# Viscous dissipation effect on heat transfer characteristics of rectangular microchannels under slip flow regime and H1 boundary conditions

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DOI: 10.1007/s00231-005-0075-5

- Cite this article as:
- Aynur, T.N., Kuddusi, L. & Eğrican, N. Heat Mass Transfer (2006) 42: 1093. doi:10.1007/s00231-005-0075-5

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## Abstract

Viscous dissipation effect on heat transfer characteristics of a rectangular microchannel is studied. Flow is governed by the Navier–Stokes equations with the slip flow and temperature jump boundary conditions. Integral transform technique is applied to derive the temperature distribution and Nusselt number. The velocity distribution is taken from literature. The solution method is verified for the case where viscous dissipation is neglected. It is found that, the viscous dissipation is negligible for gas flows in microchannels, since the contribution of this effect on *Nu* number is about 1%. However, this effect should be taken into account for much more viscous flows, such as liquid flows. Neglecting this effect for a flat microchannel with an aspect ratio of 0.1 for *Br*=0.04 underestimates the *Nu* number about 5%.

### List of symbols

*a*long side of microchannel

*A*_{n}constant defined by Eq. 25

*b*short side of microchannel

*b*_{1},*b*_{2},*b*_{3}*B*_{n}constant defined by Eq. 26

*Br*Brinkman number

*c*_{P}specific heat

*C*_{n}constant defined by Eq. 42

*C*_{1,n}^{′},*C*_{2,n}^{′},*C*_{3,n}^{′},*C*_{4,n}^{′},*C*_{5,n}^{′},*C*_{6,n}^{′},*C*_{7,n}^{′}*C*_{1,n},*C*_{2,n},*C*_{3,n},*C*_{4,n}*D*_{h}hydraulic diameter

*D*_{n}constant defined by Eq. 75

*E*_{n}constant defined by Eq. 44

*F*_{t}thermal accommodation coefficient

*F*_{v}tangential momentum accommodation coefficient

*G*constant defined by Eq. 43

*k*thermal conductivity

*k*_{1},*k*_{2},...,*k*_{14}constants defined in Eqs. 60, 61, 62, 63, 64, 65 and Eqs. 67, 68, 69, 70, 71, 72, 73, 74

*K*Kernel

*Kn*Knudsen number

*Nu*Nusselt number

*p*fluid pressure

*P*normalized pressure gradient

*Pr*Prandtl number

*q*heat flux

*R*specific heat ratio

*S*_{1},*S*_{2},*S*_{3},*S*_{4}constants introduced for simplicity

*T*temperature

- \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{T}\)
nondimensional temperature

*u*fluid velocity

- \(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\frown}$}}{u}\)
nondimensional fluid velocity

*x*,*y*,*z*nondimensional coordinates

### Greek symbols

### Subscripts

- b
bulk property

- m
mean value

*n*index

*s*fluid property near the wall

- w
wall value

- 0
inlet property