International Journal of Heat and Mass Transfer 126 (2018) 557–566
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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A numerical modeling for the steady-state performance of a micro heat pipe using thin liquid film theory Eui Guk Jung a, Joon Hong Boo b,⇑ a b
Dept. of Fire & Disaster Prevention Engineering, Changshin University, 262 Palyong-ro, Masanhoewon-gu, Changwon-si, Gyeongsangnam-do 51352, Republic of Korea School of Aerospace and Mechanical Engineering, Korea Aerospace University, Goyang, Gyeonggi-do 10540, Republic of Korea
a r t i c l e
i n f o
Article history: Received 12 October 2017 Received in revised form 11 May 2018 Accepted 12 May 2018
Keywords: Micro heat pipe Phase-change interface Thin liquid-film theory Film thickness Augmented Young-Laplace equation Heat transfer rate
a b s t r a c t A numerical analysis on heat and mass transfer in a micro heat pipe was performed in this work. Mass, energy, and momentum equations were applied to vapor and liquid phases under steady-state operation in the numerical model. As a result, the trends in mass flow, pressure distribution, and temperature distribution for the working fluid circulating inside the micro heat pipe were obtained for vapor and liquid, respectively. In particular, the vapor-liquid interface shape obtained with a thin liquid film through augmented Young-Laplace equation and areas for heat and mass transfer of liquid and vapor from the vapor-liquid interface thickness were predicted. These areas were applied to solution on the governing equations for vapor and liquid. The numerical model was validated by experimental results. The errors between the experimental and numerical results for the average temperature difference of the evaporator and condenser were found to be less than 1 °C. The errors for thermal resistance had maximum and minimum values of 15.2% and 3.8%, respectively. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction Miniaturization and compact packaging of electronic devices have significantly increased thermal energy per unit area, and thermal control means for such equipment have been actively studied [1,2]. Heat pipe (hereafter referred to as HP) is a heat transfer device that utilizes latent heat during evaporation and condensation of a working fluid charged inside a sealed container to transfer heat between the evaporator and condenser. When a thermal load is applied to the evaporator outer wall, the working fluid charged in the container is vaporized and transferred to the condenser by vapor pressure difference between the evaporator and condenser. In the condenser section, the transferred vapor is condensed to liquid phase, by discharging heat to the cooling medium surrounding the condenser wall. Then the liquid is returned to the evaporator by the capillary force generated by the surface tension and curvature of the vapor-liquid interface. As the latent heat of the working fluid is utilized, the heat pipe exhibits a high heat transfer performance per unit volume. The heat pipe has been used for various thermal control devices in electronic packaging, both in the ground and space environment,
⇑ Corresponding author. E-mail addresses:
[email protected] (E.G. Jung),
[email protected] (J.H. Boo). https://doi.org/10.1016/j.ijheatmasstransfer.2018.05.067 0017-9310/Ó 2018 Elsevier Ltd. All rights reserved.
due to its high efficiency, reliability and design flexibility [1]. Because the micro heat pipe (hereafter referred to as MHP) is very small in size, it normally adopts a polygonal geometry in the crosssection, instead of additional capillary structure, so that the liquidstate working fluid circulation is achieved by the capillary force generated by the acute-angled corners. Especially, because a hydraulic diameter of the MHP container has a typical dimension of 100 mm or less, it is suitable for the chip-level and has been applied as the thermal control means in micro-electromechanical systems (MEMS) [2,3]. In recent years, multiple MHPs in an array [4–12] have been actively applied to various industrial fields, such as renewable energy [4–10] and air conditioning systems [11,12]. Deng et al. [4] applied it to a thermal dissipation device for household solar cells and conducted experimental studies on the improvement of its thermal and electrical efficiency. Diao et al. presented experimental results on the charging/discharging thermal performance and system efficiency by employing flat type MHPs in a thermal storage system [5] and a solar water heater [6]. Wang et al. [7,8] developed the concept of an integrated system that connects the photovoltaic solar collectors, the solar thermal accumulators, and the utilization parts with MHP arrays, and obtained an efficiency of up to 94.6%. Zhu et al. [9,10] tested the system efficiency by applying an MHP array as a heat transfer device to a solar air heater and obtained a maximum system efficiency of approximately 82%.
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Nomenclature A c D g h H K k L m00 MHP N Nu P p q Q R Re r T T u V v w
cross sectional area (m2) specific heat (kJ kg1 °C1) diameter (m) gravity acceleration (m s2) enthalpy (J kg1) Hamaker constant curvature of the interface (m1) thermal conductivity (W m1 °C1) length or total length of MHP (m) mass flux (kg m2 s1) micro heat pipe number of MHP containers Nusselt number perimeter (m) pressure (Pa) heat flux (W m2) input thermal load (W) universal gas constant (J kg1 °C1) Reynolds number radius (m) temperature (°C) average temperature (°C) velocity (m s1) volume (m3) specific volume (m3 kg1) width (m)
Ling et al. [11] conducted experimental work on a cooling performance investigation using an MHP heat exchanger in the air conditioner of a telecommunication station and obtained a cooling performance of at least 3.4 kW. Diao et al. [12] developed an MHP heat exchanger for building ventilation and investigated the applicability of the MHP heat exchanger by testing its heat recovery performance. Kwon and Kim [13] and Yang et al. [14] fabricated a micro pulsating heat pipe and observed the vapor and liquid flow patterns depending on the input thermal load through visualization works; thereafter, they visually investigated the behavior of the liquid film depending on the thermal load. Moon et al. [15] fabricated micro heat pipes with triangular and rectangular shapes and measured heat transfer performance up to 7 W. A relatively small number of studies on analysis and simulation of the MHP performance have been conducted, compared to experimental studies, and steady-state analysis models have mostly developed for them. Studies on the thermal fluid analysis of the MHP have applied the energy conservation equations for the container solid wall and working fluid, and the mass and momentum conservation equations for vapor and liquid. Additionally, the Young-Laplace relation was applied to the vapor-liquid interface to predict its shape [16–22]. As a result, the ratio of liquid and vapor flow to the cross-section of the MHP could be estimated [16–19]. The thermo-fluid analysis model developed by various researchers has contributed to the prediction of the performance of MHP and the investigation of physical phenomena, and there are models that focus on specific physical mechanisms [20,21]. Furthermore, the models that assumed a constant solid wall temperature have a shortcoming in that the operating vapor temperature of the MHP must be fixed in the performance analysis [20–23]. These studies are mainly concerned with physical phenomena of the fluid flow of the MHP working fluid. On the other hand, studies focusing on vapor-liquid interface analysis [21] provide physically
z
axial coordinate (m)
Greek symbols convective heat transfer coefficient (W m2 °C1) b MHP tilt angle (degrees) l viscosity (N s m2) r surface tension (N m1) s shear stress (N m2) n liquid thin-film thickness (m)
a
Subscripts a adiabatic section e evaporator section c condenser section or capillary d disjoining fg liquid-vapor phase change h hydraulic i interface il vapor-liquid interface l liquid v vapor w wall (solid) lw wall-liquid interface vw wall-vapor interface
meaningful parameters to be considered in performance analysis of MHP. While these studies are useful for understanding complex physical behaviors, the MHP analysis has been limited in practice to predicting the heat transfer performance as a function of input heat load or temperature distribution. Nonetheless, a few numerical models capable of predicting the heat transfer performance or various temperature distributions of MHP have been developed using mass, energy, momentum conservation, and the vaporliquid interface with pressure relations. Although these models have been quite helpful in predicting the heat transfer performance of the MHP, each model has its own drawbacks. Firstly, there are models [16–26] that did not rigorously analyze the shape of the vapor-liquid interface in an MHP. Moreover, numerous studies [16–20,22–26] have applied the relations between vapor, liquid, and capillary pressures in a vapor-liquid interface analysis, but ignored the effect of disjoining pressure. Because heat and mass transfer areas for liquid and vapor are typically determined by the shape of the vapor-liquid interface inside the MHP, a precise physical approach of the interface would provide the rationality for the analysis. In particular, the disjoining pressure can provide a region of thin liquid film with a thickness less than 1 lm, and its importance in the study on the micro channel is well known [27]. Secondly, there are models [16–26] in which the conservation equations were mostly composed of a liquid phase. Under normal operation, the working fluid in MHP is separated into liquid and vapor, respectively; therefore, when the conservation equation is composed only of liquid, the validity as well as accuracy could be diminished. This study aims at presenting a numerical model of an MHP that can provide a meaningful accuracy. Jung and Boo [30] have analyzed the effects of disjoining pressure at the vapor-liquid interface for various tilt angles, but mass and energy conservation equations were primarily composed of the liquid phase and, furthermore, the operating temperatures (or temperature difference between solid
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wall and liquid) had to be assumed for a given input thermal load. To overcome the associated problems, mass, energy, and momentum conservation equations were applied respectively to both liquid and vapor, and the heat and mass transfer area for each phase was determined by the vapor-liquid thickness. Most importantly, by applying the full governing equations of the thin liquid-film shape and thermal fluid flow to the MHP model, a more precise vapor-interface area can be employed to calculate the heat transfer area with higher accuracy. This effectively provided the axial distributions of mass, energy, and momentum for both liquid and vapor, which resulted in an improved accuracy, compared to the previous model. The discussion will include detail comparison results.
Capillary pressure (pc ) is a function of the curvature of the interface (K) and the surface tension (r), as shown in Eq. (2).
pc ¼ rK
ð2Þ
Considering the geometrical shape of the liquid film, the curvature (K) of the liquid film can be defined as Eq. (3). As seen in Eq. (3), the curvature of the liquid film is given by the sum of the circumferential curvature (the first right term of Eq. (3)) and the axial curvature (the second right term of Eq. (3)), and is defined as the average curvature at the vapor-liquid interface [30]. 2
K¼
2
1 d f=dx h i1=2 þ h i3=2 2 2 1 þ ðdf=dzÞ ðr h fÞ 1 þ ðdf=dzÞ
ð3Þ
2. Numerical modeling of the MHP In this study, the MHP has a polygonal shape. Fig. 1 schematically shows the shape of the vapor and liquid distribution inside the container of the MHP under normal operation. Fig. 2 illustrates the typical mechanism of liquid flow with film flow in the MHP [13–19]. The MHP is also sensitive to gravity. In an MHP with the favorable tilt angle (upward slope toward the condenser), gravity assists liquid return: the larger the favorable tilt, the greater the heat transfer capability would be resulted. On the contrary, in the adverse tilt angle (upward slope toward the evaporator) would result in a reduced heat transfer performance. Table 1 summarizes the geometric dimensions of the MHP applied in this study, along with the working fluid and material of the container. The geometric configurations are identical in dimensions to the MHP in the associated references [15,30]. It was considered in this study that, for a more precise performance prediction, the numerical model of the MHP should properly reflect the mass, momentum, and energy equations, and reasonably integrate the physical mechanism associated with the phase-change phenomena at the liquid-vapor interface. 2.1. Thin liquid film modeling of the MHP As depicted in Fig. 2, the capillary flow in a liquid thin film can be divided into three regions. The liquid thin-film region has a typical film thickness of 1 lm or less. An area with a larger thickness than a thin film is defined as a meniscus area and a region without a vapor-liquid interface is defined as an absorbed area. The equilibrium equation of the pressure for a phase change interface with a thin-film region can be defined by an augmented Young-Laplace equation. The pressure difference between vapor and liquid at the liquid-vapor interface is mainly due to the disjoining pressure and capillary pressure, and is expressed in Eq. (1) using the augmented Young-Laplace equation.
pv ¼ pl þ pc þ pd
ð1Þ
where r h is the hydraulic radius of the MHP, which can be applied to a polygonal shape having several corners. The disjoining pressure is given by Eq. (4).
pd ¼
H 6pf3
ð4Þ
where H is the non-retarded Hamaker constant, known as 3:148 1021 J for water [32,33]. The physical descriptions of this value were presented in detail in [33]. The thin film region is represented by the disjoining pressure, which is the energy density of the surface of the liquid film. As shown in Eq. (4), when the disjoining pressure decreases, the thickness of the liquid film becomes thicker and stable. In respect of the heat transfer, however, a thicker liquid film would increase the conduction resistance and thus reduce the heat transfer across the thickness. Fig. 3 represents the effect of the Hamaker constant on the film thickness. As shown in Eq. (4), the larger the Hamaker constant, the greater the disjoining pressure becomes, and the film thickness decreases as a result. The influence of the Hamaker constant was quantitatively presented in Fig. 3 by varying it from 3.148 1021 J to 0.1 1021 J with arbitrary intervals, for a heat input of 2 W. The film thickness was calculated as 0.28 mm and 0.38 mm at the end of the condenser (z = L) in the input range of the Hamaker constant, and the film thickness increased by 26.3% when the Hammer constant decreased by 21%. This shows that the film thickness has a very sensitive to the value of the Hamaker constant. For convenience, however, the value of the Hamaker constant for water in this study adopted the value calculated in [31]. Differentiating Eq. (5) with respect to the axial direction z, one obtains:
dpv dpl dpc dpd ¼ þ þ dz dz dz dz
ð5Þ
Rearranging Eq. (5) after solving for the terms for the capillary force and disjoining pressure:
Fig. 1. Schematics of a micro heat pipe having a triangular cross-section.
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Fig. 2. Vapor-liquid interface profile with thin liquid film in an MHP at steady state.
Table 1 Design specifications of the MHP in this study. Items
Specifications
Total length (mm) Evaporator length (mm) Adiabatic length (mm) Condenser length (mm) Working fluid Cross-sectional geometry Number of corners Container material Hydraulic diameter (mm)
50 10 15 25 Pure water Triangular 3 Copper 1.5
dpv dpl dK d H þ ¼ þr dz dz dz dz 6pf3
ð6Þ
Eq. (6) was differentiated with respect to z to obtain the thirdorder ordinary differential equation of liquid film thickness over the length of the MHP.
2 !2 " 2 #1 2 ! d f df d f df df d f ðr h fÞ1 þ 3 1 þ þ 3 dz dz2 dz dz dz2 dz " 2 # df df ðr h fÞ2 1þ dz dz " 2 #3=2 1 df dpv dpl dpd 1þ ¼0 dz r dz dz dz 3
8 Pz¼Le Ai Q > > > Az¼0 > e hfg w > > > > Pz¼LLe La
dz > > P > > z¼L > A Q > > z¼LLe La i : in condenser section Ac h w
ð9Þ
fg
ð7Þ
The disjoining pressure gradient in the axial direction is shown in Eq. (8).
dpd d H ¼ dz dz 6pf3
Fig. 3. Effect of the Hamaker constant on the film thickness.
ð8Þ
8 Pz¼Le A Q > z¼0 i > > > Ae hfg w > > > > < Pz¼LLe La
in evaporator section Ai Aw av ðT w T v Þ d qv Av uv z¼LLe ¼ in adiabatic section Aa hfg w > dz > > P > > z¼L > A Q > > z¼LLe La i : in condenser section Ac h w fg
2.2. Governing equation of the MHP Numerical models have been developed using a onedimensional approach because the hydrodynamic diameter of the MHP is very small compared to its length [16–28]. The steadystate axial mass conservation equations for vapor and liquid are presented in Eqs. (9) and (10), respectively, which were developed for the evaporator, adiabatic, and condenser sections. The righthand sides of Eqs. (9) and (10) represent the axial mass flow rate of the vapor and liquid, respectively, inside the evaporator, the adiabatic section, and the condenser. As shown in both equations, thermal energy is supplied in the evaporator and removed in the condenser. The mass flow rate in the evaporation and condensation modes have positive sign and negative sign, respectively.
ð10Þ where, hfg is the enthalpy of vaporization and
P 1 df= sinðtan df=dzÞ can be determined by the geometry w ¼ z¼z z¼0 of the thin liquid film shown in Fig. 2. w is the length of the film curve and it is approximated as a straight line because of the very small size of dz. Since the capillary radius of curvature can be calculated after the thin liquid film shape is obtained by Eq. (7), the contact angle between the solid and the interface can be determined. The contact angle ½h ¼ tan1 df=dz of liquid thin film was considered in the first step and it is calculated as 12.6° from the geometry of the liquid film shown in Fig. 2. Eq. (11) shows the area (Ai ) of the vapor–liquid interface of the liquid thin film on a control volume. The vapor-liquid interfacial
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area of Eq. (11) was obtained by approximating the control volume for the vapor region to a truncated cone shape by applying a hydraulic radius (rh ) [29]. The side length of the truncated cone shape, l½¼ pdf= sinðtan1 df=dzÞ was determined by the geometry of the liquid film shape in Fig. 2.
Ai ¼
pdf
fðrh fÞ þ ½r h ðf þ dfÞg sinðtan1 df=dzÞ n o 2 þ p ðr h fÞ2 þ ½r h ðf þ dfÞ
ð11Þ
In Eq. (10), the thermal energy portion entered into the control volume of the liquid film is proportional to the area ratio (Ai =Ae ) of the entire evaporator to the control volume of the liquid thin film for the input thermal load (Q ) of the evaporator. The liquid area Al ½¼ As and vapor area Av ½¼ Að1 sÞ can be expressed respectively using s½¼ f=rh , the ratio of the film thickness ðfÞ to the hydraulic radius of the MHP container (rh ). In Eqs. (9) and (10), al and av are convective heat transfer coefficients between each phases (vapor and liquid) and the inner wall of the MHP. The liquid and vapor flows are assumed to be laminar, and the Nusselt number is basically defined as 2.68 [16–26]. For an adiabatic section of Eqs. (9) and (10), the mass flow rate was derived by considering the convective heat transfer rate owing to the temperature difference between the MHP solid wall and the working fluid is equal to the phase change heat transfer rate (the right side of the second equation of Eqs. (9) and (10)). The momentum equations for liquid and vapor are shown in Eqs. (4) and (5), respectively.
d u2 Vv m00v hv þ v dz 2 Pz¼Le 8 A QL > T T z¼0 i > k l Al w f v þ > > Ae > < T w T v ¼ k l Al f P > > z¼L > > A QL > z¼LLe La i : T w T v k l Al f Ac
in evaporator section
ð15Þ
in adiabatic section in condenser section
where, Aw is the inner surface area of the control volume on the P P z¼Le z¼L MHP solid wall. Q L z¼La Ai =Ac can be treated z¼0 Ai =Ae and Q L as heat transfer rate by phase change in length of the evaporator and condenser, respectively. If no heat loss is assumed, the values of the right term in Eqs. (14) and (15) are the same as the input thermal load. Therefore, the phase change heat transfer rate (QL) could be adjusted so that energy conservation is satisfied. For example, in the case of the evaporator section in Eq. (14), the phase change heat transfer rate (Q L ½¼ Q al Aw ðT w T l Þ) can be defined as the value obtained by subtracting the convective heat transfer rate from the input heat load (Q). The liquid and vapor temperatures can be expressed as a function of pressure and enthalpy. The energy equation for the solid wall temperature (T w ) of an MHP is shown in Eq. (16).
8 Pz¼z > A QL > z¼0 i > a A ð T T Þ þ l > l w w Ae @2T w < V w kw ¼ al Aw ðT w T l Þ > @z2 Pz¼z > > A Q > : al Aw ðT w T l Þ z¼0Ac i L
in evaporator section in adiabatic section in condenser section
d 2 Al ql ul þ pl dz ¼ sil Ai þ slw Alw ql gAl sin b dz dz
ð12Þ
ð16Þ
d Av qv u2v þ pv dz ¼ siv Ai sv w Av w qv gAv sin b dz dz
ð13Þ
where, the convective heat transfer coefficient (al ) in Eq. (16) can be modified so that the input thermal load and the right term of both equations are equal in the evaporator and condenser.
In Eqs. (12) and (13), the shear stress of the vapor-liquid inter_ v ul Þ ¼ m _ 2 ð1=qv Av 1=ql Al Þ is a flow resistance face, sil ½¼ mðu caused by the vapor-liquid contact at the interface, and the wall _ l =Dh ql , is a flow resistance caused by the shear stress, slw ½¼ 8ll m contact between the liquid and the inner solid wall of the MHP [31]. The shear stresses siv and sil for the vapor – liquid interface are equal in magnitude but of opposite signs. The shear stress between the vapor and the inner solid wall, as shown in the second term of Eq. (13), can be ignored in the area where the liquid film is present. As shown in Eqs. (12) and (13), the directions of shear stresses in liquid and vapor are opposite, and the physical parameters of the control volume are well introduced in the references [16–20,22,24–26]. The energy conservative equations for liquid and vapor are given in Eqs. (14) and (15), respectively. The first terms on the right side of Eq. (14) are the thermal energy transferred from the wall to the liquid and vapor via convection; in the condenser, the solid wall temperature (T w ) becomes lower than liquid temperature (T l ) because the thermal energy is emitted to the coolant media, resulting in a negative value. However, the first terms on the right side of Eq. (15) is the conduction heat transfer rate transferred to the vapor across the thin liquid film.
u2 d m00l hl þ l dz 2 8 Pz¼Le > A QL > z¼0 i > a A ðT T Þ þ > l w w l Ae > < ¼ al Aw ðT w T l Þ > Pz¼L > > Ai Q L > > z¼LL L e a : a A ðT T Þ
Vl
l
w
w
l
Ac
2.3. Boundary conditions The boundary conditions for solving the numerical model are as follows: Considering that the evaporator end (z = 0) is a stagnation point, the followings can be specified.
ul jz¼0 ¼ uv jz¼0 ¼ 0;
8 >
:ðr i fj Þ 1 þ ðdf=dzj Þ2 z¼0 z¼0 9 2 2 > = d f=dz H z¼0 ; þh i3=2 > þ 3 2 p f 6 ; z¼0 1 þ ðdf=dzjz¼0 Þ
in adiabatic section in condenser section ð14Þ
ð17Þ
where, T w jz¼0 was assumed to have no solid temperature gradient at the evaporator end (z = 0) [1,16–19]. The condenser end (z = L) is another stagnation point, and the boundary conditions are shown in Eq. (18).
ul jz¼L ¼ uv jz¼L ¼ 0;
in evaporator section
dT w ¼ 0; pv jz¼0 ¼ psat ðT w jz¼0 ; Þ; dz z¼0
dT w ¼ 0; dz z¼L
ð18Þ
To solve Eq. (7), three boundary conditions at the evaporator end (z = 0) need to be added, which were defined in [30], as shown in Eq. (19). However, the value of df=dzjz¼0 was obtained by iteration to satisfy the boundary condition of the MHP end (z = L)
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in Eq. (18). The thickness of the thin liquid film is strongly dependent on fjz¼0 and df=dzjz¼0 ; nonetheless, it is almost insensitive to 2 2 the d f=dz value. z¼0
Figs. 4 and 5 show the effects of the first-order differential term and second-order differential term on the liquid film thickness among the boundary conditions at the evaporator end (z = 0), for an input heat load Q = 2.0 W. Fig. 4 shows the effect of the size of the first-order differential term (df=dzjz¼0 ) of the boundary condition on the liquid film thickness. As shown in Fig. 4, the boundary value was varied from 1 1011 to 0.1 1011. The boundary value of the first derivative term is defined as a value that determines the degree of inclination of the film at the starting point. Thus, the thickness of the liquid film decreases as the boundary value decreases. At the boundary values of 1 1011 and 0.1 1011, the thickness of the liquid film was calculated to be 0.93 mm and 0.02 mm, respectively; corresponding to a decrease by 97%. This is because the first differential term of the boundary condition at the starting point is very sensitive to the thickness of the liquid film. Therefore, proper selection of the liquid film thickness is necessary depending on the size of the MHP. Fig. 5 shows the effect of the second-order differential term of the boundary condition on the liquid film thickness. As shown in Fig. 5, the value of the liquid film was constant, while the second-order differential term varied from 1 1019 to 10 1011. This means that the second derivative term of the boundary value has minimal effect on the liquid film thickness. The conventional method for selecting these boundary values is not known and depends on the experience of the researcher. In this paper, these boundary values are basically those presented in the references [28,30].
fjz¼0
2 df d f 19 ¼ 1 10 ; ; ¼ 1 10 ; dz z¼0 dz2 z¼0 9
Fig. 5. Effect of the second derivative boundary condition on the film thickness.
(1) The validity of the spatial step size (dz) was investigated based on the resultant liquid temperature distribution and liquid film thickness, as summarized in Fig. 6. The larger spatial step size is generally preferred for the less computation time. Beyond a certain size in spatial step, however, divergence problem might occur such that the calculation process of Eq. (7) becomes impossible. For comparison purposes, the spatial step was increased from 10 lm to 70 lm. In Fig. 6 (a),
ð19Þ
2.4. Numerical process The governing equations for the thermo-hydraulic behavior of the MHP consist of a set of conservation equations, from Eqs. (9) through (15). These equations were solved simultaneously by applying the boundary conditions. Eq. (16) is a partial differential equation in the form of an axial heat conduction equation. This equation is discretized by finite difference, and the discretized equations are solved using the Gauss-Seidel algorithm. Eq. (7) is solved using the fourth-order Runge-Kutta algorithm by applying the boundary condition. The solution flow on these equations is as follows:
Fig. 4. Effect of the first derivative boundary condition on the film thickness.
Fig. 6. Influence of the space step size to provide the validity of the solutions: (a) profile of the liquid film thickness; (b) axial distribution of the liquid temperature.
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it is obvious that the larger values of liquid film thickness resulted in as the larger spatial step size was used for calculation. Based on the signs of the terms in Eqs. (12) and (14), one can tell the fluid pressure (pl) and thermal energy (hl) are inversely proportional to the liquid area (Al). Therefore, as the liquid area increases toward the condenser end, the liquid film thickness (f) would increase, while the local enthalpy would decrease, resulting a decrease in the local temperature. The solutions provided in Fig. 6(a) and (b) coincide with this qualitative prediction. Fig. 6(b) exhibits that the predicted values of the liquid temperature at the end of the condenser, as well as those of the temperature distribution over the entire length, decreased as the space step (dz) increased. However, for a spatial step of 30 lm in Fig. 6(a), the vapor–liquid interface profile deviated to a curved line at about z/L = 0.6 and the liquid film thickness became thicker than that for dz = 50 lm at z/L > 0.8. Consequently, when dz increased to 50 lm in Fig. 6 (b), the liquid temperature near the condenser end became higher than that for dz = 30 lm, although the temperature increase was only about 2.1 °C. Considering the computation time and stability of the numerical process, dz was fixed at 50 lm and this value was used throughout the simulation processes. (2) Eq. (8) is solved using the finite difference method, and the inlet wall temperature of the next control volume was obtained by applying T w jz¼0 and T l jz¼0 ½¼ T l;sat ðpl jz¼0 Þ. (3) Eqs. (9) through (15) are simultaneously solved to obtain the mass flow rate, pressure and enthalpy on the liquid and vapor at the outlet of the control volume. Liquid and vapor temperatures can be obtained respectively by the pressure and enthalpy of the liquid and vapor from saturation condition as specified in the boundary conditions. (4) Eq. (7), the equation for the thin liquid-film thickness with capillary flow, can be solved by the input of boundary conditions and liquid surface tension (r). The film thickness (f) is obtained for one control volume, i.e., one space step, starting from the thin liquid film starting point (z = 0), by applying 2 2 and r½¼ rsat ðpl jx¼0 Þ. fjz¼0 , df=dzjz¼0 , d f=dz z¼0
(5) After proceeding from (1) through (4) to the MHP end (z = L), if the lengths for the condenser and evaporator are the same, Eq. (18) can be obtained automatically because a symmetrical shape is constructed based on the middle point of the MHP for the mass flow profile; however, Eq. (18) is not satisfied if the two lengths are different. As shown in Eqs. (9) and (10), the conservation of mass equations for liquid and vapor are strongly dependent on the area of the vaporliquid interface of Eq. (3). The area is determined by the liquid interface thickness, which is very sensitive to the value of df=dzjz¼0 . Therefore, df=dzjz¼0 is used as a variable of iterative solution that satisfies Eq. (19). The iterations converged by the bisection algorithm. (6) This problem was solved using the MATLAB 2010a software, and the physical properties of the working fluid were obtained by REFPROP 9.0, and The physical properties of copper, as the MHP container material, was used for calculations by generating the polynomial function only as a function of temperature.
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model of this study one-dimensional and it is possible to apply an MHP with various polygonal shapes when using the hydraulic diameter. The numerical and experimental results were compared and investigated under horizontal tilt angle. Fig. 7 and Table 2 show the comparison of the results of the experiments and numerical results with those of the previous model for the solid wall temperature distribution along the MHP axial position at the input thermal load of 2 W, 3 W, and 4 W. In the case of the input thermal load of 2 W, the average temperature difference (T e ) of the evaporator between the numerical model and the experimental result is approximately 0.2 °C. Moreover, the average temperature difference (T c ) of the condenser in the numerical model and the experimental result is approximately 1.1 °C. For the input thermal load of 3 W, the average temperature difference of the evaporator in the present model and the experimental result is approximately 0.2 °C. Furthermore, the average temperature difference of the condenser in the numerical model and the experimental result is approximately 0.7 °C. For the input thermal load of 4 W, the average temperature difference of the evaporator in the present model and the experimental result is approximately 0.1 °C. Similarly, the average temperature difference of the condenser in the numerical model and the experimental result is approximately 0.9 °C. As shown in Fig. 7 and Table 2, the maximum temperature difference between the experiment and prediction for the axial temperature distribution of the MHP was calculated to be 5.7° C, and excellent accuracy was obtained. Fig. 8 represents a comparison of the thermal resistance (Rth ½¼ ðT e T c Þ=Q ) values among the experimental result and present model. The thermal resistance is the difference between the average evaporator temperature (T e ) and the average condenser temperature (T c ) divided by the input thermal load (Q ). The thermal resistance error of the model is 22% (Q = 2.5 W) maximum, and 3.8% (Q = 0.5 W) minimum. Fig. 9 shows the liquid phase temperature of the working fluid as a function of the input thermal load along the MHP axial length. As shown in Fig. 9, the liquid temperature increases with increasing input thermal load, and the liquid temperature decreases exponentially with increasing axial length. The liquid temperature at the adiabatic region (0.2 z/L 0.5) has a nearly constant value. This is because Eqs. (14) and (15) have do not have a term for heat transfer caused by the phase change. As the input thermal load increased by 0.5 W, the average temperature increased by 11 °C. The average temperature difference between the evaporator and the condenser was calculated to be approximately 6.9 °C.
3. Result and discussions Table 1 summarizes the geometric size of the MHP considered in this study, along with the working fluid and container material. The geometric dimensions of the MHPs listed in Table 1 are identical with those in the associated reference [15,30]. The numerical
Fig. 7. Axial distributions of solid wall temperatures for input thermal loads of 2 W, 3 W and 4 W.
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Table 2 Axial distributions of solid wall temperatures for input thermal loads. Position (z/L)
Items
Input thermal load (W) 2W
3W
4W
Solid wall temperature (°C) 0.066 0.132 0.348 0.664 0.830
Experiment Prediction Experiment Prediction Experiment Prediction Experiment Prediction Experiment Prediction
84.9 85.8 84.3 83.4 84.2 83.9 78.2 78.6 75.9 73.9
109.9 110.5 109.6 109 105 107.9 103.8 104.8 100.6 101
123.7 124.5 123.9 123 120 121.9 119.2 120.8 117.1 114.2
Fig. 10. Axial distributions for liquid and vapor pressures for input thermal loads of 1 W and 2 W.
Fig. 8. Comparisons of thermal resistances with respect to input thermal loads.
Fig. 9. Liquid temperature for the axial position as function of input thermal loads.
Fig. 10 shows the liquid and vapor pressure distributions along the axial length at input thermal loads of 1.0 W and 2.0 W, respectively. As the axial length increases, the vapor pressures decrease and the liquid pressures increase. As shown in Eqs. (12) and (13), this is because the shear stress is applied in the opposite direction at the solid–liquid and vapor-liquid interface. Under normal operation of the MHP, the liquid portion in the condenser increases, and only liquid phase is present in the liquid block. If the length of the MHP is sufficient, the pressures will become equal. At the input
thermal load of 2.0 W, the vapor pressure increased by approximately 13.6 kPa and the liquid pressure increased by 5.8 kPa at the condenser end (z = L). However, the vapor pressure at the input thermal load 1 W was reduced by 2.5 kPa, and the liquid pressure was increased by 1.1 kPa, showing that an input thermal load increase results in a decrease of the vapor pressure and an increase of the liquid pressure. This is because as the input thermal load increases, the velocity of the liquid and vapor increases to induce a pressure change. Fig. 11 shows the MHP solid wall and liquid temperature distributions along the axial length at the input thermal load of 2.0 W. Starting at the end of the evaporator (z = 0), passing through the adiabatic section, and up to approximately 20% of the length of the condenser, the solid wall temperature is higher than the liquid temperature; however, it becomes lower than the liquid temperature thereafter. As in Fig. 11, the MHP solid wall and liquid temperature at the evaporator end were 89.3 °C and 86.5 °C, respectively, showing that the wall temperature was higher by approximately 3 °C. However, the wall temperature and liquid temperature at the condenser end (z = L) were 69 °C and 73 °C, indicating a liquid temperature lower by 4 °C. Fig. 12 shows the thickness of the thin liquid film as a function of input thermal load with MHP axial length. The higher the input thermal load, the lower the liquid film thickness. Because the vapor mass flow rate increases as the input thermal load increases, the liquid film thickness becomes smaller, resulting in a physically reasonable trend. As shown in Table 1, the hydraulic diameter of the MHP considered in this work is 1.5 mm and the hydraulic radius is 0.75 mm. The liquid block is the region at which the thickness of the liquid thin film reaches the size of the hydraulic radius. The lower the input thermal load, the thicker the liquid block length, and the higher the input thermal load, the thinner it gets. For input thermal loads of 0.1 W and 0.5 W, the liquid block started at approximately z/L = 0.78 and 0.85, respectively. It was found that there is no liquid block region at input thermal load 1.0 W or higher. The liquid thin film shape increased linearly at the input thermal load of 1.5 W and 2.0 W but increased exponentially at the lower thermal load. Fig. 13 shows the mass flow rate of the working fluid as a function of the input thermal load along the MHP axial length. As shown in Eqs. (9) and (10), the mass flow rates of liquid and vapor are the same; nonetheless, the velocity of liquid and vapor are different because liquid density, heat and mass transfer areas are different. As shown in the boundary condition of Fig. 12 and Eq. (18), the mass flow rate was calculated as zero at the evaporator end (z = 0) and the condenser end (z = L). As shown in Eqs. (9) and (10),
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Fig. 11. Axial distributions of the solid wall and liquid temperatures for an input thermal load of 2 W.
Fig. 12. Axial distribution of the thin liquid film thickness of an MHP for different input thermal loads.
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Fig. 13. Axial distributions for liquid mass flow rate as function of input thermal loads.
Fig. 14. Pressure distribution with respect to the axial position in an MHP.
the mass flow rate at the evaporator increases and then start decreasing at a certain point in the adiabatic section until it becomes zero at the end of the condenser. As the input thermal load increases, the mass flow rate also increases. For the input thermal load of 0.1 W, the maximum mass flow rate was 0.37 lg/s, and for the input thermal load of 2.0 W, the mass flow rate was increased to 2.13 lg/s. Fig. 14 shows the distributions of vapor, liquid, capillary and disjoining pressure on the axial length at the input thermal load of 2.0 W. As the axial length increases, the vapor pressure decreased from 36 kPa to 23 kPa while the liquid pressure increased from 10.4 kPa to 16.2 kPa. Furthermore, the capillary pressure decreased from 19.5 kPa to 16.2 kPa, and the disjoining pressure decreased from 6.8 kPa to 0.5 kPa. The pressure distribution was satisfactory within 3% of Eq. (1). Fig. 15 shows the vapor and liquid mass for the input thermal h P i load. The mass, M ¼ z¼L z¼0 qV , is the product of the density and the volume occupied by each phase for the entire length of the MHP. The total mass (M t ) is the sum of the liquid mass and the vapor mass. This can be defined as the minimum fill charge ratio of working fluid with which the MHP normally operates. Liquid and vapor masses increase with increasing input thermal load, which then increases the total mass. At the input thermal load of 0.2 W, total mass is 0.4 lg, which increases up to 17 lg when the thermal load becomes 2.0 W. The mass of the vapor was calculated
Fig. 15. Mass with respect to input thermal loads.
to be approximately 50% higher than the mass of the liquid. The densities of liquid and vapor were obtained by inputting temperature and pressure to REFPROP software. As shown in Fig. 10, the vapor pressure is calculated to be higher than that of the liquid, which causes the vapor density to be higher than that of the liquid. The vapor space occupying the interior of the MHP is relatively
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large compared to the liquid. As a result, the mass of the vapor was calculated to be higher than that of the liquid. 4. Conclusions A modified numerical model for MHP thermal fluid analysis was developed in this study. Conservation equations for vapor and liquid were derived and equations for the thickness of liquid thin film were derived. The area of the vapor-liquid interface was calculated using the shape of the thin liquid film, and it was defined as the heat and mass transfer area. The pressure, temperature, and velocity for these phases were calculated by applying conservation equations for vapor and liquid. The model was validated by the experimental results and it was confirmed that error of the average temperature difference between the evaporator and the condenser, as well as the error of the thermal resistance, were decreased when compared with the previous model. The error of the average temperature difference between the evaporator and the condenser were evaluated to be less than 1 °C. The maximum and minimum values of errors for the thermal resistance were 22% and 3.8%, respectively. The MHP performance based on the model investigated the physical rationality for the mass flow rate, the temperature, and the pressure distributions on the working fluid inside the MHP container with solid wall conduction. The pressure distributions satisfied the equilibrium equations, and the distributions of the temperature, mass flow, and pressure were found to be physically reasonable. This model was evaluated to be useful as a tool for performance prediction and MHP design. Conflict of interest The authors declare that there are no conflict of interest. Acknowledgement This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (grant number: NRF2016R1A2B4014933). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.ijheatmasstransfer. 2018.05.067. References [1] A. Faghri, Heat Pipe Science and Technology, Taylor & Francis, 1995, pp. 625– 671. [2] J. Qu, H. Wu, P. Cheng, Q. Wang, Q. Sun, Recent advances in MEMS-based micro heat pipes, Int. J. Heat Mass Transfer 110 (2017) 294–313. [3] A.U. Palkar, An Experimental Investigation of Liquid Metal MHPs (MS dissertation), University of Pune, 2007. [4] Y.T. Deng, Z. Quan, Y. Zhao, L. Wang, Z. Liu, Experimental research on the performance of household-type photovoltaic-thermal system based on microheat-pipe array in Beijing, Energy Convers. Manage. 106 (2015) 1039–1047. [5] Y.H. Diao, S. Wang, Y.H. Zhao, T.T. Zhu, C.Z. Li, F.F. Li, Experimental study of the heat transfer characteristics of a new-type flat micro-heat pipe thermal storage unit, App. Therm. Eng. 89 (2015) 871–882.
[6] Y. Deng, Y. Zhao, Z. Quan, T. Zhu, Experimental study of the thermal performance for the novel flat plate solar water heater with micro heat pipe array absorber, Energy Procedia 70 (2015) 41–48. [7] Z.Y. Wang, Y.H. Diao, L. Liang, Y.H. Zhao, T.T. Zhu, F.W. Bai, Experimental study on an integrated collector storage solar air heater based on flat micro-heat pipe arrays, Energy Build. 152 (2017) 615–628. [8] T.Y. Wang, Y.H. Diao, T.T. Zhu, Y.H. Zhao, J. Liu, X.Q. Wei, Thermal performance of solar air collection-storage system with phase change material based on flat micro-heat pipe arrays, Eerg. Convers. Manage. 142 (2017) 230–243. [9] T.T. Zhu, Y.H. Zhao, Y.H. Diao, F.F. Li, Y.C. Deng, Experimental investigation on the performance of a novel solar air heater based on flat micro-heat pipe arrays (FMHPA), Energ. Procedia 70 (2015) 146–154. [10] T.T. Zhu, Y.H. Diao, Y.H. Zhao, Y.C. Deng, Experimental study on the thermal performance and pressure drop of a solar air collector based on flat micro-heat pipe arrays, Energy Convers. 94 (2015) 447–457. [11] L. Ling, Q. Zhang, Y. Yu, Y. Wu, S. Liao, Study on thermal performance of microchannel separate heat pipe for telecommunication stations: experiment and simulation, Int. J. Refrig. 59 (2015) 198–209. [12] Y.H. Diao, L. Liang, Y.M. Kang, Y.H. Zhao, Z.Y. Wang, T.T. Zhu, Experimental study on the heat recovery characteristic of a heat exchanger based on a flat micro-heat pipe array for the ventilation of residential buildings, Energy Build. 152 (2017) 448–457. [13] G.H. Kwon, S.J. Kim, Experimental investigation on the thermal performance of a micro pulsating heat pipe with a dual-diameter channel, Int. J. Heat Mass Transfer 89 (2015), 871–822. [14] K.S. Yang, Y.C. Cheng, M.C. Liu, J.C. Shyu, Micro pulsating heat pipes with alternate microchannel widths, App. Therm. Eng. 83 (2015) 131–138. [15] S.H. Moon, G. Hwang, S.C. Ko, Y.T. Kim, Experimental study on the thermal performance of micro-heat pipe with cross-section of polygon, Microelectron. Reliab. 44 (2004) 315–321. [16] M.Y. Hung, K.K. Tio, Thermal analysis of optimally designed inclined micro heat pipes, Int. Comm. Heat Mass Transfer 39 (2012) 1146–1153. [17] K.K. Tio, Y.M. Hung, Analysis of overloaded micro heat pipes: effects of solid thermal conductivity, Int. J. Heat Mass Transfer 81 (2015) 737–749. [18] F.L. Chang, Y.M. Hung, The coupled effects of working fluid and solid wall on thermal performance of micro heat pipes, Int. J. Heat Mass Transfer 73 (2014) 76–87. [19] Y.M. Hung, Q. Seng, Effects of geometric design on thermal performance of star-groove micro-heat pipes, Int. J. Heat Mass Transfer 54 (2011) 1198–1209. [20] B. Suman, S. De, S. DasGupta, A model of the capillary limit of a micro heat pipe and prediction of the dry-out length, Int. J. Heat Fluid Flow 26 (2005) 495–505. [21] V. Sartre, M.C. Zaghdoudi, M. Lallemand, Effect of interfacial phenomena on evaporative heat transfer in micro heat pipes, Int. J. Therm. Sci. 39 (2000) 498– 504. [22] S.S. Rao, H. Wong, Heat and mass transfer in polygonal micro heat pipes under small imposed temperature difference, Int. J. Heat Mass Transfer 89 (2015) 1369–1385. [23] J. Qu, H. Wu, P. Cheng, Effects of functional surface on performance of a micro heat pipe, Int. J. Heat Mass Transfer 35 (2008) 523–528. [24] B. Suman, P. Kumar, An analytical model for fluid flow and heat transfer in a micro heat pipe of polygonal shape, Int. J. Heat Mass Transfer 48 (2005) 4498– 4509. [25] B. Suman, N. Hoda, Effect of variations in thermophysical properties and design parameters on the performance of a V-shaped micro grooved, Int. J. Heat Mass Transfer 48 (2005) 2090–2101. [26] B. Suman, S. De, S. DasGupta, Transient modeling of micro-grooved heat pipe, Int. J. Heat Mass Transfer 48 (2005) 1633–1646. [27] K.H. Do, S.J. Kim, S.V. Garimella, A mathematical model for analyzing the thermal characteristics of a flat micro heat pipe with grooved wick, Int. J. Heat Mass Transfer 51 (2008) 4637–4650. [28] H. Wang, V. Suresh, V. Garimella, Murthy. Characteristics of an evaporating thin film in a microchannel, Int. J. Heat Mass Transfer 50 (2007) 3933–3942. [29] http://keisan.casio.com/exec/system/1223372110. [30] E.G. Jung, J.H. Boo, A Numerical study on heat and mass transfer of a micro heat pipe with phase-change interface analysis, Heat Mass Transfer 53 (2017) 3367–3372. [31] P.H. Oosthuizen, D. Naylor, Convective Heat Transfer Analysis, second ed., McGraw-Hill, 1999, pp. 555–599. [32] S. Pati, S.S. Shakraborty, Combined influences of electrostatic component of disjoining pressure and interfacial slip on thin film evaporation in nanopores, Int. J. Heat Mass Transfer 64 (2013) 304–312. [33] S. Narayanan, A.G. Fedorov, Y.K. Joshi, Interfacial transport of evaporating water confined in nanopores, Langmuir 27 (17) (2011) 10666–10676.
