differential and then combined with the continuity expression. The result was a first order ordinary dif- ferential equation that related the radius of curvature of the liquid–vapor interface to the axial position along the pipe. Building upon this model, Peterson (1988a) and Babin et al. (1990) developed a steady-state model for a trapezoidal micro heat pipe using the conventional steady-state modeling techniques outlined by Chi (1976) and described earlier in this chapter. The resulting model demonstrated that the capillary pumping pressure governed the maximum heat transport capacity of these devices. The performance limitations resulting from the models presented by Cotter (1984) and by Babin et al. (1990) were compared and indicated significant differences in the capillary limit predicted by the two models. These differences have been analyzed and found to be the result of specific assumptions made in the initial formulation of the models [Peterson, 1992]. A comparative analysis of these two early models was performed by Gerner et al. (1992), who indicated that the most important contributions of Babin et al. (1990) were the inclusion of the gravitational body force and the recognition of the significance of the vapor pressure losses. In addition, the assumption that the pressure gradient in the liquid flow passages was similar to that occurring in Hagen–Poiseuille flow was questioned, and a new scaling argument for the liquid pressure drop was presented. In this development, it was assumed that the average film thickness was approximately one-fourth the hydraulic radius, result- ing in a modified expression for the capillary limitation. Micro Heat Pipes and Micro Heat Spreaders 11-11 Evaporator Condense r Heat input Heat output Liquid Side view End view Vapor 20 mm 120 µm FIGURE 11.4 Micro heat pipe operation. (Reprinted with permission from Peterson, G.P., Swanson, L.W., and Gerner, F.M. (1996) “Micro Heat Pipes,” in Microscale Energy Transport, C.L. Tien, A. Majumdar, and F.M. Gerner eds., Taylor-Francis Publishing Co., Washington D.C.) © 2006 by Taylor & Francis Group, LLC A significant contribution made by Gerner et al. (1992) was the recognition that the capillary limit may never actually be reached due to Kelvin–Helmholtz type instabilities occurring at the liquid–vapor interface. Using stability analysis criteria for countercurrent flow in tubes developed by Tien et al. (1979) and mini- mizing the resulting equations, the wavelength was found to be approximately 1cm for atmospheric water and methanol. Since this length was long with respect to the characteristic wavelength, it was assumed that gravity was the dominant stabilizing mechanism. The decision as to whether to use the traditional capillary limit proposed by Babin et al. (1990) or the interfacial instability limit proposed by Gerner (1992) should be governed by evaluating the shape and physical dimensions of the specific micro heat pipe being considered. Khrustalev and Faghri (1994) presented a detailed mathematical model of the heat and mass transfer processes in micro heat pipes that described the distribution of the liquid and the thermal characteristics as a function of the liquid charge. The liquid flow in the triangular corners of a micro heat pipe with poly- gonal cross-section was considered by accounting for the variation of the curvature of the free liquid sur- face and the interfacial shear stresses due to the liquid vapor interaction. The predicted results were compared with the experimental data obtained by Wu and Peterson (1991) and Wu et al. (1991) and indicated the importance of the liquid charge, the contact angle, and the shear stresses at the liquid vapor interface in predicting the maximum heat transfer capacity and thermal resistance of these devices. Longtin et al. (1994) developed a one-dimensional steady-state model for the evaporator section of the micro heat pipe. The governing equations were solved, assuming a uniform temperature along the heat pipe. The solution indicated that the maximum heat transport capacity varied with respect to the cube of the hydraulic diameter of the channel. An analytical model for the etched triangular micro heat pipe developed by Duncan and Peterson (1995) is capable of calculating the curvature of the liquid–vapor meniscus in the evaporator. This model was used to predict the capillary limit of operation of the heat pipe and to arrive at the optimal value of the liquid charge. In a subsequent work, a hydraulic diameter was defined, incorporating the frictional effects of the liquid and the vapor, and was used in a model for predicting the minimum meniscus radius and maximum heat transport in triangular grooves [Peterson and Ma, 1996b]. The major parameters influ- encing the heat transport capacity of the micro heat pipe were found to be the apex angle of the liquid arter- ies, the contact angle, heat pipe length, vapor velocity and the tilt angle. Ma and Peterson (1998) presented analytical expressions for the minimum meniscus radius and the maximum capillary heat transport limit in micro heat pipes that were validated with experimental data. A detailed steady-state mathematical model for predicting the heat transport capability of a micro heat pipe and the temperature gradients that contribute to the overall axial temperature drop as a function of the heat transfer was developed by Peterson and Ma (1999). The unique nature of this model was that it considered the governing equation for fluid flow and heat transfer in the evaporating thin film region. The model also consisted of an analytical solution of the two dimensional heat conduction in the macro evapo- rating regions in the triangular corners. The effects of the vapor and liquid flows in the passage, the flow and condensation of the thin film caused by the surface tension in the condenser, and the capillary flow along the axial direction of the micro heat pipe were considered in this model. The predicted axial temperature distribution was compared with experimental data, with very good agreement. The model was capable of calculating both the heat transfer distribution through the thin film region and the heat transfer- operating temperature dependence of the micro heat pipe. It was concluded from the study that the evap- orator temperature drop was considerably larger than that at the condenser and that the temperature drops increased with an increase in input power when the condenser is kept at a constant temperature. The maximum heat transfer capacity of copper–water micro heat pipes was also explored by Hopkins et al. (1999) using a one-dimensional model for predicting the capillary limitation. In this analysis, the liquid–vapor meniscus was divided into two regions depending on whether the contact angle can be treated as a constant at the evaporator or as a variable along the adiabatic and condenser sections. 11.2.1.2 Transient Modeling As heat pipes diminish in size, the transient nature becomes of increasing interest. The ability to respond to rapid changes in heat flux coupled with the need to maintain constant evaporator temperature in modern 11-12 MEMS: Applications © 2006 by Taylor & Francis Group, LLC high-powered electronics necessitates a complete understanding of the temporal behavior of these devices. The first reported transient investigation of micro heat pipes was conducted by Wu and Peterson (1991). This initial analysis utilized the relationship developed by Collier (1981) and used later by Colwell and Chang (1984) to determine the free molecular flow mass flux of evaporation. The most interesting result from this model was the observation that reverse liquid flow occurred during the start-up of micro heat pipes. As explained in the original reference [Wu et al., 1990], this reverse liquid flow is the result of an imbalance in the total pressure drop and occurs because the evaporation rate does not provide an adequate change in the liquid–vapor interfacial curvature to compensate for the pressure drop. As a result, the increased pressure in the evaporator causes the meniscus to recede into the corner regions forcing liquid out of the evaporator and into the condenser. During start-up, the pressure of both the liquid and vapor are higher in the evaporator and gradually decrease with position, promoting flow away from the evaporator. Once the heat input reaches full load, the reverse liquid flow disappears and the liquid mass flow rate into the evaporator gradually increases until a steady-state condition is reached. At this time the change in the liquid mass flow rate is equal to the change in the vapor mass flow rate for any given section [Wu and Peterson, 1991]. The flow reversal in the early transient period of operation of a micro heat pipe has also been cap- tured by Sobhan et al. (2000) using their numerical model. Several more-detailed transient models have been developed. Badran et al. (1993) developed a conjugate model to account for the transport of heat within the heat pipe and conduction within the heat pipe case. This model indicated that the specific thermal conductivity of micro heat pipes (effective thermal con- ductivity divided by the density) could be as high as 200 times that of copper and 100 times that of Gr/Cu composites. Ma et al. (1996) developed a closed mathematical model of the liquid friction factor for flow occurring in triangular grooves. This model, which built upon the earlier work of Ma et al. (1994), considered the inter- facial shear stresses due to liquid–vapor frictional interactions for countercurrent flow. Using a coordinate transformation and the Nachtsheim–Swigert iteration scheme, the importance of the liquid vapor inter- actions on the operational characteristics of micro heat pipes and other small phase change devices was demonstrated. The solution resulted in a method by which the velocity distribution for countercurrent liquid–vapor flow could be determined, and it allowed the governing liquid flow equations to be solved for cases where the liquid surface is strongly influenced by the vapor flow direction and velocity. The results of the analysis were verified using an experimental test facility constructed with channel angles of 20, 40, and 60 degrees. The experimental and predicted results were compared and found to be in good agreement [Ma and Peterson 1996a, 1996b; Peterson and Ma 1996a]. A transient model for a triangular micro heat pipe with an evaporator and condenser section was pre- sented by Sobhan et al. (2000). The energy equation as well as the fluid flow equations were solved numer- ically, incorporating the longitudinal variation of the cross-sectional areas of the vapor and liquid flows, to yield the velocity, pressure, and temperature distributions. The effective thermal conductivity was com- puted and characterized with respect to the heat input and the cooling rate under steady and transient operation of the heat pipe. The reversal in the liquid flow direction as discussed by Wu and Peterson (1991) was also obvious from the computational results. 11.2.1.3 Transient One-Dimensional Modeling of Micro Heat Pipes A flat micro heat pipe heat sink consisting of an array of micro heat pipe channels was used to form a com- pact heat dissipation device to remove heat from electronic chips. Each channel in the array served as an independent heat transport device. The analysis presented here examined an individual channel in such an array. The individual micro heat pipe channel analyzed had a triangular cross-section. The channel was fabricated on a copper substrate and the working fluid used was ultrapure water. 11.2.1.3.1 The Mathematical Model The micro heat pipe consisted of an externally heated evaporator section and a condenser section subjected to forced convective cooling. A one-dimensional model was sufficient for the analysis, as the variations in the field variables were significant only in the axial direction, due to the geometry of the channels. A transient Micro Heat Pipes and Micro Heat Spreaders 11-13 © 2006 by Taylor & Francis Group, LLC model that proceeded until steady-state was utilized to analyze the problem completely. In this problem, the flow and heat transfer processes are governed by the continuity, momentum, and energy equations for the liquid and vapor phases.A nonconservative formulation can be utilized because the problem deals with low velocity flows. As phase change occurs, the local mass rates of the individual liquid and vapor phases are coupled through a mass balance at the liquid–vapor interface. The cross-sectional areas of the vapor and liquid regions and the interfacial area vary along the axial length due to the progressive phase change occurring as the fluid flows along the channel. These variations in the area can be incorporated into the model through the use of suitable geometric area coefficients, as described in Longtin et al. (1994). The local meniscus radii at the liquid–vapor interface are calculated using the Laplace–Young equation. The friction factor, which appears in the momentum and energy equations is incorporated through appropriate models for fluid friction in varying area channels, as described in the literature. The governing differential equations can be described as follows: 11.2.1.3.2 Laplace–Young Equation P v Ϫ P l ϭ (11.30) 11.2.1.3.3 Vapor phase equations Vapor continuity equation: evaporator section d 2 Ϫ β l r 2 Ϫ 2 β l u v r ϩ β i rV il ϭ 0 (11.31) Vapor continuity equation: condenser section d 2 Ϫ β l r 2 Ϫ 2 β l u v r Ϫ β i rV il ϭ 0 (11.32) It should be noted that the vapor continuity equation incorporates the interfacial mass balance equation. ρ l V il ϭ ρ v V iv (11.33) Vapor momentum equation: ρ v d 2 Ϫ β l r 2 ϭ 2 ρ v d 2 Ϫ β l r 2 u v Ϫ 2 ρ v β l ru 2 v ϩ d 2 Ϫ β l r 2 Ϫ ρ v u 2 v f vw (3d Ϫ β lw r) Ϫ ρ 2 v u 2 v f vi β l r (11.34) Vapor energy equation: evaporator section d 2 Ϫ β l r 2 ϩ ΄ u v d 2 Ϫ β l r 2 (E v ϩ P v ) ΅ ϭ Ά µ v d 2 Ϫ β l r 2 u v ϩ k v d 2 Ϫ β l r 2 · ϩ q(3d Ϫ β lw r) ϩ h fg V il ρ l β l r ϩ ρ v u 2 v f vw u v (3d Ϫ β lw r) ϩ ρ v u 2 v f vi u v β l r (11.35) 1 ᎏ 2 1 ᎏ 2 ∂T v ᎏ ∂x ͙ 3 ෆ ᎏ 4 ∂u v ᎏ ∂x ͙ 3 ෆ ᎏ 4 4 ᎏ 3 ∂ ᎏ ∂x ͙ 3 ෆ ᎏ 4 ∂ ᎏ ∂x ∂E v ᎏ ∂t ͙ 3 ෆ ᎏ 4 1 ᎏ 2 1 ᎏ 2 ∂P v ᎏ ∂x ͙ 3 ෆ ᎏ 4 ∂r ᎏ ∂x ∂u v ᎏ ∂x ͙ 3 ෆ ᎏ 4 ∂u v ᎏ ∂t ͙ 3 ෆ ᎏ 4 ρ l ᎏ ρ v ∂r ᎏ ∂x ∂u v ᎏ ∂x ͙ 3 ෆ ᎏ 4 ρ l ᎏ ρ v ∂r ᎏ ∂x ∂u v ᎏ ∂x ͙ 3 ෆ ᎏ 4 σ ᎏ r 11-14 MEMS: Applications © 2006 by Taylor & Francis Group, LLC Vapor Energy equation: condenser section d 2 Ϫ β l r 2 ϩ ΄ u v d 2 Ϫ β l r 2 (E v ϩ P v ) ΅ ϭ Ά µ v d 2 Ϫ β l r 2 u v ϩ k v d 2 Ϫ β l r 2 · ϩ h fg V il ρ l β l r Ϫ h o (3d Ϫ β lw r)∆T ϩ ρ v u 2 v f vw u v (3d Ϫ β lw r) ϩ ρ v u 2 v f vi u v β i r (11.36) 11.2.1.3.4 Liquid Phase Equations Liquid continuity equation: evaporator section r ϩ 2u l Ϫ V il ϭ 0 (11.37) Liquid continuity equation: condenser section r ϩ 2u l ϩ V il ϭ 0 (11.38) Liquid momentum equation: ρ l r ϭ Ϫ2 ρ l ru l ϩ u 2 l Ϫ r Ϫ ρ l u 2 l f lw Ϫ ρ l u 2 l f li (11.39) Liquid energy equation: evaporator section β l r 2 ϩ [u l β l r 2 (E l ϩ P l )] ϭ ΄ µ l u l β l r 2 ϩ k β l r 2 ΅ ϩ q β lw r Ϫ h fg V il ρ l β i r ϩ ρ l u 2 l f lw u l β lw r ϩ ρ l u 2 l f li u l β i r (11.40) Liquid energy equation: condenser section β l r 2 ϩ [u l β l r 2 (E l ϩ P l )] ϭ ΄ µ l u l β l r 2 ϩ k β l r 2 ΅ ϩ h fg V il ρ l β i r Ϫ h o β lw r∆T ϩ ρ l u 2 l f lw u l β lw r ϩ ρ l u 2 l f li u l β i r (11.41) The vapor and liquid pressures can be computed as follows: 1. The ideal gas equation of state is utilized for computing the pressure in the vapor. Because the vapor is either saturated or super heated, the ideal gas state equation is reasonably correct and is used exten- sively in the analysis. 1 ᎏ 2 1 ᎏ 2 ∂T l ᎏ ∂x ∂u l ᎏ ∂x 4 ᎏ 3 ∂ ᎏ ∂x ∂ ᎏ ∂x ∂E l ᎏ ∂t 1 ᎏ 2 1 ᎏ 2 ∂T l ᎏ ∂x ∂u l ᎏ ∂x 4 ᎏ 3 ∂ ᎏ ∂x ∂ ᎏ ∂x ∂E l ᎏ ∂t β i ᎏ β l 1 ᎏ 2 β lw ᎏ β l 1 ᎏ 2 ∂P l ᎏ ∂x ∂r ᎏ ∂x ∂u l ᎏ ∂x ∂u l ᎏ ∂t β i ᎏ β l ∂r ᎏ ∂x ∂u l ᎏ ∂x β i ᎏ β l ∂r ᎏ ∂x ∂u l ᎏ ∂x 1 ᎏ 2 1 ᎏ 2 ∂T v ᎏ ∂x ͙ 3 ෆ ᎏ 4 ∂u v ᎏ ∂x ͙ 3 ෆ ᎏ 4 4 ᎏ 3 ∂ ᎏ ∂x ͙ 3 ෆ ᎏ 4 ∂ ᎏ ∂x ∂E v ᎏ ∂t ͙ 3 ෆ ᎏ 4 Micro Heat Pipes and Micro Heat Spreaders 11-15 © 2006 by Taylor & Francis Group, LLC 2. For the liquid phase the Hagen–Poiseuille equation is used as a first approximation, with the local hydraulic diameter for the wetted portion of the liquid-filled region adjacent to the corners. The val- ues of pressure obtained from this first approximation are substituted into the momentum equations and iterated for spatial convergence. 11.2.1.3.5 State Equations Equation of state for the vapor: P v ϭ ρ v R v T v (11.42) Hagen–Poiseuielle Equation as first approximation for the liquid flow 8 µ l u l ϭ Ϫ (11.43) The boundary conditions are at x ϭ 0 and x ϭ L u l ϭ 0; u v ϭ 0; ∂T/∂x ϭ 0 The initial conditions are at t ϭ 0 and for all x P l ϭ P v ϭ P sat ; T l ϭ T v ϭ T amb. At x ϭ 0 P v Ϫ P l ϭ (11.44) The value of r o , the initial radius of curvature of the interface meniscus for the copper–water system, was adopted from the literature. A numerical procedure based on the finite difference method was used to solve the above system of equations to obtain the transient behavior and field distributions in the micro heat pipe, and the results of computation have been discussed in the literature [Sobhan and Peterson, 2004]. Parametric studies were also presented in this paper. 11.2.1.3.6 Area Coefficients in the Computational Model Figure 11.5 illustrates the geometric configuration of the vapor and liquid flow in the cross-section of the micro heat pipe, along with the meniscus idealized as an arc of a circle at any longitudinal location. The definitions of the area coefficients, as derived for this configuration, are given below: Referring to Figure 11.5, the cross section is an equilateral triangle with φ ϭ π /3 Ϫ α , and η ϭ r sin φ . The total area of the liquid in the cross section is A l ϭ β l r 2 (11.45) Where, β l ϭ 3 ΄ ͙ 3 ෆ sin 2 Ϫ α ϩ 0.5 sin 2 Ϫ α Ϫ Ϫ α ΅ π ᎏ 3 π ᎏ 3 π ᎏ 3 σ ᎏ r o D 2 H ᎏ 4 ∂P l ᎏ ∂x 11-16 MEMS: Applications © 2006 by Taylor & Francis Group, LLC The total area of interface in length dx is A i ϭ β i rdx, where β i ϭ 6 Ϫ α (11.46) The total wetted perimeter for the three corners ϭ β lw r, where β lw ϭ π sin Ϫ α (11.47) 11.2.2 Testing of Individual Micro Heat Pipes As fabrication capabilities have developed, experimental investigations on individual micro heat pipes have been conducted on progressively smaller and smaller devices, beginning with early investigations on what now appear to be relatively large micro heat pipes, approximately 3 mm in diameter and progress- ing to micro heat pipes in the 30 m diameter range. These investigations have included both steady-state and transient investigations. 11.2.2.1 Steady-State Experimental Investigations The earliest experimental tests of this type reported in the open literature were conducted by Babin et al. (1990), who evaluated several micro heat pipes approximately 1mm in external diameter. The primary purposes of this investigation were to determine the accuracy of the previously described steady-state mode- ling techniques, to verify the micro heat pipe concept, and to determine the maximum heat transport capacity. The fabrication techniques used to produce these test articles were developed by Itoh Research and Development Company, Osaka, Japan (Itoh, 1988). As reported previously, a total of four test articles were evaluated, two each from silver and copper. Two of these test pipes were charged with distilled deionized water, and the other two were used in an uncharged condition to determine the effect of the vaporization– condensation process on these devices’ overall thermal conductivity. Steady-state tests were conducted over a range of tilt angles to determine the effect of the gravitational body force on the operational characteris- tics. An electrical resistance heater supplied the heat into the evaporator. Heat rejection was achieved through the use of a constant temperature ethyl–glycol solution, which flowed over the condenser portion of the heat pipe. The axial temperature profile was continuously monitored by five thermocouples bonded to the π ᎏ 3 π ᎏ 3 Micro Heat Pipes and Micro Heat Spreaders 11-17 d 2 r FIGURE 11.5 Cross-sectional geometry of the triangular micro heat pipe for determining the area coefficients. (Reprinted with permission from Longtin, J.P., Badran, B., and Gerner, F.M. [1994] “A One-Dimensional Model of a Micro Heat Pipe During Steady-State Operation,” ASME J. Heat Transfer 116, pp. 709–15.) © 2006 by Taylor & Francis Group, LLC outer surface of the heat pipe using a thermally conductive epoxy. Three thermocouples were located on the evaporator, one on the condenser, and one on the outer surface of the adiabatic section. Throughout the tests, the heat input was systematically increased and the temperature of the coolant bath adjusted to maintain a constant adiabatic wall temperature (Babin et al., 1990). The results of this experiment have been utilized as a basis for comparison with a large number of heat pipe models. As previously reported [Peterson et al., 1996], the steady-state model of Babin et al. (1990) over-predicted the experimentally determined heat transport capacity at operating temperatures below 40°C and under-predicted it at operating temperatures above 60°C. These experimental results represented the first successful operation of a micro heat pipe that utilized the principles outlined in the original con- cept of Cotter (1984) and, as such, paved the way for numerous other investigations and applications. There has been a large amount of experimental research work on micro heat pipes under steady-state operation following the early experimental studies reported by Babin et al. (1990). Fabricating micro heat pipes as an integral part of silicon wafers provided a means of overcoming the problems imposed by the thermal contact resistance between the heat pipe heat sink and the substrate material, and Peterson et al. (1991) in their first attempt to study the performance of an integral micro heat pipe, compared the tem- perature distributions in a silicon wafer with and without a charged micro heat pipe channel. The wafer with the integral micro heat pipe showed as much as an 11% reduction of the maximum chip temperature, which worked out to a 25% increase in effective thermal conductivity, at a heat flux rate of 4 W/cm 2 . More extensive experimental work and detailed discussions on triangular and rectangular micro heat pipes and on micro heat pipe arrays fabricated on silicon wafers can be found in Mallik et al. (1992) and Peterson et al. (1993). Peterson (1994) further discussed the fabrication, operation, modeling, and testing aspects of integral micro heat pipes in silicon. Steady-state experiments on a micro heat pipe array fabricated in silicon using the vapor deposition technique were also reported by Mallik and Peterson (1995). In an attempt to conduct visualization experiments on micro heat pipes, Chen et al. (1992) fabricated a heat transport device by attaching a wire insert to the inner wall of a glass capillary tube, so that capillary action is obtained at the corners formed by the two surfaces. This was also modeled as a porous medium. The device was highly influenced by gravity, as revealed by comparisons of the experimental and pre- dicted results for maximum heat flow with horizontal and vertical orientations. It appears that this device functioned more like a thermosyphon than a capillary driven heat pipe. Experimental studies were performed on triangular grooves fabricated in a copper substrate with methanol as the working fluid in order to determine the capillary heat transport limit [Ma and Peterson, 1996a]. A parameter, “the unit effective area heat transport,” was defined for the grooves (q eff ϭ q/A eff , where A eff ϭ D H /Le) to be used as a performance index. An optimum geometry was found that gave the maximum unit effective area heat transport. Further, this maximum depended on the geometrical parameters, namely the tilt angle and the effective length of the heat pipe. Modifying the analytical model for the maximum heat transport capacity of a micro heat pipe developed by Cotter (1984), a semiempirical correlation was proposed by Ha and Peterson (1998b). The method used was to compare the results predicted by Cotter’s model with experimental data and then modify the model to incorporate the effects of the intrusion of the evaporator section into the adiabatic section of the heat pipe under near dry-out conditions. With the proposed semiempirical model, a better agreement between the predicted and experimental results was obtained. Hopkins et al. (1999) experimentally determined the maximum heat load for various operating tempera- tures of copper–water micro heat pipes.These micro heat pipes consisted of trapezoidal or rectangular micro grooves and were positioned in vertical or horizontal orientations. The dry-out condition also was studied experimentally. The effective thermal resistance was found to decrease with an increase in the heat load. 11.2.2.2 Transient Experimental Investigations While the model developed by Babin et al. (1990) was shown to predict the steady-state performance limi- tations and operational characteristics of the trapezoidal heat pipe reasonably well for operating temper- atures between 40 and 60°C, little was known about the transient behavior of these devices. As a result, Wu et al. (1991) undertook an experimental investigation of the devices’ transient characteristics. This 11-18 MEMS: Applications © 2006 by Taylor & Francis Group, LLC experimental investigation again utilized micro heat pipe test articles developed by Itoh (1988); however, this particular test pipe was designed to fit securely under a ceramic chip carrier and had small fins at the condenser end of the heat pipe for removal of heat by free or forced convection, as shown in Figure 11.3. Start-up and transient tests were conducted in which the transient response characteristics of the heat pipe as a function of incremental power increases, tilt angle, and mean operating temperature were measured. Itoh and Polásek (1990a, 1990b), presented the results of an extensive experimental investigation on a series of micro heat pipes ranging in size and shape from 1 to 3 mm in diameter and 30 to 150mm in length. The investigation utilized both cross-sectional configurations, similar to those presented previously or a conventional internal wicking structure (Polásek, 1990; Fejfar et al., 1990). The unique aspect of this par- ticular investigation was the use of neutron radiography to determine the distribution of the working fluid within the heat pipes [Itoh and Polásek, 1990a; Itoh and Polásek, 1990b; Ikeda, 1990]. Using this tech- nique, the amount and distribution of the working fluid and noncondensale gases were observed during real time operation along with the boiling and/or reflux flow behavior. The results of these tests indicated several important results [Peterson, 1992]; ● As is the case for conventional heat pipes, the maximum heat transport capacity is principally dependent upon the mean adiabatic vapor temperature. ● Micro heat pipes with smooth inner surfaces were found to be more sensitive to overheating than those with grooved capillary systems. ● The wall thickness of the individual micro heat pipes had greater effect on the thermal performance than did the casing material. ● The maximum transport capacity of heat pipes utilizing axial channels for return of the liquid to the evaporator were found to be superior to those utilizing a formal wicking structure. The experimental work on the micro heat pipe array fabricated in silicon using vapor deposition tech- nique [Mallik and Peterson, 1995] was extended to also include the performance under transient condi- tions. The results of this study were presented in Peterson and Mallik (1995). 11.3 Arrays of Micro Heat Pipes Apart from theoretical and experimental research on individual micro heat pipes, modeling, fabrication, and testing of micro heat pipe arrays of various designs also have been undertaken. Significant work on these subjects is presented in the following sections. 11.3.1 Modeling of Heat Pipe Arrays The initial conceptualization of micro heat pipes by Cotter (1984) envisioned fabricating micro heat pipes directly into the semiconductor devices as shown schematically in Figure 11.6.While many of the previously discussed models can be used to predict the performance limitations and operational characteristics of indi- vidual micro heat pipes, it is not clear from the models or analyses how the incorporation of an array of these devices might affect the temperature distribution or the resulting thermal performance. Mallik et al. (1991) Micro Heat Pipes and Micro Heat Spreaders 11-19 x y z FIGURE 11.6 Array of micro heat pipes fabricated as an integral part of a silicon wafer. © 2006 by Taylor & Francis Group, LLC developed a three-dimensional numerical model capable of predicting the thermal performance of an array of parallel micro heat pipes constructed as an integral part of semiconductor chips similar to that illustrated in Figure 11.7. In order to determine the potential advantages of this concept, several different thermal loading configurations were modeled and the reductions in the maximum surface temperature, the mean chip temperature,and the maximum temperature gradient across the chip were determined [Peterson, 1994]. Although the previous investigations of Babin et al. (1990), Wu and Peterson (1991), and Wu et al. (1991) indicated that an effective thermal conductivity greater than ten times that of silicon could be achieved, addi- tional analyses were conducted to determine the effect of variations in this value. Steady-state analyses were performed using a heat pipe array comprised of nineteen parallel heat pipes. Using an effective thermal con- ductivity ratio of five, the maximum and mean surface temperatures were 37.69°C and 4.91°C respectively. With an effective thermal conductivity ratio of ten, the maximum and mean surface temperatures were 35.20°C and 4.21°C respectively. Using an effective thermal conductivity ratio of fifteen, the maximum and mean surface temperatures were 32.67°C and 3.64°C respectively [Peterson,1994].These results illustrate how the incorporation of an array of micro heat pipes can reduce the maximum wafer temperature, reduce the temperature gradient across the wafers, and eliminate localized hot spots. In addition, this work high- lighted the significance of incorporating these devices into semiconductor chips, particularly those con- structed in materials with thermal conductivities significantly less than that of silicon,such as gallium arsenide. This work was further extended to determine transient response characteristics of an array of micro heat pipes fabricated into silicon wafers as a substitute for polycrystalline diamond or other highly thermally conductive heat spreader materials [Mallik and Peterson 1991; Mallik et al. 1992]. The resulting transient three-dimensional numerical model was capable of predicting the time dependent temperature distribu- tion occurring within the wafer when given the physical parameters of the wafer and the locations of the heat sources and sinks. The model also indicated that significant reductions in the maximum localized wafer temperatures and thermal gradients across the wafer could be obtained through the incorporation of an array of micro heat pipes. Utilizing heat sinks located on the edges of the chip perpendicular to the axis of the heat pipes and a cross-sectional area porosity of 1.85%, reductions in the maximum chip temperature of up to 40% were predicted. 11-20 MEMS: Applications FIGURE 11.7 (See color insert following page 2-12.)Silicon wafer into which an array of micro heat pipes has been fabricated. © 2006 by Taylor & Francis Group, LLC [...]... grooves The experiments indicated that the maximum heat transport capacity decreased significantly as the effective length of the © 20 06 by Taylor & Francis Group, LLC 1 1-3 0 MEMS: Applications 24 .0 Methanol, =10 degree 20 .0 Exp Model 2 16.0 Qmax, W Model 1 12. 0 8.0 4. 0 0.0 0 20 40 60 80 100 120 140 160 Length of the film, mm FIGURE 11.17 Comparison of the modeling and experimental results on microchanneled... Thermophysics and Heat Transfer 15(1), pp 42 49 .] Prototype No 1 Material Working fluid Total dimension (mm) Thickness of sheet (mm) Diameter of wire (mm) Number of wires No 2 No 3 Aluminum Acetone 1 52 ϫ 1 52. 4 0 .40 0.50 43 Aluminum Acetone 1 52 ϫ 1 52. 4 0 .40 0.80 43 Aluminum Acetone 1 52 ϫ 1 52. 4 0 .40 0.50 95 Wang and Peterson (20 02a) presented an analysis of wire-bonded micro heat pipe arrays using a onedimensional... D.R., Mitchell, R.T., Tuck, M.R., Palmer, D.W., and Peterson, G.P (1998) “Ultra High Capacity Micro Machined Heat Spreaders,” Microscale Thermophys Eng., 2( 1), pp 21 29 Therm Conduct (W/cm-K) CTE (10Ϫ6/K) Cost Substrate ($/Square Inch) Scaling with Area Cost Trend 0 .25 Depends on copper 1.00 2. 00 1 .48 8.00 → 20 .00(?) 2. 37 3.98 10.00 20 .00 0.13 Ͼ8.00 2. 00 (at 70%) 6.7 13.0 4. 1 4. 7 4. 7 41 .8 28 .7 1.0–1.5... coating of the foil and the raised points of the grooved film formed the micro heat pipe channels The analysis was used to compute the capillary pumping pressure and the dynamic and frictional pressure drops in the © 20 06 by Taylor & Francis Group, LLC Micro Heat Pipes and Micro Heat Spreaders 1 1-3 1 2. 4 Capillary limit, qc, mW 2. 0 1.6 1 .2 0.8 0 .4 0.0 0 40 80 120 160 140 180 Temperature,°C (a) 20 Capillary... pp 129 – 34 © 20 06 by Taylor & Francis Group, LLC 12 Microchannel Heat Sinks 12. 1 Introduction 1 2- 1 12. 2 Fundamentals of Convective Heat Transfer in Microducts 1 2- 2 Modes of Heat Transfer • The Continuum Hypothesis • Thermodynamic Concepts • General Laws • Particular Laws • Governing Equations • Size Effects 12. 3 Single-Phase Convective Heat Transfer in Microducts 1 2- 8 Flow Structure... and Peterson, G.P (20 02c) “Capillary Evaporation in Microchanneled Polymer Films,” paper no AIAA -2 0 0 2- 2 767, 8th AMSE/AIAA Joint Thermophysics and Heat Transfer Conference, 24 27 June, St Louis, MO.) polymer film increased The experimental observations also indicated that the maximum liquid meniscus radius occurred in the microgrooves just prior to dry-out An analytical model based on the Darcy law was... arrays No 1, No 2, and No 3 were 144 6 .2 W/Km, 521 .3 W/Km, and 3 023 .1 W/Km, respectively For the micro heat pipe arrays without any working fluid, the effective conductivities in the x-direction were 126 .3 W/Km, 113.0 W/Km, and 136 .2 W/Km respectively Comparison of the predicted and experimental results indicated these flexible radiators with the arrays of micro heat pipes have an effective thermal conductivity... Arrays,” AIAA J Thermophys Heat Transfer, 16, pp 346 –55 Wang, Y.X., and Peterson, G.P (20 02b) “Optimization of Micro Heat Pipe Radiators in a Radiation Environment,” AIAA J Thermophys Heat Transfer, 16, pp 537 46 Wang, Y.X., and Peterson, G.P (20 02c) “Capillary Evaporation in Microchanneled Polymer Films,” paper no AIAA -2 0 0 2- 2 767, 8th AMSE/AIAA Joint Thermophysics and Heat Transfer Conference, 24 27 June,... state analytical model that incorporated the effects of the liquid–vapor phase interactions and the variation in the cross-section area The model was used to predict the heat transfer performance and optimum design parameters An experimental facility was fabricated, and tests were conducted to © 20 06 by Taylor & Francis Group, LLC 1 1 -2 4 MEMS: Applications 28 0 With working fluid 24 0 Temperature difference... analytical models from basic principles rather than empirical correlations been reported Peles and Haber (20 00) calculated the steady-state, one-dimensional, evaporating, two-phase flow in a triangular microchannel Although the physical model has been simplified to render the mathematical model tractable, it is the first serious attempt to calculate such a complex flow field In this chapter, the discussion . (mm) 1 52 ϫ 1 52. 4 1 52 ϫ 1 52. 4 1 52 ϫ 1 52. 4 Thickness of sheet (mm) 0 .40 0 .40 0 .40 Diameter of wire (mm) 0.50 0.80 0.50 Number of wires 43 43 95 © 20 06 by Taylor & Francis Group, LLC 1 1 -2 4 MEMS: . drops in the 1 1-3 0 MEMS: Applications Methanol, =10 degree 0.0 4. 0 8.0 12. 0 16.0 20 .0 24 .0 0 20 40 60 80 100 120 140 160 Length of the film, mm Exp. Model 2 Model 1 Q max , W FIGURE 11.17 Comparison. different thermal loading configurations were modeled and the reductions in the maximum surface temperature, the mean chip temperature,and the maximum temperature gradient across the chip were determined