The importance of viscous dissipation in micro-tube and micro-gap flows

Increasing heat flux density of modern micro-electronic devices has promoted a transition to liquid-based forced convection cooling. The miniaturization and maldistribution of micro-electronic heat generating elements (e.g. transistors and laser diodes) has promoted a similar decrease in size of cooling flow elements, specifically, micro-channels, micro-gaps and micro-jets. Convection heat transfer scaling laws do not contain a scale-factor in dimensionless form, and heat transfer coefficient (HTC) should continually increase with a decrease in size, as h∝1/d. However, extremely high HTCs are not found at tens of microns, which can be explained by the emergence of a typically neglected effect - heating by viscous dissipation. Traditionally, dissipation is only associated with high-Mach gas flows or high-viscosity oil flows. Nonetheless, it reemerges in micro-cooling, as shown here through theoretical analysis of simple cases. The extreme near-wall gradients and high L/d ratios, of these flows reintroduce dissipation as significant. When flow diameters reach a critical size, on the scale of tens of microns at Re=2,000, depending on flow configuration, rate and liquid properties, the energy generated by dissipation is sufficient to counteract the inherent increase of HTC and the trend reverses. This maximum in HTC is the absolute lower limit to the cooling element size, a matter which has not been properly addressed. The present study lays a framework of recommendations and limitations for future cooling studies, thereby curbing the ongoing trend of flow miniaturization.