Microscale Heat Transfer - Fundamentals and Applications: Proceedings of the NATO Advanced Study Institute on Microscale Heat Transfer - Fundamentals ... II: Mathematics, Physics and Chemistry)

Microscale Heat Transfer Fundamentals and Applications

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Series II: Mathematics, Physics and Chemistry – Vol. 193

Microscale Heat Transfer Fundamentals and Applications

edited by

S. Kakaç University of Miami, Coral Gables, FL, U.S.A.

L.L. Vasiliev Luikov Heat and Mass Transfer Institute, Minsk, Belarus

˘ Y. Bayazitoglu Rice University, Houston, TX, U.S.A. and

Y. Yener Northeastern University, Boston, MA, U.S.A.

Published in cooperation with NATO Public Diplomacy Division

Proceedings of the NATO Advanced Study Institute on Microscale Heat Transfer – Fundamentals and Applications in Biological and Microelectromechanical Systems Cesme-Izmir, Turkey 18 – 30 July 2004 A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 1-4020-3360-5 (PB) ISBN 1-4020-3359-1 (HB) ISBN 1-4020-3361-3 (e-book)

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Printed in the Netherlands.

Table of Contents Preface

vii

Single-Phase Forced Convection in Microchannels – A State-of-the-Art Review Yaman Yener, S. Kakaç, M. Avelino and T. Okutucv

1

Measurements of Single-Phase Pressure Drop and Heat Transfer Coefficient in Micro and Minichannels André Bontemps

25

Steady State and Periodic Heat Transfer in Micro Conduits Mikhail D. Mikhailov, R. M. Cotta and S. Kakaç

49

Flow Regimes in Microchannel Single-Phase Gaseous Fluid Flow Yildiz Bayazito÷lu and S. Kakaç

75

Microscale Heat Transfer at Low Temperatures Ray Radebaugh

93

Convective Heat Transfer for Single-Phase Gases in Microchannel Slip Flow: Analytical Solutions Yildiz Bayazito÷lu, G. Tunc, K. Wilson and I. Tjahjono

125

Microscale Heat Transfer Utilizing Microscale and Nanoscale Phenomena Akira Yabe

149

Microfluidics in Lab-on-a-Chip: Models, Simulations and Experiments Dongqing Li

157

Transient Flow and Thermal Analysis in Microfluidics Renato M. Cotta, S. Kakaç, M. D. Mikhailov, F. V. Castellões and C. R. Cardoso

175

From Nano to Micro to Macro Scales in Boiling Vijay K. Dhir, H. S. Abarajith and G. R. Warrier

197

Flow Boiling in Minichannels André Bontemps, B. Agostini and N. Caney

217

Heat Removal Using Narrow Channels, Sprays and Microjets Matteo Fabbri, S. Jiang, G. R. Warrier, V. K. Dhir

231

Boiling Heat Transfer in Minichannels Vladimir Kuznetsov, O. V. Titovsky and A. S. Shamirzaev

255

Condensation Flow Mechanisms, Pressure Drop and Heat Transfer in Microchannels Srinivas Garimella Heat Transfer Characteristics of Silicon Film Irradiated by Pico to Femtosecond Lasers Joon Sik Lee, S. Park v

273

291

vi

Microscale Evaporation Heat Transfer Vladimir V. Kuznetsov and S. A. Safonov

303

Ultra-Thin Film Evaporation(UTF)-Application to Emerging Technologies in Cooling of Microelectronics Mike Ohadi, J. Lawler and J. Qi

321

Binary–Fluid Heat and Mass Transfer f in Microchannel Geometries for Miniaturized Thermally Activated Absorption Heat Pumps Srinivas Garimella

339

Heterogeneous Crystallization of Amorphous Silicon Accelerated by External Force Field: Molecular Dynamics Study Joon Sik Lee and S. Park

369

Hierarchical Modeling of Thermal Transport from Nano-to-Macroscales Cristina H. Amon, S. V. J. Narumanchi, M. Madrid, C. Gomes and J. Goicochea

379

Evaporative Heat Transfer on Horizontal Porous Tube Leonard Vasiliev, A. Zhuravlyov and A. Shapovalov

401

Micro and Miniature Heat Pipes Leonard L. Vasiliev

413

Role of Microscale Heat Transfer in Understanding Flow Boiling Heat Transfer and Its Enhancement K. Sefiane and V. V. Wadekar Heat Transfer Issues in Cryogenic Catheters Ray Radebaugh Sorption Heat Pipe - A New Device for Thermal Control and Active Cooling Leonard L. Vasiliev and L. Vasiliev, Jr Thermal Management of Harsh-Environment Electronics Mike Ohadi and J. Qi

429 445

465 479

Thermal Transport Phenomenon in Micro Film Heated by Laser Heat Source Shuichi Torii and W. J. Yang

499

Index

507

Preface This volume contains an archival record of the NATO Advanced Institute on Microscale Heat Transfer – Fundamental and Applications in Biological and Microelectromechanical Systems held in Çesme – Izmir, Turkey, July 18–30, 2004. The ASIs are intended to be high-level teaching activity in scientific and technical areas of current concern. In this volume, the reader may find interesting chapters and various Microscale Heat Transfer Fundamental and Applications. The growing use of electronics, in both military and civilian applications has led to the widespread recognition for need of thermal packaging and management. The use of higher densities and frequencies in microelectronic circuits for computers are increasing day by day. They require effective cooling due to heat generated that is to be dissipated from a relatively low surface area. Hence, the development of efficient cooling techniques for integrated circuit chips is one of the important contemporary applications of Microscale Heat Transfer which has received much attention for cooling of high power electronics and applications in biomechanical and aerospace industries. Microelectromechanical systems are subject of increasing active research in a widening field of discipline. These topics and others are the main theme of this Institute. The scientific program starts with an introduction and the state-of-the-art review of single-phase forced convection in microchannels. The effects of Brinkman number and Knudsen numbers on heat transfer coefficient is discussed together with flow regimes in microchannel single-phase gaseous fluid flow and flow regimes based on the Knudsen number. In some applications, transient forced convection in microchannels is important. Steady, periodic and transient-state convection heat transfer are analytically solved for laminar slip flow inside micro-channels formed by parallel-plates, making use of the generalized integral transform technique, Laplace transforms and the exact analytical solution of the corresponding eigenvalue problem in terms of the confluent hypergeometric functions. A mixed symbolic-numerical algorithm is developed under the Mathematica platform, allowing for the immediate reproduction of the results and comprehension of the symbolic and computational rules developed. Analytical solutions for flow transients in microchannels are obtained, by making use of the integral transform approach, and mixed symbolic-numerical algorithm is constructed employing the Mathematica platform. The proposed model involves the transient fully developed flow equation for laminar regime and incomprehensible flow with slip at the walls, in either circular tubes or parallel plate channels. The solution is constructed so as to account for any general functional form of the time variation of the pressure gradient along the duct. In several lectures discuss the measurements of single-phase pressure drop and heat transfer coefficient in micro and mini-channels. Experimental results of pressure drop and heat transfer coefficient of flow boiling are presented in mini-channels. Many correlations for flow boiling heat transfer coefficient in mini-channels have been established. The nature of boiling heat transfer in a channel with the gap less than the capillary is also studied and presented. The condensation flow mechanisms, pressure drop and heat transfer in microchannels, role of microscale heat transfer in augmentation of nucleate boiling and flow boiling heat transfer, binary-fluid heat and mass transfers in microchannel geometries for miniaturized thermally activated absorption heat pumps, evaporation heat

vii

viii

transfer on porous cylindrical tube disposed in a narrow channel, from macro to micro scale boiling are presented in several lectures. In the applications, industrial heat exchanges are mini-and-microscale heat transfers, miniature and micro heat pipes and heat transfer issues in cryogenic catheters are presented. Nanotechnology and heat transfer including heat transfer characteristics of silicon film irradiated by pico to femtosecond lasers are also introduced and discussed. During the ten working days of the Institute, the invited lecturers covered fundamentals and applications of Microscale Heat Transfer. The sponsorship of the NATO Scientific Affairs Division is gratefully acknowledged; in person we are very thankful to Dr. Fausto Pedrazzini director of ASI programs who continuously supported and encouraged us at every phase of our organization of this Institute. Our special gratitude goes to Drs. Nilufer Egrican, Hafit Yuncu, Sepnem Tavman and Ismail Tavman for coordinating sessions and we are very thankful to the Executive and Scientific Secretary Tuba Okutucu and to the Assistant Secretary Melda Koksal for their invaluable efforts in making the Institute a success. A word of appreciation is also due to the members of the session chairmen for their efforts in expediting the technical sessions. We are very grateful to Annelies Kersbergen of Kluwer Academic Publishers for her close collaboration in preparing this archival record of the Institute, to F. Arinc, Secretary General of ICHMT, to the General Scientific Coordinator of this NATO ASI Dr. Mila Avelino, Barbaros Cetin, Ozgur Bayer, Burak Yazicioglu, Cenk Kukrer and to Dr. Wei Sun and Mr. Christian Quintanilla-Aurich for their guidance and help during the entire process of the organization of the Institute. Finally our heartfelt thanks to all lecturers and authors, who provided the substance of the Institute, and to the participants for their attendance, questions and comments.

S. Kakaç L. L. Vasiliev Y. Bayazito÷lu, Y. Yener

SINGLE-PHASE FORCED CONVECTION IN MICROCHANNELS A State-of-the-Art Review Y. YENER1 , S. KAKAC C ¸ 2 , M. AVELINO 2, 3 and T. OKUTUCU1

1

2

Northeastern University, Boston, MA, 02115-5000, USA University of Miami, Coral Gables, FL, 33124-0611, USA

1. Introduction With the recent advances in microfabrication, various devices having dimensions of the order of microns such as, among others, micro-heat sinks, micro-biochips, micro-reactors for modification and separation of biological cells, micro-motors, micro-valves and micro-fuel cells have been developed. These found their applications in microelectronics, microscale sensing and measurement, spacecraft thermal control, biotechnology, microelectromechanical systems (MEMS), as well as in scientific investigations. The trend of miniaturization, especially in computer technology, has significantly increased the problems associated with overheating of integrated circuits (ICs). With existing heat flux levels exceeding 100 W/cm2, new thermal packaging systems incorporating effective thermal control techniques have become mandatory for such applications. The recent developments in thermal packaging have been discussed by Bar-Cohen [5], and experimental, as well as analytical methods have been reported by a number of researchers in Cooling of Electronic Systems, edited by Kakac¸ et al. [16]. The need for the development of efficient and effective cooling techniques for microchips has initiated extensive research interest in microchannel heat transfer. Microchannel heat sinks have been recommended to be the ultimate solution for removing high rates of heat in microscale systems. A microchannel heat sink is a structure with many microscale channels machined on the electrically inactive face of the microchip. The main advantage of microchannel heat sinks is their extremely high heat transfer area per unit volume. Since microchannels of noncircular cross sections are usually integrated in silicon-base microchannel heat sinks, it is important to know the fluid flow and heat transfer characteristics in these channels for better design of the systems. Moreover, the key design parameters like the pumping pressure for the coolant fluid, fluid flow rate, fluid and channel wall temperatures, channel hydraulic diameter and the number of channels in the sink have further to be optimized to make the system efficient and economical. 2. Motivations The use of convective heat transfer in microchannels to cool microchips has been proposed over the last two decades. Many analytical and experimental studies, involving both liquids and gases, have been carried out to gain a better understanding of fluid flow and heat transfer phenomena at the micro level. Experimental studies have demonstrated that many microchannel fluid flow and heat transfer phenomena cannot be explained by the conventional theories of transport theory, which are based on the continuum hypotheses. For friction factors and Nusselt numbers, 3 Mechanical Engineering Department – State University of Rio de Janeiro, 20560-013, Rio de Janeiro, RJ, BRAZIL – [email protected]

1 S. Kakaç et al. (eds.), Microscale Heat Transfer, 1– 24. © 2005 Springer. Printed in the Netherlands.

2

there are a great deal of discrepancies between the classical values and the experimental data. For instance, the transition from laminar to turbulent flow starts much earlier than the classical limit (e.g. from Re=300); the correlations between the friction factor and the Reynolds number are very different from those predicted by the conventional theories of fluid mechanics; and the apparent viscosity and the friction factor of a liquid flowing through a microchannel may be several times higher than those in the conventional theories. Experimental data also appear to be inconsistent with one another. Such deviations are thought to be the results of the rarefaction and compressibility effects mainly due to the tiny dimensions of microchannels, the interfacial electrokinetic effects near the solidfluid interface and various surface conditions, which cannot be neglected in microsystems because of the large surface-to-volume ratio in these systems. These effects significantly affect both the fluid flow and the convective heat transfer. Typically, in macrochannels, fluid velocity and temperature are taken to be equal to the corresponding wall values. On the other hand, these conditions do not hold for rarefied gas flow in microchannels. For gas flow in microchannels, not only does the fluid slip along the channel wall with a finite tangential velocity, but there is also a jump between the wall and fluid temperatures. Several gaseous flow studies have been carried out for the slip flow conditions where, although the continuum assumption is not valid due to the rarefaction effects, Navier-Stokes equations were applied with some modifications in the boundary conditions. On the other hand, there does not seem to be a general consensus among the researchers regarding the boundary conditions for liquid flows. It is not clear if discontinuities of velocity and temperature exist on the channel walls. Therefore, there is still a need for further research for a fundamental understanding of fluid flow and heat transfer phenomena in microchannels in order to explore and control the phenomena in a length scale regime in which we have very little experience. 3. Fluid Flow and Heat Transfer Modeling There are basically two ways of modeling a flow field; the fluid is either treated as a collection of molecules or is considered to be continuous and indefinitely divisible - continuum modeling. The former approach can be of deterministic or probabilistic modeling, while in the latter approach the velocity, density, pressure, etc. are all defined at every point in space and time, and the conservation of mass, momentum and energy lead to a set of nonlinear partial differential equations (Navier-Stokes). Fluid modeling classification is depicted schematically in Fig. 1. Navier-Stokes-based fluid dynamics solvers are often inaccurate when applied to MEMS.

Figure 1.

Classification of fluid modeling.

3

This inaccuracy stems from their calculation of molecular transport effects, such as viscous dissipation and thermal conduction, from bulk flow quantities, such as mean flow velocity and temperature. This approximation of microscale phenomena with macroscale information fails as the characteristic length of the (gaseous) flow gradients approaches the average distance travelled by molecules between collisions - the mean path. The ratio of these quantities is referred to as Knudsen number. 3.1. KNUDSEN NUMBER The Knudsen number is defined as

λ , (1) L where L is a characteristic flow dimension (i.e. channel hydraulic diameter Dh ) and λ is the mean free molecular path, which is given, for an ideal gas model as a rigid sphere, by Kn =

λ= √

¯ kT . 2 πP σ 2

(2)

Generally, the traditional continuum approach is valid, albeit with modified boundary conditions, as long as Kn< 0.1. The Navier-Stokes equations are valid when λ is much smaller than the characteristic flow dimension L. When this condition is violated, the flow is no longer near equilibrium and the linear relations between stress and rate of strain and the no-slip velocity condition are no longer valid. Similarly, the linear relation between heat flux and temperature gradient and the no-jump temperature condition at a solid-fluid interface are no longer accurate when λ is not much smaller than L. The different Knudsen number regimes are delineated in Fig. 2. For the small values (Kn≤ 10−3 ), the flow is considered to be a continuum flow, while for large values (Kn≥ 10), the flow is considered to be a free-molecular flow. The range 10−3 2300 is the most general [17]: Nu =

( Λ//8) Re Pr

D ⎞ ⎛ ⎜1+ h ⎟ L ⎠ 1,07 12,7 7 ( Λ//8) (Pr 2 3 - 1) ⎝

23

⎛ Pr ⎞ ⎟⎟ ⎜⎜ ⎝ Prw ⎠

0 ,11

(43)

where Λ = (0,790 ln (Re) - 1,64)-2 6

for 2300 ≤ Re ≤ 5 10

(44)

and 0.5 < Prr < 2000

The entrance effects are taken into account through the term (1 + Dh / L)2/3 . In the case of a fluid flowing in a plane wall channel, transition seems to occur at higher Reynolds numbers [18]. 2.12

NUSSELT NUMBER AS A FUNCTION OF REYNOLDS NUMBER

If the Nusselt number is plotted as a function of the Reynolds number, the curve in figure 6 is obtained. One can see the constant value for laminar flow. In the turbulent regime the Gnielinski correlation is compared to a Dittus-Boelter type correlation. Nu

(d)

100 (c) (b)

1

,3 4,3 (a)

10

100

10 000

Re

Figure 6. Nusselt number as a function of Reynolds number. (a) Nu = 4.36 (Laminar flow, fixed heat flux) (b) With entrance effects in laminar regime (c) Gnielinski correlation (d) Dittus-Boelter correlation These curves will serve as a reference for the measured values. 3.

Effects involved in pressure drop and heat transfer coefficient modification

3.1

EXPERIMENTAL CONDITIONS AND EXPERIMENTAL ERROR ESTIMATION

A fine review of the experimental conditions which may lead to misinterpretation of results has been carried out by Kandlikar [9]. It is instructive to recall some of those here. 3.1.1

Accuracy of channel geometry measurement

The smaller the channel dimensions, the more the errors involved in length measurements become significant. As an example, for a rectangular channel whose hydraulic diameter is Dh =

2 wh w h

(45)

35

the uncertainty on the Darcy coefficient is

∆Λ ∆h ∆ w ∆ Dh ∆w =2 +2 + ≈7 . Λ h w Dh w

(46)

An error of 2% on a channel dimension can lead to a 14 % error in the Darcy coefficient determination. It is essential to use an adapted instrumentation to measure the geometrical characteristics of a channel. Sometimes, the cross section may not be the same from one end of a channel to the other and, if necessary, the manufacturer’s data must be verified carefully. An uncertainty analysis on the Poiseuille number determination is given by Celata [8] following the work by Holman [19] 3.1.2. Accuracy of pressure measurements The correct position of pressure taps is essential to obtain good measurements. To measure friction losses it is best to locate them far from the inlet and the outlet of the channels to avoid entrance effects. However, if the pressure is measured by means of small holes through the wall it is important to verify that the openings do not disturb the flow streamlines. If some gas or air is to be found between the liquid and the sensor, due to the high pressure reached, dissolution of the gas in the liquid can be observed which will modify the pressure value. 3.1.3

Accuracy of temperature measurements

Several effects can play a role in the temperature measurement accuracy. Due to the small channel length, the temperature difference between the channel outlet and inlet can be as small as the sensor sensitivity. Thermocouples can have a size comparable to the channel dimensions and where is measured the temperature is questionable. Moreover, the heat flow rate through the thermocouple itself can be not negligible. The importance of these effects f must be appreciated. 3.1.4. Entrance region and developing flow effects As pointed out by Kandlikar [9] the entrance conditions can play an important role. If the pressure taps are located before and after the channel in a header with a different diameter or with elbows, the singular pressure losses can be prominent compared to the regular ones. They have been forgotten in some publications. The channels can have a short length L and the ratio L/Dh can be smaller than that in conventional channels. The developing length effects can be considerable. 3.1.5. Maldistribution condition To obtain sufficient heat or mass flow, several channels in parallel can be used. A small defect or a different roughness in a given channel can strongly affect the pressure drop and the flow distribution. The header also can play an important role in flow distribution. 3.1.6

Longitudinal heat conduction

One-dimensional conduction, i-e between the external and internal wall only is the implicit assumption to calculate the Nusselt number from experimental data. In the case of minichannels whose wall thickness can be of the same order of magnitude as the hydraulic diameter this hypothesis may be questionable. 3.2

PHYSICAL EFFECTS

A lot of physical phenomena were advanced to explain the deviation from the classical theories. They will be briefly discussed here in order to compare the plausible mechanisms between them and between experimental results.

36

3.2.1

Variation of physical properties

This variation which is always taken into account for gases is often forgotten for liquids. However, very high fluxes can be obtained to or from small amounts of liquid. Reynolds numbers can be doubled between inlet and outlet of a channel, mainly due to viscosity variation [20]. Such effects could be invoked to explain the decrease in friction factors in heated channels but cannot explain results for isothermal flows. 3.2.2. Viscous dissipation Under the effect of viscosity, the fluid itself can be heated throughout the bulk. The importance of this effect can be appreciated with the help of the Brinkman number Br. It is the ratio between the mechanical power degraded in heat flow and the power transferred by conduction in the fluid. It is written as Br =

µ Vr λ∆T

(47)

where Vr is a reference velocity. If Brr 3 the convective effects are prominent and for BiiL < 0.3 the longitudinal heat flux produces an effect on the temperature profiles. The definition given by Commenge was calculated for counter-current heat exchangers and leads to different values. Evaluating these numbers would be useful in ensuring the heat flux is purely transversal. 6.

Conclusion

The main features of the various theories have been recalled in order to facilitate the understanding of the presented results. The theories invoked to explain the discrepancy between experimental results and conventional theories were listed. To extract the proper interpretation of the different experiments, new experimental work was carried out to eliminate parasitic effects. Concerning the friction factor, the experiments aim to eliminate (i) the entrance effects (ii) the effects of the pressure tap positioning (iii) the effect of the ion concentration of the fluid. It was shown that, down to the characteristic dimension of 7 µm and for the fluids used, the hydrodynamics obey the conventional theories deduced from the Navier - Stokes equations. The effect of roughness on the flow behaviour needs complementary work. Concerning the heat transfer, the experimental difficulties must be underlined. When dimensions become smaller the heat flow does not go directly through the walls. For very small dimensions and temperature difference, the heat transfer coefficients are subject to large uncertainties. The use of a longitudinal Biot number can be of help to estimate the heat flow which may not be used to calculate the heat transfer coefficient. Acknowledgments.. The author thanks B. Agostini, F. Ayela, R. Bavière, S. Le Person, M. FavreMarinet for their results.

47

REFERENCES 1. Papautsky, I., Brazzle, J., Ameel, T., Bruno Frazier, A., Laminar fluid behavior in microchannels using micropolar fluid theory, Sensors an Actuators, Vol. 75, pp. 101-108, (1999). 2. Agostini, B. Watel, B., Bontemps, A. and Thonon. B., Effect of geometrical and thermophysical parameters on heat transfer measurements in small diameter channels. GRETh Grenoble. Internal report 2003. (Unpublished). 3. Gravesen, P., Branebjerg,J., Jense, O.S., Microfluidics – A review, J. Micromech. Microeng., Vol. 3, pp. 168 – 182, (1993). 4. Sabry, M.-N., Scale effect on fluid flow and heat transfer in microchannels, IEEE Transactions on components and packaging technologies, Vol. 23, N°. 3, pp. 562 – 567, (2000). 5. Gad-el-Hak, M., The fluid mechanics of microdevices, J. Fluid Engineering, Vol. 121, pp. 5 – 33, (1999). 6. Obot, N.T., Towards a better understanding of friction and heat/mass transfer in microchannels – A literature review., Proc. Int. Conf. On Heat Transfer and Transport Phenomena in Microscale, Banff, Canada, October 15-20, (2000). 7. Palm, B., Heat transfer in microchannels, Microscale Thermophys. Engineering, Vol. 5, pp. 155 – 175, (2001). 8. Celata, G.P., Single-phase heat transfer and fluid flow in micropipes. 1stt Int. Conf. on Microchannels and Minichannels, Rochester, N.Y., April 24-25, (2003). 9. Kandlikar, S.G., Microchannels and Minichannels – History, Terminology, classification and current research needs. 1st Int. Conf. on Microchannels and Minichannels, Rochester, N.Y., April 2425, (2003). 10. Mehendale, S.S., Jacobi, A.M., Shah, R.K., Fluid flow and heat transfer at micro- and mesoscales with application to heat exchanger design, Appl. Mech. Rev., Vol. 53, pp. 175 – 193, (2000). 11. Anduze, M., Etude expérimentale et numérique de microécoulements liquides ddans les microsystèmes fluidiques, Ph. D. Thesis, Institut National des Sciences Appliquées, Toulouse, (2000). 12. Mills A.F., Heat transfer, Irwing, Boston, USA, (1992). 13. Deissler, R.G., An analysis of second-order d slip flow and temperature--jump boundary conditions for rarefied gases, Int. J. Heat Mass Transfer, Vol. 7, pp. 681-694, (1964). 14. Shah, R.K., London, A.L., Laminar forced convection in ducts, Advanced heat transfer, Academic Press, New York, (1978). 15. Sparrow, E.M., N.A.C.A. note N° 3331, (1955). 16. Idelchick, I.E., Handbook of hydraulic resistance, Hemisphere Publishing Corporation, 2d edition, (1986). 17. Gnielinski, V., New equations for heat and mass transfer in turbulent pipe and channel flow, Int. Chemical Engineering, Vol. 16, N° 2, pp. 359 – 368, (1976). 18 Carlson, D.R., Widnall, S.E., Peeters, M.F., A flow visualisation study of transition in plane Poiseuille flow, J. Fluid Mech. Vol. 121, pp. 487 – 505, (1982).

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19. Holman, J.P., Experimental methods for engineers, Mc Graw Hill, (1978). 20. Wang, B.X., Peng, X.F., Experimental investigation on liquid forced convection heat transfer through microchannels, Int. J. Heat Mass Transfer, Vol. 37, pp. 73 –82, (1994). 21. Tso, C.P., Mahulikar, S.P., The role of the Brinkman number in analysing flow transitions in microchannels, Int. J. Heat Mass Transfer, Vol. 42 pp. 1813 – 1833, (1999). 22. Mala, G.M., L.I, D., Werner, C., Jacobasch, H.J., Ning, Y.B., Flow characteristics of water through a microchannel between two parallel plates with electrokinetic effects, Int. J. Heat Fluid Flow, Vol. 18, pp. 489 – 496, (1997). 23. Choi, C.-H., Westin, K.J.A., Breuer, K.S., Apparent slip flows in hydrophilic and hydrophobic microchannels, Physics of fluids, Vol. 15, N° 10, pp. 2897 – 2902, (2003). 24. Yang, J., Kwok, D.Y., Electrokinetic slip flow in microfluidic -based heat exchangers with rectangular microchannels, Int. J. Heat Exchangers, Vol. 5, pp. 201 – 220, (2004). 25. Gao, P., Le Person, S., Favre-Marinet, M., Scale effects on hydrodynamics d and heat transfer in two-dimensional mini and microchannels. Int. J. Thermal Sciences, Vol. 41, pp. 1017 – 1027, (2002). 26. Bavière, R., Ayela, F., Le Person, S. and Favre-Marinet, M., An experimental study of water flow in smooth and rectangular micro-channels. To be published. 27. Agostini, B., Watel, B., Bontemps, A. and Thonon, B., (2004), Liquid flow friction factor and heat transfer coefficient in small channels: an experimental investigation. Experimental Thermal and Fluid Science, Vol. 28, pp. 97-103 28. Hu, Y., Werner, C., Li, D., Influence of three-dimensional roughness on pressure-driven flow through microchannels, J. Fluids Engineering, Vol. 125, pp. 871 – 879, (2003). 29 Bavière, R., Ayela, F., Micromachined strain gauges for the determination of liquid flow friction coefficients in microchannels, Measurements science and technology, Vol. 15, pp. 377 – 383, (2004). 30. Agostini, B., Etude expérimentale de l’ébullition de fluide réfrigérant en convection forcée dans les mini-canaux, Ph. D., Thesis, Université Joseph Fourier, Grenoble, (2002). 31. Commenge, J.M., Réacteurs microstructurés : hydrodynamique, thermique, transfert de matière et applications aux procédés, Ph.D. Thesis, INP Lorraine, Nancy, (2001). 32. Morini, G. L., Laminar-to-turbulent flow transition in microchannels, Microscale Thermophysical Engineering, Vol. 8, pp. 15-30, (2004).

STEADY STATE AND PERIODIC HEAT TRANSFER IN MICRO CONDUITS

M. D. MIKHAILOV, R. M. COTTA, S. KAKAÇ Mechanical Engineering Department, Universidade Federal do Rio de Janeiro Rio de Janeiro, Brasil  Department

of Mechanical Engineering - University of Miami

Coral Gables, Florida, USA

1.

Introduction

The modern microstructure applications led to increased interest in convection heat transfer in micro conduits. Fluid transport in micro channels has found applications in a number of technologies such as biomedical diagnostic techniques, thermal control of electronic devices, chemical separation processes, etc. Experimental results have been published for micro tubes [1], micro channels [2], and micro heat pipe [3]. The micro scale experimental results differ from the prediction of conventional models. Some neglected phenomena must be taken into account in micro scale convection. One of them is the Knudsen number defined as the ratio of the molecular mean free path to characteristic length of the micro conduit. In the paper by Barron et al. [4], a technique developed by Graetz in 1885 [5] is used to evaluate the eigenvalues for the Graetz problem extended to slip-flow. The first 4 eigenvalues were found with precision of about 4 digits. The method used appears to be unstable after the fifth root, so that only the first 4 eigenvalues are reliable. The authors of the paper [4] concluded that an improved method with enhanced calculation speed would be of future interest. In reality the extended to slip-flow egenproblem has exact solution in terms of hypergeometric function and more efficient numerical methods for its solution are also available [6, 7, 8]. As demonstrated by Mikhailov and Cotta [9] the eigenvalues could be computed with specified working precision by using Mathematica software system [10], but the Mathematica rule given in [9] needs a small modification to by applicable for high-order eigenvalues. Heat transfer by forced convection inside micro tube, generally referred as the Graetz problem, has been extended by Barron et al. [11] and Larrode and al. [12] to include the velocity slip described by Maxwell in 1890 [13] and the temperature jump [14] on tube surface, which are important in micro scale at ordinary pressure and in rarefied gases at low-pressure. The paper by Barron et al. [11] use the first 4 eigenvalues from the above mentioned communication by Barron et al. [4] to analyze the heat convection in a micro tube. The temperature jump, although explicitly mentioned in the text, is ignored in the calculation of the eigenvalues. Therefore the temperature distribution didn't take into account the temperature jump. The correct solution of heat convection in circular tubes for slip flow, taking into account both - the velocity

49 S. Kakaç et al. (eds.), Microscale Heat Transfer, 49 – 74. © 2005 Springer. Printed in the Netherlands.

50

slip and the temperature jump, is given by Larrode, Housiadas, and Drossinos [12]. These authors introduce a scaling factor that incorporate both rarefaction effect and gas-surface interaction parameters and develop uniform asymptotic approximation to high-order eigenvalues and eigenfunctions. Heat transfer in microtubes with viscous dissipation is investigated by Tung and Bayazitoglu [15]. The temperature jump, is ignored in the calculation of the temperature distribution, but taken into account in determination of the Nusselt number. Conventional pressure driven flow requires costly micro pumps giving significant pressures [16]. A micro scale electro-osmotic flow is a viable alternative to pressure-driven flow, with better flow control and no moving part [17]. Liquid is moved relative to a micro channel do to an externally applied electric field. This phenomena is first reported by Reuss in 1809 [18]. The fully developed velocity distribution in micro parallel plate channel and micro tube are well known [19]. Using a fully developed velocity one could investigate thermally developing heat transfer and its limiting case - thermally developed heat transfer. The corresponding solutions for electro-osmotic flow in micro parallel plate channel and micro tube are special cases from the general results given in the book by [20]. Thermally fully developed heat transfer do to electro-osmotic fluid transport in micro parallel plate channel and micro tube has been recently investigated by [21]. The dimensionless temperature profile and corresponding Nusselt number have been determined for imposed constant wall heat flux and constant temperature. The complement paper [22] study the effect of viscous dissipation. These two papers gives important physical details and references. The analyses of both papers is based on the classical simplifying assumptions that are avoided in the book by Mikhailov and Ozisik [20]. The conventional laminar forced convection in conduits at periodic inlet temperature is investigated mainly by Kakaç and coworkers [23, 24, 25, 26, 27]. The periodic heat transfer in micro conduits, to the our knowledge, is not investigated. The solutions given here, are special cases from the general results for temperature distribution, average temperature and Nusselt numbers presented in the book [20]. Nevertheless all formulae have been derived again by using Mathematica software system [10]. Mathematica package is developed that computes the eigenvalues, the eigenfunctions, the eigenintegrals, the dimensionless temperature, the average dimensionless temperature, and the Nusselt number for steady state and periodic heat transfer in micro parallel plate channel and micro tube taking into account the velocity slip and the temperature jump. Some results in form of tables and plots are given bellow. For electro-osmotic flow only the limiting Nusselt numbers for thermally fully-developed flow in parallel plate channel and circular tube are obtained as a special case from the solution for thermally developing flow.

2.

Slip Flow Velocity in Parallel Plate Micro-Channel

Consider a fully developed steady flow of an incompressible constant property fluid inside a micro-channel. Let z (0†z0 is zero i. e. without electric field there is no osmotic movement. For curiosity let us find normalized velocity usually used in conventional heat transfer analyses. W#R'

U#R' s Uav

(49)

Introducing eq. (46) and eq. (48) into eq. (49) we obtain: W#R'

+1  I0 #R Z' s I0 #Z'/ s +1  2 I1 #Z' s +Z I0 #Z'//

(50)

For Z=ˆ the normalized velocity profile correspond to slug flow W[R]=1. For Z=0 the limit gives W[R]m 2 (1-R2 ). At Z=0 the osmotic movement is zero, but if the average velocity exist it is the Poiseuille parabola.

6.

Steady State Heat Transfer in Micro Conduits

Consider steady-state heat transfer in thermally developing, hydrodynamically developed forced laminar flow inside a micro conduits (parallel plate micro channel or micro tube) under following assumptions: Ë The fluid is incompressible with constant physical properties. Ë The free heat convection is negligible. Ë The energy generation is negligible. Ë The entrance temperature is uniform. Ë The surface temperature is uniform. The temperature T[r,z] of a fluid with velocity profile u[r], diffusivity D along the channel 0†z†ˆ in the region 0†r†r1 is described by the following problem [ 20 ]: ™ T#r, z' u#r' cccccccccccccccccccccccc ™z

™2 T#r, z' n ™ T#r, z' \ L M cccccccccccccccc ] DM ccccccccccc  cccc cccccccccccccccccccccccc ] M ] 2 ™ r r ™r N ^

(51)

where n=0 for parallel plate micro channel and n=1 for micro tube. The boundary conditions at the center of the micro conduits is: ™ T#0, z' cccccccccccccccccccccccc ™r

0

for

n

0,

™ T#r, z' \ L M Mr cccccccccccccccccccccccc ] ] ™r N ^r!0

0

for

n

1

(52)

The boundary condition (52 a) is commonly used for both - parallel plate channel and tube. The correct boundary condition for cylindrical geometry is given by eq. (52 b) [31].

59

The surface temperature of the micro conduits is Ts. As result of the temperature jump on the surface the boundary condition at r1 becomes:

T#r1, z'

™ T#r1, z' Ts  2 Kn r1 Et cccccccccccccccccccccccccc ™r

(53)

where Et is ((2-Dt )/Dt )(2 J/(J+1))/Pr Dt is the thermal accommodation coefficient. O

is the molecular mean free path.

J

is the ratio of specific heat at constant pressure cp and specific heat at constant volume cv .

Kn is the Knudsen number. The entrance temperature is: T#r, 0'

Ti

(54)

To simplify eqs. (51) to (54) we define the dimensionless velocity W[R] and dimensionless temperature T[Y,Z] as: W#R'

u#r' s uav,

T#R, Z'

+T#r, z'  Ts/ s +Ti  Ts/

(55)

where R is the transverse coordinate, Z is the axial coordinate: R

r s r1,

Z

z D s +C0 r12 uav/

(56)

The eqs. (51) in dimensionless form becomes: W#R' ™ T#R, Z' ccccccccccccc cccccccccccccccccccccccc C0 ™Z

™2 T#R, Z' n ™ T#R, Z' ccccccccccccccccccccccccccc  cccc cccccccccccccccccccccccc R ™ R2 ™R

(57)

The dimensionless velocity for parallel plate micro channel, eqs. (10), and micro tube eqs. (22 ) could be unified as: W#R'

C0 +1  4 KnEv  R2 /,

where

C0

1 s +2 s +n  3/  4 KnEv/

(58)

where n=0 for parallel plate micro channel, and n=1 for micro tube. After introducing the velocity (58) into eq.(57) we obtain: ™ T#R, Z' +1  4 KnEv  R2 / cccccccccccccccccccccccc ™Z

™2 T#R, Z' n ™ T#R, Z' ccccccccccccccccccccccccccc  cccc cccccccccccccccccccccccc ™ R2 ™R R

(59)

The eqs. (52) to eq.(54) in dimensionless form become: ™ T#R, Z' \ L M MRn cccccccccccccccccccccccc ] ] ™R N ^R!0

0,

™ T#1, Z' T#1, Z'  2 KnEv E cccccccccccccccccccccccc ™R

(60) 0,

T#R, 0'

1

60

In eq.(60 b) the term Kn*Et is replaced by KnEv*E, where E=Et/Ev. The problem given by eq.(59) subject to the conditions (60) is referred to as extended Graetz problem in honor of the pioneering work [5]. To solve this problem we need the eigenvalues m and the eigenfunctions y[R] of the eigenproblem: n y…… #R'  cccc y… #R'  +1  4 KnEv  R2 / m2 y#R' 0, R y#1'  2 KnEv E y… #1' 0 +Rn y… #R'/R‘0 0,

(61)

The solution of eq.(61 a) that satisfies the boundary condition (61 b) is y#R'

Exp#m R2 s 2' 1 F1#+n  1  +1  4 KnEv/ m/ s 4; +n  1/ s 2; m R2 ' (62)

where 1F1[a;b;c] is the Kummer confluent hypergeometric function. Introducing the eq. (62) in the boundary condition (61 c) we obtain the eigencondition: +n  1/ +1  2 KnEv E m/ 1 F1#+n  1  +1  4 KnEv/ m/ s 4; +n  1/ s 2; m'  2 KnEv E m +n  1  +1  4 KnEv/ m/ 1 F1#+n  5  +1  4 KnEv/ m/ s 4; +n  3/ s 2; m' 0

(63)

The roots of (63) gives the desired eigenvalues. The FindRoot function of Mathematica software system calculates these roots starting from the values given by the asymptotic formula on p.113 of the book [20]. Fig. 5 shows the seconds per eigenvalue spend on 3 Gz computer to find 100 roots of a slightly modified eq.(63). The first 50 roots are computed much faster than the last 50 roots.

Sec 3 2.5

2

1.5 1

0.5 i 20

Fig. 5

40

60

80

100

CPU time in seconds per root of eq. (62) on 3 Gz PC for n=1, KnEv=0.1 and E=10.

The solution of the extended Graetz problem, eqns. (59, 60), is a special case from the solution given by Mikhailov and Ozisik in the book [20]:

61

n

T#R, Z'

Å A#i' y#i'#R' Exp#Z m#i' ^ 2'

(64)

i 1

The dimensionless axial coordinate defined by eq. (56 b), after taking into account eq.(58 b) could be rewritten as: Z

4 +2 s +3  n/  4 KnEv/ X

(65)

where X is the axial distance expressed through Pecklet number Pe = uav*d/D with characteristic length d=2 r1. zsd X m cccccccccccc Pe

(66)

Than the dimensionless temperature given by eq.(64) could be rewritten as: n

T#R, X'

Å A#i' y#i'#R' Exp#4 +2 s +3  n/  4 KnEv/ X m#i' ^ 2'

(67)

i 1

The constants A[i] in the solution (67) are given by: 1

A#i'

¼0 Rn +1  4 KnEv  R2 / y#i'#R'Å R cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccccccc 1 2 ¼0 Rn +1  4 KnEv  R2 / y#i'#R' Å Y

(68)

The dimensionless average temperature Tav[Z] is defined as: 1

¼0 Rn +1  4 KnEv  R2 / T#R, X'Å R cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc ccccccccccccc 1 ¼0 Rn +1  4 KnEv  R2 / Å R

Tav#X'

(69)

Introducing T[R,X] from eq. (67) into eq.(69) we obtain: n

Tav#X'

Å Aav#i' Exp#4 +2 s +3  n/  4 KnEv/ X m#i' ^ 2'

(70)

i 1

where 1

Aav#i'

¼0 Rn +1  4 KnEv  R2 / y#i'#R'Å R cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccccccc A#i' 1 ¼0 Rn +1  4 KnEv  R2 /Å R

(71)

The heat transfer coefficient h[z] is determined from the balance equation:

h#z' +Tav#z'  Ts/

™ T#r1, z' k cccccccccccccccccccccccccc ™r

The Nusselt number Nu[X]=h[z]*(2r1)/k is given by:

(72)

62

™ T#1, X' 2  cccccccccccccccccc cccccccccccccccccccccccc Tav#X' ™R

Nu#X'

(73)

Introducing eqs. (67) and (70) into eq. (73) we obtain the Nusselt number: Nu#X' ½ni 1 A#i' y#i'… #1' Exp#4 +2 s +3  n/  4 KnEv/ X m#i' ^ 2' 2 cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc ccccccccccccccccc ½ni 1 Aav#i' Exp#4 +2 s +3  n/  4 KnEv/ X m#i' ^ 2'

(74)

For large Z only the first terms of both sums in eq. (74) has to be taken into account. Than we obtain: 1

Nu#ˆ'

¼0 Rn +1  4 KnEv  R2 / Å R 2 cccccccccccccccccccccccccccccccc cccccccccccccccccccccccccccccccc ccccccccccccccccc y#1'… #1' 1 ¼0 Rn +1  4 KnEv  R2 / y#1'#R' Å R

(75)

The integrals in eq. (75) have exact solutions and the limiting Nusselt number becomes:

Nu#ˆ'

1 \ L 1  4 KnEv 2M M ccccccccccccccccccccccccc  cccccccccccc ] ] m#1'2 1n 3n ^ N

(76)

5 4 3 Nu#ˆ' 2 1 0

0 0 05 0.0 0.05

2.5

0.1 KnE Ev v 0.15

5 E 7.5 10 0.2

Fig. 6 The limiting Nusselt number as function of KnEv and E.

Nu#ˆ' E 0 5

E 0.05

4

E 0.2

3

E 0.5 E 1

2 1

E 3 E 10 KnEv 0.2

0.4

0.6

0.8

1

Fig. 7 The limiting Nusselt number as function of KnEv and parameter E .

63

The limiting Nusselt number is of great practical interest. For n=0 (parallel plate micro channel) and n=1 (micro tube) the limiting Nusselt number depend on 2 parameters: KnEv and E. The KnEv control mainly the velocity slip and have influence on the temperature jump. The parameter E control only the temperature jump. The limiting Nusselt number is shown on Fig 6. In the paper [12] is discovered that for E=0 Nusselt number increases with increasing of KnEv. At large E=10 this behavior is reversed. To understand this phenomena we plot on the Fig. 7 the Nu[ˆ] versus KnEv from 0 to 1. We see that the curve pass through a maximum. In the interval of practical interest, 0