oscillatory flow forced convection in micro heat spreaders

Numerical Heat Transfer, Part A, 42: 685±705, 2002 Copyright # 2002 Taylor & Francis 1040-7782 /02 $12.00 + .00 DOI: 10.1080/1040778029005976 5

OSCILLATORY FLOW FORCED CONVECTION IN MICRO HEAT SPREADERS Cuneyt Sert and Ali Beskok Mechanical Engineering Department, Texas A&M University, College Station, Texas, USA A new micro heat spreader (MHS) concept for ef cient transport of large, concentrated heat loads is introduced. The MHS is a single-phase, closed micro uidic system, which utilizes reciprocating ¯ow forced convection. This micro uidic device has the potential to control transient heat loads by active control strategies. Hence, it can be utilized in various thermal management applications, mainly in electronic cooling. The MHS concept is validated by numerical solutions of two-dimensional Navier–Stokes and heat transport equations via a recently developed arbitrary Lagrangian Eulerian spectral element algorithm. Analytical relations for power consumption and heat removal capacities of the MHS are presented.

INTRODUCTION Thermal management of microelectronic components is a challenging problem. Increase in the central processing unit (CPU) speeds is required for faster computers. However, the amount of heat generation also increases with the increased CPU speeds. Ine cient thermal design leads to large chip surface temperatures, which severely a ect the chip performance and commonly result in chip malfunction. The thermal management problems of microelectronic components will worsen with further miniaturization. Reduction in size brings severe limitations to the conventional cooling techniques. For example, the small size of palm computers prohibits fan-based cooling, which in return limits the CPU speeds. Table 1 shows the heat ¯ux and the maximum allowable surface temperature of some of the commonly used microprocessors. As seen from the table, increase in the clock speeds results in increased heat generation, which makes the cooling process more challenging. Microscale thermal=¯uidic devices, such as the microchannel-base d forced convection heat exchangers, can be used in electronic packaging to develop innovative cooling strategies. The microchannel heat exchangers have great advantages for high heat-¯ux applications due to their high surface-area-to-volum e ratio, low thermal resistance, and the small coolant volume. Unfortunately, the small

Received 19 October 2001; accepted 2 March 2002. This work was supported by Texas Higher Education Council, Advanced Research Program Grant number 000512-0418-1999 . Address correspondence to Ali Beskok, Mechanical Engineering Department, Texas A&M University, TAMU 3123, College Station, TX 77843, USA. E-mail: [email protected] 685

C. SERT AND A. BESKOK

686

NOMENCLATURE A Cp Dh h H k l L Nu p P Pr q Q Re St t T

amplitude of membrane oscillation constant pressure speci®c heat hydraulic diameter channel height reservoir height thermal conductivity channel length reservoir length Nusselt number pressure power Prandtl number heat ¯ux volumetric ¯ow rate Reynolds number Strouhal number time temperature

Uo w m u r t o

reference velocity width of the MHS dynamic viscosity kinematic viscosity density period of membrane oscillation frequency of membrane oscillation

Subscripts a property of the ambient ¯uid b bulk quantity o reference parameter w property at the MHS top wall Superscripts * nondimensional parameter 0 per unit membrane width Ð channel-averaged quantity

hydraulic diameter of the microchannels, which allows very high heat ¯ux dissipation, leads to large pressure drops. Therefore, steady forced convection micro heat transfer requires advanced micro-pumping technologies. As an alternative to these active heat transfer devices, micro heat pipes, based on capillary pumping of a multiphase ¯uid in microchannels, have been developed [1]. Most of the ongoing e orts in microscale heat transfer research are experimental studies that compare thermal=¯uidic transport characteristics observed in microscales with the corresponding classical continuum theories. Wang and Peng conducted experiments to investigate single-phase forced convection of water and methanol ¯owing through microchannels with rectangular crosssection [2]. They also investigated transition to turbulence in microchannels. Peng and Peterson studied single-phase forced convection heat transfer and ¯ow characteristics of water in microchannels, both in the laminar and turbulent regimes [3]. They investigated the importance of microchannel geometry on thermal=¯uidic transport and developed several empirical correlations for both heat transfer and pressure drop. Adams et al. performed experimental studies of turbulent single-phase forced convection of water in micro-capillaries [4]. They have shown that the Nusselt numbers in the turbulent

Table 1. Maximum heat ¯ux and allowable temperatures of typical processors

Processor DEC=Compaq Alpha 22264 A 750 MHz AMD Athlon 650 MHz Intel Pentium-III 550 MHz Intel Celeron 466 MHz

Max. heat ¯ux [W=cm2]

Max. allowable temperature [¯ C]

39 29 26 17

60 65 65 70

FORCED CONVECTION IN MICRO HEAT SPREADERS

687

regime are higher than their macro-scale counterparts. Tso and Mahulikar proposed that the Brinkman number (product of the Eckert and Prandtl numbers) is more appropriate for correlation of convective heat transfer parameters in microchannels [5]. Based on this, they developed new empirical correlations, that agree with the experimental data in the laminar ¯ow regime. Mala and Li performed pressure drop and ¯ow rate experiments in microtubes [6]. They showed that for lower Reynolds numbers the results are similar to the Poiseuille ¯ow theory. However, the friction factor increases above the predictions of the classical theory for increased Reynolds numbers. A review of the micro and mesoscale heat exchanger studies is given in [7]. Weisberg and Bau analyzed microchannel heat exchangers by numerical solution of a conjugate heat transfer problem, which enabled simultaneous determination of the solid and ¯uid temperature ®elds [8]. They provided a design algorithm for selection of the heat exchanger dimensions. In a recent numerical study, Ambatipudi and Rahman investigated the e ects of channel aspect ratio, Reynolds number, and the number of channels on thermal performance of a microchannel heat sink [9]. Most of the previous forced convection studies in microchannels have been performed for steady unidirectional ¯ows. Here, we introduce the MHS concept, which utilizes reciprocating ¯ow forced convection in microchannels. This article is organized as follows: In the next section we describe the MHS and its working principle. Then we de®ne the simpli®ed two-dimensional problem with the governing equations, boundary conditions, and nondimensional parameters. Next we give the details of our numerical formulation and discuss eight speci®c cases that we simulated. This is followed by the velocity, pressure, and temperature ®eld results. We present velocity and temperature pro®les inside the channel, phase lag between the pressure and velocity oscillations, average Nusselt number, and surface temperature distributions. We discuss two alternative designs that enhance performance of the MHS. We also present analytical relations for power consumption and heat removal capacities of the device. Finally, we present further discussions about this study. DEVICE DESCRIPTION The MHS concept presented in this study consists of two reservoirs connected by microchannels as shown in Figure 1. A heat source is located on top of the microchannels. Micro membranes are fabricated at the bottom sides of the reservoirs. A pumping action is generated by actuating the membranes with a phase di erence of p radians. Actuation of the membranes can be achieved either electrostatically or piezoelectrically [10]. The membranes pump the ¯uid from one reservoir to the other in a continuous cycle. In the meantime, heat generated by the source is conducted to the ¯uid. Heating of the ¯uid is very fast due to the small channel height. The heated ¯uid is pumped toward the exit of the microchannel, where the ¯ow suddenly expands and starts to recirculate in the receiving reservoir. This is desired in order to enhance mixing of exiting hot ¯uid with the colder ¯uid in the reservoirs. Then the pumping direction reverses and the process repeats cyclically. This design allows two forced convection passes per membrane cycle, increasing the cooling e ciency. The ¯uid ¯ow and heat transfer in this conceptual design is time periodic. The MHS concept incorporate s microchannels expanding into much larger reservoirs. This device is designed to minimize temperature variations within the entire system by

688


Figure 1. Schematic view of the MHS.

e ectively transporting heat from the source, through the microchannels, into the reservoirs by forced convection and mixing. Once heat is e ectively spread over the MHS, cooling is achieved by exposing the MHS side walls to ambient conditions. For enhanced performance of the MHS, its side wall surface area should be larger than that of the heat source. Hence conventional cooling techniques with smaller heat ¯ux rates, such as ®ns, can be utilized on the MHS side walls. However, we must note that this ®nal cooling stage requires a complete ®n design (selection of ®n material and geometry, and the ambient conditions), which is not included in the current study.

PROBLEM DEFINITION We performed two-dimensional simulations of heat transfer and ¯uid ¯ow in the MHS device described in the previous section. The geometric parameters and discretization of the solution domain is shown in Figure 2. Geometry of the device is determined by speci®cation of height (h) and length (l) of the microchannel, and height (H) and length (L) of the reservoirs. Since the numerical simulations are performed for nondimensional parameters, all dimensions of the MHS are described in terms of the microchannel height (h). This enables us to utilize geometric similarity in determining the appropriate dimensions of the MHS. Two other important parameters in MHS operation are the membrane oscillation amplitude (A) and frequency (o). Governing equations are the conservation of mass, incompressible Navier± Stokes, and heat transport equations, presented in the following nondimensional form: H¤ ° ~ u¤ ˆ 0

…1†

q~ u¤ 1 ¤2 ¤ ‡ …~ u ¤ ° H ¤ †~ u ¤ ˆ ¡H ¤ p ¤ ‡ H ~ u ¤ qt Re

…2†


689

Figure 2. Geometric parameters and spectral element discretization of the solution domain. The domain is divided into 68 quadrilateral and triangular elements, each employing seventh-order expansion. The elements are shown by thick lines, and the collocation points for selected elements are indicated by thin lines.

qT ¤ 1 ‡ …~ u ¤ ° H ¤ †T ¤ ˆ H¤2 T ¤ qt ¤ Re Pr

…3†

where ~ u ¤ ; t ¤ ; p ¤ ; T ¤ are the nondimensional velocity vector, time, pressure, and temperature, respectively. Re is the Reynolds number and Pr is the Prandtl number. Fluid properties are assumed to be constant, and viscous work terms in the heat transport equation are neglected. Reynolds and Prandtl numbers are the two important parameters in determining thermal=¯uidic performance of the system. A third important nondimensional parameter is the Strouhal number (St ˆ oh=Uo ), which is based on the characteristic length (h), velocity (Uo), and time scale (1=o). Due to the lack of a free stream velocity in the MHS system, we selected the reference velocity to be Uo ˆ ho, making the Strouhal number unity, and reducing the nondimensional parameters to Re and Pr. Nondimensionalizatio n of the physical parameters are summarized below: t ¤ ˆ ot ~ u¤ ˆ q¤ ˆ

q kDT=h

u ~ oh

p¤ ˆ

Re ˆ

p ro2 h2

oh2 v

Pr ˆ

T¤ ˆ rvCP k

T ¡ To DT …4†

where r is the density, v is the kinematic viscosity, Cp is the constant-pressur e speci®c heat, k is the thermal conductivity of the ¯uid, and q ¤ is the nondimensional heat ¯ux. To is the constant side wall temperature and DT is a parameter that will be used to determine the dimensional maximum heat ¯ux dissipated by the MHS. DT can be calculated using the maximum allowable temperature di erence between the chip surface and the side walls …Tmax ¡ To † and the computed maximum nondimensional ¤ ) using DT ˆ …T ¤ temperature (Tmax max ¡ To †=Tmax . Correspondingly, the heat dis¤ sipation rate is found by q ˆ q kDT=h. Since the maximum allowable temperature …Tmax ¡ To † is a design parameter, normalization of temperature in this manner


690

Figure 3. Boundary conditions for the MHS: The side walls are at ambient conditions (at To¤ ), the microchannel top surface has speci®ed heat ¯ux, and all other surfaces are adiabatic.

makes utilization of dynamic similarity in obtaining the dimensional temperature values easier. Boundary conditions used in the problem are shown in Figure 3. Nondimensional side-wall temperature, To¤ , is taken to be a constant. Constant heat ¯ux is applied on top of the microchannel. The heat ¯ux toward the ends of the channel is reduced to zero smoothly with a tangent-hyperboli c function. All the other boundaries, except from the side walls, are assumed to be adiabatic in order to minimize the overall heat loss and to simulate the worst possible scenario. Time periodic motion of the membranes are speci®ed in the following form: ´ x¤ v ˆ A cos…t † sin p ¤ : L ¤

¤

¤

³

…5†

Here we discuss the importance and implications of constant temperature sidewall boundary conditions used in our analysis. For complete thermal analysis of the MHS system, one must also consider the e ectiveness of heat transfer to the environment using ®ns. Although this is a vigorous approach for overall device design, it requires solution of the conjugate heat transfer problem in the ®ns, which necessitate speci®cation of the ®n material, shape, ambient temperature (Ta), and the heat convection coe cient between the ambient and the ®n surface (ha). This approach signi®cantly complicates the overall design process. It is also counterproductive for this ®rst study, since we want to investigate the e ectiveness of reciprocating forced convection in microchannels and mixing in the reservoirs. For these reasons we preferred to simplify the side-wall boundary condition by specifying a ®xed side-wall temperature. NUMERICAL FORMULATION AND SIMULATION PARAMETERS The numerical simulations are performed using an unstructured h=p-type spectral element algorithm. Fluid ¯ow and thermal transport equations are solved in moving domains using a recently developed arbitrary Lagrangian Eulerian (ALE) algorithm [11]. Our method allows up to third-order time accuracy for transient


691

problems and delivers spectral spatial resolution. The details of the algorithm and formal numerical convergence studies, including case studies of phase and di usion errors of the scheme, are presented in [11]. In the spectral element method, the solution domain is divided into elements, similar to the ®nite element method. However, each element is further discretized using an Nth-order polynomial approximation per direction. The polynomial bases are chosen from the general Jacobi polynomials, which are the eigenfunctions of a singular Sturm±Liouville equation. This choice of the approximation polynomials results in exponentiall y fast convergence (for su ciently smooth problems) upon successively increasing the polynomial expansion order (p-type re®nement) [12, 13]. In our simulations, we employed a third-order time integration scheme and performed successive p-type grid re®nements. The computational domain shown in Figure 2 is divided into 68 quadrilateral and triangular elements, where thick lines denote the elemental discretization and thin lines show the collocation points in selected elements. Grid independent velocity and temperature solutions are established using this twodimensional computational domain, employing 7th±11th-orde r discretization, depending on the Re and Pr. We present here ®ve main and three supporting simulations. Parameters used in these simulations are shown in Table 2. Here we must emphasize that the geometry becomes more realistic with higher l=h, L=h, and H=h ratios. In this sense, only the last case of Table 2 resembles an actual MHS in terms of its aspect ratio (l=h). For the actual MHS dimensions, we would select the microchannel height to be in the order of 100 mm, while the channel length should be determined by the chip size (typically about a few centimeters). The mixing chamber length, L, also should be a few centimeters. These realistic dimensions create a very large aspect ratio device. The ¯ow domain is asymmetric due to the out-of-phas e oscillations of the membranes. Hence, the entire MHS needs to be simulated. We integrate the governing equations (in time) until a time-periodic solution in velocity, pressure, and temperature ®elds is achieved. The velocity and temperature boundary layers get sharper with increased Reynolds and Prandtl numbers, respectively. Resolution of these sharp gradients requires higher-order approximations in our spectral element algorithm. Hence, the numerical complexity of the problem increases signi®cantly. Also, the computationa l time to reach the time-periodic state takes longer when the Prandtl number is increased. Typically, Pr ˆ 10 ¯ow simulation takes about 10 times as much integration time to reach the time-periodic state, compared with

Table 2. Parameters used in the MHS simulations Simulation

Re

Pr

l=h

L=h

H=h

A=h

Notes

1 2 3 4 5 6 7 8

2p 2p 2p 4p 8p 2p 2p 2p

1 10 25 1 1 1 1

6 6 6 6 6 6 6 50

10 10 10 10 10 10 10 50

5 5 5 5 5 5 5 25

1 1 1 1 1 1 1 1

± ± ± ± ± obstacles guidance ±


692

a Pr ˆ 1 case. Due to these restrictions, we selected a low aspect ratio geometry as shown in Table 2, with the exception of case 8. Our current e ort is concept veri®cation, and we believe that our results serve this purpose. Once the MHS geometry is decided upon, there are only two other nondimensional parameters to be chosen, namely, the Reynolds and Prandtl numbers. As shown in Table 2, we used di erent combinations of Re ˆ 2p, 4p, 8p and Pr ˆ 1, 10, 25. Simulations 1, 2, and 3 will enable us to understand the e ects of Pr, and simulations 1, 4, and 5 will show the importance of Re. These simulations provide us with the velocity, pressure, and temperature ®elds within the device. Each of these will be explained in detail in the following sections. The velocity ®eld is used to analyze the ¯ow pro®le inside the microchannel. Pressure distribution on the membranes is used to obtain the actuation force, which is further utilized to predict the power consumption of the MHS. The maximum temperature on the MHS surface is the most valuable output of the simulations since the ultimate objective of the MHS is to keep the chip surface temperature as low as possible while maximizing the heat dissipation. After observing the velocity and temperature ®elds in the reservoirs of the main simulations (cases 1±5 in Table 2), we developed two alternative designs with mixing promoters and ¯ow guides. The parameters used for these cases (6 and 7) are also shown in Table 2. Our aim in these simulations is either to achieve enhanced mixing inside the reservoirs or to guide the hot ¯uid coming out of the microchannel toward the side walls as e ectively as possible. Finally, we simulated a high aspect ratio MHS geometry (case 8 in Table 2) to identify the aspect ratio e ects on performanc e of the device. VELOCITY AND PRESSURE FIELD RESULTS The velocity and pressure ®elds depend on the Reynolds number, and they converge to a time-periodic solution much faster than the temperature ®eld. It is our experience that for Re < 25 ¯ows, time-periodic velocity, and pressure solutions are obtained after a few membrane cycles. Velocity Field Velocity pro®les at the channel entrance, midpoint and the channel exit are shown in Figure 4 for three di erent Re cases. The y* in the ®gure is the nondimensional vertical distance measured from the bottom wall of the channel. Velocity pro®les presented in Figure 4 correspond to time of the maximum ¯ow rate obtained when membranes pass through the zero de¯ection position. At the channel entrance, deviations from the parabolic pro®le are observed. These deviations are due to the asymmetric entry of the microchannel. Flow asymmetry increases with increased Reynolds number. At the channel exit, there are small deviations from the parabolic velocity pro®le due to the downward-biased expansion of the ¯uid into the mixing reservoir. The maximum nondimensional velocity inside the channel for all Reynolds number cases is about the same. Although the velocity pro®les at the mid-channel section are parabolic, this does not imply fully developed ¯ow conditions. This is because Figure 4 corresponds to the instant of maximum ¯ow rate,


693

Figure 4. Velocity pro®les at the entrance (dashed), center (solid), and exit (dashed-dotted) of the microchannel at the time of the maximum ¯ow rate.

and the velocity pro®les change by time. Figure 5 shows the velocity pro®les at di erent instances in a quarter cycle for Re ˆ 8p ¯ow. Solid lines show the numerical solution, and dashed lines denote the analytical solution for two-dimensional fully developed oscillatory ¯ow [14]. The details of this analytical solution will be discussed in the next section. As seen from Figure 5, velocity pro®les always have a downward bias and never reach a fully developed state, because of the small microchannel length. Another important observation is that the analytical solution

Figure 5. Four instances of velocity pro®les at four di erent times at the channel-center for Re ˆ 8p ¯ow. Dashed lines are the analytical solution obtained for fully developed oscillatory ¯ows, and solid lines show the numerical simulation results. The discrepancy between these two results is due to the inlet and exit e ects.


694

for this oscillatory ¯ow results in velocity overshoots close to the walls. This is a known behavior of high-frequency oscillatory ¯ows [14]. Pressure Field We are interested in the pressure ®eld in order to obtain the power requirements for MHS actuation. There is pressure drop inside the channel due to the ¯ow. There are also entry and exit e ects, as well as developing ¯ow conditions. The pressure distribution on each membrane is almost uniform at a given time. In Figure 6 we present nondimensional pressure di erence between the two membranes for three di erent cases. Here t¤ ˆ 0 corresponds to the time of zero de¯ection of the membranes and t¤ is the nondimensional period of the membrane oscillation. The thick line is a cosine curve representing variation of the channel-average d velocity included to identify the phase di erence between velocity and pressure ¯uctuations. The maximum magnitude of the channel-average d velocity is 4. Figure 6 shows the maximum nondimensional pressure di erence between the membranes to be 49.1, 35.0, and 30.0 for Re ˆ 2p, Re ˆ 4p, and Re ˆ 8p cases, respectively. These pressure drop values include the channel entry, exit, and ¯ow development e ects. The phase lag between pressure and channel-average d velocity are 8.8%, 14.0%, and 18.8% for Re ˆ 2p, Re ˆ 4p, and Re ˆ 8p cases, respectively. Although the nondimensional pressure di erence between the membranes decreases with increased Reynolds number, the dimensional value of pressure will increase, since the pressure is normalized by the dynamic head. (See Eq. (4).) We compared the pressure results of our simulations with the analytical solution of ¯ow between parallel plates driven with a time harmonic pressure gradient of amplitude [14]: ¡

1 qp ˆ ae¡iot r qx

…6†

Figure 6. The nondimensional pressure di erence between the membranes (solid line: Re ˆ 2p; dashedline: Re ˆ 4p; dashed-dotted line: Re ˆ 8p). The thick solid line shows average velocity variation in the channel during one membrane period.


695

Under these conditions the average velocity over a cross section is given as » µ ¶¼ ia ¡iot tan…kh=2† u· ˆ Real e 1¡ o kh=2

…7†

where

where i ˆ

1‡i k ˆ p 2v=o

…8†

p ¡1. Using Eq. (4), We can nondimensionalize Eqs. (6)±(7) to result in qp ¤ ¤ ˆ a ¤ e¡it ¤ qx » ¤ µ ¶¼ ia ¡it ¤ tan…k ¤ h ¤ =2† u· ¤ ˆ Real e 1 ¡ o¤ k ¤ h ¤ =2 ¡

…9† …10†

where normalized frequency o ¤ is unity and a a ˆ 2 o h ¤

r Re k h ˆ …1 ‡ i† 2 ¤ ¤

…11†

Using Eqs. (9)±(10), we ®nd the phase lag between velocity and pressure as 8.9%, 14.1%, and 18.6% for Re ˆ 2p, Re ˆ 4p, and Re ˆ 8p cases, respectively. These values are almost identical to the ones obtained from the simulations. For nondimensional pressure-drop comparisons, we ®rst calculate the volumetric ¯ow rate per unit channel width using the membrane motion given by Eq. (5): Q

0¤

ˆ

Z

L¤ 0

v ¤ dx ¤ ˆ 4 cos…t ¤ †

…12†

The nondimensional average velocity inside the channel becomes u· ¤ ˆ

Q0 ¤ ˆ Q0 ¤ ˆ 4 cos…t ¤ † h¤

…13†

Therefore, the magnitude of the maximum nondimensional channel-average d velocity is 4.0, regardless of the Reynolds number. Using this value in Eq. (10), we can ®nd values of a¤ for di erent Reynolds numbers. Finally substituting a¤ into Eq. (9), we can determine the nondimensional magnitude of the maximum pressure drop as Dpmax º ¡

qp ¤ l ¤ ˆ a ¤l ¤ qx ¤ max

…14†

696


We found the maximum nondimensional pressure drop in the channel to be 34.11, 23.25, and 19.54 for Re ˆ 2p, Re ˆ 4p, and Re ˆ 8p cases, respectively, while the simulation results gave 34.1, 23.7, 19.7. The analytical results for fully developed ¯ows and the corresponding simulation results for developing ¯ows are surprisingly close to one another. One can examine the velocity distribution at the channel center, shown in Figure 5, to assess the shear stress on the top and bottom walls of the channel. The shear stresses obtained from the numerical solutions are smaller than the analytical solution at the top wall, while they are larger on the bottom wall. Hence, we observe a canceling e ect, which yields comparable results between the analytical solution and the simulations. TEMPERATURE FIELD RESULTS In heat transfer analysis and determination of the temperature distribution within MHS, both the Reynolds and Prandtl numbers play an important role. Temperature pro®les across the channel midpoint, corresponding to the maximum ¯ow rate conditions, are given in Figure 7. This ®gure is helpful in understanding the general e ects of Reynolds and Prandtl numbers on the instantaneous thermal boundary layer. Constant heat ¯ux and adiabatic wall conditions are speci®ed at the channel top and bottom surfaces, respectively. Hence, the temperature is calculated relative to the side-wall temperature, which is ®xed at zero. The nondimensional temperatures decrease with increased Reynolds and Prandtl numbers. Furthermore, the thermal boundary layer becomes thinner as the Prandtl number increases. A similar trend is also noticed for increased Reynolds number. The slope of temperature variation at the upper wall is exactly 1.0 in all of the simulations. This value is equal to the constant heat ¯ux boundary condition enforced on this wall.

Figure 7. Temperature pro®les at the center of the microchannel, corresponding to the time of the maximum ¯ow rate.


697

Table 3. Maximum nondimensional temperature inside the microchannel Re

Pr

¤ Tmax

2p 2p 2p 4p 8p Pure conduction

1 10 25 1 1

4.31 1.71 1.29 2.86 2.45 12.61

Values of maximum nondimensional temperature for the ®ve main cases are shown in Table 3. In this table, we also included the result of the nondimensional temperature obtained by pure conduction (no membrane actuation). For all cases, the maximum temperature occurs on the upper wall of the channel. Here we should note that the maximum temperatures on the MHS surface do not occur at the center of the microchannel but closer to the two ends. However, the center of the channel is exposed to equally high temperatures for longer times. The maximum temperature decreases with increasing Re and Pr. Low values of the maximum nondimensional temperature indicate that time-periodic forced convection is an e ective means of heat dissipation. The maximum nondimensional temperature for the Re ˆ 2p, Pr ˆ 25 case shows almost a 10-fold decrease, compared with the pure conduction result. This comparison of nondimensional temperatures is equally applicable to the comparison of dimensional temperatures provided that the thermal conductivities of the coolant and the conductive material are the same. However, one must be careful in obtaining the dimensional temperatures if the thermal conductivity of the two materials are di erent. Bulk temperature is an important parameter that is used in the calculation of the Nusselt number. Since we are studying time-periodic reciprocating ¯ows, the channel-average d velocity alternates sign and becomes zero twice in a cycle. At these instances the bulk temperature and Nusselt number can not be de®ned. In order to avoid negative bulk temperatures, we followed the work of Guo and Sung for oscillatory ¯ows and de®ned the bulk temperature as [15]: Tb¤ …x ¤ ; t ¤ † ˆ

1 ‰u· ¤ …x ¤ ; t ¤ †Š2

Z

1

T ¤ …x ¤ ; y ¤ ; t ¤ †‰u· ¤ …x ¤ ; t ¤ †Š2 dy ¤

0

…15†

where u· ¤ is the channel-average d velocity de®ned as u· ¤ …x ¤ ; t ¤ † ˆ

1 h¤

Z

h¤ 0

u¤ …x ¤ ; y ¤ ; t ¤ †dy ¤

…16†

The Nusselt number is de®ned as Nu…x; t† ˆ

hc …x; t†Dh k

…17†


698

where k is the thermal conductivity of the ¯uid, Dh is the hydraulic diameter, and h c is the convection coe cient, which can be expressed in terms of the wall heat ¯ux, wall temperature, and bulk temperature in the following form: hc …x; t† ˆ

qw Tw …x; t† ¡ Tb …x; t†

…18†

For parallel plates the hydraulic diameter Dh ˆ 2h. Substituting this value and Eq. (18) into Eq. (17), we obtain the Nusselt number as Nu…x; t† ˆ

qw 2h Tw …x; t† ¡ Tb …x; t† k

…19†

We substitute the de®nition of nondimensional heat ¯ux and temperature from Eq. (4) Nu…x; t† ˆ

qw¤ kDT=h 2h ¤ ‰Tw …x; t† ¡ Tb¤ …x; t†ŠDT k

…20†

Hence, we obtain the Nusselt number in the following form: Nu…x; t† ˆ

2 ‰Tw¤ …x ¤ ; t ¤ † ¡ Tb¤ …x ¤ ; t ¤ †Š

…21†

The channel-average d velocity, bulk temperature, and Nusselt number are functions of the time and streamwise location. We divided the period of the MHS oscillation into 40 time slices and numerically integrated the known nodal values of velocities and temperatures at 25 di erent cross sections inside the microchannel for each time slice. We calculated the space and time variations of bulk temperature and Nusselt number. Entrance e ects, nonuniform heating at the channel inlet and exit, and the developing ¯ow conditions a ect the Nusselt number value considerably. In order to make a comparison for di erent Reynolds and Prandtl numbers, we present in Table 4 the minimum, maximum, and time-averaged Nusselt numbers at midcross section of the channel. The minimum Nu values in Table 4 always correspond to the time of the minimum ¯ow rate (i.e., the instant when the membranes are at their maximum de¯ection positions). The maximum values, however, correspond to di erent times for di erent cases. Here we observe that the time-averaged Nusselt

Table 4. The minimum, maximum, and time-averaged Nusselt numbers at mid-cross section of the channel Re

Pr

Min. Nu

Max. Nu

Ave. Nu

2p 2p 2p 4p 8p

1 10 25 1 1

3.62 5.34 6.96 2.30 0.66

5.92 9.88 11.82 6.96 8.00

5.50 8.42 10.20 5.92 6.31


699

number is not the arithmetic mean of the maximum and minimum Nu values. Its value is closer to the maximum Nu value. Figure 8a shows the temperature distribution at the upper wall of the MHS. Boundary conditions on the top wall are speci®ed as q¤ ˆ 0

at 0 < x¤ < 10

q¤ ˆ 1

at 10 < x¤ < 16

16 < x¤ < 26 …22†

The top and bottom plots show the a ects of the Re and Pr, respectively. The pure conduction result shows the steady state temperature distribution when there is no membrane motion. For this case, the hot ¯uid is trapped under the heat source and the top-wall temperature makes a peak at the middle of the channel. For other cases, the top-wall temperature varies with time and we show two instances during a cycle for each case. One of these instances corresponds to the time of maximum top-wall temperature. It is clearly seen that the temperature peak is eliminated, and more uniform top wall temperature distribution is obtained. The results show an overall decrease in temperature inside the microchannel by increasing Reynolds and=or Prandtl numbers.

SUPPORTING SIMULATIONS There are two di erent methodologies to cool the hot ¯uid convected from the microchannels into the reservoir. First, by e ectively mixing the exiting ¯uid with the colder ¯uid in the reservoirs using mixing promoters, second, by bringing the hot ¯uid in contact with the colder side walls using guidance plates. We simulated one case for each of these ideas. The parameters used are shown in Table 2 as cases 6 and 7. The modi®ed MHS geometries used in these simulations are shown in Figure 9, while the top-wall-temperatur e variations are shown in Figure 8b (top). Figure 9 shows the temperature contours corresponding to the time of the minimum ¯ow rate (i.e., maximum temperatures). The top plot is for the case without any mixing promoter or guidance plate (case 1 in Table 2). The second plot has two small blocks inside the reservoirs placed close to the channel. The idea is to divide the hot ¯uid stream coming out of the channel into smaller streams and enhance mixing and cooling. Even this simple con®guration works well enough to reduce the maximum nondimensional temperature from 4.00 (top plot) down to 3.24 (second plot). The third plot shows the temperature contours for the case with guidance plates. The idea here is to carry the hot ¯uid coming out of the channel to the cold side walls from the corridor above the guidance plate during suction to the reservoir and to feed the colder ¯uid under the guidance plate into the channel during the ejection stage. The guidance plate design shows a 20% drop in the maximum surface temperature, as shown in Figure 8b (top). Although the maximum temperature drop is similar to the mixing promoter design, this idea did not perform as well as we ®rst anticipated. For this very low Re, the ¯uid coming out of the channel does not reach the side walls completely, and the hot ¯uid that came out of the channel is mostly fed back during the ejection stage. We expect this idea to perform better for higher Re cases. It is obvious that one can perform design optimization for a ®xed Reynolds number by

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Figure 8. (a) Nondimensional top-wall-temperature variations for various Reynolds (top) and Prandtl (bottom) number simulations.

changing the geometry and position of the blocks and plates. Considering that the Reynolds number can be changed by adjusting the membrane oscillation frequency, an Re-speci®c geometric optimization may not be very useful. Finally, we simulated a case with a more realistic aspect ratio. The parameters used are shown as the last entry of Table 2. Temperature contours at the instant of minimum ¯ow rate, obtained for this high aspect ratio geometry, can be seen in the last plot of Figure 9. Note that although the Reynolds number based on the channel height and the membrane frequency is still 2p, the velocity inside the microchannel is higher than that of the low aspect ratio cases. This is due to the increase in the membrane length, which also increases the swept volume in a half-period. The importance of this case is that it shows interesting ¯ow dynamics inside the reservoirs. Since the Prandtl number is unity, temperature contours closely follow ¯uid vorticity for this two-dimensional simulation. The temperature contours indicate the presence of vortex rolls due to a shear layer instability of the jet exiting to the reservoir, along


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Figure 8. (b) Nondimensional top-wall-temperature variations obtained in modi®ed designs (top) and in the high aspect ratio MHS (bottom). Simulation results for Re ˆ 2p, Pr ˆ 1 are shown.

with two primary counter-rotatin g zones. Preliminary discussions about this case can be found in [16]. The bottom plot of Figure 8b shows a nondimensional temperature distribution at the top surface at four di erent times. Constant heating is applied in the region 50 < x ¤ < 100. The e ect of vortex rolls on the top-wall temperature can be observed around x ¤ º 35 and 135. Although it is not shown here, we also simulated pure conduction in this high aspect ratio geometry, and we obtained a maximum nondimensional temperature of 400. As seen from Figure 8b (bottom), the maximum temperature on the top surface is 3.7, which indicates a more than 100fold decrease compared with pure conduction (provided that the thermal conductivities of the coolant and the conductive material are the same). This shows that time-periodic forced convection in more realistic geometries is more e ective than the low aspect ratio geometries presented in cases 1±5.

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Figure 9. Nondimensional temperature contours obtained at the minimum ¯ow rate conditions, where the membranes are at maximum de¯ection. Simulation results for Re ˆ 2p, Pr ˆ 1 are shown.

DETERMINATION OF THE OPERATING CONDITIONS In this section, we will develop equations to demonstrate conversion of the previously obtained nondimensional results to dimensional quantities. In order to do this one should ®rst select the coolant. In coolant selection, low viscosity, light weight, and high thermal conductivity are desired. Low viscosity and small density of the coolant will reduce the power consumption, and high thermal conductivity will


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increase the heat dissipation rate. In order to match the simulation parameters, one should choose the coolant by matching the Prandtl number. Then the channel height (h) should be selected, and the oscillation frequency should be calculated from the de®nition of Reynolds number o ˆ v Re=h2

…23†

Therefore, decreasing the channel height by keeping Re ®xed increases the actuation frequency quadratically. For low aspect ratio geometry, nondimensional channel-average d velocity, u· ¤ ˆ 4, regardless of the Reynolds number. This value can be converted into a dimensional form by multiplying it with oh. However, the average ¯ow rate per unit width gives a better idea of the operating conditions of the device, which can be calculated as Q0 ˆ u· ¤ oh2 ˆ 4oh2 ˆ 4v Re

…24†

Hence, the average ¯ow rate in the MHS increases linearly with the Reynolds number. Dimensional values for the maximum pressure di erence between the two membranes can be obtained by using the numerical simulation results and Eq. (4) as Dpmax ˆ

¤ Dpmax r

³

´ v Re 2 h

…25†

The power necessary to drive the membranes is an important design criterion in determining the feasibility of the MHS for electronic cooling. The power per unit width is predicted by P0 ˆ

Z

L

Dp v dx 0

…26†

where v is the velocity of the membrane. Since the pressure ¯uctuates with time (see Figure 6), the power requirement of the MHS also varies with time. Here we note that only the pumping membrane is doing work, while the suction membrane is inactive. The power requirement increases with Reynolds number. For a ®xed Re operation, the coolant with the smallest kinematic viscosity and density requires the least power. The dimensional value of the maximum power per unit width is found by P0max ˆ P0¤ max

r …v Re†3 h2

…27†

The heat removal capacity of the MHS is the most important design criteria. The dimensional value of the heat ¯ux can be obtained by q ˆ q¤

kDT h

…28†

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Here we need to determine DT in order to calculate the maximum heat dissipation. For this we use DT ˆ

Tmax ¡ To T¤max

…29†

¤ is the maximum nondimensional temperature di erence between the chip where Tmax surface and the side walls (see Table 3). The parameter (T 7 To)max can be determined by choosing a safe chip surface temperature from Table 1. The dimensional value of the heat dissipation rate depends on the thermal conductivity of the ¯uid ¤ obtained by numerical simulations. Hence, it is essential to use and the value of Tmax a high thermal conductivity coolant.

DISCUSSIONS We presented concept validation for a new MHS device using numerical simulations. The new MHS device utilizes reciprocating ¯ow forced convection for e cient transport of concentrated heat loads from small areas. Hence, its primary use will be in electronic cooling applications. We performed parametric studies of ¯uid ¯ow and heat transfer in moving domains as a function of Re and Pr. However, the design space of MHS is diverse, due to many possible combinations of geometric and thermo=¯uidic parameters. Hence, device optimization requires further studies. Our analysis has shown that a ¯uid with low viscosity, low density, and high thermal conductivity is desired in coolant selection. Low viscosity and low density of the coolant will reduce power consumption, and high thermal conductivity will increase heat dissipation rate. We have demonstrated also that the aspect ratio of the MHS has signi®cant e ects on the performance of the device. Time-periodic forced convection is more e ective in high aspect ratio (realistic) MHS geometries. Potential advantages of the MHS: ° Transient control: The MHS o ers the ability to control the maximum surface temperature with closed loop control strategies. For example, variation of computational loads on the microprocessors results in periodic spikes in the heat load. One point temperature measurement on the chip surface via a thermocouple can be used to create a feedback signal to the MHS to vary the operation frequency. Active cooling strategies are potentially a great advantage of the MHS compared with the passive cooling devices. ° Integration to microelectronics: Since the MHS is a micro¯uidic system, it can be integrated into the microchip components at the design and fabrication stages, enabling a compact chip with an onboard cooling system. This feature of the MHS becomes important for e cient thermal packaging of miniaturized electronic components in compact designs, where more conventional cooling strategies (such as fan-based cooling) cannot be employed. We expect these potential advantage s enable employment of the MHS systems in electronic cooling applications, either as a stand-alone device or in addition to existing technologies, such as heat sinks and microheat pipes. Future studies on the


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MHS systems require systematic simulation of more realistic device aspect ratios and e cient ®n design by solving a conjugate heat transfer problem. REFERENCES 1. G. P. Peterson, Micro Heat Pipes and Micro Heat Spreaders, in M. Gad-El-Hak (ed.), The MEMS Handbook, chap. 31, CRC Press, Boca Raton, FL, 2002. 2. B. X. Wang and X. F. Peng, Experimental Investigation on Liquid Forced-Convection Heat Transfer Through Microchannels, Int. J. Heat Mass Transfer, vol. 37, suppl. 1, pp. 73±82, 1994. 3. X. F. Peng and G. P. Peterson, Convective Heat Transfer and Flow Friction for Water in Microchannel Structures, Int. J. Heat Mass Transfer, vol. 39, no. 12, pp. 2599±2608, 1996. 4. T. M. Adams, S. I. Abdel-Khalik, S. M. Jeter, and Z. H. Qureshi, An Experimental Investigation of Single Phase Forced Convection in Microchannels, Int. J. Heat Mass Transfer, vol. 41, no. 6±7, pp. 851±857, 1998. 5. C. P. Tso and S. P. Mahulikar, The Use of the Brinkman Number for Single Phase Forced Convective Heat Transfer in Microchannels, Int. J. Heat Mass Transfer, vol. 41, no. 12, pp. 1759±1769, 1998. 6. G. M. Mala and D. Li, Flow Characteristics in Microtubes, Int. J. Heat and Fluid Flow, vol. 20, pp. 142±148, 1999. 7. S. S. Mehendale, A. M. Jacobi, and R. K. Shah, Heat Exchangers at Micro and MesoScales, Compact Heat Exchangers and Enhancement Technology for the Process Industries, in R. K. Shah, K. J. Bell, H. Honda, and B. Thonon (eds.), Proc. Int. Conference for the Process Industries, pp. 55±74, 1999. 8. A. Weisberg and H. H. Bau, Analysis of Microchannels for Integrated Cooling, Int. J. Heat Mass Transfer, vol. 35, no. 10, pp. 2465±2474, 1992. 9. K. K. Ambatipudi and M. M. Rahman, Analysis of Conjugate Heat Transfer in Microchannel Heat Sinks, Numerical Heat Transfer, Part A, vol. 37, pp. 711±731, 2000. 10. J. G. Smits, Piezoelectric Micropump with Three Valves Working Peristatically, Sensors and Actuators A, vol. 21±23, pp. 203±206, 1990. 11. A. Beskok and T. C. Warburton, An Unstructured H=P Finite Element Scheme for Fluid Flow and Heat Transfer in Moving Domains, J. Computational Physics, 174, pp. 494±509, 2001. 12. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods in Fluid Dynamics, Springer-Verlag, New York, 1988. 13. G. M. Karniadakis and S. J. Sherwin, Spectral=hp Element Methods for CFD, Oxford University Press, New York, 1999. 14. L. D. Landau and E. M. Lifshitz, Course of Theoretical Physics Volume 6ÐFluid Mechanics, 2nd ed., Pergamon Press, Oxford, UK, 1987. 15. Z. Guo and H. J. Sung, Analysis of the Nusselt Number in Pulsating Pipe Flow, Int. J. Heat Mass Transfer, vol. 40, no. 10, pp. 2486±2489, 1997. 16. C. Sert and A. Beskok, Shear Layer Instability and Mixing in Micro Heat Spreaders, (Heat Transfer Photo Gallery paper), J. Heat Transfer, vol. 123, no. 4, p. 621, 2001.