flux boundary condition. The authors correlated the Nusselt number in the fully developed region with the Brinkman number. For gases in laminar flow, the thermal entry length and the viscous dissipation are quite negligible and the theoretical approach must be different. This paper is aimed at presenting a three-dimensional numerical analysis concerning a laminar gas flow in the circular microchannels of a heat sink. As a fundamental hypothesis the axial distribution of the heat flux is considered constant and, consequently, the temperature between the inlet and the outlet is linear. The thermal conductivity of the walls (in polymeric material) is not so high, hence the axial distribution can be reasonably assumed constant. Temperature distribution, thermal resistance and pressure drop are presented as functions of mass flow rate and dissipated power, they are compared with experimental data available in literature.
2. Analysis The heat sink under consideration is schematically depicted in Fig. 1, where the Cartesian coordinates x, y, z and the relevant dimensions are reported. The gas passes through a number of adjacent microchannels and takes heat away from an electric heater attached below, simulated as a constant heat flux at the bottom wall of the sink. The circular tubes are in polymeric materials, embedded in a conductive glue. The horizontal surface at the coordinate y = H receives the uniform heat flux, while the surface at y = H is adiabatic. The analysis is conjugate since convection between fluid and walls is simultaneously considered together with conduction in the solid substrate. The flow is forced, laminar, fully developed and steady. Viscous dissipation, natural convection, rarefaction effects, interfacial electrokinetic and rough effects
near the fluid–solid interface are negligible. Thermal sources are absent. The fluid is Newtonian and incompressible, the temperature changes are moderate, hence the thermophysical properties can be assumed constant for all the materials. The channel walls are rigid and nonporous. Under these hypotheses, the temperature distribution in the micro heat sink can be determined by solving the conjugate problem concerning heat conduction in the solid and convection in the gas region. The suitable energy balance equations, for each material, must be solved, and continuity (for temperature and heat flux) must be imposed on all the interfaces between different materials. Because of geometrical symmetry, a unit cell consisting of a microtube and the surrounding solid is considered. The relevant geometry parameters are the height 2H, the unit cell width De (representing the outer diameter of the tube) and the tube inner diameter D. The 3-D domain is divided in three regions, respectively the gas region inside the tube, the polymeric circular tube, the glue substrate. In the gas region, the Navier–Stokes equation for continuum flow can be applied to the whole length of the microtube. The velocity profile is then represented by the classical second order equation for hydrodynamically developed flow x2 þ y 2 vðx; yÞ ¼ 2W 1 4 ð1Þ D2 In the heated region, the hypothesis of H boundary conditions leads to the usual result dT w dT b oT 4q ¼ ¼ ¼ oz qcWD dz dz
ð2Þ
where the last passage is obtained considering the heat transfer balance in a control volume of the duct of length dz. Assuming constant heat flux on the bottom side of the sink, and o2T/oz2 = 0 (as suggested by the fully developed condition and by the negligible fluid axial conduction), the energy balance equation reads as 2 4qvðx; yÞ o T o2 T ¼k þ ð3Þ DW ox2 oy 2 The average wall temperature Tw is then a linear function of the longitudinal coordinate z; in any point of the wetted perimeter the local temperature is a linear function of z. Be Tw,0 the local wall temperature in an arbitrary point 0 of the wetted perimeter. Following Eq. (2), this temperature reads as T w;0 ¼
Fig. 1. Heat sink and cross-section of its unit cell.
789
4q z þ T w;0;in qcWD
ð4Þ
This result suggests to represent the unknown function T(x, y, z) as the sum of two functions Tw,0 and Txy, the former depending on the longitudinal coordinate z only,
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T max T in Q
the latter depending on the transversal coordinates x, y, and solution to the energy equation
Rth ¼
4q vðx; yÞ ¼ kr2 T xy D W
where Tmax is evaluated in the bottom corner of the rectangular exit cross-section and represents the temperature in the hot spot of the whole heat sink. The pressure drop is ! qW 2 L X Dp ¼ f þ Ki þ K1 ð11Þ D 2 i
ð5Þ
Under these assumptions, the temperature profile Txy in the cross-section is the same throughout all the channel length, i.e., for any value of the longitudinal coordinate z. Analogous considerations are valid for the two solid regions (polymeric material and substrate) where the temperature profiles Txy is the solution to the conduction equation r2 T xy ¼ 0
ð6Þ
The classical energy equations are solved by assigning no slip at the boundary. For any region, the longitudinal contribution is a linear function of the coordinate z, while the transverse profile Txy is deduced by the finite element code FlexPDETM. This software package performs the operation necessary to turn a description of a partial differential equations system into a finite element model and solve the system, resorting to the Rayleigh–Ritz–Galerkin method. The cross-section of the micro heat sink is divided into triangular elements, and an iterative refinement of the grid is performed, until the user-defined error tolerance, correlated with the maximum local residual value, is achieved. The thermal flux at the bottom wall of the heat sink is known, as well as the bulk fluid temperature in the inlet section. Continuity of temperature and heat flux are imposed at all the fluid–solid and solid–solid interfaces. The entrance effects are neglected because the gas flow is assumed hydraulically and thermally fully developed; as a consequence the heat transfer coefficient is constant. The knowledge of the fluid velocity and temperature allows to determine all the characteristics of the thermal problem. Then, in the gas region, the average wall temperature is valuated as I 1 Tw ¼ T xy dP ð7Þ pD w The gas bulk temperature is the surface integral ZZ 4 Tb ¼ vðx; yÞT xy dx dy ð8Þ pD2 W x;y Hence the Nusselt number is Nu ¼
hD qD ¼ k kðT w T b Þ
ð9Þ
The thermal resistance of the whole micro heat sink is defined as
ð10Þ
where f represents the classical Darcy Weisbach friction P factor (f = 64/Re) and K i represents the minor losses due to the inlet and exit, while K1 represent the pressure coefficient in the hydrodynamically developing region, accounting for momentum change and accumulated increment in wall shear between developing flow and developed flow [26]. The rectangular cross-section of Fig. 1 is chosen as the computation domain, numerical solutions of the above model are obtained by the finite element approach; solutions are considered converged when the sum of the absolute dimensionless residuals of the balance equations are less than 105.
3. Model validation and results A first validation of the numerical results is given by the Nusselt number deduced by Eq. (9); for any numerical simulation, the Nusselt number is 4.3636, the typical value corresponding to the classical H boundary condition with fully developed flow in circular channels [26]. An useful benchmark for the results obtained in this paper is given by the experimental data presented in [22] (provided the assumptions on which the model is based are satisfied), where the authors investigated heat transfer characteristics of helium and nitrogen flow in circular Nylon channels, with the following geometrical and physical data: D = 200 lm, De = 380 lm, N = 74, kp = 0.250 W/m K. Experimental uncertainties of the data are not indicated in [22]. The tube length is L = 55 mm, the heated length is Lh = 25 mm (at the centre of the test section), the thickness of the Nylon heat sink is 2H = 1.38 mm. The length of the hydrodynamical entrance region is valuated as [26] Lhy 0:60 þ 0:056Re ¼ 0:035Re þ 1 D
ð12Þ
It results less than about 12 mm, hence in the heated region the velocity profile is fully developed and well represented by Eq. (1). The tubes are glued with a silicon paste having ks = 3 W/m K, that constitutes the substrate of the heat sink. Among the large amount of experimental data presented in [22], only those concerning fully developed
A. Barba et al. / Applied Thermal Engineering 26 (2006) 787–794
laminar flow are considered, where rarefaction and compressibility effects are negligible. The gas is considered to behave as a perfect gas over the variations in temperature attainable in subsonic flow, pressure is always less than 2.7 bar. For gas flows in circular microtubes, the laminar—to turbulent transition occurs for Reynolds numbers from 1700 to 6000 [15]. In all the tests performed and described in this paper, the gas flows with Reynolds numbers less than 1200, the Prandtl numbers for helium and nitrogen are respectively 0.68 and 0.71. The gas flow is laminar and the hydrodynamical and thermal entrance regions are very short (being the Reynolds and Prandtl numbers very low, and the ratio of length to hydraulic diameter very large), hence the flow can be considered fully developed. The flow is forced by a pressure gradient. The effects of viscous dissipation are absent because the Brinkman number is always less than 103 [27]. The mean free paths kfp = m (p/2RTK)0.5 are 0.2 and 0.07 lm, respectively for helium and nitrogen. As a consequence, the Knudsen numbers are respectively 0.001 and 0.0003 and the effects of rarefaction can be neglected [27]. The flow of the gases in a tube may be considered incompressible if the Mach number is less than about 0.3 and if the pressure drop is also less than about 5% of the of the gas pressure in the inlet section [28,29]. The numerical results have been obtained by the finite element model, assuming constant physical properties, evaluated for the gas at the average temperature between inlet and outlet sections. The physical gas properties are indicated in Table 1 as a function of temperature (expressed in Celsius degrees); the gas constant is R = c cv and the density is deduced as q = p/RTK. The experimental results obtained in [22] are referred to helium flows with mass flow rates in the range 3 and 77 mg/s, while the dissipated power by the heat sink is 0.6 < Q < 9 W. For nitrogen flows the test were performed with 39 < m < 312 mg/s and 0.66 < Q < 7.5 W. The numerical solutions requires the knowledge of inlet fluid pressure and temperature, the power Q given to the fluid in the heated length, the mass flow rate. The average heat flux at the fluid–solid wall is then
Hence the longitudinal bulk temperature profile is given by the equation T b ðzÞ ¼
4q z þ T in qcWD
ð13Þ
ð14Þ
and the temperature distribution in the 2-D cross-section is given by the finite element solution of the conjugate problem. The pressure drop deduced by Eq. (11) is shown in Fig. 2, for nitrogen flow, as a function of the mass flow rate in the single channel of the heat sink; the experimental results are indicated by small squares. The gas density has been determined by the state equation q = p/RTK, assuming pressure equal to the average pressure between inlet and outlet sections; the temperature is the average gas temperature. The minor losses due to the inlet and P exit (through two collectors) are valuated assuming K i ¼ 22 in Eq. (11). This high loss coefficient (suggested by the authors of the experimental tests) accounts for the local and distributed pressure drop in two collectors (each 30 mm long), having approximately the same length of the heat sink. The losses in the hydrodynamic entrance region are determined as [26] K 1 ¼ 1:20 þ
1.8 bar 1.6
38 Re
ð15Þ
p
Nitrogen
1.4 1.2 1 0.8 0.6 0.4 0.2
msc (mg/s)
0 0.5
Q q¼ NPLh
791
1
1.5
2
2.5
3
3.5
4
Fig. 2. Pressure drop in the heat sink for nitrogen flow.
Table 1 Physical properties of helium and nitrogen Helium
Nitrogen
c = 5234 J/kg K cv = 3155 J/kg K k = 7 · 107T2 + 0.0003T + 0.1459 W/m K l = 2 · 1011T2 + 4.5 · 108T + 1.86 · 105 Pa s
c = 0.029T + 1039.2 J/kg K cv = 0.03 T + 742.3 J/kg K k = 2 · 107T2 + 8 · 105T + 0.0239 W/m K l = 4 · 1011T2 + 4.6 · 108T + 1.66 · 105 Pa s
4.5
792
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The Reynolds number in Eq. (15) has been valuated at an average temperature of 43 C, based on the experimental data in [22], where pressure was measured by a high-accuracy transducer, between the inlet section of the first collector and the outlet section of the second collector. The measured pressure drop accounts for the losses in the two collectors (60 mm long) and in the test section (55 mm long). The comparison between numerical results and experimental data is satisfactory, even if a deeper insight in the collector losses should be appreciated. The agreement is satisfactory, confirming the reliability of the continuum model, when compressibility, rarefaction, viscous dissipation and electrostatic effects are absent. The relative difference between the numerical results and the measured pressure values is less than 10%. The 2-D temperature distribution, given by the numerical solution of Eq. (3), is depicted in Fig. 3, for the gas region, the polymer annular section and the substrate, assuming nitrogen flow with m = 210.3 mg/s, Q = 6.46 W, Tin = 33 C. The average heat flux at the fluid-wall interface (at the inner tube surface) is q = 2.134 kW/m2, the exit gas bulk temperature is 65.2 C. In the exit section of the heated length, the bottom substrate surface is the hottest region of the heat sink, the hot spot is located in the left (or right) bottom corner of the unit cell; the maximum temperature is Tmax = 78.7 C, while the minimum temperature (in the top surface) is 57.6 C (Fig. 3). The temperature in the hot spot of the heat sink is determined by the numerical solution to Eq. (3); it
Fig. 3. Temperature distribution in the cross-section of the unit cell.
250
T max-T in (˚C)
A: B: C: D: E: F: G: H:
Helium
H
200
G
150 100
F
Q=0.58 W Q=1.26 W Q=1.73 W Q =2.29 W Q =3.05 W Q=6.39 W Q=7.89 W Q=8.79 W
E D
50
C B
0
A
5
F
D A
10
B
15 20
G
C
25
30
35
H E
40 45
50
55
60
65 70
75
80
m (mg/s)
Fig. 4. Maximum temperature of the heat sink and substrate temperature (helium flow).
100 90
T max-T in (˚C)
A: B: C: D: E F: G: H:
Nitrogen
H
80
E
70 60 50
G
D F
C
40
Q=0.67 W Q=1.44 W Q=2.09 W Q=2.32 W Q=3.09 W Q=3.95 W Q=4.04 W Q=6.46 W
H
B
30
E F
20
C
A A
10
G
D B
0 30
60
90
120
150
180
210
240
270
300
330
m (mg/s)
Fig. 5. Maximum temperature of the heat sink and substrate temperature (nitrogen flow).
depends on the gas mass flow rate and the dissipated power. The difference Tmax Tin is an almost linear function of the dissipated power and decreases for increasing mass flow rates. The maximum heat sink temperatures are shown in Figs. 4 and 5 (for helium and nitrogen, respectively) as a function of the mass flow rate, and compared with some experimental data [22], related to a point in the bottom surface of the substrate. Each experimental result is referred to a different dissipated power; it is indicated by small squares in Fig. 4; the eight letters, from A to H, refer the experimental point and the theoretical line to the corresponding dissipated power. For helium flow the agreement between experimental data and numerical results is quite satisfactory. For nitrogen flow a small discrepancy is apparent (with a maximum relative difference of 12%), it can be reasonably justified by some uncertainty in the exact positioning of microthermocouples. The inner wall temperature is not uniformly distributed on the wetted perimeter, it presents a minimum near the top of the sink and a maximum near the bot-
A. Barba et al. / Applied Thermal Engineering 26 (2006) 787–794
tom, where heating occurs. This temperature on the wetted perimeter is shown in Fig. 6, where for symmetry conditions the range of the azimuthal angle is 0–p (h = 0 at the top, h = p at the bottom of the perimeter). A nitrogen flow is considered, with m = 7.8 mg/s, Q = 2.48 W, Tin = 34.6 C. The knowledge of the temperature distribution allows to determine the thermal resistance. The numerical results highlight that the resistance decreases significantly with increasing mass flow rate, while it depends very slightly on the dissipated power; for helium flows the thermal resistance is practically independent of the power. The large amount of results obtained by the numerical solutions can be correlated obtaining a simple and compact representation of the thermal resistance for helium and nitrogen flows in the heat sink Rth ¼ 0:001m0:8654 ðheliumÞ Rth ¼ 0:0163m0:7208 Q0:0224 ðnitrogenÞ
ð16Þ
Fig. 7 presents the thermal resistance for both gas flows in the Nylon microtubes. For nitrogen flow two lines are indicated, for different values of Q, with the intention of proving the negligible effect of the dissipated power on the thermal resistance of the heat sink.
70.2
local T W
Nitrogen
70
793
Considering a mean gas temperature of 43 C, the thermal conductivities are 0.158 and 0.027 W/m K, for helium and nitrogen respectively; the convective heat transfer coefficients at the inner wall of the microducts are then 3436 and 588 W/m2 K. This justifies the higher thermal resistance for nitrogen flows.
4. Conclusions The paper has presented a 3-D analysis of a heat sink with a gas running in parallel circular microchannels. Pressure drops and temperatures are numerically determined and thermal resistance are obtained. The classical theory of continuum flow gives numerical results in very good agreement with the experimental data, obtained in a polymeric heat sink with helium and nitrogen flows at low Reynolds, Prandtl, Knudsen, Mach, and Brinkman numbers. For same mass flow rates and dissipated powers, helium flows allow to obtain lower temperatures in the heat sink, with a lower thermal resistance. The development of strips of polymeric micro heat sinks will require technological efforts, in order to improve the thermal conductivity of materials (polymers and substrate).
Acknowledgements
69.8 69.6
The financial support of the Italian Space Agency (Grant I/R/266/02) and the Ministry of University and Scientific Research (Grant COFIN 2003) is gratefully acknowledged. The authors would like to thank Prof. Marengo and Dr. Chignoli, of the University of Bergamo, for the profitable discussions and suggestions.
69.4 69.2 69 68.8 68.6 68.4
θ degrees
68.2 0
30
60
90
120
150
180
Fig. 6. Wall temperature distribution in the wetted perimeter of the exit section.
30 Rth
25
˚C /W
20 Q=1 W
15 N2
Q=8 W
10 He
5 m (mg/s)
0 0
30
60
90
120
150
180
210
240
270
300
Fig. 7. Thermal resistance for helium and nitrogen flows.
330
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