3-D TRANSIENT SIMULATION OF A FLAT CAPPILARY HEAT

Transcription

1 3-D TRANSIENT SIMULATION OF A FLAT CAPPILAR HEAT PIPE VIA AN INTERFACE TRACKING METHOD H. A. Machado Centro Técnico Aeroespacial - CTA Instituto de Aeronáutica e Espaço - IAE Pr. Mal Eduardo Gomes, 5, Vila das Acácias , São José dos Campos, SP, Brasil humbertoam@iae.cta.br Universidade do Estado do Rio de Janeiro Faculdade de Tecnologia de Resende Estrada Resende-Riachuelo, s/n, Morada da Colina , Resende, RJ, Brasil machado@at.uerj.br ABSTRACT In capillary and micro heat pipes the internal porous media is replaced by a capillary groove, yielding a meniscus ormed by the liquid phase. The meniscus height varies along the groove length, rom the condenser section to the evaporator section, and the dierence between the pressures in each extremity is responsible or pumping the liquid. Such devices have been employed in electronic cooling, due its small dimensions. This system provides excellent thermal control, assuring uniorm temperature distribution. In previous works related to capillary heat pipe simulation, empirical or simple 1-D models have been used, but not taking into account the complete unsteady phenomena. In this work, the 3-D unsteady simulation o a lat micro heat pipe is presented, where the meniscus radius is considered constant along the transversal section. The phase change problem o a heated liquid meniscus in a groove is simulated via an interace tracking method, which is an hybrid Lagrangean-Eulerian method or moving boundary problems. A variation o the one-dimensional Stephan problem is used to validate the numerical code. Keywords: Capillary and micro heat pipes, Moving boundary, Phase change, Interace tracking NOMENCLATURE c speciic heat, J/kg.K G gravity, m/s 2 I indicator unction K thermal conductivity, W/m.K L latent heat o phase change, J/kg P pressure, Pa Pr Prandtl number, Pr = ν/α t time, s T temperature u velocity vector, m/s u,v,w velocity components, m/s x position vector, m x,y,z Cartesian coordinates, m Greek symbols α thermal diusivity, m 2 /s ρ density, kg/m 3 µ viscosity, Pa.s w relaxation coeicient Subscripts interace l liquid phase v vapor phase reerence or initial value SAT saturation condition ree stream INTRODUCTION Phase change processes through capillary grooves are commonly incorporated in design o heat exchangers, increasing the interacial areas among liquid-vapor phases or evaporation and vapor liquid phases or condensation. These grooves allow the liquid phase to coalesce and move rom the condensation region or the evaporator, promoting the reduction o thickness o the liquid ilm. Recently, the use o such systems in micro scale devices or cooling o electronic equipment has deserved attention, due its high heat transer rates and excellent thermal control, improving the uniormity o temperature distribution, specially when employed as micro heat pipes, that are built straight over the chips structure. Mini and micro heat pipes have been employed in high perormance computers, laptops and workstations, satellites and space probes, ine surgery instruments, etc. The development o this kind o heat pipe has been done using empirical methods or simpliied models. Most o the experimental works deal with a speciic aspect o micro heat pipes, such as a particular shape or phenomenon involved in the pipe operation, or accuracy o any theoretical approximation used in the model. In these cases, some inormation is available about thermocappilary eects in a heated meniscus (Prat et al, 1998) or about the inluence o geometry in the heat exchange process (Ravigururajan, 1998). In all experiments, several diiculties were ound in reproducing the realistic dimensions and conditions o operation o a 72 Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

2 easible M.H.P (Faghri, 1995). Huang et al (1998) studied a constrained vapor bubble behavior, and characterized it as a large version o a micro heat pipe. A strong dependence o its perormance with the orientation with respect to gravity was observed. Sun & Tien (1972) presented a simple conduction model or steady state operation with an enclosure condition, based in the local wall temperature and the mass low rate, and compared with some experimental data available in literature. The results demonstrated the high heat transer capacity o such devices, and its dependence to Biot number and relative contribution o radial conduction through the wick and axial conduction to the pipe wall. Duncan and Peterson (1994) summarized the conclusions and limitations o each model available in a opened literature review. Krushtalev and Faghri (1994) have shown a one-dimensional model to predict the maximum heat exchange capacity and thermal resistance o a polygonal M.H.P, which was developed rom some approximations based in empirical observations. Hopkins et al (1998) compared the one-dimension model to experimental data, obtained in test with trapezoidal micro heat pipes, in order to veriy the maximum heat lux. Theoretical prediction o the capillary limitation in the horizontal orientation agreed reasonably with the experimental data. Ma & Peterson (1998) investigated the minimum meniscus radius and the heat exchange limit in M.H.P through the conservation laws or momentum and Laplace-oung equation, providing analytical expressions or parameters o interest. Ha & Peterson (1998) have been developed a model where the dierential expression is similar to Bernoulli s equation. An approximate analytical solution or the axial variation o meniscus curvature is presented. Peterson & Ma (1999) developed a more detailed model or a M.H.P through a 3 rd order ordinary dierential equation, that is able to estimate the heat transport capacity and the temperature gradients along longitudinal coordinate, as a unction o heat lux. However, all these works are 1-D models, just applicable to steady state processes. The present work is an extension o the work presented by Machado & Miranda (21), where a mathematical model and a computational algorithm or simulation o the two-dimensional unsteady heat transer and phase change processes in an opened square cavity, was proposed. Machado (23) applied the model to a closed square cavity, representing a lat capillary heat pipe. In this work, a threedimension closed cavity is used to simulate the actual behavior o a capillary heat pipe. The phase change problem is solved via the interace tracking method proposed by Unverdi and Trygvason (1992), that is an hybrid Lagrangean- Eulerian method or moving boundaries, and has been employed to simulate usion/solidiication o crystals and metal alloys and bubble ormation in nucleate boiling. This method has been successully extended to three-dimensional problems o phase change (Esmaeeli & Arpaci, 1998; Shin & Juric, 22) and bubble low (Esmaeeli & Tryggvason, 1998, 1999; Arruda, 1999). Preliminary results show the behavior o the computer code or a simple case. The model proposed here may be easily extended to problems with irregular geometries. PHSICAL PROBLEM AND MATHEMATICAL MODEL Consider a square cavity, illed with a pure Newtonian luid with constant properties and isochoric behavior, shown in Fig. (1). Thermodynamic equilibrium is assumed in both phases, liquid and vapor, inside. The lateral and superior sides are thermally insulated. Lateral and superior walls insulated y VAPOR Initial position o the interace x Q (evaporator) Blocked liquid -Q (condenser) LIQUID Lateral view (meniscus) Figure 1. Schematic o geometry and physical process. The set o equations in dimensionless orm used to represent the simple physical problem is written in an appropriate orm to be solved by the interace tracking method, as proposed by Juric (1996). Terms in bold are vector quantities. Continuity equation: this orm is equivalent to: ρ +.w = (1) t. w = M (2) M = mδ( x x ) da (3) A ρ v m = V.n (4) ρl where w = ρu (mass lux), V is the interace velocity, n is the interace normal vector, m is the mass lux through the interace per unit o area and δ(x - x ) is a 3-D delta unction, that is non-zero only at the interace. Subscripts l and v reers to liquid and vapor Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

3 phases, respectively. Tangential eects are neglected, as the luid properties are assumed constant and uniorm in each phase. Thereore, the interace velocity is supposed not having a tangential component. Such assumptions should be veriied careully, in a more detailed model. Conservation o momentum: w) 1 +.( wu) = P +. µ ( u + u t Re ( T ) + F (5) where P is the pressure and F is a source term that takes into account the interacial surace orces: F = δ x x ) da (6) A ( where x is the interace position vector, is the interace pressure due to the surace tension: = (κ Ι + κ ΙΙ ). n/we, and κ is twice the mean curvature o the interace, and subscripts I and II are related to transversal and longitudinal directions, respectively. Conservation o energy: ( ρct) 1 +.( w ct) =.K T + Q (7) t Pe where c is the speciic heat at constant pressure, K is the thermal conductivity and Q is a source term that takes into account the absorption or liberation o latent heat during phase change: Q = qδ( x x ) da (8) A q is the source term o energy per unit o surace o the interace: ρll q = ( ρv w).n (9) ρ L v where L is the latent heat o phase change and L is its corrected orm, taking into account dierent speciic heats or each phase: L = L + (c l c v )T v (1) The equations above, with the source terms included, satisy the jump conditions automatically, assuming a thickless interace. Although speciic volumes are dierent in the phases, the low is assumed to be uncompressible, and both densities are considered constant. The total volume occupied or each phase remains constant during the heating. Thereore, the rates o condensation and evaporation must be the same, in order to satisy the mass conservation within the cavity. During the transient process o heat transer, the reerence pressure inside the cavity rises and yields a correspondent increase o the local saturation temperature in every interace point, until the low reaches steady state. The interace is supposed to be at thermal but not thermodynamic equilibrium (which supposes no temperature jump at the interace). This hypothesis added to the Clausius-Clapeyron relation applied to a curved surace, but neglecting the kinect mobility eects yields the temperature condition at the interace as a unction o the reerence pressure: T T SAT ρ v cv + ρl cl σκ + We.T 1 ρ 1 ρ (P P ) 2 ( T T ) + ϑ( ρv w). n = SAT SAT l v (11) where T is the local interace temperature, T SAT is the local saturation temperature (rom the equation o state), P is the local interace pressure, P oo is the reerence pressure within the cavity, σ = c l T o γ/ρ v L 2 o y o. T o is the reerence temperature, y o is the reerence length and γ is the surace tension. Last term includes the non-equilibrium eects in the molecular kinetics: ϑ = ρlc lu / ρvlϕ, where ϕ is the molecular kinetics coeicient. Small values o ϕ suppress the growth o protrusions in the interace. The non-dimensional numbers resulting rom the mathematical model are: Reynolds number, Re = ρ l U l/µ l, Peclet number, Pe = Re.Pr, Weber number, We = ρ l U 2 l/γ and Froude number, Fr = U 2 /G.y (G is the local acceleration o gravity). METHOD OF SOLUTION The moving boundary problem was solved by the Interace Tracking Method, introduced by Unverdi & Trygvason (1992), and employed by Juric (1996) in the solution o phase change problems. In this method, a ixed uniorm Eulerian grid is generated, where the conservation laws are applied over the complete domain. The interace acts as a Lagragean reerential, where a moving grid is applied. The instantaneous placement o the interace occurs through the constant remeshing o the moving grid, and each phase is characterized by the Indicator Function, represented by I(x,t), which identiies the properties o vapor and liquid phases. The interace surace is built considering the transversal radius (R I ) constant (according to the ormal deinition o a micro heat pipe), and its longitudinal shape is represented as a parametric curve, R(u), where the normal and tangent vectors, longitudinal radius (R II ) and curvature are extracted rom. The interace discretization is shown in Fig (2). The positions o interace points in transversal direction are interpolated by a Lagrange polynomial, 74 Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

4 which allows to obtain the geometric parameters and remeshes the curve, keeping the distance d between curve points within the interval.9 < d/h < 1.1, where h is the distance among the ixed grid points, as shown in Fig. (2). 1(x) (x) = 1/ 2 (2 x ) 1 i i x 1 i 1 < x < 2 x 2 (14.a) x x 4x 1(x) = (14.b) (x) x Figure 3. Probabilistic distribution proile - (x). The vector G(x) is obtained rom: G = D ( x ). n( x ).ds( ) (15) i, j,k i, j,k m,n m,n xm, n m,n Figure 2. Eulerian and Lagrangean meshes. The Indicator Function varies rom 1 (vapor) to (liquid), and is numerically constructed using the interace curve to determine a source term G(x). The jump across the interace is distributed over the ixed grid points, yielding a gradient ield in the mesh: G(x) = I = nδ( x x )da (12) A which should be zero, except over the interace, as represented by the Dirac Delta Function. However, such representation is not convenient or a discrete number o points. The Distribution Function is used to represent the interace jump. Such Function is a curve similar to a Gaussian one, as shown in Fig (3), and its value depends on the distance x ij - x k among the Lagrangean and Eulerian points: D i, j,k [(x ( x m,n m,n ) = x ) / h].[(y i m,n y ) / h].[(z h 3 j m,n z ) / h] k (13) where D ij is the Distribution Function or a point k in the Lagrangian mesh in relation to a Eulerian point. One should note that increasing h, the interace becomes thicker. The unction is the probability distribution, related to the distance h as: where ds(x m,n ) is the surace element, shown in Fig.(4), and given as ds = dx.dy. rii dy Figure 4. Surace element, ds. The divergence o the gradient ield is ound by numerical derivation o Poison s equation: 2 n ri dx I =.G (16) Despite o being considered constants in each phase, the properties inside the domain must be treated as variable in the ormulation. A generic property φ (ρ, µ, c or K ) is expressed as: φ(x) = φ l + (φ v - φ l ) I(x,t) (17) The coupling between the moving mesh and the ixed grid is done at each time step, through the Distribution Function, that represents the source terms in the balance equations and interpolates the ields with ininitesimal discontinuities into a inite Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

5 thick region at the interace. The momentum equation at the interace surace becomes: I = V. I (18) t The initial interace shape is irst speciied and then the Indicator Function is constructed. The property and temperature ields are determined. The iterative process targets the reerence pressure and its correspondent interace velocity at each time step. The steps to be ollowed are: 1. Using the current value o V, the interace points are transported to a new position, calculated explicitly through the equation V n = (dx /dt).n; 2. Density and speciic heat are calculated at the new interace position; 3. Surace tension is calculated and distributed into the ixed grid; 4. In the irst iteration, reerence pressure is estimated. Ater that, its value is corrected according to the mass lux residual. 5. V n+1 is estimated via Newton iterations, using a numerical relaxation schedule, and used to calculate the mass lux m crossing the interace. The interace mass lux is distributed in the ixed gird; 6. According to boundary conditions, w, u e P are obtained rom Eqs. (2) and (5); 7. Density ound in step 2 and lux w are interpolated via distribution unction in order to ind the mass lux w and density ρ at the interace; 8. Heat lux q is calculated through Eq. (9) and distributed into the ixed grid; 9. According to boundary conditions, energy equation - Eq. (7) is used to obtain the temperature at time n+1; 1. Temperature and pressure are interpolated to ind T and P at the interace; 11. The equilibrium condition is tested. I it is not satisied, a new estimate or V n+1 is calculated and returns to step The mass lux residual is calculated, and i it is lower than the reached tolerance, the ields o viscosity and conductivity are updated or the new position, advance one step time, otherwise returns to step 4. are repeated until R(T) in every point become smaller than the tolerance. The optimum value or ω is ound by tentative, at the beginning o the calculation. A similar schedule is used to obtain the reerence pressure, where the mass lux residual is the algebraic summation o the source term M, Eq. (3), along the interace. RESULTS AND DISCUSSION FORTRAN language was used to construct the numerical code and it runs in a PC microcomputer. The transport equations were solved in a regular displaced mesh, via inite volume method, where the P-V coupling schedule used was the PRIME algorithm, and the discretization was done under the FTCS schedule. Validation o the numerical method The numerical results were validated by a variation o the one-dimensional Stean problem, within a < x < 1 domain, with a constant heat lux at x =, and constant temperature at x = 1. The initial interace position is x =.5, with the liquid phase at x <.5 and vapor phase at x >.5. Both phases are at saturation temperature when a heat lux q is imposed at x =. The values used were K v /K l =.33, ρ v /ρ l =.5, c v /c l =.5, q =.1, T sat = 1, Pe =.4, Re=.8 and We =. The results or the interace velocity are compared to those obtained via the Generalized Integral Transorm Technique - GITT (Cotta et al, 1998). In Fig. (5), good agreement between both methods o solution is observed. The numerical solution improves as the number o points o Langragian mesh (ni) increases. This improvement is remarkable at the initial and steady state times. In this case, the number o points o the Eulerian mesh also increases, due the meshes coupling by the Distribution Function. The convergence criterion used in step 11 is the residual in Eq. (11). Once it has reached the desired tolerance, convergence or interace velocity is assumed. Otherwise, the velocity is corrected via Newton Iterations, given as: n+ 1 n V = V ω.r(t) (19) where ω is a constant and R(T) is the residual or the temperature jump condition at the interace. Iteration Figure 5. Comparison between Interace Tracking Method and GITT. 76 Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

6 In Fig. (6) the eect o the interace thickness may be observed in the proiles o pressure and vertical velocity component. The thickness reduces as the number o points used increases. With ni = 1, interace thickness occupies a considerable raction o the domain. With ni = 4, the discontinuity at the interace is represented with a good precision. When ew points are used in the Lagragian mesh, the results out o the interace are very close to the exact values, that are reached when ni = 2. (a) Pressure proile 2 T SAT.1.(P + P ) + = T (2) where T v is the saturation temperature or the initial reerence pressure, equal to 1. For this case, the initial value or the reerence pressure is taken to zero. Figure (7) shows the variation o the reerence pressure along the time or several values o the meniscus radius, and its comparison with the results or the two-dimensional case. Reerence pressure varies strongly with the radius. When R < 1.2, steady state is not reached beore the interace touch the superior wall (when the calculations are stopped). The curve or R = 1 1 (ininite) is similar to the 2-D simulation, but the reerence pressure is twice that or two-dimensional calculations. The dierence between the ininite radius and 2-d cases is that the irst is a 3-D case, but the interace is plane in the z- direction. Interace shape at t=2.2 (were the steady state has been reached or ininite radius), Fig. (8), presents little variation, but its right extremity seems to be more stable than the 2-D interace or t = 1.1. v (b) Vertical velocity proile Figure 6. Solutions according with the number o points used in the Lagrangean mesh (ni). Results or a square cavity Reerence pressure.12 8e-5 4e-5 Transversal meniscus radius R = oo R = 1.5 R = 1.2 R = 1. R =.8 2-D Preliminary results or 3-D unsteady simulation were obtained or a lat square cavity representing a micro heat pipe, considering the eects o gravity and surace tension. The domain considered is the region deined as < x < 1 and < y < 1 and < z <.5, under prescribed temperatures in the inerior wall, T w = 11, in < x < 5 and T w = -9, in.75 < x < 1., considering t = 1 the dimensionless saturation temperature at the initial conditions. In the intermediary section, 5 < x <.75, the wall is thermally insulated, q w =. Lateral and superior walls are also insulated. The values used in the test case, or the physical parameters, were: K v /K l =.5, ρ v /ρ l =,9 c v /c l =.5, µ v /µ l =.5, Pe = 2, Re= 1, We = 3.96, Fr = 3.92, σ =.99 e ϑ =, and the interace is placed at y =.5 at t =. A 1 x 1 x 5 points grid was used. The schedule or correcting the reerence pressure was tested using a simpliied second order polynomial as the equation o state: Time Figure 7. Reerence pressure variation with time. Figure 8. Interace shape at t = 2.2 (t = 1.1 or 2-D case). Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

7 In Fig. (9), the streamlines or R =oo and R = 1.2 are similar. The interace perturbation in the 2-D case appears as a displacement o the center o recirculation within the cavity. Figure (1) shows the 3-D view o the low iled or R =.8 and ininite. There is no apparent inluence o the transversal meniscus radius in low patterns. Transversal low is not observed, even or R = Figure 1. Three-dimensional view o streamlines and velocity vectors at t = 2.2, or R =.8 and ininite. a) R = oo In Fig. (11), the evolution o interace position with the time (starting as an horizontal line) is shown or R =, demonstrating the good behavior o the grid reconstruction schedule. At t = 1, the interace displacement is still small, although it indicates the tendency or its inal proile. Steady state proile is shown at t = 2.2. b) R = 1.2 Figure 11. Evolution o interace along the time or R =. c) 2-D Figure 9. Streamline in z = 5 or the 3-D simulation and 2-D simulation, ater steady state has been reached. In Fig. (12), interace velocity, temperature and pressure over the interace surace are shown. Velocity and temperature distributions are homogeny and similar or R =.8 and ininite. Interace temperature varies according to the jumping conditions, rom Eq. (12). Variation o pressure is inluenced by the increase o the hydrostatic pressure and transversal radius o curvature o the meniscus. Such inluence yields the negative values observed when R = Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

8 (a) Interace velocity (b) Interace temperature c) Interace pressure.8 Figure 12. Interace velocity, temperature and pressure and temperature at = 2.2, or R =.8 and ininite. Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

9 Table (1) shows the values or dimensionless thermal resistance at t =2.2 or several values o transversal meniscus radius. The values rises with meniscus radius, what may indicate that thinner heat pipes would be relatively more eective or the same luid, operating under same conditions. As it would be expected the value closest to the two-dimensional case is that or ininite radius, when the interace has no transversal variation in its shape. Table 1. Dimensionless thermal resistance as a unction o transversal meniscus radius at t = 2.2, compared with two-dimensional result at t = 1.1. Meniscus Thermal Obs. Radius Resistance Ininite Steady state not reached Steady state not reached 2-D at t = CONCLUSIONS In this work, the Interace Tracking Method was employed to simulate the unsteady 3-D heat transer and phase change process inside a lat capillary heat pipe, represented by a square closed cavity. The code was validated comparing its results to an exact solution o a one-dimensional problem, demonstrating its capacity to represent the interace discontinuity. Such representation improves as more points are added to the Lagrangean mesh. The complete physical model was tested, considering a constant transversal meniscus radius, and the method shown to be able to represent the physical process end capture the dierences due to the variation o the transversal meniscus radius. The radius rising yields a delay in reaching the steady state and increases thermal resistance and reerence pressure. When compared with two-dimensional results, the low patterns or 3-D simulation seem to be more stable, and the reerence pressure or steady state gets higher. Although a rough mesh was employed, the results present physical coherence and are qualitative close to that expected or a heat pipe. Complete test or luids and dimensions ound in easible heat pipes in reined meshes are necessary or comparison to the one-dimensional models and experimental data available in literature, in order to veriy the accuracy o the physical model. The methodology described in this work can be extended easily to irregular geometries. ACKNOWLEDGEMENTS The author would like to acknowledge to FAPESP or the inancial support during this work. REFERENCES Arruda, J. M., 1999, Simulação Numérica dos Processos de transporte, Deormação e Captura de Interaces tridimensionais, Msc. Thesys, Universidade Federal de Uberlândia, Uberlândia Brazil. Cotta, R. M., 1998, The Integral Transorm Method in Thermal and Fluids Science and Engineering, Begell House. Duncan, A. B., and Peterson, G. P., 1994, Review o Microscale Heat Transer, Applied Mechanics Review, Vol. 47, No.. 9, pp Esmaeeli, A., and Arpaci, V., 1998, Numerical Modeling o Three-dimensional Fluid Flow with Phase Change, in: Proc. 4 th Microgravity Fluid Physics & Transport Phenomena Conerence, Cleveland, Ohio, pp Esmaeeli, A., and Tryggvason, G., 1998, Direct Numerical simulation o Bubble Flows - Part 1 Low Reynolds Number Arrays, J. Fluid Mechanics, Vol. 377, pp Esmaeeli, A., and Tryggvason, G., 1999, Direct Numerical simulation o Bubble Flows - Part 2 Moderate Reynolds Number Arrays, J. Fluid Mechanics, Vol. 385, pp Faghri, A., 1995, Heat Pipe Science and Technology, Taylor&Francis. Hopkins, R., Faghri, A., and Khrustalev, D., 1999, Flat Miniature Heat Pipes with Micro Capillary Grooves, J. Heat Transer, Vol. 121, pp Huang, J., Karthikeyan, M., Plawsky, J., Wayner Jr., P.C., 1998, Constrained Vapor Bubble, in: Proc. 4 th Microgravity Fluid Physics & Transport Phenomena Conerence, Cleveland, Ohio, pp Juric, D., 1996, Computations o Phase Change, PhD. Thesis, University o Michigan. Juric, D. and Tryggvason, G., 1998, Computations o Boiling lows, Int. J. o Multiphase Flow, Vol. 24, No. 3, pp Khruhstalev, D., and Faghri, A., 1994, Thermal Analysis o a Micro Heat Pipe, Journal o Heat Transer, Vol. 116, pp Ma, H. B., and Peterson, G. P., 1998, The Minimum Meniscus Radius and Capillary Heat Transport Limit in Micro Heat Pipes, Journal Heat Transer, Vol. 12, pp Machado, H. A., and Miranda, R. F., 21, Simulation o Micro Heat Pipes Using an Interace Tracking Method, in: Proceedings o the VI COBEM - Brazilian Congress o Mechanical Engineering, Uberlândia, Brazil. Machado, H. A., 23, Simulation o a Flat Capillary Heat Pipe Using an Interace Tracking 8 Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p

10 Method, in: Proceedings o the VII COBEM - Brazilian Congress o Mechanical Engineering, São Paulo, Brazil. Peterson, G. P., 1994, An Introduction to Heat Pipes Modeling, Testing and Applications, John Wiley & Sons. Peterson, G. P., Ha, J. M., 1998, Capillary Perormance o Evaporating Flow in Micro Grooves: An Approximate analytical Approach and Experimental Investigation, Journal Heat Transer, Vol. 12, pp Peterson, G. P., and Ma, H. B., 1999, Temperature Response o Heat Transport in a Micro Heat Pipe, Journal Heat Transer, Vol. 121, pp Prat, D. M., Brown, and J. R., Hallinan, K. P., 1998, Thermocapillary Eects on the Stability o a Heated, Curved Meniscus, Journal Heat Transer, Vol. 12, pp Ravigururajan, T. S., 1998, Impact o Channel Geometry on Two-Phase Flow Heat Transer Characteristics o Rerigerants in Microchannel Heat Exchangers, Journal Heat Transer, Vol. 12, pp Shin, S., and Juric, D., 22, Modeling threedimensional Multiphase low Using a Level contour Reconstruction Method or Front Tracking Without Connectivity, J. Comp. Physics, Vol. 18, pp Sun, K. H., and Tien, C. L., 1972, Single Conduction Model or Theoretical Steady-State Heat Pipe Perormance, AIAA Journal, Vol. 1, pp Unverdi, S. O., and Tryggvason, G., 1992, A Front-Tracking Method or Viscous, Incompressible, Multi-luid Flows, Journal o Computational Physics, Vol. 1, pp Engenharia Térmica (Thermal Engineering), Vol. 5 N o 2 December 26 p