International Journal of Photoenergy

Volume 2018, Article ID 8126318, 12 pages

https://doi.org/10.1155/2018/8126318

## Preliminary Estimate of Coriolis Force of Vapor Flow in Rotating Heat Pipes Based on Analytical Solution

School of Electrical and Power Engineering, China University of Mining and Technology, Xuzhou 221116, China

Correspondence should be addressed to Huanguang Wang; nc.ude.tmuc@3102ghw

Received 25 August 2017; Revised 16 January 2018; Accepted 30 January 2018; Published 3 April 2018

Academic Editor: Ben Xu

Copyright © 2018 Huanguang Wang and Qi Yu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In current theory models for rotating heat pipes, the temperature field of the vapor phase is often supposed to be homogenous, and as a result of such simplification, the experiment result of the heat transfer performance for high rotating speed has some discrepancy with that predicted by theory models. In this paper, the analytical solution of the vapor flow in rotating heat pipes was obtained on the hypothesis of potential flow and with the method of variable separation. A specific rotating heat pipe was examined under three kinds of boundary conditions: linear distribution, uniform but asymmetric distribution, and uniform as well as symmetric distribution of heat load. The flow field was calculated, the Coriolis force is estimated, and it is found that (1) for a rotating heat pipe with high speed or large heat load, it is necessary to consider the Coriolis force, as its magnitude can be that of the gravitational acceleration; (2) the maximum Coriolis force is located at the vapor-liquid interface at the evaporator and condenser sections, and the directions in these two sections are opposite; (3) the Coriolis force is closely related with working conditions and working fluids, and it decreased with working temperature and increased with the heat load; and (4) the maximum viscous shear stress is located at the adiabatic section.

#### 1. Introduction

A rotating heat pipe is a special kind of heat pipe which rotates with the object it cools, and the liquid fluid in it flows back under the centrifugal force. The structure of a rotating heat pipe includes a cylindrical rotating heat pipe [1], a parallel axis rotating heat pipe [2, 3], a radial rotating heat pipe [4], a truncated cone rotating heat pipe [1], and an even fan-shaped rotating heat pipe [5]. On account of the stable and strong driving ability caused by the centrifugal force, the advantage of heat pipes to get more even temperature field can be brought into full play by rotating heat pipes, and an important application of it is the thermal and cold protection for rotary machines, such as generators [6], rotor of gas turbines [4, 5, 7], aeroengine nose cones [8–10], milling machines [11, 12], and drills [13]. For such equipment, due to friction, to electromagnetic induction, and to being heated or even cooled [8–10], the temperature of the rotating shaft, blade, or nose cone is often uneven, causing thermal stress [4], deformation, and even failure of the machine. Thus, thermal and cold protection is required; however, there are some problems with the currently used thermal or cold protection method more or less, like economic or security ones, for instance, hydrogen cooling in generators [14] and film cooling in gas turbines. Therefore, the passive dredging thermal protection scheme of rotating heat pipes is now gradually revealing its advantage.

Current researches on rotating heat pipes contain two directions: experiment and numerical simulations, the former focuses on the relationship between thermal resistance and parameters like rotation speed and liquid charging rate, such as the research of Xie et al. [15], Ponnappan et al. [16], Song et al. [17], and so on. Ponnappan et al. [16] researched on the case of high rotation speed and found that the variation of thermal resistance was opposite to the result got by the Nusselt condensation model. While numerical simulation mainly studies on the four processes of evaporation, condensation, vapor flow, and liquid flow happened in the rotating heat pipe. According to recent researches, the thermal resistances in the four processes are of different magnitudes. The resistance in the evaporation process is usually small and can be ignored, while for the condensation process, the modified Nusselt model is widely adopted which is now deemed as sophisticated, and when it comes to vapor and liquid flow, the vapor phase is usually simplified as a system with uniform temperature and pressure, and the interior boundary condition of the liquid flow can be set based on such hypothesis. Thus, liquid flow can be solved under different types of models, according to their dimensions. For example, Daniels and Al-Jumaily [18] and Uddina et al. [19] ignored the inertia term of the liquid phase and obtained its temperature distribution by the zero dimension model. Song et al. [17], Bertossi et al. [1], and Hassan and Harmand [2, 20] considered the liquid phase as a one-dimensional flow; Li et al. [21] treated it as a two-dimensional flow. However, to discuss the performance of the heat pipe clearly under different working conditions, the effect of the vapor phase should be taken into full consideration. Faghri et al. took full researches on who resolved originally the velocity field of the vapor phase in rotating heat pipes in [22] and who got the complex distribution of vapor phase parameters in high rotating speed. In their article, the effect of the radial Reynolds number Re_{r} on the vapor flow was intensively discussed, and it is found that when Re_{r} is small, the circumferential velocity is proportional to the radius and the field takes on the feature of plane flow; when the Re_{r} is large enough, the linear relationship between the circumferential velocity and the radius was broken, with the velocity becoming larger at the evaporation section and smaller at the condensation section, which indicated some circumferential forces existing and becoming obvious at high rotation speed, and it is just the Coriolis force. An Indian scholar, Solomon et al. [23], pointed out that vapor flow in the rotating heat pipe is affected by the Coriolis force to some extent, and this is the first literature to date pointing out the existence of the Coriolis force in rotating heat pipes.

In summary, it can be seen that vapor flow in a rotating heat pipe is very complicated, and its flow pattern has a significant impact on the whole performance of a rotating heat pipe; however, theoretical models aiming for rotating heat pipe simulating and mechanism revealing are insufficient. It is worth studying whether the heat transfer characteristics of a rotating heat pipe found by Ponnappan et al. at high rotation speed are affected by the vapor flow pattern, and the dynamics and thermodynamics law of vapor flow of a rotating heat pipe at high rotation speed or at high heat load is also needed to be further clarified. Thus, in this article, it is aimed to obtain the flow of vapor in rotating heat pipes by an analytical method based on the hypothesis of potential flow and especially explore the Coriolis force inflicted to the vapor phase.

#### 2. Establishment of Physical Model for Calculation

For the sake of convenience of theoretical research and viability of the analytical method to get the solution of vapor flow, a two-dimensional rotating heat pipe model was established, which is also the mainstream model for rotating heat pipe research, and the structure of which is shown as the following figure: the rotating heat pipe has a rotating shell which was installed on the object it cools, heat was sucked on the left side—the evaporator, and released on the right side—the condenser, and the middle part of the pipe corresponds to the adiabatic section. The liquid film is attached to the inner wall of the shell, under the function of the centrifugal force. It is advisable and reasonable to suppose that the centrifugal force is large enough, so the variation of the thickness of the liquid film along the direction of the axis can be neglected. The vapor phase flow is driven by the phase change at the liquid-vapor interface, which constitutes the boundary condition of the vapor phase flow. The length of the evaporator is denoted as *l*_{1}, and the length of the condenser is denoted as *l*_{2}. The heat flux at the evaporator is denoted as *q*_{1}, and the heat flux at the condenser is denoted as *q*_{2}. Accordingly, the velocity boundary condition of the vapor flow at the evaporator is denoted as *u*_{1}, and that at the condenser is denoted as *u*_{2}. For the sake of convenience, the inner diameter of the heat pipe is denoted as 2*b*, and the radius of the liquid-vapor interface is denoted as *a*, so the thickness of the liquid film is *b*-*a*.

#### 3. Mathematical Descriptions of the Model and Simplification of the Governing Equations

To solve the vapor flow in the physical model shown in Figure 1, it should be based on the NS functions coupled with the energy function and state function under rotating coordinates. In order to obtain the analytical solution, the flow of the vapor is supposed to be the potential flow; under which hypothesis, the above functions can be simplified to a great extent. The universal form of NS functions is the vector form, as shown below: