# Numerical investigation of the effect of second order slip flow conditions on interfacial heat transfer in micro pipes

## Abstract

Heat transfer that occurs in micro scale devices has a very important place among the engineering applications that cooling or heating. This heat transfer mechanism in devices having dimensions at micron level is a completely different problem in the macro level analysis. Therefore, in the calculations made, the flow events and heat transfer in micron scale pipes are calculated by using more realistic expressions. For this reason, in this study heat transfer in a circular micro pipe with wall and fluid conjugation for laminar rarefied gas flow in transient regime is investigated under the second order slip boundary conditions at the interface. Patankar’s control volume method is used here to solve the problem numerically. This analysis includes of axial conduction, viscous dissipation and rarefaction effects which are indispensable in micro-flow structure. From the results, it is seen that the values that are indicating heat transfer are excessively affected by wall thickness, viscous heating and gas rarefaction especially in transient regime.

## Keywords

Heat transfer transient regime micro-pipe slip flow rarefied gas viscous heating## List of symbols

- \( a \)
discretization constants

- \( b \)
residual term

- \( Br \)
Brinkman number [\( = \frac{{\mu_{f} u_{m}^{2} }}{{k\left( {T_{1} - T_{0} } \right)}} \)]

- \( c_{p} \)
specific heat [J/kgK]

- \( d \)
wall thickness [m]

- \( D \)
hydraulic diameter [m]

- \( k \)
thermal conductivity [W/mK]

- \( Kn \)
Knudsen number [=λ/D]

- \( Nu \)
Nusselt Number [\( = \frac{hD}{k} \)]

- \( Pe \)
Peclet number [=\( \text{Re} \Pr = \tfrac{{2r_{wi} u_{m} \rho_{f} c_{pf} }}{{k_{f} }} \)]

- \( Pr \)
Prandtl number [=µc

_{p}/k]- \( Po \)
Poiseuille number

- \( q \)
heat flux [W/m

^{2}]*R*radial coordinate [m]

*Re*Reynolds number [=u

_{m}D/ν]- \( t \)
time [s]

- \( T \)
temperature [K]

- \( T_{s} \)
slip fluid temperature [K]

- \( u \)
axial velocity [m/s]

- \( u_{s} \)
slip velocity [m/s]

- \( v \)
radial velocity [m/s]

- \( x \)
axial coordinate [m]

## Greek symbols

- \( \alpha \)
thermal diffusivity [m

^{2}/s]*γ*specific heat ratio

*λ*molecular mean free path [m]

- \( \delta r \)
radial grid difference [m]

- \( \delta x \)
axial grid difference [m]

- \( \kappa \)
kappa [\( = \tfrac{2\gamma }{\gamma + 1}\frac{1}{\Pr } \)]

- \( \Delta r \)
radial step [m]

- \( \Delta t \)
time step [s]

- \( \Delta x \)
axial step [m]

- \( \rho \)
density [kg/m

^{3}]- σ
_{m} tangential momentum accommodation coefficient

- σ
_{T} thermal accommodation coefficient

- \( \mu \)
dynamic viscosity [Pa.s]

## Subscripts

- \( b \)
bulk

- \( f \)
fluid

- \( i \)
inner

- \( i,j \)
at nodal point

*i*,*j*- \( m \)
mean

- \( w \)
wall

## Superscripts

- \( ' \)
dimensionless quantity

^{0}at previous time step

## Notes

### Acknowledgements

The Authour is thankful to Selçuk University for the support.

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