Jump in the conduction heat flux at the gas/solid interface in micro-channels

doi:10.1016/j.ijthermalsci.2020.106599

International Journal of Thermal Sciences

Volume 159, January 2021, 106599

https://doi.org/10.1016/j.ijthermalsci.2020.106599 Get rights and content

Abstract

In micro-channel gas flows, velocity slip and temperature jump must be applied at the fluid/solid interface to take the thermodynamic non-equilibrium into account in the vicinity of the wall. In this work, simulations at molecular scale have been carried out and shown a jump in the conduction heat flux, in agreement with the boundary condition proposed by Maslen (1958) [1] in continuum mechanics.

Introduction

The understanding of fluid flows and heat transfer at scales of few microns is of paramount importance since the development of Micro-electro-mechanical Systems (MEMS). When the characteristic length scale of the flow decreases to reach about 10 times the free mean path λ of the gas molecules, the usual Navier-Stokes and energy equations remain valid in the core flow, but careful attention must be paid on the gas/solid boundary conditions. In a fluid layer of order λ close to the wall, the so-called Knudsen layer, the gas molecules are no more in a local thermodynamic equilibrium: the macroscopic model based on the continuity of the velocity, temperature and heat flux at the interface between the gas and the wall falls and slip or jump conditions have to be used, like those independently proposed by Navier [2] and later on by Maxwell [3]. Whereas the boundary conditions applied on the velocity and temperature are quite well established in the ‘gaseous microfluidic’ community, that associated to the jump in the conduction heat flux at the wall due to the viscous friction, and first proposed by Maslen [1] in 1958, has often been neglected, forgotten or misunderstood, including in late papers. In the recent works by X. Nicolas et al. [4], a review of few papers using or ignoring the Maslen's flux condition is performed, followed by its proof which relies on the energy flux conservation principle at the macro-scale. The same authors also showed that the use of the Maslen's flux condition explains the very small Nusselt number values in experiments performed on microfluidic gaseous flows [5]. However, their mathematical approach is questionable since it relies on the assumption of the continuum mechanics which is valid far from the boundaries but seems no more applicable in the Knudsen layer.

This short contribution aims to check if the Maslen's flux condition at the gas/solid interface is correct by getting rid of the continuum model and by simulating the fluid flow and energy transfer at the molecular scale.

Section snippets

Continuum description

An established argon (Ar) fluid flow between two infinite $(x, y)$ -parallel solid walls of platinum (Pt) is considered (Fig. 1). Walls are distant from H and their thickness is e. The flow is created by a volume force $\vec{g} = g_{x} {\vec{e}}_{x}$ acting as a uniform pressure gradient in x-direction. The platinum wall starting at $z = H / 2$ is thermally isolated, i.e. the conduction heat flux $q_{z} = 0$ , whereas the one ending at $z = - H / 2$ is isotherm, at the temperature $T_{1}$ .

For the compressible fluid flow in the micro-channel of

Microscopic approach

Solid walls are composed of platinum atoms disposed on a FCC111 lattice. The dynamics of all atoms, either argon confined between the solid walls or platinum, relies on the classical Newton's law. The trajectory of each atom i, or particle, is governed by ${\vec{F}}_{i} = m_{i} {\ddot{\vec{r}}}_{i}$ where ${\ddot{\vec{r}}}_{i}$ is the acceleration vector, $m_{i}$ is the mass and ${\vec{F}}_{i}$ is the resultant of external forces acting on particle i. With ${\vec{r}}_{i}$ is the particle position, $\vec{g}$ is the volume force which drives the fluid flow, and assuming only binary

Results and discussions

The molecular dynamics simulations are based on a Lennard-Jones potential interactions. Therefore, the pressure, the temperature and the density are solutions of a specific equation of state [14]. Likewise, the dynamic viscosity μ and the thermal conductivity $k_{f}$ are estimated through the correlations by Galliéro et al. [7], and Bugel and Galliéro [8]. The results are then presented as a function of the physical quantities, namely the mean density ρ and the volume force $g_{x}$ . The magnitude order

Conclusion

Results of molecular dynamics simulation, obtained for a Lennard-Jones fluid, have shown that the Maslen's boundary condition [1], demonstrated in the framework of the continuum mechanics (see [4]), although questionable in the Knudsen layer, is proved to be efficient to connect the conduction heat fluxes between the wall and the fluid domain. Therefore, the total energy flux conservation at the fluid/solid interface introduces effectively a jump in the conduction heat flux which is equal to

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

References (16)

X. Nicolas et al.
Thermal boundary conditions for convective heat transfer of dilute gases in slip flow regime
Int. J. Therm. Sci.
(2019)
X. Nicolas et al.
Revisited analysis of gas convection and heat transfer in micro channels: influence of viscous stress power at wall on Nusselt number
Int. J. Therm. Sci.
(2018)
M. Bugel et al.
Thermal conductivity of the Lennard-Jones fluid: an empirical correlation
Chem. Phys.
(2008)
J. Kolafa et al.
The Lennard-Jones fluid: an accurate analytic and theoretically-based equation of state
Fluid Phase Equil.
(1994)
S.H. Maslen
On heat transfer in slip flow
J. Aero. Sci.
(1958)
C. Navier
(1823)
J.C. Maxwell
VII. on stresses in rarified gases arising from inequalities of temperature
Phil. Trans. Roy. Soc. Lond.
(1879)
D.C. Rapaport
The Art of Molecular Dynamics Simulation
(2004)

There are more references available in the full text version of this article.

Cited by (1)

Surface wettability effect on heat transfer across solid-water interfaces
2022, Chemical Engineering Science
Citation Excerpt :
With the development of nanotechnology, the demand for miniaturization has significantly boosted the utilization of nano-micro devices (He et al., 2021); which highlights the importance of understanding and regulating heat transfer efficiency at nano-micro scale (Zhai et al., 2016). Indeed, due to the small size of nanochannels in devices, the interfacial effect on heat transfer becomes significant (Chibouti et al., 2021), and various abnormal phenomena have been found. For examples, Aalilija et al. (2021) and Yenigun and Barisik (2021) observed abrupt temperature jumps across solid–solid and solid–liquid interfaces.
Although it has been found that the heat transfer across solid–liquid interface can be affected by the surface wettability, the underlying mechanism remains ambiguous. Herein non-equilibrium molecular dynamics simulations are conducted to unravel the microscopic mechanisms of heat transfer across solid-water interfaces. Specifically, two aspects of the solid surface wettability are investigated including modifying surface chemistry and introducing molecular roughness. We find that enlarging the solid-water interaction strength facilitates the formation of hydrogen-bonded network of water, hence enhancing the interfacial heat transfer. In addition, introducing molecular roughness can promote heat transfer by increasing the solid-water contact area. When the contact area remains the same, the substrates with multiple and shallow vacancies are more beneficial to heat transfer than those with deep but fewer vacancies. This work sheds insights into the mechanistic understanding of the heat transfer process and provides theoretical guidance for the development of efficient heat transfer devices.

View full text

Jump in the conduction heat flux at the gas/solid interface in micro-channels

Abstract

Introduction

Section snippets

Continuum description

Microscopic approach

Results and discussions

Conclusion

Declaration of competing interest

Int. J. Therm. Sci.

Int. J. Therm. Sci.

Chem. Phys.

Fluid Phase Equil.

On heat transfer in slip flow

J. Aero. Sci.

VII. on stresses in rarified gases arising from inequalities of temperature

Phil. Trans. Roy. Soc. Lond.

The Art of Molecular Dynamics Simulation