No.07-5 日本機械学会熱工学コンファレンス 2007 講演論文集〔2007.11.23-24, 京都〕

Copyright©2007 社団法人 日本機械学会

熱流動のスケール効果とナノ・マイクロマシン Scale Effect in Heat and Fluid Flow and Nano- Micro Machines

正 円山 重直（東北大 流体科学研究所）

Shigenao Maruyama, Institute of Fluid Science, Tohoku University, Katahira 2-1-1, Aoba-ku, Sendai 980-8577 Micro and nano-scale systems are explained and these are categorized in terms of heat and fluid flow phenomena. Some typical application of nano and micro machines are presented. Heat and fluid flow phenomena of nano-micro spatio-temporal systems are described as follows Micro-system： • The size of the machine element is from 1 mµ to 1 mm. • The fluid motion and heat transfer can be treated as continuum, and is described by Navier-Stokes equations and

Fourier’s law. Nano-system： • The fluid can not be treated as continuum any more. • Discontinuity and wave characteristics of fluid and energy have to be considered.

Key Words: Nano-system, Micro-scale, MEMS, NEMS, Scale Effect, Heat and Fluid Flow

1. Introduction Recently remarkable development has been achieved in nano-technologies. The governments of Japan and other countries set special budgets to research the nano-technologies. This is the one of the reason which many researches have been carried on in nano-technology areas. The nano-technologies cover the biotechnology, materials science, energy, and environments. They also cover very wide ranges of phenomena, i.e. nano-scale phenomena comprised of small number of atoms and particles such as fullerenes and carbon nano-tubes, and ultra fast phenomena such as a femto-seconds pulse laser.

Nano and micro-scale phenomena are also related many areas in heat and fluid flow researches. These phenomena are not special in small scale technologies but affect the normal scale and large scale heat and fluid flow systems. In the present article, micro and nano-scale phenomena are explained. These are categorized in terms of heat and fluid flow phenomena. 2. Micro Scales and Their Transport Mechanism Figure 1 shows the scale effect of the movement mechanism of a big fish and a spermatozoon [1]. The fish moves by ejecting the momentum of the fluid backward and thereby obtains thrust. On the other hand, water is too viscid to eject the fluid backward for the spermatozoon.

The Reynolds numbers are 11 orders of magnitude different from each other. Hence, the spermatozoon moves by utilizing the friction between the liquid and its body like a snake.

Figure 1 Transport mechanism of a spermatozoon and

a big fish [1].

This is an example of the scale effect between normal scale and micro scale. Namely, the appearances of the heat and fluid flow may be similar in the different scales; however, these mechanisms are completely different. As for the optimum shape or mechanism, heat and fluid flow systems have different shapes and mechanisms for each scale.

Figure 2 Optimum airfoil shape and scale effect of

various wing sections [2]. Figure 2 shows the maximum lift-drag ratios of various wings [2]. The Reynolds number of a large airliner is of the order of 108. Smooth airfoils show high performances at high Reynolds number; however, the performance deteriorates when the Reynolds number decreases. The airfoil optimized at high Reynolds number sometimes shows worse performance

than that of a simple flat plate. We do not need a streamlined airfoil for a small size wing, such as that of an insect.

Figure 3 Micro turbine operated at pressure ratio 5

and at 2.6x106 rpm, (rotating velocity 500m/s) [3].

Due to abrupt development in micro-fabrication

technology, micro machines can be available. A micro gas turbine, as shown in Fig. 3, was made using device technology [3]. The Reynolds number at operation condition is 43 104.2106.1 ××= ～Re . According to Fig. 2, the performance of the airfoil deteriorates at the Reynolds numbers. The performance of the gas turbine may not be high if the same shape as that of a conventional large gas turbine is adopted. A micro system shows an interesting feature in heat and mass transfer phenomena. Consider heat transfer from a sphere in a gas flow. The heat transfer is characterized by the heat transfer coefficient. At the smallest limit of the sphere diameter, the Nusselt number approaches 2, as shown in Eq. (1). This can be derived from the analogy of heat conduction [4].

2)/)(/[/ 0 →−== dTTkAQkhdNu & (1)

According to Eq. (1), the heat transfer coefficient becomes very large when the diameter of the sphere decreases. Table 1 shows the heat transfer coefficient of spheres of various diameters placed in air flow. The heat transfer coefficient of a sphere of 10 µ m shows a value similar to that of one produced by boiling or condensation heat transfer. Table 1 Heat transfer coefficient of a sphere in air

flow at 1 m/s [1].

Porous media or fibrous media are comprised of a micro structure consisting of fiber or small particles of a few µ m. These media have a large surface area per unit volume. Accordingly, porous media show good heat transfer performances. These characteristics were applied to active thermal insulation [5] as shown in Fig. 4. Fibrous medium comprised of ceramic fibers of a few µ m protects very large radiative heat flux

within 5 mm [6]. This active thermal insulation can be applied to the thermal protection of the leading edges of a Scramjet [7] and re-entry vehicles, as shown in Fig. 5.

Figure 4 Concept and performance of active thermal

insulation comprised of fibrous micro structures and an injection of gas.

Figure 5 Application of an active thermal insulation

layer to the thermal protection of the leading edges of a hyper sonic airplane.

Heat transfer enhancement in a micro structure

sometimes deteriorates the performance of micro machines. As an example, a micro gas turbine, as shown in Fig. 3, suffers heat leakage during the expansion process, which is usually an adiabatic process in a normal size gas turbine. Moreover, combustion in a micro gas turbine has more serious problems in achieving an effective chemical reaction. The flame may suffer quenching in the combustor and the residual time is too short to complete the chemical reaction.

Transient heat transfer can be characterized by Fourier number; 2/ LtFo α= , and the characteristic time constant is inversely proportional to the square of length scale L. In other words, a micro machine can react very quickly. One of the successful examples of micro-machines is an ink-jet printer. As shown in Fig. 6, ink in a micro-channel is heated and boiled very rapidly [1]. Then a drop of the ink is ejected from the nozzle. The scale of the nozzle is 1-10 µ m, the volume of the drop is a few pico-liters. The time-scale of the micro channel is very short, because the Fourier number is an inverse of the square of the characteristic length. Then

(a) Concept of activethermal Insulationinsulation

(b) Performance of active thermal Insulation compared with that of a conventional one

the ink-drop is ejected at more than 510 times per second. If the ink-jet has a scale of mm, this system cannot work.

Figure 6 Mechanism of ink-jet printer and free

nucleation boiling.

Figure 7 shows multiple scales phenomena in heat and mass transport. Diffusion fields around a crystal under normal and micro gravity conditions were observed by a phase-shifting Mach-Zehnder interferometer [8], [9]. The view areas were less than 1 mm2, and the mass diffusion boundary was less than 20-30 µ m. Double diffusive convection occurs in a small scale after a short duration of microgravity (10 sec). The transport phenomena under normal and microgravity conditions can be described by conventional Navier-Stokes equations; however, nano-scale consideration is needed at the boundary between a crystal surface and solution [10] to describe the crystal growth. As an example of mega-scale heat and mass transfer, upwelling of deep seawater by Stommel’s perpetual salt fountain is depicted in Fig. 7. The author succeeded in measuring the upwelling velocity in an ocean experiment [11]. The pipe diameter was 0.3 m and the length was 300 m. This mega-scale natural convection can be described by conventional transport equations by introducing a turbulent diffusion coefficient which is five orders of magnitude larger than the molecular diffusion coefficient and viscosity [12]. Figure 7 shows DNA molecule selection using the ratcheting electrophoresis [13] in a microchip. Micro-scale phenomena, such as the diffusion of big molecules in solution, are described by the Langevin equation [14], by which the Brownian motion of DNA can be predicted.

Figure 7 Multi scale modeling in micro to mega-scales

phenomena. It is important to realize that a micro system may

not work by a simple scaledown of conventional machines. Micro machines or micro systems have to have appropriate principles and optimum shapes applied in order to make them work. All the systems of micro heat and fluid flow discussed above, however, follow the classic laws of fluid dynamics. Namely, the non-slip condition of fluid on a body is applicable, and fluid can be treated as continuous. We can still use Navier-Stokes equations in the micro heat and fluid flow system. 3. Nano Scales and Their Transport Mechanism In Micro heat and fluid flow, fluid can be treated as continuous. On the other hand, an assumption of continuous fluid cannot be applied to nano-scale heat and fluid flow. One has to take into account the slip of the molecules on the solid body. In addition, the characteristics of particles and discontinuity of energy appear in nano-systems.

One of the familiar examples of nano-machines is the hard disk drive of a computer. The minimum distance between the flying head and the hard disk is 10 nm [15]. Since the mean free path of air at atmospheric pressure is 64 nm, the distance of the hard disk is much smaller than the mean free path. This system has to be considered the slip of the molecules on the surface. Nano-scale phenomena are not unique to small scale technologies but also affect normal scale and large scale heat and fluid flow systems.

Figure 8 Multi-scale phenomena of flow dynamics

between solid interfaces.

Flow interaction between surfaces shows different features according to the scales of the phenomena. In the scale of cm to m, flow interaction can be described by the boundary layer and aerodynamic interaction. As shown in Fig. 8, the ground effect of aero-rain [16] is one of the typical examples of normal-scale flow interaction. CVD polycrystalline diamond can be easily polished to make flat surface with random nano-texture [17]. The friction coefficient between polished diamond and metal counterparts becomes low [18] as shown in Fig. 8. At relatively high slipping velocity, air plays a lubricant fluid and a slider hovers on the metallic plate. In this region, the flow can be treated as a continuum fluid. When the distance between each surfaces decreases, the sliding of atomic-scale can be found at the boundary between solid and water molecules. The thickness of the molecule layer is less than nm.

Figure 9 shows the velocity and temperature distributions of a shear flow in a very narrow channel [19]. Analysis was carried out by molecular dynamic simulation. The liquid is water and the wall is platinum crystal. The thickness of the channel is a few nm. Temperatures in terms of molecular transition and rotation are depicted in Fig. 9. It is noted that the velocity and temperature distributions change according to the surface condition of the solid. Large temperature and velocity jumps can be observed in the case of a (111) solid surface, however, the water can be treated as almost continuous in the case of a (110) surface.

Figure 9 Velocity and temperature distributions in a

nano slider [19].

It is known that the beautiful blue Morpho butterfly does not have blue color on the surface of its wings. The scales on the wings have a nano-structure [20]. The scales reflect shorter wavelength visible light more than longer wavelength because the regular nano-structure is optimized for reflection of blue light. Light behaves as waves against this nano-structure. Most engineering machine elements are much larger than the wavelength of thermal radiation. The radiation can be treated as energy rays in the radiative energy transfer of such a system. When the structure becomes small and the size becomes comparable with the wavelength, we cannot treat the radiation as energy rays, and wave characteristics appear [21]. The well-known Planck’s law of black body emission introduces the quantum hypothesis of energy level to derive the law theoretically [21]. It is not very well-known that Planck introduced another assumption which Rayleigh and Jeans also introduced. The assumption is that “the cavity is much larger than the wavelength of thermal radiation”. What happens to the thermal emission if the cavity is smaller or of comparable size to the wavelength of thermal radiation? The authors made accurately constructed highly ordered arrays of small cavities using device technology, as shown in Fig. 10. The thermal radiation emitted form the array of small cavities was measured by a specially constructed FT-IR spectrometer [22], [23].

Figure 10 Black body emission from a nano-structure

0

0.1

0.2

0.3

0.4

0 5 10 15u [m/s]

µ

Ra_m= 0.36 [µm] 　　 　0.43　 　　0.50　　　 0.57

Figure 11 (a) SEM photographs of the nano-structures

of highly ordered cavities, (b) spectral emissivity of nano-cavities with various cavity depths and (c) spectral emissivity with normalized cavity widths.

The Spectral emissivity from the cavities is depicted in Fig. 11. Figure 11-(b) compares with the case of a flat surface. The spectral emissivity shows a larger value at the wavelength of cavity resonance. The result of Fig. 11-(c) shows that the spectral emissivity can be controlled by changing the size of the cavities, and that it does not depend on the characteristics of the material. The structure is comprised of 5 µ m cavities; however, the wavelength of the thermal radiation is of a similar order. It is noted that the wave characteristics can appear even if the structure is larger than 1 µ m if the wavelength of the radiation is comparable with the structure. After our findings, Yugami et al. followed up on our work, and nano-cavities of a high temperature resisting material were utilized for the effective emission of relatively short wavelength radiation [24]. Recently, similar investigation and numerical analysis was carried out by Hanamura and Kameya [25] The heat conduction of electrical non-conducting material or dielectric material can be described by phonon transfer. Heat conduction in a normal scale or micro scale system can be described by Fourier’s law; however, when the size of the element becomes of the order of the phonon wavelength, Fourier’s law breaks down. The phonon wavelength of the non-conducting material and gas can be expressed, as follows:

3/Λ= Cvk (2)

where C is volumetric heat capacity, v is velocity of phonon or gas-molecules, and Λ is the mean free path lengths of phonon or gas molecules.

The order of Λ for silicon crystal and water at normal temperature, are the order of 260 nm and 0.3 nm, respectively. The Λ for air at standard condition is 64 nm. The scale of the system compared with the mean free path is the characteristic parameter regardless of whether a continuum condition, such as Fourier’s law or Navier-Stokes equations, can be applied to the system. This parameter is the Knudsen number LKn /Λ= . When the Knudsen number is large, conventional continuum mechanics cannot be applied to the system. The Boltzmann equation is usually applied to the system of a large Knudsen

number [14].The Boltzmann equation is not easy to solve for the case of arbitrary configuration. It is noted that the phonon transport equation is similar to the radiative transport equation, i.e. the photon transport equation [26].

The authors have developed a new method to solve radiative heat transfer: the Discrete Ordinate Radiation Element Method (DREM) [27]. This method is applied to solve the phonon transport equation. Figure 12 shows a comparison of the temperature distributions of a 3-dimensional complicated nano-structure with various Kundsen numbers. When the Kundsen number increases, the temperature profile becomes different from the one by Fourier’s law.

Figure 12 Dimensionless temperature distributions in nanostructure compared with the one by Fourier’s law.

4. Categories of Micro-system and Nano-system As has been discussed in the preceding sections, micro and nano systems show different features and characteristics in heat and fluid flow. The characteristics are decided not only by the size but also the relation with the phenomena of fluid dynamics and heat transfer. In terms of heat and fluid flow, micro and nano systems can be categorized as follows:

Micro-system： • The size of the element or component is generally

from 1 mµ to 1 mm. • The fluid motion can be treated as continuum, and is

described by Navier-Stokes equations. • The diffusion of heat and momentum is dominant. • Turbulent advection and momentum of fluid motion

is less important in a micro-system. • A micro system has its own optimal shape and

function, and simple size reduction to a micro-machine may not work properly.

Nano-system： • The fluid cannot be treated as continuum anymore. • Discontinuity and wave characteristics of fluid and

energy have to be considered. • The Boltzmann transport equation and molecular

dynamics is a more effective tool for describing the nano heat and fluid flow than Navier-Stokes equations.

• A nano-system is sometimes described by comparison of the mean free path length of phonon, photon and gas molecules and the characteristic length of the element.

• Nano-phenomena may appear in a system which is

larger than 1 µ m, and may not appear in one which is smaller than 1nm.

References (1) Maruyama, S., “Heat and Fluid Flow in

Micro-Machines,” Science of Machine, Vo. 52, (2000), pp. 335-341.

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(3) Jacobson, S.A., “Aerohermal Challenges in the Design of a Microfabricated Gas Turbine Engine,” 29th AIAA Fluid Dynamics Conference, Albuquerque, June 15-18, (1998), AIAA 98-2545, pp.1-11.

(4) Maruyama, S. et al., JSME Text Series: Heat Transfer, Chap. 1 and 8, Maruzen Publishing Co., Tokyo, (2005).

(5) Maruyama, S., Viskanta, R. and Aihara, T., “Active Thermal Protection System Against Intense Irradiation,” Journal of Thermophysics and Heat Transfer, Vol.3, No.4, (1989-10), pp.389-394.

(6) Maruyama S. and Shimizu, N., “Temperature Distribution in a Layer of an Active Thermal Insulation System Heated by a Gas Burner,” Heat Transfer Japanese Research, Vol.22, No.3, (1993), pp.224-237.

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(8) Maruyama, S., Shibata, T., Tsukamoto, T., “Measrurement of Diffusion Fields of Solutions Using Real-time Phase-shift Interferometer and Rapid Heat-transfer Control System,” Experimental Thermal and Fluid Science, Vol. 19, No. 1, (1999), pp.34-48.

(9) Maruyama, S., Takahashi, K., Komiya, A. and Behnia, M., “Measurement of Transient Double Diffusive Convection and Crystal Growth Using Real-time Phase-shifting Interferometer,” JSME International Journal, Ser. B, Vol. 44, o. 4., (2001), pp.561-567.

(10) Maruyama, S., Ohno, K., Komiya, A. and Sakai, S., “Description of the Adhesive Crystal Growth Under Normal and Micro-gravity Conditions Employing Experimental and Numerical Approaches,” Journal of Crystal Growth, Vol. 245, (2002), pp.278-288.

(11) Maruyama, S., Tsubaki, K., Taira, K. and Sakai, S., “Artificial Upwelling of Deep Seawater Using the Perpetual Salt Fountain for Cultivation of Ocean Desert,” Journal of Oceanography, Vol. 60, (2004), pp. 563-568.

(12) Zhang, X.R., Maruyama, S., Sakai, S., Tsubaki, K. and Behnia, M.,” Deep-Sea Research, Part 1, Vol. 51, (2004), pp. 1145-1157.

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Micro/Nano Thermofluid, Ed. Shigeo Maruyama, (2006) NTS, Tokyo, pp. 40-46.

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(17) Abe, T., Takagi, T., Sun, Z.M., Uchimoto, T., Makino, J. and Hashimoto, H. “Machinable Ceramic Substrate for CVD Diamond Coating”, Diamond and Related Materials, Vol. 13, (2004) pp. 819–822.

(18) Takeno, T., Komoriya, T., Nakamori, I., Miki, H., Abe, T., Uchimoto, T. and Takagi, T. “Tribological Properties of Partly Polished Diamond Coatings,” Diamond & Related Materials, Vol. 14, (2005), pp. 2118 – 2121.

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(23) Maruyama, S., Kashiwa, T., Yugami H. and Esashi, M., “Thermal Radiation from Two-dimensionally Confined Modes in Microcavities,” Applied Physics Letters, Vol. 79, No. 9, (2001), pp1393-1395.

(24) Yugami, H., Sasa, H. and Yamaguchi, H., “Thermophotovoltaic Systems for Civilian and Industrial Applications in Japan,” Semiconductor Science and Technology, Vol. 18, (2003), pp. 239-246.

(25) Hanamura, K. and Kameya, Y., “Spectral Control of Thermal Radiation Using Micro-structured Emitter-surface for Thermophotovoltaic Generation of Electricity, Proc. ASME-JSME 2007 Thermal Engineering and Summer Heat Transfer Conference, Vancouver July 8-12, DVD, pp. 1-5.

(26) Majumdar, A., “Microscale Heat Conduction in Dielectric Thin Films,” ASME Journal of Heat Transfer, Vol. 115, (1993), pp.1-16.

(27) Maruyama, S., Sakurai, A. and Komiya, A., “Discrete Ordinates Radiation Element Method for Radiative Heat Transfer in Three-Dimensional Participating Media, Nuverical Heat Transfer, Part B, Vol. 51, (2007), pp.121-140.