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Sci Rep. 2017; 7: 43125.
Published online 2017 March 3. doi:  10.1038/srep43125
PMCID: PMC5335653

Scale effect of slip boundary condition at solid–liquid interface

Abstract

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; An external file that holds a picture, illustration, etc.
Object name is srep43125-m1.jpg, 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.

Table 1
Nusselt and Poiseuille numbers for a fully developed laminar flow in various channels6.

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.

Models and Methods

Slip boundary condition at solid–liquid 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:

An external file that holds a picture, illustration, etc.
Object name is srep43125-m2.jpg

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:

Figure 1
Slip boundary condition at the solid–liquid interface: slip length and thermal slip length.
An external file that holds a picture, illustration, etc.
Object name is srep43125-m3.jpg

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.

Forced convection for fully developed laminar flow under slip boundary condition

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:

Figure 2
Parallel-plate Poiseuille flow under slip boundary condition.
An external file that holds a picture, illustration, etc.
Object name is srep43125-m4.jpg

where μ is the viscosity and dP/dx is the pressure drop. For a fully developed laminar flow, Po is

An external file that holds a picture, illustration, etc.
Object name is srep43125-m5.jpg

where ls* = ls/(2h) and f is the fanning friction factor. For ls = 0 or ls [dbl greater-than sign] 2h, Eqs (3) and (4) agree with the theoretical predictions under the continuum assumption and Po is a constant (=24). Nu can be obtained as

An external file that holds a picture, illustration, etc.
Object name is srep43125-m6.jpg

where lk* = lk/(2h). For ls = 0 or ls [dbl greater-than sign] 2h and lk = 0 or lk [dbl greater-than sign] 2h, 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.

Experiment

Si-based microchannel test section

Si-based microchannels (70 mm (length) × 15 mm (width); 4 channel depths: 30, 50, 100, 150 μm) 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 2.0 kV 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).

Figure 3
Si microchannel test section.

To measure the pressure drop, two holes spaced 50 mm 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.

Experimental apparatus

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 = 18 MΩcm) was used as the working fluid. The fluid temperatures were measured using a T-type thermocouple (diameter = 0.2 mm) 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 40 RH% by using a data logger and then transmitted to a computer.

Figure 4
Schematic of experimental apparatus.

Results and Discussions

Experimental Poiseuille Number

The friction factor f is obtained from the pressure drop An external file that holds a picture, illustration, etc.
Object name is srep43125-m7.jpg, 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).

An external file that holds a picture, illustration, etc.
Object name is srep43125-m8.jpg

Then, the Po number can be obtained as follows.

An external file that holds a picture, illustration, etc.
Object name is srep43125-m9.jpg

where the microchannel width w is 15 mm, An external file that holds a picture, illustration, etc.
Object name is srep43125-m10.jpg is the mass flow rate, and An external file that holds a picture, illustration, etc.
Object name is srep43125-m11.jpg 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 60 μm, 100 μm, 200 μm, and 300 μm, 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.

Figure 5
Experimental Poiseuille numbers Po vs. Reynolds number Re in microchannels.

Experimental Nusselt number

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,

An external file that holds a picture, illustration, etc.
Object name is srep43125-m12.jpg

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.

An external file that holds a picture, illustration, etc.
Object name is srep43125-m13.jpg
An external file that holds a picture, illustration, etc.
Object name is srep43125-m14.jpg

where An external file that holds a picture, illustration, etc.
Object name is srep43125-m15.jpgis 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 60 μm, 100 μm, 200 μm, and 300 μm, 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.

Figure 6
Experimental Nusselt numbers Nu vs. Reynolds number Re in microchannels.

Scale effects of interfacial resistances

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:

An external file that holds a picture, illustration, etc.
Object name is srep43125-m16.jpg

Similarly, lk can be estimated using the experimental mean Nu and Eq. (5):

An external file that holds a picture, illustration, etc.
Object name is srep43125-m17.jpg

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 = 1 μm, for which the error is less than 2%.

Figure 7
Scale effect of slip length on hydrodynamic resistance in microchannels.
Figure 8
Scale effect of thermal slip length on convective heat transfer in microchannels.

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 = 0 μm and lk = 150 μm, while those reported by Qu et al.8 and Gao et al.9 agree well with the theoretical prediction of ls = 0 μm and lk = 50 μm. 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.

Summary

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.

Additional Information

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.

Acknowledgments

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.

Footnotes

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.

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