The fluid flow and heat transfer characteristics of micro heat pipes are analyzed theoretically, in order to understand the physical phenomena and quantify the influence of various parameters on overall thermal performance of these devices. A one-dimensional model is utilized to solve the governing equations for the liquid/vapor flow and the heat transfer in the heat pipe channel. Variations in the liquid and vapor cross-sectional areas along the axial length of the heat pipe are included and the equations are solved using an implicit finite difference scheme. Appropriate models for fluid friction in small passages with varying cross-sectional areas have been incorporated to yield the axial distribution of the meniscus radius of curvature and the velocity, temperature and pressure in both the liquid and the vapor phases. Using this information, the effective thermal conductivity of the micro heat pipe is modeled, and parametric studies are performed by changing the heat load and cooling rate. The results of the analysis are discussed and compared with other theoretical models and experimental results found in the literature. By so doing, this analysis provides greater insight into the physical phenomena of flow and heat transfer in micro heat pipes and identifies a methodology for optimizing the design of these devices.
Introduction
Although the early steady-state models, and the later transient numerical models were developed to calculate the velocity, temperature and pressure distributions within individual micro heat pipes, they have not provided the level of detail necessary to understand the complicated and interrelated physical phenomena taking place in micro heat pipes, hence, many aspects of the operation and performance characteristics of these devices are still not clearly understood. In addition, because these devices have found their place in many key industrial and scientific applications, the current level of understanding of the effect of variations in the geometric and thermophysical parameters is of significant importance to present and future technological development.
The first steady-state model specifically designed for use in modeling of micro heat pipes was proposed by Cotter (1984). Building upon this model, Peterson (1988) and Babin et al. (1990) developed steady-state models for trapezoidal micro heat pipes, using the conventional steady-state modeling techniques outlined by Chi (1976). Khrustalev and Faghri (1994) presented a detailed mathematical model of the heat and mass transfer processes in micro heat pipes. The analytical results were compared with the experimental data obtained by Wu and Peterson (1991) and Wu et al. (1991). The results of this comparison, confirmed the relative importance of the liquid charge, the contact angle, and the shear stresses at the liquid vapor interface in assessing the maximum heat transfer capacity and thermal resistance of these devices as predicted by Peterson (1991).
Analytical models for triangular micro heat pipes etched in silicon wafers have been developed by Duncan and Peterson (1995), Peterson and Ma (1996), Ma and Peterson (1998) and Peterson and Ma (1999). The maximum heat transfer capacity of copper-water micro heat pipes was explored by Hopkins et al. (1999) using a one-dimensional model for predicting the capillary limitation.
The first reported transient investigation of micro heat pipes was conducted by Wu and Peterson (1991). The most interesting result from this model was the observation that reverse liquid flow occurred during the startup of micro heat pipes. Ma et al. (1996) developed a closed mathematical solution for the liquid friction factor occurring in triangular grooves. The results of the analysis were verified using experimental test data.
Longtin et al. (1994) developed a one-dimensional, steady-state model for the evaporator section of micro heat pipes, assuming a uniform temperature along the heat pipe. A transient model for a triangular micro heat pipe with distinct evaporator and condenser sections was presented by Sobhan et al. (2000). The effective thermal conductivity was computed, and characterized with respect to the heat input and cooling rate, under steady and transient operation. The reverse liquid flow initially predicted by Wu and Peterson (1991) was also apparent in the computational results.
The transient formulation described in Sobhan et al. (2000) is also used in the present study. The primary aim of the current work is to perform extensive parametric analysis, treating the heat input at the evaporator and the heat transfer coefficient at the condenser as the major parameters. As it is of interest to understand the influence of a change in the relative lengths of the evaporator and condenser sections, a different combination of these is treated in the present study, keeping the overall length of the heat pipe the same as in the earlier work. A major limitation in the numerical solution presented in Sobhan et al. (2000) was the use of an explicit scheme in the calculation of the field variables. This imposed stringent limits on the grid size and time step used in the computation and limited the scope of the numerical studies due to limitations on input parameters. In order to overcome this, the current investigation employs an implicit scheme, which allows more extensive parametric studies and identifies the influence of the operational parameters, namely the heating level (input heat at the evaporator) and the heat transfer coefficient at the condenser, on the effective thermal conductivity.
Model Formulation
A flat micro heat pipe heat sink consisting of an array of micro heat pipe channels is used to form a compact heat dissipation device that can effectively remove heat from an electronic chip. Each channel in the array serves as an independent heat transport device. The analysis presented here examines an individual channel in such an array, as shown in Fig. 1(a). More details of the construction of these micro heat pipe heat sinks have been presented previously by Sobhan et al. (2000). The individual micro heat pipe channel analyzed in the present work is triangular in cross-section and has a characteristic length (side) of 0.3 mm. The channel is fabricated on a copper substrate and the working fluid is ultrapure water. The construction details are given in the cross-sectional view shown in Fig. 1(b).
Fig. 1(a) - External view of the micro heat pipe heat sink. 40 Individual micro heat pipes are arranged in an array inside. (Dimensions are in mm.), Click to Enlarge
Fig. 1(b) - Construction details of the micro heat pipe channels, fabricated on a copper substrate. The side of triangle is 300 microns. (Dimensions are in mm.), Click to Enlarge
The Mathematical Model
The micro heat pipe considered in the present study consists of an externally heated evaporator section and a condenser section subjected to forced convective cooling. A one-dimensional model is expected to be sufficient for the analysis, as the variations in the field variables are significant only in the axial direction, due to the geometry of the channels. Computation is performed for heat inputs in the range 1 W/cm2 to 4.5 W/cm2 in the evaporator and three values of heat transfer coefficients in the condenser, ranging from 50 W/m2K to 160 W/m2K and obtained through forced air convection with an average temperature of 300 K. A transient model was used and assumed an initial overall temperature of 300 K. The computational process proceeded until steady-state was attained, as indicated by constant temperature values throughout. It was assumed that the thermophysical properties of the working fluid both in the liquid and vapor phases did not vary significantly with temperature, within the operating range investigated. However, the variation of vapor density with temperature was incorporated in the computational scheme by interpolating from available thermophysical property data.
The flow and heat transfer processes are governed by the continuity, momentum and energy equations for the liquid and vapor phases. 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 were incorporated into the model through the use of suitable geometric area coefficients, as described in Sobhan et al. (2000). The local meniscus radius at the liquid-vapor interface was calculated using the Laplace-Young equation. The friction factor, which appears in the momentum and energy equations was incorporated through appropriate models for fluid friction in varying area channels, as described in the literature (Longtin et al., 1994; Sobhan et al., 2000).
The governing differential can be described as follows:
Laplace-Young Equation:
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(1)
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Vapor phase equations:
Vapor continuity equation: evaporator section
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(2)
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Vapor continuity equation: condenser section
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(3)
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It should be noted that the vapor continuity equation incorporates the interfacial mass balance equation
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(4)
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This equation serves as the connecting link between the vapor flow and the liquid flow in the micro heat pipe, as it established the mass balance between the two phases. The mass addition and depletion in the two phases, enters the computation through the incorporation of this interfacial mass balance into the continuity equations.
Vapor momentum equation:
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(5)
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Vapor energy equation: evaporator section
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(6)
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Vapor Energy equation: condenser section
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(7)
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Liquid phase equations:
Liquid continuity equation: evaporator section
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(8)
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Liquid continuity equation: condenser section
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(9)
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Liquid momentum equation
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(10)
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Liquid energy equation: evaporator section
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(11)
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Liquid energy equation: condenser section
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(12)
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The vapor and liquid pressures are computed as follows
- 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 extensively in the analysis.
- 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 values of pressure obtained from this first approximation are substituted into the momentum equations and iterated for spatial convergence.
State Equations
Equation of state for the vapor:
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(13)
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Hagen- Poiseuielle Equation for the liquid flow
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(14)
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The boundary conditions are
The initial conditions are
At x = 0
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(15)
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The value of r0, the initial radius of curvature of the interface meniscus for the copper-water system, was adopted from the literature (Longtin et al., 1994).
Numerical Solution
The governing differential equations were solved using a fully implicit finite difference scheme. Central difference approximations were used for the first and second derivatives. A time step size of 10-7 seconds was used, and stable and converged solutions were obtained by successive grid refinement. The steps involved in the numerical computation scheme have been discussed in Sobhan et al. (2000). The use of an implicit scheme in the numerical solution makes the model capable of analyzing the performance of the heat pipe for a much wider range of parametric variation in the heat input and the condenser heat transfer coefficients, without facing computational instabilities. This also provides faster computing compared to the explicit model used in Sobhan et al. (2000) which means that finer grid sizes could be used, with coarser time step. From a comparison of the nature of the results given in Sobhan et al. (2000) and the present work, it is obvious that the present computation has been capable of capturing the field variations better, as seen in the temperature distributions which show a more gradual and complete field variation, at the evaporator-condenser junction.
Results and Discussion
The primary results obtained from the numerical simulation were the instantaneous local velocity, the temperature, and the pressure distribution throughout the domain. In order to make a comparative analysis, an effective thermal conductivity was defined using a one-dimensional heat conduction analogy in a solid rod.
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(16)
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In this expression, Qin represents the heat input in the evaporator section and Ac is the overall cross-sectional area of the heat sink. A dimensionless effective thermal conductivity ratio (keff*) was also defined as the ratio of the effective thermal conductivity to the thermal conductivity of copper.
Distributions of the field variables:
The first step in computing the variations of the field variables is the calculation of the radius curvature of the liquid-vapor meniscus as a function of the longitudinal distance along the micro heat pipe channel. As described above, the Laplace-Young equation was used for this purpose and was solved along with the other governing equations incorporating the computed pressures from the model at each step. The results were then stepped forward in time to yield the steady-state distribution of the meniscus radius of curvature. The resulting distribution of the meniscus radius of curvature is illustrated in Fig. 2. The nature of the distribution is in accordance with those presented previously for copper-water systems by Khrustalev and Faghri (1994).
Fig. 2 - Distribution of the radius of curvature of the liquid-vapor interface meniscus in the micro heat pipe, Click to Enlarge
The numerical solution procedure involved marching in the time domain to obtain the steady-state results as characterized by constant temperature values along the length of the heat pipe. The vapor temperature in the evaporator was then used as the indicator for the attainment of steady-state and the computational process was terminated when the value of the mid-point temperature did not vary more than 0.1% over a time period of 20 seconds. A further check for steady-state was performed by ensuring that the input heat at the evaporator matched the heat transfer at the condenser. A typical transient variation of the vapor temperature at the evaporator mid-point is shown in Fig. 3.
The nature of the velocity, pressure and temperature distributions in the vapor and the liquid flowing along the micro heat pipe was obtained using an explicit solution of the governing equations and has been presented and discussed in earlier publications by Sobhan et al. (2000). In the present work, a more powerful implicit solution technique was used and hence, the results are considerably more accurate in terms of computational errors. The field variables follow the same trend as discussed in the previous work.
Fig. 3 - Transient variation of the vapor temperature at the mid-point of the evaporator section, showing the attainment of steady-state (corresponds to a heat input of 3.5 W/cm2 and a condenser heat transfer coefficient of 120 W/m2K), Click to Enlarge
Fig. 4 - Normalized vapor velocity distribution along the micro heat pipe (corresponds to a heat input of 2.5 W/cm2 and a condenser heat transfer coefficient of 120 W/m2K, Click to Enlarge
Fig. 5 - The normalized liquid velocity distribution in the micro heat pipe for the case corresponding to that presented in Fig. 4, Click to Enlarge
Typical steady-state distributions of the normalized vapor and liquid velocity are shown in Figs. 4, and 5. The velocity distribution depicts the mass addition and depletion along the evaporator and condenser sections of the micro heat pipe. The liquid velocity distribution also signifies the impact of the area variation of the liquid flow stream as it moves along the heat pipe adjacent to the corners of the triangular channel. This is evident from the non-linear nature of the longitudinal distribution in the condenser section.
Fig. 6 - Vapor temperature distributions in the micro heat pipe corresponding to various input heat fluxes at the evaporator section of the heat sink. The condenser heat transfer coefficient is 120 W/m2K for all cases shown, Click to Enlarge
The typical steady-state temperature profiles shown in Fig. 6, correspond to three different heat flux rates at the surface of the 40-channel heat sink and a heat transfer coefficient of 160 W/m2 at the condenser. The temperature variation is much smaller along the heat pipe than would be expected in a solid copper rod of the same dimensions, assigning a high value for the effective thermal conductance. Similar trends are observed for all the cases analyzed. The temperature distributions presented here represent the field better than those given in Sobhan et al. (2000), as the present computations were based on a finer spatial grid, which better captures the effect at the junction of the evaporator and condenser, as shown by the gradual change in values in the vapor, across the junction.
Parametric studies:
The overall effect of the temperature distribution was consolidated in terms of the effective thermal conductivity of the heat pipe, using an analogy with heat conduction in a solid rod with the same dimensions as the heat sink, as illustrated in equation (16). The effective thermal conductivity is presented in the form of a normalized value, k*, with respect to the thermal conductivity of copper, which can be used as a performance index for the system analyzed. The influence of two major operating parameters of the heat pipe, namely the input heat and the heat transfer coefficient at the condenser are studied and quantified below.
The variation of the effective thermal conductivity ratio with respect to the heat flux at the evaporator section is shown in Fig. 7, and the increasing trend is expected until the heat transfer limit is reached. In this case, the heat input is increased and the heat transfer coefficient at the condenser is held constant.
Fig. 7 - Variation of the effective thermal conductivity ratio with respect to the input heat flux to the heat sink. The case corresponds to a heat transfer coefficient of 160 W/m2K at the condenser section, Click to Enlarge
These results clearly indicate the function of the heat pipe as a phase change device, where the effects of the circulation of the vapor and liquid on heat transport improve as the device operates at higher power levels and hence higher operating temperatures. The increase in the performance index of the heat pipe with an increase in the heat input is expected to continue until the operating limit is reached, which indicates that the operating conditions close to the maximum operating limits, is best characterized by the optimal performance of the heat pipe.
Fig. 8 - Variation of the effective thermal conductivity ratio with respect to the heat transfer coefficient at the condenser section of the heat sink (input heat flux is 4.5 W/cm2), Click to Enlarge
The effect of variations in the heat transfer coefficient at the condenser section on the effective thermal conductivity ratio is shown in Fig. 8. In the present study, increases in the condenser heat transfer coefficient, with constant coolant temperature, yields a larger temperature drop between the evaporator and condenser, for a given heat input. Thus the calculated values of the effective thermal conductivity as defined in equation (16) decrease with respect to increases in the condenser heat transfer coefficients. The rate of the decrease in the effective thermal conductivity ratio becomes smaller as the heat transfer coefficient increases as shown in Fig. 8.
Fig. 9 - Comparison of the results from the present computational study with results presented in the literature, Click to Enlarge
In Fig. 9, the dependence of the vapor temperature on the applied heat input, Qin*, is compared with results previously obtained and presented in the literature. The vapor temperatures plotted in this graph are at the mid-point of the evaporator section and are representative of the temperature level of the micro heat pipe. The variation of temperature with respect to the heat input is linear for all of the cases investigated. The temperatures obtained from the present analysis are greater than those from Wu et al. (1991), for the same heat input levels as expected, due to much smaller dimensions of the channel and the higher heat fluxes used in the present analysis. Further, a quantitative comparison is not possible, as the analysis in the present study does not closely represent the experimental case presented by Wu et al. (1991). The present model is for a heat pipe with only the evaporator and condenser sections, and for a case where the heat transfer coefficient at the condenser is kept constant, allowing the condenser temperature to attain the equilibrium value. The analysis by Longtin et al. (1994) is different in that only the evaporator section was analyzed, assuming a uniform temperature for the vapor, and hence the effects due to liquid and vapor flow velocities and the external cooling conditions of the condenser were not reflected in the vapor temperature. Because of the differences in the cases shown here, the comparison presented is primarily qualitative in nature, and depicts only the general trends. However, experimental study of the physical system described in the present work is required for accurate benchmarking of the computational model.
Conclusions
A computational study of the vapor and liquid flow and the heat transfer in a micro heat pipe designed for use in the cooling of electronic components, is presented and discussed. The transient governing equations describing the physical processes in the micro heat pipe are solved using an implicit numerical scheme, to yield the distributions of flow variables and temperature in the one-dimensional domain. The resulting temperature distributions are used to calculate an effective thermal conductivity of the heat sink. The performance of the heat sink is then evaluated with respect to the heat input at the evaporator and the external heat transfer coefficient at the condenser. The results indicate that the effective thermal conductance increases significantly with increases in the heat input, suggesting that the ideal operating conditions for the heat pipe could be expected to occur close to the operating limits. An increase in the heat transfer coefficient with a constant coolant temperature was found to decrease the effective thermal conductivity for a given heat input. The computational results are discussed, compared and contrasted with results previously reported in the literature.
Nomenclature
References
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Acknowledgement
The authors acknowledge the support of the National Science Foundation (CTS-0312848) and the Office of Naval Research (N140010454).
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