A general form of capillary rise equation in micro-grooves

Micro-grooves are a crucial feature in many applications, such as microelectro-mechanical systems, drug delivery, heat pipes, sorption systems, and microfluidic devices. Micro-grooves utilize capillary action to deliver a liquid, with no need for an extra pumping device, which makes them unique and desirable for numerous systems. Although the capillary action is well studied, all the available equations for the capillary rise are case-specific and depend on the geometry of the groove, surface properties, and the transport liquid. In this study, a unified non-dimensional model for capillary rise is proposed that can accurately predict the capillary rise for any given groove geometry and condition and only depends on two parameters: contact angle and characteristic length scale, defined as the ratio of the liquid-vapor to the solid-liquid interface. The proposed model is compared against data from the literature and can capture the experimental results with less than 10% relative difference. The effect of the grooves' height, width, and contact angle is investigated and reported. This study can be used for a unified approach in designing heat pipes, capillary-assisted evaporators for sorption systems, drug delivery micro-fluidic devices, etc.