Experimental and theoretical study of a water-vapor chamber thermal diode

International Journal of Heat and Mass Transfer 138 (2019) 173–183

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Experimental and theoretical study of a water-vapor chamber thermal diode M.Y. Wong a, B. Traipattanakul b, C.Y. Tso c, Christopher Y.H. Chao d,⇑, Huihe Qiu a a

Department of Mechanical and Aerospace Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong, China Asian Institute of Technology, Pathum Thani, Thailand c School of Energy and Environment, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kong, China d Department of Mechanical Engineering, The University of Hong Kong, Hong Kong, China b

a r t i c l e

i n f o

Article history: Received 30 November 2018 Received in revised form 21 March 2019 Accepted 10 April 2019

Keywords: Diodicity Effective thermal conductivity Heat transfer Phase change Thermal diode Thermal rectification

a b s t r a c t Similar to an electrical diode, a thermal diode/switch is a device which allows heat to flow to a preferential direction. While a number of different types of thermal diodes/switches exist, a phase-change thermal diode/switch yields a greater diodicity performance compared to other types. However, most phasechange thermal diodes/switches have complex designs, complicated fabrication processes, and sophisticated working principles. Moreover, some materials used in thermal diodes/switches are rare, expensive and toxic. Thus, in this study, a simple water-vapor chamber thermal diode using latent heat of vaporization is designed, assembled and investigated, both experimentally and theoretically. The effects of the temperature difference between the hot side and the cold side and the water-air volume ratio on the effective thermal conductivity and diodicity of the water-vapor chamber thermal diode are also investigated. Mathematical models are also developed, not only for predicting one-dimensional phase change heat transfer performance in a rigid enclosure, but also for verifying the results from the experiment. This is the first study in which the water-vapor chamber thermal diode is studied both experimentally and theoretically. The experimental results show that an increase in temperature at the hot side of the thermal diode significantly enhances the performance of the thermal diode. The effective thermal conductivity at the hot side temperature of 70 °C shows a 50% improvement when compared with that at the hot side temperature of 40 °C in the forward direction. Additionally, with the water-air volume ratio of 0.5, the maximum diodicity of 1.43 is reported. Moreover, it is also found that the water-vapor volume ratio affects heat transfer performance and diodicity of the water-vapor chamber thermal diode. The results of these findings can further advance knowledge on phase-change heat transfer and can be applied to several applications, including heat transfer enhancement, waste heat power systems, thermal management and solar thermoelectric power generation. Ó 2019 Published by Elsevier Ltd.

1. Introduction Analogous to an electrical diode, a thermal diode and a thermal switch are devices that allow heat to flow in a preferential direction (called a forward direction) and at the same time prevent heat to flow in the other direction (called a reverse direction) [1,2]. A thermal diode operates with two terminals, and the effectiveness of a thermal diode, also known as the rectification coefficient or diodicity, is defined as the ratio of the difference in the effective thermal conductivities in the forward direction and the reverse

⇑ Corresponding author at: Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong, China. E-mail address: [email protected] (C.Y.H. Chao). https://doi.org/10.1016/j.ijheatmasstransfer.2019.04.046 0017-9310/Ó 2019 Published by Elsevier Ltd.

direction to the effective thermal conductivity in the reverse direction as shown in Eq. (1) [1]:

gd ¼

kf  kr ; kr

ð1Þ

where gd is the diodicity of a thermal diode, kf is the effective thermal conductivity in the forward direction (W/m-K) and kr is the effective thermal conductivity in the reverse direction (W/m-K). Unlike a thermal diode, a thermal switch consists of a third terminal which is used to control heat flow [2]. A thermal switch is in the ON state when it performs as a conductor, and it is in the OFF state when it acts as a thermal insulator. The effectiveness of a thermal switch, also known as an ON/OFF ratio, is defined as the ratio of the effective thermal conductivity during the ON state to that during the OFF state, and it can be written as shown in Eq. (2) [2]:

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Nomenclature A c g h hfg k L M Nu P Pr Q_ q} R Ra Rg T b DT Dx

l m q r

gs ¼

kON ; kOFF

t

area [m2] constant [–] gravitational acceleration [m/s2] heat transfer coefficient [W/m2K] enthalpy of vaporization [J/kg] thermal conductivity [W/mK] characteristic length [m] molar mass [kg/mol] Nusselt number [–] pressure [Pa] Prandtl number [–] heat transfer rate [W] heat flux [W/m2] thermal resistance [K/W] rayleigh number [–] universal gas constant [J/molK] temperature [K] volume expansivity [K1] temperature difference [K] length of the thermal diode [m] dynamic viscosity [kg/ms] kinematic viscosity [m2/s] density [kg/m3] accommodation coefficient [–]

specific volume [m3/kg]

Subscripts a air c condensation cd conduction cr critical convection cv d diode e evaporation eff effective f forward direction lv liquid-vapor interface off OFF state on ON state r reverse direction s switch ss stainless steel t total v vapor w water wall water tank wall

ð2Þ

where gs is the ON/OFF ratio of a thermal switch, kON is the effective thermal conductivity during the ON state (W/m-K) and kOFF is the effective thermal conductivity during the OFF state (W/m-K). A thermal diode/switch can be used in thermal management systems [3,4], energy systems [5–7], microelectronic systems [8,9] and space technology [10,11]. Thermal diode panels for use in cooling in buildings [4], convective thermal diodes with passive solar heating [5], silicon diode temperature sensors integrated with electronic circuits [9] and heat pipe thermal diodes/switches for cryogenic temperature operation [11] are some of the many sample engineering applications which require thermal diodes/switches. First discovered in 1936, thermal diodes/switches, consist of various types, each of which is equipped with a specific heat transfer working principle and thermal effectiveness which can vary between below one and hundreds have been investigated, and many of these studies have shown satisfactory results [12–27]. In this manuscript, four major groups of thermal diodes/switches are categorically presented based on their states and working principles, which consist of solid-state thermal diodes/switches [13– 16], near-field radiation-based thermal diodes/switches [17–19], fluid-based thermal diodes/switches [20–22] and phase-changebased thermal diodes/switches [23–27]. Solid-state thermal diodes/switches have been extensively studied and have reportedly shown various thermal rectifications. Kobayashi et al. [13] reported a diodicity of 1.43 at a temperature difference of 58.9 K from the investigation of an oxide thermal diode made of two cobalt oxides with different thermal conductivities. Heat current density and rectifying coefficient using thermal conductivity data were theoretically determined and were in line with the experimental data. It was also observed from the study by Kobayashi et al. that a combination of two different materials with two different temperature dependences leads to a diodicity of the thermal diode [13]. Additionally, Wang et al. [14] experimentally investigated monolayer graphene thermal rectifiers, and

found diodicity of 26. Although solid-state thermal diodes/ switches described above showed the diodicity of less than 30, a few studies revealed greater thermal rectification performance with almost or over a hundred. Tso and Chao [15] experimentally studied a shape memory alloy solid-state thermal diode, and a diodicity of 93 was reported. However, it is noted that the shape memory alloy thermal diode consists of several moving and nonmoving parts which might result in difficulty in maintenance and a great deal of energy loss at each moving joint [15]. Moreover, a hybrid device combining normal metals tunnel-coupled to superconductors with the diodicity of up to 140 was experimentally investigated by Martínez-Pérez et al. [16]. In addition to solidstate thermal diodes/switches, near-field radiation-based thermal diodes/switches have also been investigated. Basu and Francoeur [17] used a film doped silicon and a bulk doped silicon separated by a nanometer vacuum gap for studying a near-field thermal rectifier. The findings of the study revealed the thermal rectification to be greater than 0.5 when a 10-nm thick film was utilized at the gap widths between 1 nm and 50 nm [17]. Moreover, Yang et al. [18] studied radiative heat transfer between silicon dioxide (SiO2) and vanadium oxide (V2O5) in a nanometer vacuum gaps. The results of the study showed that if silicon dioxide thin film was utilized, thermal rectification of 3 was achieved at the vacuum gap of 100 nm [18]. Additionally, Wang et al. [19] investigated nearfield thermal radiation between intrinsic silicon and doped silicon, and intrinsic silicon and silicon dioxide (SiO2). The study showed that at a 5-nm vacuum gap, and at temperatures of 1000 K and 300 K, thermal rectification of intrinsic silicon and silicon dioxide (SiO2) reached 9.9 [19]. Apart from solid-state thermal diodes/ switches and near-field radiation-based thermal diodes/switches, fluid-based thermal diodes/switches have also been studied experimentally. Gaddam et al. [20] reported a liquid-state thermal diode using thermal expansion of mercury in air space with a diodicity of 1. Catarino and Paine [21] further investigated the gas gap heat switch using Helium as the heat-transfer fluid. At 1.7 K, the ON/ OFF ratio was 1.67 [21]. Furthermore, Neon and Hydrogen were

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also used in the study of gas gap heat switch with the ON/OFF ratios of 220 at 20 K and 420 at 10.7 K, respectively [22]. Besides thermal diodes/switches using solid or liquid materials, phasechange materials have also frequently been utilized in thermal diode/switches. Ito et al. [23] investigated a radiative thermal diode using a silicon wafer deposited with a thin film of vanadium oxide (V2O5), and a diodicity of 2 was reported. Moreover, a thermal diode based on poly(N-isopropylacrylamide) (pNIPAM) and polydimethylsiloxane (PDMS) was studied by Pallecchi et al. with the reported diodicity of 2 [24]. Additionally, paraffin wax was utilized in a phase-change thermal switch in a study by Wang et al. [25] with the observed ON/OFF ratio of 30. The ON mode took place when the paraffin wax melted to a liquid phase, while the OFF mode occurred when the solid-state paraffin wax formed. Additionally, Boreyko et al. [1] reported a very high diodicity of 150 in the study of a planar phase-change vapor chamber thermal diode using water evaporation on a superhydrophilic surface and using dropwise condensation as well as coalescing-jumping droplets on a superhydrophobic surface. The diodicity of 150 reportedly increased to 250 if a proper degassed process of the vapor chamber was conducted [26]. However, due to progressive flooding caused by limited jumping heights of coalescing-jumping droplets and accumulation of large-sized water droplets on the superhydrophobic surface in phase-change vapor chamber thermal diodes in Refs. [1] and [26], heat transfer performance of the thermal diode can drastically drop [27]. Nevertheless, Traipattanakul et al. [27] found that a parallel electric field can address the heat transfer degradation issue of the phase-change vapor chamber thermal diode. This is because utilizing the parallel electric field can both enhance the jumping height of coalescing-jumping droplets [28–29] and remove macro-sized droplets from the superhydrophobic surface [30]. With the applied electric field, the study by Traipattanakul et al. showed that diodicity of a phase-change thermal diode using electrostatic-induced coalescing-jumping droplets reached 325, the result of which has been considered one of the best-performing thermal diodes/switches operating in a normal working condition (25–80 °C) [27]. Table 1 shows the effectiveness of the thermal diodes/switches reviewed in this manuscript. It is clearly portrayed that when compared with other thermal diodes/switches, in normal working conditions, phase-change thermal diodes/switches yield greater effectiveness due to having a high latent heat. However, the above-reviewed phase-change thermal diodes/ switches contain (i) complex designs and complicated fabrication processes such as depositing a thin film of vanadium oxide on a silicon wafer [23] and fabrication of a paraffin-wax-based thermal

switch [25]; (ii) sophisticated working principles such as dropwise condensation and coalescing-jumping droplets on a superhydrophobic surface in the water-vapor chamber thermal diodes [2,26,27]; (iii) rare and/or expensive materials such as pNIPAM [24], and 11-mercapto-1-undecanol for fabricating a superhydrophilic surface [27]; and (iv) toxicity such as vanadium oxide [23], 11-mercapto-1-undecanol [27] and heptadecafluoro-1decathiol [27], leading to difficulty in fabrication, higher cost in production and increased operational and maintenance risks. Moreover, although the previous studies involved a number of thermal diodes with various materials, there is still a lack of understanding of the thermal rectification performance of a thermal diode utilizing latent heat from a simple substance, liquid water. Additionally, there is no theoretical model predicting the effective thermal conductivities in the forward and the reverse modes. Thus, the current study aims to build a water-vapor chamber thermal diode, with the characteristics of a simple design, low cost and the usage of non-toxic materials, for determining the effects of temperature differences on the effective thermal conductivity and the thermal rectification. Mathematical models for predicting the thermal diode performance will also be developed. The water-vapor chamber thermal diode will be designed, fabricated and studied, both experimentally and theoretically. This is the first study of a phase-change thermal diode in which water is utilized in the thermal diode, and in which mathematical models predicting the heat transfer performance are developed. The findings can serve as fundamental knowledge for the future design of a highly efficient water-vapor chamber thermal diode, and can shed some lights on a number of phase-change heat transfer engineering applications. The working principle of the water-vapor chamber thermal diode will be described in Section 2, the thermal diode fabrication and the experimental setup will be explained in Section 3, the developed mathematical models will be shown in Section 4, and the findings of the study will be reported and discussed in Section 5. The Section 5 is divided into 5 subsections, discussing the forward and reverse heat transfer performances, the thermal rectification performance, the effect of water-air volume ratio to the performance, and the limitation of the proposed water-vapor chamber thermal diode. 2. Working principle of the water-vapor chamber thermal diode Before describing thermal diode fabrication and the experimental setup of the study, the working principle of the water-vapor chamber thermal diode is discussed in this section. Fig. 1 shows the working principle of the water-vapor chamber thermal diode.

Table 1 Diodicity of various kinds of thermal diodes/switches. Thermal diodes/switches

Type

Mechanism

Diodicity

ON/OFF ratio

Cobalt oxide thermal diode [13] Monolayer graphene thermal diode [14] Solid-state shape memory alloy thermal diode [15] Metals tunnel-coupled to superconductors thermal diode [16] Film doped silicon and bulk doped silicon thermal diode [17] Silicon dioxide and vanadium dioxide thermal diode [18] Intrinsic silicon and silicon dioxide thermal diode [19] Mercury-based thermal diode [20] Helium-based thermal switch [21] Neon-based / Hydrogen-based thermal switch [22] Vanadium-oxide radiative thermal diode [23] PNIPAM-PDMS thermal diode [24] Paraffin wax-based thermal switch [25] Planar jumping-drop thermal diode [1] Jumping-drop vapor chamber thermal diode [26] Electrostatic-induced coalescing-jumping droplets [27] Water-vapor chamber thermal diode [This Work]

Solid Solid Solid Solid Radiation Radiation Radiation Fluid Fluid Fluid Phase-change Phase-change Phase-change Phase-change Phase-change Phase-change Phase-change

Bulk effect Asymmetric phonon scattering Surface contact Tunnel junction Near-field radiation Near-field radiation Near-field radiation Thermal expansion Adsorption Adsorption Insulating state to metallic state Junction of two materials Thermal expansion Jumping droplets Jumping droplets Jumping droplets with electric fields Thermal convection and phase change

1.43 26 93 140 0.5 3 9.9 1 – – 2 2 – 150 250 325 1.43

– – – – – – – – 1.67 220/420 – – 30 – – – –

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to the bottom plate, and heat conducts through the chamber and four bars. In brief, the thermal rectification ratio of the thermal diode is contributed with thermal convection (thermal conduction and fluid movement) and phase change heat transfer (evaporation and condensation) in the forward heat transfer mode, and with thermal conduction in the reverse mode. 3. Experimental procedures 3.1. Thermal diode fabrication

Fig. 1. The working principle of the water-vapor chamber thermal diode in the (a) forward heat transfer direction and (b) reverse heat transfer direction.

Due to having a high latent heat, water, the most common liquid substance on earth, is used as a working fluid in this device and is stored in the chamber. Each metal plate is installed at the top and the bottom of the chamber. Four cylindrical bars connecting the top and the bottom plates are used to support the device. During the forward mode, heat is supplied to the bottom plate of the thermal diode, and water inside the chamber evaporates and carries latent heat to the top. Due to the evaporation and the heat transfer between water and air and water vapor, water on the water surface is cooled down, and moves to the bottom to be heated up again due to the density difference. As a result, water circulation occurs in the water boundary. In the vapor space of the thermal diode, the hot water vapor and air inside flow up because of the buoyancy effect. Once water vapor and air touch the cold top plate, energy is released due to condensation. Due to the density difference, the cold air and water vapor circulate down for reheating purposes. In contrast, when heat is supplied from the top plate during the reverse mode, there is no water carrying heat

Fig. 2 shows the water-vapor chamber thermal diode. The cylindrical chamber is made of Teflon because of its light weight, low thermal conductivity (k = 0.35 W/mK) [31] and easy fabrication. The Teflon chamber is a hollow cylinder with a 0.075 m outer diameter, a 0.0125 m thickness and a 0.0149 m length. The wall thickness in the middle of the water tank changes to 0.015 m, creating a protruded part. As shown in Fig. 2, two stainless steel halfrings and four stud bolts are used to prevent the movement of the chamber and to avoid water leakage. Two square stainless steel plates (k = 16.6 W/mK) [31], with a width of 0.1 m and a thickness of 0.015 m, are selected, because of their lighter weight when compared with copper plates. A 3-mm diameter hole is drilled on one side of each stainless steel plate for inserting a thermocouple sensor. These two stainless steel plates are placed at the top and the bottom of the Teflon chamber and supported by four Polyether ether ketone (PEEK) bars which are rigid and have high thermal resistance (k = 0.25 W/mK) [32]. It is noted that the dimensions of the PEEK bars are 0.0125 m in diameter and 0.145 m in length. Four holes, 5 mm in dimeter and with a counterbore (5 mm in depth), are drilled at the corners of the plates for connecting the stainless steel plates and the PEEK bars. At one face of the stainless steel plates, a circular notch (0.075 m in diameter and 0.003 m in depth) is made, in order to attach to the chamber. A circular groove (inner diameter of 0.065 m, thickness of 0.005 m and depth of 0.0025 m) concentric with the notch is drilled and an O-ring is compressed into the groove. The O-ring is used to seal the interface between the stainless steel plate and the Teflon chamber. For the stainless steel plate placed at the bottom of the thermal rectifier, four through holes and threads are prepared for fastening the stud bolts in order to fix the position of the water chamber. To assemble the thermal diode, four PEEK bars are first fastened at the corners of the bottom stainless steel plate using hex socket head cap screws. Then, the Teflon water chamber is placed on the notch of the bottom plate. The male ends of stud bolts are tightened to the side of the bottom plate. Two stainless steel halfrings are put above the protruded part of the water chamber, along with the stud bolts, for preventing both the movement of the water chamber and water leakage. After pouring water into the water chamber, the other stainless steel plate (top plate) is placed on top of the water tank and PEEK bars with four hex socket head cap screws. 3.2. Experimental setup Although there are a number of methods used for measuring the effective thermal conductivity and diodicity of the thermal diode [33,34], the cut bar method [35] is selected in this study because it is one of the most common and reliable methods to measure the thermal conductivity of an unknown sample material. A sample is sandwiched between two reference materials with known thermal conductivities. With heat flux crossing the two reference materials and the temperature gradient across the sample, the thermal conductivity of the sample can be obtained. The experimental setup is shown in Fig. 3. Two copper heat exchangers

M.Y. Wong et al. / International Journal of Heat and Mass Transfer 138 (2019) 173–183

177

ples are attached to the thermal diode; three of which are placed at the outer surface of the water chamber, two of which are inserted to the 3-mm holes on the side of the stainless steel plates, and the other two are put at the two heat exchangers. Besides this, two heat flux sensors (Omega HFS-3) for recording the amount of heat flow per unit area are placed between the heat exchangers and the thermal diode, on both sides. To reduce thermal contact resistance and the thermal resistance due to the air gap between the stainless steel plates and the heat exchangers, thermal grease (Silicone Thermal Grease, RS, 5 W/mK) is utilized. A data acquisition (DAQ) device (National Instruments NI9213) is used for collecting and transferring the electric signals from the thermocouples and the heat flux sensors to LabVIEW. The experiment contains two parts, namely the forward heat transfer test and the reverse heat transfer test. In the forward heat transfer, the isothermal water circulator connecting to the bottom plate of the thermal diode is set at 40 °C, 50 °C, 60 °C, and then 70 °C, while the isothermal water circulator linking to the top plate of the thermal diode is set near the room temperature of 20 °C throughout the experiment. Heat is supplied to the thermal diode through the heat exchanger. In the reverse heat transfer test, the isothermal water circulator connecting to the bottom plate of the thermal diode was maintained at 20 °C while the other isothermal water circulator at the top plate of the thermal diode is varied between 40 °C, 50 °C, 60 °C, and 70 °C. With the cut bar method, average heat flux is measured from two heat flux sensors, which are installed on each side of the water chamber. Temperatures at the hot side and at the cold side are measured from thermocouples inserted in the stainless steel plates. Effective thermal conductivity (keff) can be determined from

keff ¼

q00 Dx ; DT

ð3Þ

where q00 is average heat flux (W/m2) measured from two heat flux sensors, Dx is the length of the thermal diode (m) and DT is the temperature difference between the heat source and the heat sink (K). Then, effective thermal conductivities in both forward and reverse heat transfer modes are used to calculate the experimental diodicity of the thermal diode as shown in Eq. (1). With a total volume of 296 ml, 148 ml of water is poured into the water chamber, implying that the water-air volume ratio is 0.5. As a result, the heat transfer distances (characteristic length) of water, air, and vapor are the same. Moreover, apart from the water-air volume ratio of 0.5, in order to investigate the effects of water-air volume ratio on heat transfer and thermal rectification performances, several water-air volume ratios including 0 (0 ml of water), 0.25 (74 ml of water), 0.75 (222 ml of water) and 1 (296 ml of water) are tested where the heat sink temperature is 20 °C and the heat source temperature is 70 °C. 3.3. Uncertainty analysis

Fig. 2. The water-vapor chamber thermal diode at (a) side view and (b) symmetric view. The water-vapor chamber thermal diode consists of two stainless steel plates, a Teflon water chamber, and four PEEK bars. Two stainless steel half-rings and four stud bolts are also used to prevent the movement of the chamber and to avoid water leakage.

(k = 393 W/mK) [31] are placed on the top and the bottom plates of the thermal diode for providing heating and cooling to the device. Each heat exchanger is connected to an isothermal water circulator with a digital temperature controller (PolyScience 9102AA2P) through heat transfer tubes. Seven K-type thermocou-

For each testing condition (both forward and reverse modes), the experiment is conducted three times to ensure the accuracy of the measurement results. The uncertainties of the experiment are associated with the cooling power of the water circulator (±1%), the heat flux sensor (±5%), thermocouples (±0.4%), datalogger (±3%) and the dimensional measurement (±5%). The uncertainty of the measurement is estimated at about 8.39% from the experiment based on the error propagation calculation [36]. From the experimental results, the temperature variation is less than 1 °C, and the heat flux sensor variation is less than 5%. Based on several experiments under various temperature difference conditions, the average heat loss obtained from the different readings between the two heat flux sensors is 8.13%.

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Fig. 3. A schematic diagram showing the experimental setup for measuring the effective thermal conductivities of the water-vapor chamber thermal diode.

4. Mathematical method Apart from the experiment, mathematical models for predicting heat transfers in the forward mode and the reverse mode of the water-vapor chamber thermal diode were developed. These models can also serve as a tool for determining the heat transfer performance of other phase-change heat transfer applications, such as thermal management systems and solar power systems. The results from the experiment and from the mathematical models will be compared in the next section. In this mathematical model, it is assumed that (i) the thermal diode is well-insulated so that heat loss is negligible, (ii) heat transfers occur in the onedimensional steady state condition, (iii) radiation heat transfer is neglected, (iv) the thermal diode is well-sealed resulting in no water leakage, (v) the cross-sectional areas of the water tank and the stainless steel plates are uniform, (vi) air and water vapor behaves as the ideal gas, and (vii) the thermal properties of the two stainless steel plates and the Teflon chamber are constant. In this model, heat only transfers through the thermal diode with thermal conduction, thermal convection and the phase-change heat transfer. Fig. 4(a) shows the thermal resistance circuit of the water-vapor chamber thermal diode in the forward heat transfer direction. The thermal diode is divided into six parts: namely, stainless steel plates, the Teflon chamber, water, air, water vapor, and water-vapor interfaces (evaporation and condensation). While thermal conduction occurs through the stainless steel plates and the Teflon chamber, thermal convection takes place inside the tank with water, water vapor and air. Moreover, heat transfer due to phase transformation exists at the water-vapor interfaces. The thermal contact resistance between the stainless steel plates, the Teflon chamber and the thermal resistance through the PEEK bars are negligible. The total thermal resistance in the forward mode can be written as

Fig. 4. A schematic diagram of the thermal resistance circuit of the water-vapor chamber thermal diode in (a) forward and (b) reverse heat transfer modes.

1
 Rf ;t ¼  1=Rwall;cd þ 1= Rw;cv þ Re;lv 1
  þ 2Rss;cd ; þ 1=Rwall;cd þ 1= Rv ;cv þ Rc;lv þ 1=Ra;cv

ð4Þ

M.Y. Wong et al. / International Journal of Heat and Mass Transfer 138 (2019) 173–183

where Rf,t is the total thermal resistance (K/W) in the forward heat transfer direction; Rwall,cd and Rss,cd are the thermal resistances (K/ W) of the chamber wall and stainless steel plate by heat conduction, respectively; Rw,cv, Rv,cv and Ra,cv are the thermal resistances (K/W) of water, water vapor and air by heat convection, respectively; and Re,lv, and Rc,lv are the thermal resistances (K/W) of the water–vapor interface during evaporation and during condensation, respectively. The thermal resistances of the chamber wall and stainless steel plates due to heat conduction can be calculated with Eq. (5) [37]

Rcd ¼

Lcd ; kAcd

ð5Þ

where Lcd, k, and Acd are the thickness (m), thermal conductivity (W/ mK) and cross-section area (m2) of the materials, respectively. Given the fact that thermal convection is a combination of thermal conduction and fluid movement, heat transfer from thermal convection is always higher than that from thermal conduction. When the fluid is at rest, only thermal conduction takes place. When fluid is in motion and fluid movement cannot be ignored, the heat transfer process is defined as thermal convection. In order to determine whether thermal convection can be ignored in an enclosure, a Rayleigh number representing the ratio between the buoyancy force, and the fluid viscosity and heat diffusion, and a Nusselt number showing the ratio of thermal convection to thermal conduction are used for determining the dominated transport method in the fluid layer. The Rayleigh number, heat transfer coefficient and thermal resistance of the water, water vapor and air due to thermal convection in the enclosure are shown in Eqs. (6), (7) and (8), respectively [32,37–39]

Racv ¼

gbðT 1  T 2 ÞL3cv

m2

ð6Þ

Pr;

hc v ¼

kNucv ; Lcv

ð7Þ

Rc v ¼

1 ; h c v Ac v

ð8Þ

where Racv is the Rayleigh number of the fluid; g is the gravitational acceleration (m/s2); b, m, and Pr, are the volume expansivity (1/K), kinematic viscosity (m2/s) and Prandtl number of the fluid, respectively; T1 is the hot side temperature (K) and T2 is the cold side temperature (K) of the fluid layer, respectively; hcv represents heat transfer coefficient (W/m2K); Nucv is the Nusselt number of the fluid; and Rcv is the thermal resistance due to thermal convection (K/W). Unlike with external natural convection, Lcv, the characteristic length (m) in the enclosed space, is determined from the thickness of the enclosure, and Acv is equal to cross sectional area of the enclosure. To calculate the Nusselt number of the water, air and water vapor, the correlation proposed by Hollands et al. (based on the natural convection heat transfer across fluid layers inside a closed cylinder while heating at the bottom) is applied [38] and shown in Equation (9)

 Nucv ¼ 1 þ 1  " þ

Racr Racv

þ

2 4c 1 þ 2 #þ

1=3

Racv 5830

1

Rac1=3 v c2

3 !1ln ðRa1=3 cv =c 2 Þ 5 (

" #þ )! 1=3 Racv ; 1  exp 0:95 1 Racr ð9Þ

where Racr and Racv are the critical Rayleigh number and the fluid Rayleigh number, respectively; and c1 and c2 are the parameters depending on the Prandtl number [38]:

 0:018 0:00136 ; c1 ¼ 1:44= 1 þ þ Pr Pr2

 c2 ¼ 75 exp 1:5Pr1=2 ;

179

ð10Þ ð11Þ

where Pr is the Prandtl number. Eq. (9) will be valid if and only if the ratio of the diameter to the length of the enclosure is between 0.2 and 5 [38]. It should be noted that when the quantity in [ ]+ is negative, the term inside becomes zero. Since thermal convection is always equal to or greater than thermal conduction, a Nusselt number is always equal to or greater than 1. By considering the geometric condition of the enclosure and the thermal property of the enclosure wall, there is a critical Rayleigh number which indicates the initiation of the thermal convection. When a Rayleigh number is less than the critical value, a Nusselt number always equals to 1, implying that the effect of thermal convection on heat transfer of the fluid layer can be neglected. When a Rayleigh number is greater than the critical value, a Nusselt number is greater than 1, meaning that the effect of thermal convection must be considered. The critical Rayleigh number referred to this study is 11,323 [38]. Apart from thermal convection, the water-vapor interfaces at the surface of the water (evaporation interface) and at the top stainless steel plate (condensation interface) also play important roles in the heat transfer performance of the thermal diode. During the evaporation process, while more water molecules enter the interface, more water-vapor molecules leave the interface. This phenomenon is the opposite during the condensation process. It is assumed that the number of molecules entering and leaving the interface are equal, leading to an equilibrium condition. It is also assumed that the thickness of the interface is zero, and that the interface temperature is uniform. To determine the flux of molecules entering and leaving the interface, accommodation coefficients during the evaporation process and the condensation process are introduced. When all the molecules enter the interface but no molecules leave the interface, the accommodation coefficient equals to 1. On the other hand, the accommodation coefficient equals to 0 when all the incoming molecules leave the interface [40]. The relationship between the evaporation and condensation coefficients is assumed to remain unchanged during the phase change process. At the equilibrium state, the net flux of the molecules at the interface is zero. The accommodation coefficient can be defined as [40]

r¼



rffiffiffiffiffiffi 1 1   pffiffiffiffiffiffiffiffi tw 1 3 e 2 3tv =tw 1 ;

tv

ð12Þ

where tw and tv are the specific volumes (m3/kg) of water and water vapor, respectively. In addition, the pressure of the vapor interface and the water vapor is assumed to be equal to the saturation pressure at the interface temperature. The heat transfer coefficient of the water-vapor interface (hlv ) and its thermal resistance (Rlv ) are shown in Eqs. (13) and (14), respectively [40]

! 12  2 hfg 2r M Pv tlv ; hlv ¼ 1 2pRg T v 2  r T v tlv 2hfg 

Rlv ¼

1 ; hlv Alv

ð13Þ

ð14Þ

where hfg, and tlv are the enthalpy of vaporization (J/kg) and the specific volume difference (m3/kg) between water and water vapour, respectively; M, Pv, and Tv represent the molar mass (kg/mol), pressure (Pa), and temperature (K) of the water vapour, respectively; Rg is the universal gas constant (J/molK); and hlv, Rlv, and Alv are heat transfer coefficient (W/m2K), thermal resistance (K/W) and cross-sectional area (m2) of the phase change interface,

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respectively. By substituting Eqs. (5), (8) and (14) into Eq. (4), the total thermal resistance in the forward heat transfer direction can be obtained. The effective thermal conductivity in the forward heat transfer direction can then be calculated by

keff ;f ¼

Dx ; Rf ;t A

ð15Þ

where keff,f is the forward effective thermal conductivity (W/mK), Dx is the length of the thermal diode (m), Rf,t is the forward thermal resistance (K/W), and A is the heat flux sensor area (m2) in the experiment. On the other hand, the thermal resistance circuit of the watervapor chamber thermal diode in the reverse heat transfer condition is shown in Fig. 4(b). The stainless steel plates, the Teflon chamber, and the water and air layers are considered. In this case, heat is supplied from the top of the enclosure. Given the fact that lowdensity hot fluid stays above high-density cold fluid, no buoyancy force-driven fluid movement can be observed, implying that only thermal conduction is assumed to have taken place in the fluid layer. The total thermal resistance in the reverse heat transfer direction is

Rr;t ¼

1 1 þ þ 2Rss;cd ; 1=Rwall;cd þ 1=Rw;cd 1=Rwall;cd þ 1=Ra;cd

ð16Þ

where Rwall,cd, Rw,cd, Ra,cd and Rss,cd are the thermal resistances of the chamber wall, water, air and a stainless steel plate, respectively. Thermal resistances of the stainless steel plates, Teflon chamber, water and air layers through heat conduction can be calculated by applying Eq. (5) one by one. Similar to the effective thermal conductivity in the forward mode, the effective thermal conductivity in the reverse mode can be obtained using Equation (15). After calculating the forward and reverse effective thermal conductivities, the diodicity of the thermal rectifier can be determined based on Eq. (1). For the constant parameters used in the mathematical model, please refer to Table 2. 5. Results and discussions 5.1. Forward heat transfer performance of the water-vapor chamber thermal diode The experimental and calculation results of the effective thermal conductivities in the forward and reverse directions are shown in Fig. 5. The findings from the experiment illustrate that when the hot side temperature increases, the effective thermal conductivity in the forward mode increases. The mechanism behind the phenomenon is because of (i) enhanced buoyancy effects in the fluid and gas regions, and (ii) an increase in latent heat due to the temperature increase. It is noted that thermal convection is a combina-

Table 2 Constant parameters used in the mathematical model. Parameter

Value

Lss,cd Lwall,cd Ass,cd Awall,cd kss,cd kwall,cd Lw,cv,La,cv, Lv,cv Aw,cv, Aa,cv, Av,cv, Alv La,cd, Lw,cd Dx A

0.012 m 0.149 m 0.00442 m2 0.00245 m2 16.8 W/m K 0.35 W/m K 0.0755 m 0.00196 m2 0.0755 m 0.175 m 0.001 m2

tion of thermal conduction and fluid motion. The hot surface at the bottom first transfers heat to adjacent fluid molecules through thermal conduction. The temperature of the fluid molecules near the surface thus increases, creating a temperature difference with the fluid molecules far away from the hot surface. Temperature difference across the fluid layer causes a density gradient of the fluid in a vertical direction. Thus, fluid molecules below with high temperature are forced to move up. As a result, hot fluid molecules can transfer heat to cold fluid molecules through contact, leading to a greater thermal conductivity of the fluid layer in the forward mode when compared to pure conduction. Besides, an increase in fluid temperature can also increase kinetic energy of fluid molecules resulting in further collisions of molecules which leads to an increase in heat transfer between the molecules. During the phase change process, water molecules at the water surface evaporate due to high kinetic energy. With a higher temperature at the hot surface, more water molecules at the surface overcome the attraction force of nearby water molecules and escape the surface in the form of vapor. In this context, it implies that if the water temperature increases, then the evaporation rate also increases. More vapor in the gas region above the water surface enhances not only thermal convection but also condensation on the cold surface at the top, leading to an increase in effective thermal conductivity in the forward mode. The maximum experimental effective thermal conductivities in the forward mode at the hot side temperature of 70 °C are 2.86 W/m-K, showing a 50% improvement when compared with that at the hot side temperature of 40 °C. In addition to the outcome from the experiment, the results calculated from the developed mathematical models show a similar trend. At the hot side temperature of 40 °C, the calculated effective thermal conductivity is 2.58 W/m-K. When the hot side temperature increases to 70 °C, a calculated effective thermal conductivity of 2.82 W/m-K is obtained. This shows a 9.30% improvement. However, it is noted that the results from the experiment and those from the mathematical models are different due to heat loss and the assumptions made in the model. While perfect insulation is assumed in the mathematical models, heat loss is inevitable when conducting the experiment. Additionally, it can be observed that when the hot-side temperature increases, the difference between the experimental and calculated results are reduced. This is because the axial effective thermal conductivity of the thermal diode increases with temperature, but the radical thermal conductivity (related to the heat loss) of the Teflon chamber wall remains unchanged. Therefore, heat tends to transfer in the axial direction with an increase in temperature, and the effect of heat loss to the thermal diode is reduced. As a result, if better insulation is applied to the water-vapor chamber thermal diode, the results of the effective thermal conductivity in the forward mode enhances and the above-mentioned difference decreases. Moreover, in this study, since the Rayleigh numbers of water and air are greater than the critical value of 11,323, the heat transfer in the water and air layers is considered as thermal convection. On the other hand, the Rayleigh number of vapor in this study is only greater than the critical value of 11,323 when the temperature difference exceeds 30 °C. Thus, thermal convection in the vapor layer at the condition of temperature difference greater than 30 °C is considered, which is in line with the findings of this work demonstrating the greater enhancement of the forward effective thermal conductivity when the temperature difference is larger than 30 °C. Besides, the Nusselt number of water in this study is dramatically larger than those of air and water vapor. This implies that thermal convection in the water region has a significant effect on the heat transfer enhancement in the forward mode. It should be noted that when the experiment is repeated three times, the maximum standard deviation of the forward effective thermal conductivity is ±0.075 at the temperature difference of 30 °C, as shown in Fig. 5.

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181

Fig. 5. Simulated and experimental effective thermal conductivities of the water-vapor chamber thermal diode in both heat transfer directions where the cold-side temperature is 20 °C and the water-air volume ratio is 0.5.

5.2. Reverse heat transfer performance of the water-vapor chamber thermal diode On the other hand, in the reverse mode, the effective thermal conductivity is comparatively constant when the temperature changes. This confirms that heat transfer only takes place through thermal conduction since the thermal conductivities of all the materials (i.e. stainless steel, PEEK, Teflon) are temperatureindependent properties. While the average calculated effective thermal conductivity in the reverse mode reaches 1.46, the experimental results show an average effective thermal conductivity of 1.12. Although the calculated result and the experimental result are in line with each other, the difference between the two results is caused by heat loss when conducting the experiment. On the other hand, the performance of the water-vapor chamber thermal diode filled with air (without water) has also been examined. It can be seen from Fig. 5 that the effective thermal conductivities in both forward and reverse modes, no matter at whatever temperature difference values, are close with each other, implying that there is no diodicity effect. The diodicity effect only exists when the thermal diode is filled with water. It is noted that when the experiment is repeated three times, the highest standard deviation of the reverse effective thermal conductivity is ±0.059 when the temperature difference is 20 °C, as shown in Fig. 5. 5.3. Thermal rectification performance of the water-vapor chamber thermal diode Besides the effective thermal conductivity, the diodicity of the water-vapor chamber thermal diode from the experiment is also reported in Fig. 6. The results show that when temperature increases, the diodicity also increases. A maximum diodicity of 1.43 with the water-air volume ratio of 0.5 at the hot-side temperature of 70 °C is reported in this study. This shows that when the temperature increases by 30 °C, the diodicity improves by 25.6%. In the current study, the diodicity of the water-vapor chamber thermal diode increases with an increase in temperature difference. This macro rectification phenomenon is caused by the phase change process and fluid motion. However, a similar phenomenon at the micro level has been reported in many nanostructures through the asymmetry of phonon motion across the interface between two different materials or cross-sectional areas. For instance, the rectification performance of graphene/hexagonal boron nitride heterojunctions [41–43] varies with temperatures due to the asymmetric phonon scattering in different heat transfer

directions. Besides, the change in kinetic energy of the phonon in the graded nanowires [44] leads to a temperature-dependent rectification behaviour across the asymmetric thermal contact boundary. In order to improve the diodicity of the water-vapor thermal diode, there are several methods to enhance the forward effective thermal conductivity and reduce the reverse effective thermal conductivity. Firstly, the degassing process can be performed to remove all non-condensable gases which contribute to high thermal resistance inside the chamber. As a result, the heat transfer in the forward mode can be enhanced, and the heat transfer in the reverse mode can be reduced at the same time. Secondly, utilizing proper insulation can also enhance the performance of the thermal diode. Replacing the stainless steel plates at the top and the bottom of the thermal diode with copper plates can also potentially enhance the effective thermal conductivity since the thermal conductivity of copper is greater than that of stainless steel by almost 10 times. It is also observed that heat flow crossing the Teflon chamber plays a significant role in the reverse heat transfer condition. As a result, in order to further reduce the effective thermal conductivity in the reverse mode, the thickness of the Teflon chamber must be minimized. 5.4. The effects of water-air volume ratio on heat transfer and thermal rectification of the water-vapor chamber thermal diode Apart from experimental results shown in the previous sections with the water-air volume ratio of 0.5, this section shows the

Fig. 6. Experimental diodicity of the water-vapor chamber thermal diode where the cold-side temperature is 20 °C and the water-air volume ratio is 0.5.

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experimental investigation of the effects of the water-air volume ratio on the heat transfer performance of the water-vapor chamber thermal diode where the hot side temperature is 70 °C and the cold side temperature is 20 °C. When the water-air volume ratio is equal to 1, the chamber is full of water. On the other hand, the chamber is empty (without water) when the ratio is 0. Fig. 7 shows the effects of water-air volume ratio on the effective thermal conductivities and diodicity of the water-vapor chamber thermal diode where the hot side temperature is 70 °C. Due to the fact that thermal conductivity and Rayleigh number of water are much greater than those of air and vapor, the experimental results demonstrate an increase in the effective thermal conductivity in the forward mode when the water-air volume ratio increases, especially when the water-air volume ratio is greater than 0.5. In contrast, the findings show that the effective thermal conductivity in the reverse mode is relatively constant with an increase in the water-air volume ratio, implying that heat transfer in the reverse mode mainly takes place through thermal conduction at the chamber containers. As a result, the diodicity calculated from Eq. (1) also increases with the water-air volume ratio when the air-volume ratio is higher than 0.5. Moreover, Fig. 7 also shows that the effective thermal conductivities of the thermal diode without water (water-air volume ratio of 0) in both forward and reverse directions are similar, meaning that only thermal conduction occurs and there is no thermal diode effect. Thus, it can be concluded that due to greater thermal properties of water over air and vapor, the more water in the water-vapor chamber, the better the heat transfer performance and the diodicity of the thermal diode. When comparing the diodicity of the proposed water-vapor chamber thermal diode with other thermal diodes/switches listed in Section 1, the diodicity of the proposed water-vapor thermal diode at a high

Fig. 7. The effects of water-air volume ratio on (a) the effective thermal conductivities in forward and reverse modes, and (b) the diodicity of the water-vapor chamber thermal diode where the cold-side temperature is 20 °C and the hot-side temperature is 70 °C.

water-air volume ratio is higher than many of them (such as the cobalt oxide thermal diode, the mercury-based thermal diode and the Vanadium-oxide radiative thermal diode). In addition to the effective thermal conductivity and the diodicity, it is worth noting that this thermal diode offers a simple design, low cost and uses non-toxic materials. 5.5. Limitation of the water-vapor chamber thermal diode Additionally, it is noted that the water-vapor chamber thermal diode in this study has several limitations. Firstly, since liquid water is one of the working fluids inside the chamber, the operating temperature on both sides should not be less than 0 °C to avoid freezing. Secondly, based on the results obtained above, the heat transfer and thermal rectification performances of the watervapor chamber thermal diode depends on the operating temperature and the temperature difference across the diode. In order to obtain the expected heat transfer and thermal rectification performance, the hot and cold temperatures need to be carefully selected. Thirdly, the hot side temperature of the water-vapor chamber thermal diode cannot be too high in order to protect the O-ring and prevent drying out. Lastly, the water-vapor chamber thermal diode cannot be operated upside down since water is always on the bottom of the chamber. In other words, the proposed thermal diode is an orientation-dependent device. 6. Conclusion In this study, a water-vapor chamber thermal diode was designed, built and investigated, both experimentally and theoretically. It must be noted that this is the first study of a phase-change thermal diode in which water is utilized in the thermal diode, and in which mathematical models predicting the heat transfer performance are developed. The effective thermal conductivities in the forward mode and the reverse mode are reported, and the diodicity of the thermal diode is also revealed and discussed. The results show that with the water-air volume ratio of 0.5, the effective thermal conductivity in the forward mode increases with an increase in the hot-side temperature. In contrast, the effective thermal conductivity in the reverse mode remains unchanged regardless of the temperature. This conclusion is true for the results from both the experiment and the calculations, implying that the mathematical models were developed successfully. However, due to heat loss when conducting the experiment, differences in the results from the experiment and from the calculations were observed. This shows that if better insulation is applied in the water-vapor chamber thermal diode, the heat transfer performance from the experiment can be enhanced. With the water-air volume ratio of 0.5, the diodicity of 1.43 is reported in this study. Moreover, the findings also reveal the effects of water-air volume ratio on the heat transfer performance of the thermal diode, showing that when water-air volume ratio increases, the effective thermal conductivity in the forward mode increases, but the effective thermal conductivity in the reverse mode is relatively constant, resulting in an increase in the diodicity of the water-vapor chamber thermal diode. The findings of this study can not only offer new insights into phasechange thermal diodes, but also shed new light on water-vapor heat transfer phenomenon. Moreover, the developed models can serve as a general tool for phase-change thermal diode design in other engineering applications. Conflict of interest We declare that we do not have any actual or potential conflict of interest including any financial, personal or other relationships

M.Y. Wong et al. / International Journal of Heat and Mass Transfer 138 (2019) 173–183

with other people or organizations within three years of beginning the work submitted that could inappropriately influence (bias) our work.

Acknowledgement The funding sources for this research are provided by the Hong Kong Research Grant Council via General Research Fund (GRF) account 16202517 and Collaborative Research Fund (CRF) C602216G.

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