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Investigating Thermally Develo** Gas Slip Flow Inside a Micro-annulus Including Viscous Dissipation and Axial Conduction Effects Using the Lattice Boltzmann Method

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Abstract

In the present paper, forced convection of a laminar gas flow inside two concentric micro-cylinders with constant wall heat flux condition is numerically investigated. For this purpose, the energy equation is solved in the continuum and slip flow regimes for the thermally develo** condition using the lattice Boltzmann method. To the authors’ best knowledge, the simultaneous effects of viscous dissipation, rarefaction, axial conduction, and radius ratio in the thermally develo** region of a micro-annulus channel have not been considered in the literature. In the present work, the effects of the mentioned parameters on the heat transfer characteristics are studied in detail. Furthermore, the lattice Boltzmann method is developed to apply the viscous dissipation source term in axisymmetric slip flows under constant wall heat flux condition. The results show that in the absence of viscous dissipation, due to energy balance in the fluid, the bulk fluid temperature is independent of the Knudsen number, and its value increases linearly along the microchannel. However, the bulk fluid temperature changes with the Knudsen number by including the viscous dissipation. Also, increasing the rarefaction effect, reduces the impacts of Brinkman number and radius ratio on the local Nusselt number. Moreover, the influence of viscous dissipation is more significant at higher radius ratios.

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Abbreviations

\(A\) :

Cross-sectional area of micro-annulus (m2)

Br:

Brinkman number

\(c\) :

Lattice streaming speed (m s−1)

\({c}_{\mathrm{p}}\) :

Specific heat (J kg−1 K−1)

\({c}_{\mathrm{s}}\) :

Lattice speed of sound (m s−1)

\({D}_{\mathrm{h}}\) :

Hydraulic diameter (m)

\(\overrightarrow{e}\) :

Lattice velocity vector (m s−1)

\({g}_{\alpha }\) :

Energy distribution function in the lattice direction

\(h\) :

Local convective heat transfer coefficient (W m−2 K−1)

\(k\) :

Thermal conductivity (W m−1 K−1)

Kn:

Knudsen number

\(L\) :

Microchannel length (m)

Ma:

Mach number

\(P\) :

Pressure (N m−2)

Nux :

Local Nusselt number

Pe:

Peclet number

Pr:

Prandtl number

\({Q}_{\alpha }\) :

Viscous heating source term in the lattice direction (s−1)

\(q\) :

Wall heat flux (W m−2)

\({r}^{*}\) :

Radius ratio

\(R\) :

Dimensionless radial direction

Re:

Reynolds number

\({S}_{\alpha }\) :

Source term for the energy equation in the lattice direction (s−1)

\(T\) :

Temperature (K)

\(t\) :

Time (s)

\(\overrightarrow{u}\) :

Velocity vector (m s−1)

\({u}_{m}\) :

Mean velocity (m s−1)

\(U\) :

Dimensionless axial velocity

\({w}_{\alpha }\) :

Weight coefficients in the lattice direction

\(X\) :

Dimensionless axial direction

\(\overrightarrow{x}\) :

Space vector (m)

\(\gamma\) :

Specific heat ratio

\(\mu\) :

Dynamic viscosity (kg m−1 s−1)

\(\Delta t\) :

Lattice time step (s)

\(\Delta x\) and \(\Delta r\) :

Lattice spacing in axial and radial directions (m)

\(\theta\) :

Dimensionless temperature

\(\lambda\) :

Molecular mean free path (m)

\(\upnu\) :

Kinematic viscosity (m2 s−1)

\(\rho\) :

Density (kg m−3)

\({\rho }_{s}^{2}\) :

Slip radius in the circular microchannel

\({\sigma }_{T}\) :

Thermal accommodation coefficient

\({\sigma }_{v}\) :

Tangential momentum accommodation coefficient

\({\tau }_{g}\) :

Relaxation time for the energy distribution function

\(\chi\) :

Thermal diffusivity (m2 s−1)

\({\omega }_{g}\) :

Collision frequency for the energy equation

\(\varphi\) :

Dissipation function (s−2)

b:

Bulk

i:

Inner wall

in:

Inlet

jump:

Temperature jump

\(m\) and\(n\) :

Last nodes in the radial and axial directions

o:

Outer wall

s:

Properties of fluid at the wall

w:

Wall

\(\alpha\) :

Lattice direction

\(0\) :

First node in the radial direction

eq:

Equilibrium

′:

Equivalent value

*:

Dimensionless value

LBM:

Lattice Boltzmann method

FDM:

Finite difference method

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, University of Guilan, P.O. Box 3756, Rasht, Iran

    Masoud Hami

  2. Department of Mechanical Engineering, Faculty of Engineering, Golestan University, Gorgān, Iran

    Mohammad Kalteh

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  1. Masoud Hami
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  2. Mohammad Kalteh
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Correspondence to Mohammad Kalteh.

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Hami, M., Kalteh, M. Investigating Thermally Develo** Gas Slip Flow Inside a Micro-annulus Including Viscous Dissipation and Axial Conduction Effects Using the Lattice Boltzmann Method. Iran J Sci Technol Trans Mech Eng 48, 49–64 (2024). https://doi.org/10.1007/s40997-023-00643-z

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