Abstract
In twophase miniature and microchannel flows, the meniscus shape must be considered due to effects that are affected by condensation and/or evaporation and coupled with the transport phenomena in the thin film on the microchannel wall, when capillary forces drive the working fluid. This investigation presents an analytical model for microchannel condensers with a porous boundary, where capillary forces pump the fluid. Methanol was selected as the working fluid. Very low liquid Reynolds numbers were obtained (Re~6), but very high Nusselt numbers (Nu~150) could be found due to the channel size (1.5 mm) and the presence of the porous boundary. The meniscus calculation provided consistent results for the vapor interface temperature and pressure, as well as the meniscus curvature. The obtained results show that microchannel condensers with a porous boundary can be used for heat dissipation with reduced heat transfer area and very high heat dissipation capabilities.
Microchannels; thermal control; capillary pumped loop; condensation and meniscus
Effects of condensation in microchannels with a porous boundary: analytical investigation on heat transfer and meniscus shape
R. R. Riehl^{I}; J. M. Ochterbeck^{II}; P. Seleghim, Jr^{III}
^{I}Universidade Federal de Santa Catarina Departamento de Eng. Mecânica, C.P. 460 88040900 Florianópolis, SC. Brazil Email: rriehl@cet.ufsc.br Clemson University, Mechanical Engineering Department Clemson, SC 29634 USA
^{II}Clemson University, Mechanical Engineering Department Clemson, SC 29634 USA
^{III}Universidade de São Paulo EESC Departamento de Eng. Mecânica.C.P. 359 13560970 São Carlos, SP. Brazil Email: seleghim@sc.usp.br
ABSTRACT
In twophase miniature and microchannel flows, the meniscus shape must be considered due to effects that are affected by condensation and/or evaporation and coupled with the transport phenomena in the thin film on the microchannel wall, when capillary forces drive the working fluid. This investigation presents an analytical model for microchannel condensers with a porous boundary, where capillary forces pump the fluid. Methanol was selected as the working fluid. Very low liquid Reynolds numbers were obtained (Re~6), but very high Nusselt numbers (Nu~150) could be found due to the channel size (1.5 mm) and the presence of the porous boundary. The meniscus calculation provided consistent results for the vapor interface temperature and pressure, as well as the meniscus curvature. The obtained results show that microchannel condensers with a porous boundary can be used for heat dissipation with reduced heat transfer area and very high heat dissipation capabilities.
Keywords: Microchannels, thermal control, capillary pumped loop, condensation and meniscus
Introduction
Previous investigations addressing singlephase and twophase flows in small diameter and microchannel passages have demonstrated tremendous potential for high heat flux systems with increased heat transport capability. The vast majority of these investigations were directed toward singlephase flow and convective boiling in small diameter and microchannel passages. The basic phenomenon of convective condensation in small diameter/microchannel passages has not been fully addressed experimentally or analytically, especially in the case where the convective condensation phenomenon is coupled with a porous boundary. Better understanding of condensing twophase flows in small diameter or microchannel passages is an enablingtechnology for the development of efficient and reliable twophase thermal management systems employing small diameter and microchannel passages, especially for electronics cooling, miniature heat exchangers, and capillary pumped loops.
A literature survey on singlephase and twophase flow heat transfer coefficients was presented by Riehl et al. (1998), who reviewed the available analytical models and experimental data obtained for microchannels. Of these investigations, Tuckemann and Pease (1981) were the first to investigate systems using microchannel heat exchangers and forced singlephase liquid cooling through microchannels for cooling of electronic devices. The technology demonstrated a potential for more compact arrangements. A model was proposed by Weisberg and Bau (1992), which could predict the temperature distribution in a microchannel evaporator for water as working fluid and laminar flow.
Few previous experimental and analytical investigations have been performed regarding microchannels condensation. Smirnov and Buz (1995), Buz and Smirnov (1997) and Smirnov et al. (1997) presented an approach for condensation in small channels. This model was able to predict condensation in channels where the gravitational forces do not influence the flow. Also, the model could predict the liquid film thickness along the channel, which is important for analysis of the condensation capability and is an important factor for determining the meniscus shape.
The meniscus problem was studied for fluid flows in small diameters and primarily for capillary flow. During an investigation of an evaporating meniscus on a flat copper plate, Mirzamoghadam and Catton (1988) found that the interline wall superheat, needed to obtain a converged solution from the steadystate integral model, was in agreement with experimental observations of the wall superheat effect on the meniscus shape. Later, Swanson and Herdt (1992) developed an analytical model to predict the evaporating meniscus shape with interest in capillary wick structures. The model showed that the maximum capillary pressure cannot be used to evaluate the local interfacial mass transfer rate, which is an intermediate boundary condition between the vapor and liquid phases.
Nomenclature
DasGupta et al. (1993, 1994) found that the YoungLaplace equation could be used to determine the interfacial properties of the system and then describe the process of an evaporating meniscus. This equation showed to accurately predict the meniscus slope.
A model for determining the minimum meniscus radius in micro heat pipes was developed by Ma and Peterson (1998). The obtained equations showed that there is an optimum hydraulic radius for the grooves that have a maximum capillary heat transport capability, which directly affects the meniscus shape. Pratt et al. (1998) experimentally investigated the thermocapillary effects on a curved meniscus and observed that capillary instabilities can result from thermocapillary instabilities of the contact line region of the evaporating meniscus within capillary grooves. This may affect the near contact line region of the extended meniscus formed within the capillary pores.
Much information regarding the singlephase and twophase (boiling) convective flows in microchannels and the meniscus characteristics in capillary structures have been developed. However, few investigations have attempted to address the condensation phenomena in condensing flows in small diameter/microchannels. This paper presents an analytical investigation of microchannel condensation including liquid film distribution and meniscus shape. A detailed description of this model is presented by Riehl (2000).
Analytical Modeling in Microchannel Flow
It is important to note that the mathematical formulation developed below for both liquid film distribution and meniscus shape in microchannels assumes that capillary forces pump the condensed liquid throughout the channel. This is especially important to consider because microchannel condensers have a potential to be used on the heat dissipation of capillary pumped loops (CPL) and loop heat pipes (LHP) for electronics, structures and satellites thermal control.
The model formulation for both liquid film thickness and meniscus shape has the following assumptions: 1) channel symmetry, 2) steady state twodimensional laminar flow, 3) incompressible flow, 4) convective terms, with the exception of the axial convection in the energy equation, are negligible, 5) radial pressure gradient is negligible, 6) temperature in the bulk liquid is equal to the interfacial liquid temperature, 7) pressure in the bulk liquid is constant, 8) no slip at the wall, 9) the channel wall is smooth and the fluid is pure, 10) surface tension (s) and the dispersion coefficient () are not affected by the interfacial curvature, 11) retardation effects in the dispersion coefficient are negligible, 12) hydrostatic pressure is negligible and Marangoni effects (ds/dt) are important only in the thinfilm region.
Liquid Film Distribution Formulation
Figure 1a presents a general microchannel condenser with a porous boundary, which was used in the development of this model. For the case when only liquid is drained by a porous structure, the model geometry is shown in Figs. 1b and 2. The microchannels were taken to be of rectangular shape, with heat being removed from the bottom surface.
As it can be observed, the proposed model is solved for a twodimensional problem as a three dimensional formulation would lead to complicated interactions and heavy computational efforts. Using the model described and applying the conservation equations of mass, energy and momentum, the resulting equation set for the fluid condensing on surfaces x and y, when the porous boundary is not included is given by
where
The pumping intensity represents the maximum power that can be obtained by the difference between the liquid and vapor pressure and transport parameters, and is defined as
The dimensionless heat removal is represented as the ratio between the heat transfer rate per unit length and transport properties of the fluid, defined as
The parameter A is the maximum heat transport rate per unit length, i.e.,
For the case of liquid flowing through the porous boundary (zaxis) two other equations are required. These equations are
where J is the Darcy number defined as
The boundary conditions used for the solution along the x and yaxes are
and for the surface at the zaxis, the boundary conditions are
To avoid numerical discontinuity between the y and xaxes and between the x and zaxes, the following boundary conditions are required
After solving the system of equations, the local Nusselt number can be determined as
Meniscus Shape Formulation
Since the liquid film thickness describes a circular shape along the microchannel, the meniscus shape determination is conducted using circular coordinates related to the hydraulic diameter. This assumption avoids complicated iterations between the microchannels wall and the liquid film. The meniscus shape formulation describes the fluid mechanics, heat transfer and interfacial phenomena characteristic of a single, isolated condensing meniscus. This formulation includes the threedimensional YoungLaplace equation, Marangoni convection, LondonVan Der Waals dispersion forces, and nonequilibrium interface conditions. The model used for this formulation is represented by Fig. 3, which shows the transport processes for the meniscus.
Using the above assumptions, the system in Fig. 3 and a basic formulation following closely the model of Swanson and Herdt (1992), the equation set that describes the meniscus shape in microchannels condensation is represented as
where D_{1} is a dimensionless constant equal to , which resulted from the derivation procedure. The meniscus curvature, on its dimensional form, can be calculated from the relation given by Philip (1977) as
The set of equations was transformed into dimensionless form by using the following parameters:
where s_{w} is the fluid surface tension at the wall temperature. The variables G_{1}, G_{2}, G_{3}, G_{4}, G_{5}, G_{6} and G_{7}, are dimensionless functions, defined as
Scaling for this analysis used the variables p_{1}, p_{2}, p_{3}, p_{4} and p_{5}, which are defined as
where C is a constant usually equal to one. The physical meaning of the dimensionless variables are: p_{1} is the dimensionless pumping resistance, p_{2} is the dimensionless subcooling; p_{3} is the Crispation number; p_{4} is the mass flux at the interface and p_{5} is the dispersion number, which represents the magnitude of the dispersion forces in the thin film. The Crispation number is a dimensionless parameter used frequently in convection current analysis, obtained by dividing the product of the dynamic viscosity of a fluid and its thermal diffusivity by the product of the undisturbed surface tension and layer thickness, which is very important when capillary forces drive the fluid. The mass flux (p_{4}) was derived from the kinetic theory, which resulted from the derivation of Eq. (24) (Riehl, 2000).
Solution Method for the Proposed Model
The solution method used for solving the set of equations required constant iteration between both models. First, the equation set derived for solving the liquid film distribution is solved using the RungeKutta 4^{th} Order Method. Equations (1)(3) are then solved separately for the x and yaxis and Eqs. (2), (11) and (12) are solved for the zaxis (porous boundary) using the boundary conditions (14)(19). An iterative matching solution is applied in order to avoid numerical discontinuities between y and xaxis and between x and zaxis.
For the meniscus shape equation set, the initial values of the dependent variables are selected as
At each value of film thickness resulting from the calculation, the equation set for the meniscus shape, represented by Eqs. (21)(25), is solved using both partial linearization (using Taylor series) and backward finite differentiation. The nonlinear terms of Eq. (21) are first linearized around the previous iteration. Then, with the transformed form of Eq. (21), the resulting equation set is solved by a backward firstorder finite differentiation method. The resulting linear equation set is then solved using a GaussSeidel numerical method.
Despite the nonlinearity of the equations, the system presents good convergence. For the liquid film thickness, the equation set calculation presented stable solutions. For the meniscus shape equations, the system converged within three to five iterations for each axial position with a relative error of 10^{10} and a step size of 10^{30}. The solution could be reached over the interval 0.115 < < 1.0. For < 0.115 the equation set became unstable and D_{1} approaches negative infinity. At this point, the meniscus shape presents almost no change, which results could be neglected as it approaches HagenPoiseuille flow. Using a Pentium II 300 MHz computer, the overall solution time was less than five minutes.
Results and Discussion
The proposed model is applied for a microchannel condenser with channel size of D_{h}=1.5 mm, d= 0.01 mm, vapor pressure of 30 kPa, and methanol as the working fluid. It is assumed that T_{sat}=55 ^{o}C, =3 mm/s, L=150 mm, a porous boundary thickness of 5 mm and b=0. The thermophysical properties used in this solution were obtained from Peterson (1994). Other characteristics of the porous boundary are: mean pore radius of 15 µm and G= 10^{12} m^{2}. Figures 4, 5 and 6 show the liquid film distribution along the x, y and zaxes, respectively.
When comparing the amount of liquid at each surface, it is observed that less is presented close to the porous boundary (z axis). This behavior is due to the draining capability of the porous boundary, promoting a liquid return to the system and allowing the proper operation of the capillary evaporator. More or less draining capability will be dependent only upon the porous boundary used, which could lead to a better performance of the porous wall.
The results presented show that when the liquid has been removed for high values of pumping intensity, the condensing film is characterized by a considerable and nonmonotonous change of pressure gradient along the liquid film. The insignificant influence of inertial forces on Nusselt number is also observed on Figs. 7 and 8. Very high Nusselt numbers could be achieved even for low Reynolds number, characterizing that laminar liquid flow can lead to high heat transfer capabilities when using microchannel condensers. This is especially important where there is a restricted area for heat dissipation and limited flow rate, as presented in Fig. 7. Such behavior is also verified experimentally, as presented by Riehl (2000). Other microchannel sizes were considered to calculate the Nusselt number, as presented in Fig. 8, in order to evaluate their influence on the heat transfer capability of such microchannels.
The dependency of the average intensity of heat exchange on the pumping intensity shows that there is a tendency for a maximum for heat removal. When this maximum is reached, even for a higher pumping intensity, the heat exchange rate will not increase. As it is shown in Fig. 8, the Nusselt number increases for higher pumping intensities. The heat transfer capabilities also increase when the microchannel size decreases, showing that better heat dissipation can be achieved in very reduced areas. This is especially important in satellite applications, where the area for heat dissipation is restricted and its thermal control must be as accurate as possible.
Figure 9 shows that the dispersion number (p_{5}) has a large influence on the interface temperature. For a dispersion number of 2 x 10^{10}, the dispersion forces in the thin film present a strong influence and result in a temperature difference between the center of the channel and the wall of around 4.5 K. A greater difference is not observed for dispersion numbers of 1 x 10^{4} and 1 x 10^{9}. On a microscopic level, the characteristic of the menisci is that for larger dispersion numbers, the thin film extends further down along the channel due to the attractive forces between the vapor and the solid substrate. Such attraction forces can be observed in Fig. 9. A larger difference on the interface temperature is observed at higher saturation temperatures. Higher heat transfer capabilities are expected when higher temperatures are used, which is also improved by using microchannel heat exchangers, because greater temperature differences between the condenser inlet and outlet are obtained (Riehl, 2000).
The meniscus interface pressure is constant for any channel dimension, depending only on the saturation temperature, as shown by Fig. 10. This was expected because the heat transfer process with a volatile fluid must occur at constant pressure, being only influenced by a slightly pressure drop at the interface. The same results are obtained for different dispersion numbers, which means that the interface pressure is not affected by this parameter. Although, as the interface approaches the wall, there is a pressure drop due to the attraction forces.
Figure 11 presents the dimensionless meniscus curvature, which is highly influenced by the dispersion number by the same reasons explained above. On the other hand, the dispersion number has a higher influence on the meniscus curvature when the channel size decreases. Such influence can be significant for very small channels (D_{h} < 0.5 mm), which can compromise the proper operation of the capillary evaporator. Thus, careful consideration of such parameter must be performed during the design.
Conclusions
An analytical model is used to predict the liquid film thickness and the meniscus curvature in microchannel condensers. The conception of such model was motivated by the growing need of dissipating high heat fluxes from electronics, capillary pumped loops and loop heat pipes. The use of a porous boundary showed to be especially important to enhance the heat transfer capability, as the boundary drains the condensed liquid back to the system. Although higher pressure drops can be expected when comparing to straight channels, the use of such porous boundary presents an improvement on the overall system heat transport capability. The model presents good agreement with experimental data when a macroscopic comparison is performed.
The current model has been proposed as a tool to aid in the design of microchannel condensers with possible applications to microelectronics cooling, micro heat exchangers, and condensers in capillary pumped loops (CPL) and loop heat pipes (LHP) with restricted heat dissipation area. Validation of the model with experimental investigation still needs more investigations, although a macroscopic validation was performed by Riehl (2000). The analytical results presented here showed to be in agreement with the experimental results for microchannel condenser with 1.5 mm of hydraulic diameter. Further investigations on microchannel condensers with smaller sizes are still required, which should focus on the influence of the dissipation number on the entire system performance (CPL or LHP plus microchannel condenser) and not only on the condenser. This will lead to new designs of CPL and LHP, which is expected to improve the heat dissipation capabilities.
Acknowledgements
This work was supported in part by the Fulbright Foundation, CAPES (Brazil), and the National Science Foundation (USA).
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Publication Dates

Publication in this collection
27 Aug 2003 
Date of issue
July 2002