Convective Heat Transfer for Single-Phase Gases in Microchannel Slip Flow: Analytical Solutions

Abstract

Heat transfer in microchannels has gained more interest in the last decade due to developments in the aerospace, biomedical and electronics industries. It has been a critical issue since the performance of the devices is primarily determined by temperature. As the size decreases, more efficient ways of cooling are sought due to the reduction in the heat transfer area. Convection and conduction are the two major heat transfer mechanisms that have been investigated at microscale. Convective heat transfer in microchannels has been intensively analyzed by both experimental and analytical means. Conduction studies have focused mostly on thin films in recent years to address such questions as: How is the heat transferred? How does it differ from largescal ec onduction? As far as convective heat transfer is concerned, liquid and gaseous flows must be considered separately. Liquid flow has been investigated experimentally, whereas analytical, numerical and molecular simulation techniques have been applied to understand the characteristics of gaseous flow and heat transfer. While the Navier-Stokes equations can still be applied, due to the small size of microchannels, some deviations from the conventionally sized applications have been observed. Flow regime boundaries are significantly different, as well as flow and heat transfer characteristics. Gaseous flow has usually been investigated by theoretical means. Some experiments were also performed to verify the theoretical results. When gases are at low pressures, or are flowing in small geometries, the interaction of the gas molecules with the wall becomes as frequent as intermolecular collisions, which makes the boundaries and the molecular structure more effective on flow. This type of flow is known as rarefied gas flow. The Knudsen number (Kn) is used to represent the rarefaction effects. It is the ratio of the molecular mean free path to the characteristic dimension of the flow. For Knudsen numbers close to zero, flow is still assumed to be continuous. As the Knudsen number takes higher values, due to a higher molecular mean free path by reduced pressure or a smaller flow dimension, rarefaction effects become more significant and play an important role in determining the heat transfer coefficient. The commonly used slip boundary conditions are called Maxwellian boundary conditions [1]. Since they are first order in accuracy, an extended set of boundary conditions was proposed by [2], which can be used in early transition of the slip flow regime. To do so, the velocity and temperature gradients at the boundary are written in terms of the Taylor expansion of the gradients within the layer one mean free path away from the boundary (called the Knudsen Layer).