|Home | About | Journals | Submit | Contact Us | Français|
Rapid advances in microelectromechanical systems have stimulated the development of compact devices, which require effective cooling technologies (e.g., microchannel cooling). However, the inconsistencies between experimental and classical theoretical predictions for the liquid flow in microchannel remain unclarified. Given the larger surface/volume ratio of microchannel, the surface effects increase as channel scale decreases. Here we show the scale effect of the boundary condition at the solid–liquid interface on single-phase convective heat transfer characteristics in microchannels. We demonstrate that the deviation from classical theory with a reduction in hydraulic diameters is due to the breakdown of the continuum solid–liquid boundary condition. The forced convective heat transfer characteristics of single-phase laminar flow in a parallel-plate microchannel are investigated. Using the theoretical Poiseuille and Nusselt numbers derived under the slip boundary condition at the solid–liquid interface, we estimate the slip length and thermal slip length at the interface.
Heat and mass transfer in microchannels has been extensively investigated over the past several decades. Continuum theory is applicable to single-phase liquid flow in microchannels, which implies that the equations developed for macroscale applications, such as the Navier–Stokes equation and convective heat transport equations, will be applicable to even small channels1,2,3,4,5. In a fully developed laminar channel flow, the classical theoretical solutions for the Nusselt number (hereafter, Nu; , where α is the convective heat transfer coefficient, Dh is the hydraulic diameter of a channel, and λ is the thermal conductivity) and Poiseuille number (hereafter, Po; Po=fRe, where f is the Fanning friction factor and Re is the Reynolds number) are constants and are independent of the Reynolds number. Because αDh is nearly a constant, the forced convection heat transfer coefficients in fully developed laminar channel flows are expected to increase with a decrease in the microchannel size.
Table 1 lists the typical Nu and Po values for a fully developed laminar flow in various channels6, where NuT is the Nu at a uniform surface temperature, and Nuq is the Nu at a uniform heat flux. The ratio of Nuq to NuT is larger than 1. The NuT listed in the table is the lower limit of Nu because the axial conduction effect is absent when the surface temperature is uniform. In other words, Nu cannot be less than NuT in a thermally developing flow or in a flow under uniform surface heat flux accompanied by axial conduction. However, the experimental Nu for the forced convection of a single-phase liquid flow in Si microchannels differs significantly from that predicted using continuum theory7,8,9,10,11,12,13,14,15,16,17,18,19,20. These significant differences have been mainly attributed to experimental errors pertaining to the effects of axial conduction16,17,18 and roughness19,20. However, because most studies report similar behaviours (i.e., Nu in microchannels decreases with decreasing Re), the discrepancies between the theoretical and experimental results cannot be completely attributed to experimental errors. Davis and Gill21, who were among the first to examine the axial conduction effect in laminar flow between parallel plates, concluded that the axial conduction effect reduced Nu. Other researchers12,16,17,18 reported that the difference between experimental and theoretical Nu increased with decreasing Re in microchannels could be attributed to the effects of axial heat conduction. On the other hand, the analytical results reported by Maranzana et al.17 and Lin et al.18 for the effects of axial heat conduction on single-phase microchannel flows yielded a much lower value of Nu than the theoretical results, which is in contrast to the conventional predictions of Nuq>NuT as the axial conduction increases. Thus, whether Nu can be less than NuT, which is the lower limit of Nu without the axial conduction effect under uniform surface temperature, remains unresolved.
Here we focus on the scale effect on single-phase convective heat transfer in microchannels. We demonstrate that the deviation of Nu from that in classical theory with a reduction in the hydraulic diameters of the microchannels is due to solid–liquid interfacial resistance, which can be expressed in terms of the slip length and the thermal slip length (i.e., the Kapitza length). In addition, the convective heat transfer characteristics of single-phase laminar flow in parallel-plate microchannels are investigated experimentally. Finally, using the theoretical Po and Nu numbers derived under the slip boundary condition at the solid–liquid interface, we estimate the slip length and thermal slip length at the interface.
The boundary condition at the solid–liquid interface is a factor that strongly influences the thermohydraulic characteristics of single-phase liquid flow in microchannels. The continuum boundary condition (i.e., no-slip boundary condition) may fail because of the molecular interactions at the solid–liquid interface, and the slip boundary condition may be significant in nanochannel flow22,23,24,25,26. According to the Navier’s model, the slip velocity at solid–liquid boundaries is linearly proportional to the velocity gradient at the surface:
where ls is the hydrodynamic slip length. Slip length ls can be obtained by extrapolating the velocity profile from the position at the solid–liquid interface in the fluid to the position at which the velocity becomes zero, as shown in Fig. 1. Analogously, the slip thermal boundary condition can be determined using the “thermal slip length”, that is, the position at which the temperature difference between the liquid and solid is zero. The physical meaning of thermal slip length, also known as the Kapitza length lk, is the thickness of the thermal resistance at the solid–liquid interface:
Here, ΔT is the temperature jump of the first layer of liquid located at the interface, dT/dz is the temperature gradient of the liquid, Ri is the thermal resistance at the solid–liquid interface, and λl is the thermal conductivity of the liquid.
Consider a parallel-plate Poiseuille flow subjected to constant heat flux at one channel wall in the steady state (Fig. 2). If the spacing between the parallel plates 2h is small relative to the size of the parallel plates, the hydraulic diameter is Dh=4h. Assuming that the flow is incompressible and that all of the thermophysical properties are constant, the velocity profile of a hydrodynamic fully developed laminar flow under the slip boundary condition at both channel walls can be derived as follows:
where μ is the viscosity and dP/dx is the pressure drop. For a fully developed laminar flow, Po is
where ls=ls/(2h) and f is the fanning friction factor. For ls=0 or ls2h, Eqs (3) and (4) agree with the theoretical predictions under the continuum assumption and Po is a constant (=24). Nu can be obtained as
where lk=lk/(2h). For ls=0 or ls2h and lk=0 or lk2h, Eq. (5) agrees with the theoretical prediction under the continuum assumption, and Nu is a constant (=5.38).
When the critical dimension of the flow decreases to a size comparable with that of the liquid molecule, ls and lk can no longer be ignored, and the slip boundary condition begins to strongly influence the momentum transfer and heat transfer characteristics in the microchannels. In particular, the solid–liquid interfacial resistance is dependent on the molecular interaction, that is, the contact condition between the liquid and the channel wall. Therefore, for a macroscopic smooth wall or a nanostructured wall, the scale effect of the interfacial resistance due to surface roughness and surface wettability becomes increasingly apparent.
Si-based microchannels (70mm (length)×15mm (width); 4 channel depths: 30, 50, 100, 150μm) were prepared through KOH wet-etching of p-type Si wafers in the <100> orientation. The etched microchannels had a rectangular cross-section, as shown in Fig. 3. A Pyrex glass cover was anodically bonded to the Si wafer substrate at 350°C and 2.0kV to seal the microchannel, after which the parallel-plate microchannel test section was fabricated. Given that the depths of the microchannels were small relative to their widths and lengths, the channel hydraulic diameter Dh was nearly twice the channel depth (i.e., Dh=60, 100, 200, and 300 for the four aforementioned channel depths).
To measure the pressure drop, two holes spaced 50mm apart were fabricated in the cover glass in order to connect to a differential pressure sensor. On the backside of the microchannel, an aluminium thin film heater was sputtered, rendering the Si-based microchannel surface a heated wall subject to constant heat flux. The initial water contact angle at the fresh and clean Si surface was 58°±3°, but it decreased to 36°±3° because of the oxidation of the thin SiO2 film. This surface served as the microchannel surface which the slip boundary condition was applied. The test section was finally assembled, and the bottom surface of the microchannel substrate and the top cover glass surface were well insulated to reduce heat loss from the test section.
The experimental apparatus is shown in Fig. 4, which is consisted of a tank, a pump, valves, the test section of the Si-based microchannel, and a balance. Pure water (Kishida Chemical; electrical resistivity=18MΩcm) was used as the working fluid. The fluid temperatures were measured using a T-type thermocouple (diameter=0.2mm) at both the inlet and outlet. The wall temperatures were measured using eight T-type thermocouples. All of the pressure and temperature data were collected at 25°C and 40RH% by using a data logger and then transmitted to a computer.
The friction factor f is obtained from the pressure drop , the distance over which the pressure is measured L in the fully developed flow region, the fluid density ρ, and the mean velocity of the working fluid U, as shown in Eq. (6).
Then, the Po number can be obtained as follows.
where the microchannel width w is 15mm, is the mass flow rate, and is the dynamic viscosity of the working fluid.
Figure 5 shows the experimental results for the Poiseuille number in the microchannels with hydraulic diameters Dh of 60μm, 100μm, 200μm, and 300μm, respectively. The results obtained for the Poiseuille number (from more than 3 different independent experiments) agree well with the theoretical values, based on the continuum boundary condition. This could be explained by the surface being covered in a thin, hydrophilic SiO2 film and the slip velocity being negligible in the studied cases.
The heat flux supplied to the heater includes the heat flux through forced convection for heat exchange between the Si microchannel surface and the working fluid, as well as the heat flux through axial conduction inside the Si microchannel substrate. To avoid the axial conduction effect, the method whereby heat flux is supplied to the heater has not been used in the present study. The heat flux exchanged at the microchannel surface, q is obtained from the temperature difference at the fully developed flow region,
where Cp is the specific heat of the liquid and A is the equivalent surface area for heat transfer. The mean heat transfer coefficient and the Nusselt number are obtained as follows.
where is the mean temperature difference between the channel wall and the working fluid.
Figure 6 shows the experimentally obtained Nusselt numbers for the microchannels with hydraulic diameter Dh of 60μm, 100μm, 200μm, and 300μm, respectively. The experimental Nu numbers are much lower than the theoretical values of both Nuq (constant heat flux) and NuT (constant surface temperature) based on the continuum boundary condition. The deviations between the experimental Nu and theoretical Nu increase as the hydraulic diameter of the microchannel decreases.
The interfacial resistance (i.e., ls and lk) can be estimated from the difference between continuum theory and the experimental results. Using Eq. (4) and the experimental mean Po (=fRe), ls can be estimated as follows:
Similarly, lk can be estimated using the experimental mean Nu and Eq. (5):
Next, the forced convective heat transfer characteristics of the single-phase laminar flow in a parallel-plate microchannel are investigated experimentally. Figures 7 and and88 illustrate the experimental results and theoretical predictions, respectively, to clarify the scale effect of the hydraulic diameter on forced convection in microchannels. The theoretical Po is (fRe)th=24, 64, and 57 for the parallel-plate channel, circular tube of refs 27 and 28, and the rectangular channel of Ref. 27, respectively (Fig. 3)27,28. The slip length of the water and silicon oxide interface in the present study can be assumed to be 0 because the experimental results agree well with the theoretical predictions. However, the experimental results obtained by Judy et al.27 in rectangular channels decrease with decreasing hydraulic diameter, which agrees fairly well with the theoretical prediction of ls=1μm, for which the error is less than 2%.
In contrast to the foregoing results, the experimental Nu in Fig. 8 is significantly lower than the theoretical Nu under the no-slip boundary condition. The experimental Nu decreases with decreasing hydraulic diameter, whereas the discrepancy decreases with increasing hydraulic diameter, which is consistent with the trends reported in the literature8,9. The experimental Nu obtained in this study agrees well with the theoretical prediction of ls=0μm and lk=150μm, while those reported by Qu et al.8 and Gao et al.9 agree well with the theoretical prediction of ls=0μm and lk=50μm. In other words, the slip length and thermal slip length can no longer be ignored when these lengths are comparable with the hydraulic diameter. Therefore, we conclude that the scale effect explains the difference between the predictions of continuum theory and the experimental results.
Surface roughness19,20 exerts significant effects on forced convection heat transfer in microchannels. Moreover, surface wettability strongly affects convective heat transfer in microchannels29, and effective slip and friction reduction in nanograted superhydrophobic microchannels have been reported30. The effects of roughness and wettability, which are types of interfacial resistance, can be expressed using slip length and thermal slip length when the continuum boundary condition fails. Additional theoretical, molecular dynamics simulation26,31,32,33,34, and experimental studies35 on interfacial resistance are warranted to further clarify the mechanism.
We studied the scale effect of the boundary condition at the solid–liquid interface on the single-phase convective heat transfer characteristics in microchannel or nanochannel flow. We have shown that the increasing inaccuracy of the predictions of classical theory with a decrease in the hydraulic diameter is due to the breakdown of the continuum solid–liquid boundary condition in microchannels. In other words, the solid–liquid interfacial resistance, which can be expressed as the slip length and thermal slip length, cannot be ignored when these lengths are comparable with the hydraulic diameter. Using the theoretical Po and Nu derived under the slip boundary condition at the solid–liquid interface, we can estimate the slip length and thermal slip length at the solid–liquid interface.
How to cite this article: Nagayama, G. et al. Scale effect of slip boundary condition at solid–liquid interface. Sci. Rep. 7, 43125; doi: 10.1038/srep43125 (2017).
Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is partly supported by the Ministry of Education, Science and Culture of the Japanese Government through the Grant-in Aid for Scientific Research, Project No. 21360099. The Si microchannel fabrication in this study was supported by the Kitakyushu Foundation for the Advancement of Industry Science and Technology.
The authors declare no competing financial interests.
Author Contributions G.N. performed the structural determination and theoretical analyses, conceived the experimental design, and wrote the paper. T.M. and K.F. conducted the experiments and theoretical analyses under the supervision of G.N. All of the authors have discussed the results and have commented on the manuscript.