Capillarity of Rectangular Micro Grooves and Their Application to Heat Pipes

Transcription

1 Tamkang Journal of Science and Engineering, Vol. 8, No 3, pp (2005) 249 Capillarity of Rectangular Micro Grooves and Their Application to Heat Pipes Horng-Jou Wang, Hsin-Chang Tsai, Hwang-Kuen Chen and Tai-Kang Shing* MEMS R&D Department, Research Center, Delta Electronics, Inc. Taoyuan, Taiwan 333, R.O.C. Abstract The capillarity of micro grooves with rectangular cross-section is studied theoretically and experimentally in this paper. The Helmholtz free energy method is used to predict the capillarity as the groove is placed vertically and inserted the bottom end into the liquid. In the experiment, micro grooves are first constructed by the thick photoresist patterned by photolithography method and then a thin copper layer is deposited on their surface to improve the hydrophilic property of liquid-solid interface. It is shown that the theoretical and experimental results are in good agreement. Furthermore, the capillary limits of micro grooved heat pipes are investigated. The effects of groove s width and number on the capillary limit contributed from the maximum capillary pumping pressure and the pressure drops due to the friction and gravitational force are calculated. A workable geometry range of micro grooves for a heat pipe designed to transport a specific heat rate can be determined by these developed tools. Key Words: Capillarity, Micro Grooves, Heat Pipes, Capillary Limit 1. Introduction Micro groove, the groove with micrometer characteristic sizes, is one of the basic elements in the microelectro-mechanical systems (MEMS) and micro-fluidic devices. Micro grooves are widely used in many fields, such as the gas detection in the environmental monitoring, chemical analysis in biochemistry, drug delivery in the medical instrument, the inkjet print head and micro heat dissipation devices in the electronic industry. Bulk and surface micromachining and the molding technology can be used to fabricate the micro grooves [1]. One reason why the micro groove becomes the basic structure in the micro-fluidics devices is that micro groove can transport liquid without any additional power input. The self-pumping ability of micro grooves is due to the capillary force and pressure. In general, the grooves with smaller effective pore radius usually have *Corresponding author. tk.shing@delta.com.tw larger capillarity [2,3]. However, the friction between the structure of grooves and liquid will increase rapidly as the effective pore radius of grooves shrinks. The smaller characteristic size of micro grooves will not always have the better ability in transporting liquid. Besides, the gravitational field also affects the behavior of micro grooves. The well-designed micro groove devices must overcome the effect of gravitation and can be operated under any inclination angles from horizontal. The heat pipe is a passive two-phase heat transfer device capable of transmitting large quantities of heat with a minimal temperature drop [4]. The heat pipe combined with heat sink has been widely used in electronics cooling applications [5,6]. As the heat is applied to the evaporator of the heat pipe, the localized heat source could be spread equally in the vapor space of heat pipe and the vapor temperature is nearly isothermal. The overall efficiency of heat transfer and cooling for the heat sink is thus enhanced. The major physical behaviors of heat pipe are the evaporation, condensation and liquid trans-

2 250 Horng-Jou Wang et al. port. Micro grooves could be the micro fluid passages for transporting liquid in heat pipes [7 10]. In designing such devices, the capillary limit is one of the most important factors. The capillary limit, the ability for wick structures to pump the liquid back to the evaporator section, is dependent upon the capillary force of wick structures, the friction between fluid and wick structures and the gravitational force. Although the maximum capillary pumping pressure of micro grooves is smaller than that of conventional metal sintered powder wick structures, the features of higher liquid permeability and higher thermal conductivity make micro grooves become a good candidate for heat pipe devices. It is regarded as an emerging technique in the thermal management products, especially for high heat flux electronic devices. With the development of MEMS process, the micro grooves can be fabricated in the smaller width, larger aspect ratio and a precise size. Hence, the capillary pumping pressure of the micro grooves could be further improved. The sequential issue is how to design a suitable geometry size of micro grooves. And this topic will be discussed herein. In this paper, the capillarity of micro rectangular grooves is studied first in Section 2. The capillary force, capillary pressure and the height of liquid column are derived theoretically. An experiment is then carried out to verify these analyses. In Section 3, examples using micro rectangular grooves on heat pipes are investigated. The capillary limits of flat heat pipes with axially rectangular grooves are analyzed and a workable range of grooves can be determined for a specific engineering specification. Finally, the conclusions are drawn in Section Capillarity Capillarity is a significant design factor in micro groove-related products. In this section, the basic capillary phenomena of a rectangular groove, including the capillary force, capillary pressure and height of liquid column pulled by the capillarity are derived first. An experiment on measuring the height of liquid column is then carried out to compare with that of the theory. 2.1 Theoretical Analysis As shown in Figure 1, a liquid droplet at rest on a solid surface surrounded by a pure vapor is considered. Figure 1. A single drop at rest on a solid surface surrounded by a pure vapor. The droplet system has three interfaces, one between the solid and vapor, another one between the solid and liquid, and the third one between the liquid and vapor. Three surface tensions exist in these interfaces and satisfy the Young s equation under static equilibrium in the horizontal direction [11] sv = sl + lv cos, (1) where sv, sl and lv are the surface tensions between the solid-vapor, solid-liquid, and liquid-vapor interfaces respectively, and is the contact angle. The contact angle is dependent only upon the physical properties of the contacting media and is independent of the solid shape and gravity. Assuming the volume, number of moles, and temperature of the system are constant, the surface tensions between these interfaces can be expressed by the Helmholtz free energy [11] de ij =, i, j = s, v, l and i j, (2) da ij where A ij represents the interfacial area. Hence, the change in Helmholtz free energy between three interfaces can be written as de = lv da lv + sl da sl + sv da sv. (3) From the energy perspective and using the above equation, the capillary force, capillary pressure and the height of liquid column in a micro groove will be derived further. As shown in Figure 2(a, b), a groove with a rectangular cross-sectional area is placed in the vertical direction and its bottom end is inserted in the liquid. Assuming that the initial height of water in the groove is x and the capillarity of groove pulls the water forward a distance

3 Capillarity of Rectangular Micro Grooves and Their Application to Heat Pipes 251 where the capillary force F c is shown in Eq. (8) and the gravitational force F g is F g = m l g = l (LWH) g, (11) and m l and l are the liquid mass and density respectively. Hence, the height of liquid column H can be expressed as Figure 2. (a) Capillary phenomenon in a rectangular groove, (b) top view of groove, and (c) the free body diagram of liquid column. dx, the area changes of the liquid-vapor, solid-liquid and solid-vapor interface areas can be expressed as da lv = Wdx, (4) da sl =(2L + W)dx, (5) da lv = (2L + W)dx, (6) where W and L are the width and depth of groove respectively. The meniscus front effect of liquid is neglected here. Substituting Eqs. (4 6) into Eq. (3) and combining with Eq. (1), the change in the Helmholtz free energy de is de = lv [W (2L + W) cos ]dx. (7) The capillary force F c applied on the liquid column along the vertical direction can be obtained by taking the derivative of the Helmholtz free energy with respect to x [12], that is, de F c = = lv [(2L + W)cos W_]. (8) dx The capillary pressure P c is equal to the capillary force F c divided by the cross-sectional area of groove lv [(2L + W)cos W] P c =. (9) LW To calculate the height of liquid column, the free body diagram as shown in Figure 2(c) is considered. The forces acted on the liquid column are the capillary fore F c (upward) and gravitational force F g (downward). As the static equilibrium is reached and the height of liquid column is equal to H, we have F c = F g, (10) lv [(2L + W)cos W] H =. (12) l glw 2.2 Experimental Testing In the experiment, the rectangular micro grooves are constructed by the JSR thick photoresist. A thin copper film with a thickness of 3000 Å is first sputtered on the silicon wafer, then the JSR thick photoresist is coated on the copper film and patterned by photolithography process, and finally a thin copper film is deposited on the photoresist groove structures. The cross-section of grooves is shown in Figure 3 schematically. The deposited copper film is used to improve the hydrophilic property between the liquid and solid interfaces. The thickness of photoresist layer is 430 m. Various widths of rectangular grooves from 20 m upto 700 m are made to evaluate capillarity effects. Lengths of all grooves are 60 mm. The thick photoresist layer is made by the multiple coating processes as described in the reference [13] to obtain high aspect ratio. Water is chosen as the working liquid in the capillarity testing. Open micro grooves are placed vertically and inserted into the liquid. The heights of liquid column are measured after the static equilibrium is reached. Figure 4 plots comparisons between theoretical and experimental results. Since the achievable heights of liquid column are greater than the length of micro grooves as the width of grooves W < 200 m, the plotted experimental results are only in the range W > 200 m. In the calculation, the surface tension lv, the contact angle between copper and water interfaces, and the density of Figure 3. Cross-section of rectangular grooves (not in scale).

4 252 Horng-Jou Wang et al. factor in designing these kinds of heat transfer components. Besides, in order to minimize the component size and save the cost, the sizes of grooves must also be taken into consideration. By neglecting the pressure drops due to the evaporation and condensation at the liquid-vapor interface, the general expression for the capillary limit is given by [3] P cap,max P l P v P g 0, (13) Figre 4. Comparisons between theoretical and experimental results. water l are taken from the literature [14] and at the room temperature (20 C) they are N/m, 33, and 999 kg/m 3 respectively. It is seen that the theoretical results match well with those of experiment. The small difference may be resulted from the surface roughness, contact angle variation from copper oxidization, geometry dimension variation of micro grooves, and the meniscus front effect of liquid. 3. Applications on Heat Pipes Since micro grooves can transport liquid by their own capillary pressure without any external energy, it is proper to apply micro grooves as the wick structure in heat transfer components, such as heat pipes. Although the capillary pressure of micro grooves is smaller than that of the conventional metal sintered powder wicks, the micro grooves still become the key elements in the heat transfer components because of their higher permeability and higher effective thermal conductivity. In the following, the application of micro grooves on flat heat pipes is discussed. 3.1 Capillary Limit of Heat Pipe The capillary limit is the most commonly encountered limitation of low-temperature heat pipes. It occurs when the sum of the liquid and vapor pressure drops exceed the maximum capillary pressure that the wick can sustain, i.e., the pumping rate is not sufficient to provide enough liquid to the evaporation section. Any attempt to increase the heat dissipation above the capillary limit will cause dryout in the evaporator section and then lead to a sudden increase in wall temperature along the evaporator section. Hence, the heat rate is a significant restraint where P cap,max is the maximum capillary pressure in the wick structure, P l and P v denote the liquid and vapor pressure drops along the heat pipe respectively, and P g is the pressure drop in the liquid due to the effect of gravitational force in the direction of heat pipe axis. By letting = 0 in Eq. (9), the maximum capillary pressure P cap,max for rectangular grooves can be calculated as follows 2 lv P cap,max =. (14) W Assuming that the heat pipe needs to transport a heat rate Q and has uniform heat flux distributions along its evaporator and condenser sections, the liquid and vapor pressure drops P l and P v can be written as P l = F l QL eff, (15) P v = F v QL eff, (16) where F l and F v are the frictional coefficient for liquid and vapor flows respectively, and L eff is the effective length of heat transfer. The detailed formulations of these parameters are listed in Appendix A. The pressure drop in the liquid due to the gravitational force is P g = l gl t sin, (17) where L t is the length of heat pipe and is the inclination angle of heat pipe from horizontal. The equal sign in Eq. (13) means the point that the capillary limit exists and the greater sign means that the maximum capillary pumping pressure of micro grooves can provide enough liquid to the evaporator section to avoid dryout. For heat pipes operated in cryogenic and moderate-temperature (< 750 K) conditions, the vapor pressure drop is much smaller than the liquid pressure drop when the vapor pressure is high and the hydraulic radius for vapor flow is much larger than that for liquid flow [2]. The third term in Eq. (13) can be neglected in this situation.

5 Capillarity of Rectangular Micro Grooves and Their Application to Heat Pipes 253 Figure 6. Plot of pressures versus groove width. 3.2 Results and Discussion As shown in Figure 5, a flat heat pipe with axially rectangular micro groove wick structures is considered herein. The length of the condenser, adiabatic and evaporator sections are 34.4, 70.0 and 15.6 mm respectively. The heat pipe is designed to transport a heat rate 100 W and operate at 60 C. Based on the limit discussed above, the maximum capillary pumping pressure P cap,max, the liquid pressure drops due to the friction P l, and the pressure drop in the liquid due to the gravitational force P g versus the width of grooves W are plotted in Figure 6, where the depth of grooves is 300 m, the porosity W/S is 0.5, and the wick cross-sectional area A w is m 2. Water is chosen as the working fluid and the density l, viscosity µ l, surface tension lv and latent heat of water at the operation temperature (60 C) are kg/m 3, kg/m-sec, N/m and J/kg respectively. It is seen that the maximum capillary pumping pressure P cap,max increases as groove width shrinks, and the trend is more obvious for the liquid pressure drop due to the friction. It means that the smaller width of grooves will not always induce higher net pressure ( P cap,max P l P g ). Since the smaller width of grooves will cause the higher friction, the dryout phenomenon will occur eventually when the liquid cannot return to the evaporator section sufficiently. A workable width range of micro grooves could be determined using the above analysis and the detailed flowchart is drawn in Appendix B. Furthermore, we can also determine the number of micro grooves required in the flat heat pipe as the width and depth of micro grooves are determined by fabrication methods. Another example is shown in Figure 7. The width and depth of micro grooves are selected to be 75 and 250 m respectively, and other parameters are the same as in the previous example. It is seen that the maximum capillary pumping pressure P cap,max is greater than the sum of liquid pressure drop due to friction P l, and Figure 5. A flat heat pipe with axially rectangular micro grooves. Figure 7. Plot of pressures versus number of grooves.

6 254 Horng-Jou Wang et al. liquid pressure drop due to the gravitational force P g as the number of micro grooves is larger than 350. The minimum number of grooves required is obtained. 4. Conclusion The capillarity of rectangular grooves is investigated first in the paper. The capillary force, capillary pressure, as well as the height of liquid column are derived theoretically, and the verification experiment is carried out. It is shown that the theoretical and experimental results are in agreement, and it provides designers useful insight. The application of rectangular micro grooves on the flat heat pipe is also discussed. The capillary limit, including the maximum capillary pumping pressure and the pressure drops due to the friction and gravitational force, is analyzed. From the analyses, the workable width range and required number of rectangular grooves can be determined for a specific heat rate heat pipe. The present results can provide designers the necessary information in designing micro groove related products. Appendix A f l Re l = 24( * * * * * 5 ) (A.6) *=W/L. (A.7) and N is the number of grooves. The frictional coefficient for vapor flow is given by (f v Re v )µ v F v =, (A.8) 2r 2 h,v A v v where f v Re v,µ v, r h,v, A v and v are the coefficient of drag, vapor viscosity, hydraulic radius, vapor flow area and vapor density respectively. For a uniform heat flux in the evaporator and condenser sections, the effective length is given as the following L eff = 0.5 L c + L a L e, (A.9) where L c, L a and L e are the length of condenser, adiabatic and evaporator sections respectively. Appendix B Flowchart for determining the workable geometry range of micro grooves. The frictional coefficient for liquid flow is [3] µ l F l =, (A.1) KA w l where µ l denotes the liquid viscosity, K is the wick permeability, A w represents the wick cross-sectional area, is the latent heat of vaporization, and l is the liquid density. The wick permeability and wick cross-sectional area are calculated using the following equations 2 r 2 h,l K =, (A.2) (f l Re l ) A w = NSL, (A.3) where the wick porosity, the hydraulic radius r h,l, and the coefficient of drag f l Re l for the rectangular micro grooves are respectively expressed as = W/S, (A.4) 2LW r h,l =, (A.5) 2L + W

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