Analytical Solution of Micro-/Nanoscale Convective Liquid Flows in Tubes and Slits

Abstract

Analytical solutions examining heat transport in micro-/nanoscale liquid flows were developed. Using the energy equation coupled with fully developed velocity, we solved developing temperature profiles with axial conduction and viscous dissipation terms. A comprehensive literature review provided the published range of velocity slip and temperature jump conditions. While molecular simulations and experiments present constant slip and jump values for a specific liquid/surface couple independent of confinement size, non-dimensional forms of these boundary conditions were found appropriate to calculate non-equilibrium as a function of flow height. Although slip and jump conditions are specific for each liquid/surface couple and hard to obtain, we proposed modeling of the slip and jump as a function of the surface wetting, in order to create a general, easy to measure methodology. We further developed possible correlations to calculate jump using the slip value of the corresponding surface and tested in the results. Fully developed Nu showed strong dependence on slip and jump. Heat transfer stopped when slip and jump coefficients became higher than a certain value. Strong variation of Nu in the thermal development length was observed for low slip and jump cases, while an almost constant Nu in the flow direction was found for high slip and jump coefficients. Variation of temperature profiles was found to dominate the heat transfer through the constant temperature surface while surface and liquid temperatures became equal at heat transfer lengths comparable with confinement sizes for no-dissipation cases. In case of non-negligible heat dissipation, viscous heating dominated the Nu value by enhancing the heating while decreasing the heat removal in cooling cases. Implementation of proposed procedure on a micro-channel convection problem from a micro-fluidics application showed the dominant effect of the model defining the slip and jump relationship. Direct use of kinetic gas theory resulted in an increase of Nu by an increase in non-equilibrium, while models developed from published liquid slip and jump values produced an opposite behavior.