# Forced convection in microchannels

Due to recent advances in micro fabrication and manufacturing, various devices having the order of microns, such as micro pumps, micro-heat sinks, micro-biochips, micro-reactors, micro-motors, micro-valves, and micro-fuel cells, have been developed. These micro-devices found their applications in microelectronics, microscale sensing and measurement, power systems, spacecraft thermal control, biotechnology, and microelectromechanical systems. For example, microchannel heat sinks are one of the ultimate heat removal solutions in most microscale devices. The need for efficient and effective heat transfer techniques in miniature devices has, in recent years, fostered extensive research interest in microscale heat transfer with an emphasis on microchannels with both circular and rectangular cross-sections [1][2].

Several investigators have reported significant deviation from classical theory (e.g. with respect to the solution of the Navier-Stokes and energy equations with simplifying assumptions like no-slip boundary conditions for velocity and temperature) used in macroscale applications, while others have reported general agreement, especially in the laminar region.

The common channel flow classification based on the hydrologic diameter divides the range from 1 to 100 µm as microchannels, 100 µm to 1 mm as meso-channels, 1 mm to 6 mm as miniature channels, and greater than 6 mm as conventional channels. It is convenient to differentiate the flow regimes for experimental and theoretical predictions as a function of Knudsen number (Kn). Kn is a parameter that physically indicates the relative importance of rarefaction or non-continuum effects. It is the ratio of the flux gas mean free path, λ, to the characteristic dimension of flow field, D. The following classification is commonly accepted:

• For Kn < 10-3, the flow is a continuum flow and is accurately modeled by the Navier-Stokes and energy equations with classical no-slip boundary conditions for velocity and temperature.

• For 10-3 < Kn < 10-1, the flow is a slip flow and the Navier-Stokes and energy equations remain applicable, provided a first order velocity slip and a temperature jump are taken into account at the walls. These new boundary conditions indicate the rarefaction effects at the walls.

• For 10-1 < Kn < 10, the flow is a transition flow and the continuum approach of the Navier-Stokes equations is no longer valid. However, the intermolecular effects are not yet negligible and should be taken into account.

• For Kn > 10, the flow is a free molecular flow and the occurrence of intermolecular collisions is negligible compared with the collisions between the gas molecules and the walls.

As noted above, some special effects or conditions that are typically neglected at the macroscale should be included at the microscale. One such condition is slip flow when the fluid is rarefied or the geometry is at the microscale level. In contrast to continuum flow phenomena, the fluid no longer reaches the surface velocity or temperature. Two major characteristics of slip flow are velocity slip and temperature jump at the surface. These can be determined using the kinetic theory of gases. For a cylindrical microtube, the velocity slip condition is:

 ${{u}_{S}}=-\frac{2-F}{F}\lambda {{\left. \frac{\partial u}{\partial r} \right|}_{r={{r}_{o}}}}$ (1)
Configuration for the slip boundary condition in a microchannel.

where us is the slip velocity, as shown in figure to the right, λ is the molecular mean free path, and F is the tangential momentum accommodation coefficient; and the temperature jump is

 ${{T}_{S}}-{{T}_{w}}=-\frac{2-{{F}_{t}}}{{{F}_{t}}}\frac{2\gamma }{\gamma +1}\frac{\lambda }{\text{Pr}}{{\left. \frac{\partial T}{\partial r} \right|}_{r={{r}_{o}}}}$ (2)

where Ts is the temperature of the fluid at the wall, Tw is the wall temperature, and Ft is the thermal accommodation coefficient. To make the analysis simpler, F and Ft are usually denoted by F and assumed to be 1. The above relations clearly indicate that with an increase in the mean free path value, the slip velocity condition at the walls, as well as the temperature jump, increases.

Detailed investigations were made to compare the conventional (continuum) theories with the experimental data in microscale flows and heat transfer problems. Hetsroni, et al. [3] compared the experimental data from the literature with small Knudsen numbers (0.001 – 0.4) and Mach numbers (0.07 – 0.84) that correspond to continuum models in circular, rectangular, triangular, and trapezoidal microchannels with hydraulic diameters ranging from 1.01 μm to 4010 μm and noted, in general, a good agreement with the conventional theories. The no-slip boundary conditions are used for these models for both velocity and temperature.

Hetsroni et al. [3] concluded that the existing experimental friction factor in the literature agrees quite well with that of the conventional continuum for fully developed laminar gas flow for $0.001\le \text{Kn}\le 0.38$. There are, however, contradictory results despite the existence of significant experimental and theoretical investigations in microchannels.

For the microchannels with small Knudsen numbers which lie within the no-slip flow region, the conventional solutions apply with high accuracy. The solutions for the circular tube fully developed flow and temperature profile are thus Nu = 3.66 for the constant wall temperature and Nu = 4.36 for the constant wall heat flux boundary conditions. The Nusselt number for a rectangular microchannel has a dependence on the channel aspect ratio.

Myong et al. [4] adopted Langmuir’s slip model to characterize the slip boundary conditions. They developed a physical approach to account for the interfacial interaction between the gas molecules and surface molecules. In this approach, the gas molecules are assumed to interact with the surface of the solid via long range attractive forces. Consequently, the gas molecules can be adsorbed into the surface and then desorbed after some time lag. They found that, for most physical applications, this model always predicts the reduction of heat transfer with increasing gas rarefaction.

Several studies have also examined the transition point from laminar to turbulent flow in microchannel passages and found that the critical Reynolds number is still approximately 2300. The classical flow and thermal regions such as: fully developed flow and temperature profiles, fully developed flow profile but developing thermal profile, fully developed thermal profile but developing flow profile, and simultaneously developing flow and temperature profiles, are also applicable to the analysis of microchannels. For the microchannels with the Knudsen numbers within the slip flow condition, the flow and heat transfer is characterized by Knudsen number (Kn), Peclet number (Pe), and Brinkman number (Br).