Fluid flow and heat transfer in a dual-wet micro heat pipe

Abstract

Micro heat pipes have been used to cool micro electronic devices, but their heat transfer coefficients are low compared with those of conventional heat pipes. In this work, a dual-wet pipe is proposed as a model to study heat transfer in micro heat pipes. The dual-wet pipe has a long and narrow cavity of rectangular cross-section. The bottom-half of the horizontal pipe is made of a wetting material, and the top-half of a non-wetting material. A wetting liquid fills the bottom half of the cavity, while its vapour fills the rest. This configuration ensures that the liquid–vapour interface is pinned at the contact line. As one end of the pipe is heated, the liquid evaporates and increases the vapour pressure. The higher pressure drives the vapour to the cold end where the vapour condenses and releases the latent heat. The condensate moves along the bottom half of the pipe back to the hot end to complete the cycle. We solve the steady-flow problem assuming a small imposed temperature difference between the two ends of the pipe. This leads to skew-symmetric fluid flow and temperature distribution along the pipe so that we only need to focus on the evaporative half of the pipe. Since the pipe is slender, the axial flow gradients are much smaller than the cross-stream gradients. Thus, we can treat the evaporative flow in a cross-sectional plane as two-dimensional. This evaporative motion is governed by two dimensionless parameters: an evaporation number E defined as the ratio of the evaporative heat flux at the interface to the conductive heat flux in the liquid, and a Marangoni number M. The motion is solved in the limit E→∞ and M→∞. It is found that evaporation occurs mainly near the contact line in a small region of size E−1W, where W is the half-width of the pipe. The non-dimensional evaporation rate Q* ~ E−1 ln E as determined by matched asymptotic expansions. We use this result to derive analytical solutions for the temperature distribution Tp and vapour and liquid flows along the pipe. The solutions depend on three dimensionless parameters: the heat-pipe number H, which is the ratio of heat transfer by vapour flow to that by conduction in the pipe wall and liquid, the ratio R of viscous resistance of vapour flow to interfacial evaporation resistance, and the aspect ratio S. If HR≫1, a thermal boundary layer appears near the pipe end, the width of which scales as (HR)−1/2L, where L is the half-length of the pipe. A similar boundary layer exists at the cold end. Outside the boundary layers, Tp varies linearly with a gradual slope. Thus, these regions correspond to the evaporative, adiabatic and condensing regions commonly observed in conventional heat pipes. This is the first time that the distinct regions have been captured by a single solution, without prior assumptions of their existence. If HR ~ 1 or less, then Tp is linear almost everywhere. This is the case found in most micro-heat-pipe experiments. Our analysis of the dual-wet pipe provides an explanation for the comparatively low effective thermal conductivity in micro heat pipes, and points to ways of improving their heat transfer capabilities.

Heat and mass transfer in polygonal micro heat pipes under small imposed temperature differences

“…where only the reference parameters T 0 and q 0 are left in c and E Instead of solving the coupled system of equations for p g and T p , as in Zhang et al [25], we find it simpler to solve for the evaporation rate M.…”

“…Due to the small size of micro heat pipes, surface tension forces dominate inertia and gravity forces [25]. Therefore, convective currents are negligible and so is convective heat transfer.…”

“…The result in (2.21) agrees with his solution in the limit of zero contact angle. In the dual-wet micro heat pipe, b M $ E À1 ln E, because the interface is perpendicular to the wall [25].…”

“…where H is the dimensionless heat-pipe number that measures the ratio of evaporative heat transfer to conductive heat transfer [25], and q c is the conduction heat rate in the liquid and wall in the absence of vapor flow. Eq.…”

“…However, the flow of vapor away from the hot end, which induces evaporation at the interface, is important in correctly modeling these devices. Zhang et al [25] considered the vapor flow in their model of a dual-wet micro heat pipe where the interface is initially flat and pinned at the pipe wall. Here we extend their work by including the bubble geometry to model the evaporation, fluid flow, and heat transfer in polygonal micro heat pipes of triangular, square, hexagonal, and rectangular cross sections.…”