Energy Equation of Gas Flow With Low Velocity in a Micro-Channel

The energy equation for incompressible flow with the viscous dissipation term is often used for the governing equations of gas flow with low velocity in micro-channels. However, the results which are obtained by solving these equations do not satisfy the first law of the thermodynamics. In the case of ideal gas with low velocity, the inlet and the outlet temperatures of an adiabatic channel are the same based on the first law of the thermodynamics. However, the outlet temperature which is obtained by solving the energy equation for incompressible flow with the viscous dissipation term is higher than the inlet gas temperature, since the viscous dissipation term takes positive value. This inconsistency arose from wrong choice of the relation between the enthalpy and temperature that resulted in neglecting the substantial derivative of pressure term in the energy equation. In this paper the correct energy equation which includes the substantial derivative of pressure term is proposed. Some samples of physically consistent results which are obtained by solving the proposed energy equation are demonstrated.