A NUMERICAL ANALYSIS ON THE EFFECT OF DEVIATION FROM A CENTERED WICK STRUCTURE IN AN ULTRA-THIN FLATTENED HEAT PIPE

For ultra-thin heat pipes, a centered wick structure is often used. In this study, a numerical analysis is performed on an ultra-thin heat pipe in which the wick structure’s position has deviated from the center. A 3D heat pipe model, developed by the author in a previous study, is extended, and numerical calculations are conducted to determine any differences in performance because of an off-center wick. The major findings are as follows: (1) a completely centered wick structure is recommended for optimum performance; (2) accurate central positioning of the wick structure is important in the fabrication process of ultra-thin heat pipes.


INTRODUCTION
Many research studies have been conducted on heat pipes (e.g., Faghri, 2016;Sohel Murshed and Nieto de Castro, 2017;Mochizuki and Nguyen, 2019;Sobhan and Peterson, 2019;Guo et al., 2020). Among them, in recent years, an attention has been paid on ultra-thin heat pipes for thermal management of thin electronics devices. Recently, Shi et al. (2019) fabricated an 80-mm-long, 50-mm-wide, and 0.65-mm-thick ultra-thin heat pipe. The height of the vapor flow channel within the heat pipe was 0.2 mm. The heat pipe's effective thermal conductivity was 800 W/(mK) under a power input of 15.9 W, evaporator section temperature of 60C, and a vertical orientation mode. Tang et al. (2019) examined the thermal performance of ultra-thin heat pipes with mesh wicks in various structures. The length, width, and thickness of the heat pipe was 150, 8.5, and 1.0 mm, respectively. Zhou et al. (2019) developed an ultra-thin heat pipe of 120  9  0.75 mm with a centered wick structure. The height of the vapor flow channel was 0.35 mm. Their experiments compared different widths of the wick structure. Li et al. (2019) fabricated an ultrathin heat pipe of 100  50  0.5 mm. The effective thermal conductivity of 2.88  10 4 W/(mK) was then obtained at a power input of 7.37 W. All the abovementioned studies used a capillary wick structure to circulate a working fluid within their ultra-thin heat pipes. Furthermore, the selfexcited oscillation phenomena of a working fluid were used for fabricating ultra-thin heat pipes. Liu et al. (2019) conducted simultaneous visual and thermal investigations to better understand the thermo-hydrodynamic characteristics of a micro oscillating heat pipe. Their heat pipe had a five-turn meandering rectangular channel with a hydraulic diameter of 550 m. The other recent studies on ultra-thin heat pipes were reviewed in the author's previous papers (Koito, 2019a;Koito, 2019b). Usually, ultra-thin heat pipes are used in the thermal management of smartphones. However, they are also used in proton exchange membrane fuel cell stacks (Luo et al., 2020) and solar collectors in a solar energy system (Zhong et al., 2020).
A flattening technique is a typical approach, i.e., a capillary wick structure is inserted within a metal circular pipe, which is finished by flattening the pipe to a desired thickness, to fabricate an ultra-thin heat pipe. The wick structure is positioned at the center. Vapor flow spaces are retained on both sides of the wick structure ( Fig. 1). This inner structure is very different from that of a conventional capillary-driven heat pipe of normal size in which a wick structure is attached to the inner wall of the heat pipe container (Sadeghinezhad et al., 2016;Tharayil et al., 2017;Naemsai et al., 2019). The vapor flow channel of an ultra-thin heat pipe is extremely narrow; thus, it is vital to consider the effects of vapor flow friction in the thermal design process of ultra-thin heat pipes (Koito, 2019a).
As mentioned previously, flattening technique is used in fabricating ultra-thin heat pipes. However, it requires accurate positioning of the wick structure; otherwise, its position may deviate from the center of the heat pipe. Because the effect of vapor flow friction can be considerable, it is recognized that any wick structure deviation from the center would influence the thermal performance of the heat pipe. Even for a small deviation, the effect would be considerable because of the narrowness of the vapor flow channel. Thus, for rational thermal design and fabrication of ultra-thin heat pipes, quantitative information concerning deviation effects is required.
In this study, therefore, numerical analyses were conducted to investigate the thermal-fluid transport characteristics of ultra-thin heat pipes in which the position of the wick structure had deviated from the center. To date, although many experimental research studies have been conducted on ultra-thin heat pipes, the deviation effect has not been clarified. A mathematical model developed by the author (Koito, 2019a) was extended to consider the deviation. Figures for the velocity, pressure, and temperature distributions within the ultra-thin heat pipe were obtained for different positions of the wick structure, including where it was positioned at the side of the vapor flow channel.

Frontiers in Heat and Mass Transfer
Available at www.ThermalFluidsCentral.org Figure 1 shows a cross-section of an ultra-thin heat pipe; it is perpendicular to a heat transport direction from an evaporator to a condenser section. This study's numerical analyses considered an ultrathin heat pipe in which the position of a wick structure had deviated from the center of the heat pipe. As shown by the dotted lines in Fig. 1, a computational domain included two vapor flow channels and a wick structure between them. The deviation, , was then defined by the difference between the centers of the wick structure and the heat pipe container. Figure 2 shows a 3D mathematical model of the ultra-thin heat pipe. The heat pipe model comprised two vapor regions (VI region and VII region) and a liquid-wick region (L region). These three regions had the same total length, lt, and height, h. The widths of VI, VII, and L regions are denoted by wv,I, wv,II, and wl, respectively. The abovementioned deviation, , was then calculated by

MATHEMATICAL MODEL AND NUMERICAL CONDITIONS
The governing equations and boundary conditions presented in the author's previous papers (Koito, 2019a;Koito, 2019b) were applied to the present heat pipe mode; therefore, only the main features are described below. The following assumptions were applied to the mathematical model: a) Steady state is established within the model b) Vapor flow is laminar c) Wick structure is isotropic and filled with a working fluid d) Liquid evaporates and vapor condenses only at two interfaces between the vapor and liquid-wick regions For three regions: VI (i = v,I), VII (i = v,II), and L (i = l ), the governing equations were expressed as follows: where V is the velocity vector (= (u, v, w)), p is the pressure, T is the temperature,  is the density,  is the viscosity,is the porosity, K is the permeability, cp is the specific heat at constant pressure, k is the thermal conductivity, and keff is the effective thermal conductivity. One end (length: lh, width: wl) of the bottom surface of the L region was heated at prescribed heat flux, q, whereas the other end (length: lc, width: wl) was cooled. The boundary conditions at the heated and cooled sections are given by the following equations: 0, The present heat pipe model had two vapor-liquid interfaces, which were located at x = wv,I and wv,I + wl, respectively. Using the Clausius-Clapeyron equation, the temperatures at these interfaces were calculated by where Tref is the reference temperature, pref is the reference pressure, Rg is the gas constant, and hfg is the latent heat. The vapor and liquid velocities at these vapor-liquid interfaces were calculated from the following equations: Except for the abovementioned boundaries, the following fluid-flow and thermal boundary conditions were applied: where n is the coordinate normal to the boundary surface. Moreover, because only second kind of boundary condition, i.e., temperature gradients were given on the outer surface of the heat pipe model, the temperature at x = wv,I, y = lt /2, and z = h/2 was prescribed as the operating temperature, To. The control volume method and SIMPLE algorithm (Patankar, 1980) were employed for numerical calculation. The calculation was iterated until the changes in u, v, w, p and T values became smaller than 1.0 %. Under the physical dimensions shown in Table 1   4 distribution was axisymmetric at the center (x = 1.5 mm) of the crosssection. For  = 0.4 mm, however, the vapor velocity distribution was asymmetric, i.e., the vapor velocity in the VII region was higher than that in the VI region because of the difference in the xz cross-sectional area between VI and VII regions: the cross-sectional area of the VII region was 57% lesser than the VI region. Moreover, the vapor velocity in the VII region for = 0.4 mm was higher than that for  = 0 mm, whereas the vapor velocity in the VI region for = 0.4 mm was lower than that for the  = 0 mm. Note that the drop in vapor pressure because of friction in the heat pipe model changed with the vapor velocity. Furthermore, as expressed by Eqs. (7) and (8), the temperatures at the vapor-liquid interfaces were determined by the local vapor pressures acting on these interfaces. This indicates that the vapor temperature distribution in the heat pipe model changed with the vapor velocity. Thus, as shown in Fig.  4 (b), because the vapor velocities in the VI and VII regions were different, asymmetric temperature distribution of the heat pipe model was obtained for  = 0.4 mm. However, for  = 0 mm, as shown in Fig. 4 (a), the temperature distribution was axisymmetric at the center (x = 1.5 mm) of the heat pipe model. From the abovementioned results, the thermal-fluid transport characteristics of the heat pipe model changed with the deviation from the centered wick structure.

RESULTS AND DISCUSSION
The vapor temperature differences over VI region, Tv,I, and VII region, Tv,II, were evaluated by the following equations: , , ℎ/2 (13) Figure 5 shows the Tv,I and Tv,II values with respect to  for h = 0.2, 0.4, and 0.8 mm. As expected, Tv,II increased with  because of increase in the vapor velocity in the VII region. The increase in Tv,II with  was less for h = 0.8 mm, whereas it was very large for h = 0.2 mm. The increase in Tv,II caused an increase in the thermal resistance between the evaporator and condenser sections, resulting in a decrease in the thermal performance of heat pipe. The height of the vapor flow channel within the recent ultra-thin heat pipe was 0.2 mm (Shi et al., 2019). Thus, in the fabrication process of ultra-thin heat pipes, pinpoint accurate positioning of the wick structure is essential to avoid an unnecessary increase in thermal resistance.
The heat transport rates through VI region, Qv,I, and VII region, Qv,II, were evaluated by the following equations: where max [A, B] stands for the maximum of A and B. Because steadystate calculations were conducted in this study, the sum of Qv,I and Qv,II was essentially equal to the amount of heat applied to, or discharged from, the heat pipe model. Figure 6 shows the Qv,I and Qv,II values with respect to  for h = 0.2, 0.4, and 0.8 mm. Because the vapor velocity in the VII region was higher than that in the VI region, the vapor pressure at the evaporator section of the VII region was higher than that in the VI region. Thus, the boiling point of the working fluid at the evaporator section of the VII region was higher than that in the VI region. Therefore, Qv,II was smaller than Qv,I, except for  = 0 mm; moreover, the difference between Qv,I and Qv,II became larger with increase in  and decrease in h. However, compared with the difference between Tv,I and Tv,II (Fig. 5), such a large difference was not reported between Qv,I and Qv,II. Moreover, numerical analyses were conducted for the case in which the wick structure was positioned at the heat pipe model's side. In this calculation, only the VI region was considered with wv,I = 2.0, wl = 1.0, and wv,II = 0 mm; the deviation, , was 1.0 mm. Rather than the thermal boundary condition of Eq. (8), the adiabatic condition: fourth equation of Eq. (11) was applied to the boundary at x = wv,I + wl. Figures 7 and 8 show the vapor velocity distribution on the xy cross section at z = h/2 and the temperature distribution of the heat pipe model, respectively, under the numerical conditions of  = 1.0 and h = 0.4 mm. The boundaries between the VI and L regions, as well as the VII and L regions are indicated by dashed lines. By comparing Fig. 7 with Fig. 3 (a), the vapor velocity for  = 1.0 mm was slightly lower than that for the case of no deviation ( = 0 mm) from the centered wick structure. Under h = 0.4 mm, the vapor temperature difference over VI region, Tv,I, was 1.8C for  = 1.0 mm, whereas Tv,I = Tv,II = 2.0C for = 0 mm. Because of the abovementioned lower vapor velocity, Tv,I for  = 1.0 mm was somewhat smaller than that for = 0 mm. By comparing Fig. 8 with Fig.  4 (a), the temperature gradients in the L region near the heated and cooled sections for  = 1.0 mm were much larger than those for the case of no deviation. Note that contour levels shown in Fig. 8 are different from those in Fig. 4 (a).
The temperature difference on the VI side of the L region near the heated section, , , was evaluated using the following equation: , , , , , 0, ℎ/2 Moreover, except for  = 1.0 mm, the temperature difference on the VII side of the L region near the heated section, , , was evaluated by the following equation: