The extended Graetz problem for a gaseous slip flow in micropipe and parallel-plate microchannel with heating section of finite length: Effects of axial conduction, viscous dissipation and pressure work

The extended Graetz problem for a gaseous slip flow through a micropipe and a parallel-plate microchannel, with an isothermal heating section of finite length, is analytically investigated. The simultaneous effects of the axial heat conduction, viscous dissipation and pressure work are all taken into account and discussed. The solution obtained is based on a powerful method using self-adjoint formalism, resulting from a decomposition of energy equation into a system of the first-order partial differential equations. This solution, which is applicable for finite and semi-infinite heating section, represents an improvement and extension of those obtained in the earlier works, by considering the slip boundary conditions at the fluid-wall interface for the velocity and temperature. This extension has been done by using a new matrix operator of three dimensions and a suitable scalar product between two vectors in the Hilbert space. The analytical results are compared for simplified limiting cases with available analytical and numerical calculations and a good agreement is found. The results of the effects of different dimensionless parameters involved in the problem, namely Péclet, Knudsen, Brinkman numbers and the length of the heating section, on the heat transfer characteristics are illustrated and discussed. Furthermore, some useful correlations of these characteristics are provided for some values of Péclet number. It is shown particularly that for non-zero values of Brinkman number, when the heat flow is established, the sum of the enthalpy and the energy which results from the friction and pressure work is conserved through cross-sections of microchannels, and the heat transfer is mainly governed by the shear work at the wall. Among the most important applications of this analytical solution is its potential to simulate an isothermal hot film sensor of finite size, mounted on the wall of microchannel, which can be serve to measure a heat flux between the gas and wall and hence heat transfer coefficient in the slip flow regime.