Dropwise condensation heat transfer model considering the liquid-solid interfacial thermal resistance

International Journal of Heat and Mass Transfer 112 (2017) 333–342

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International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt

Dropwise condensation heat transfer model considering the liquid-solid interfacial thermal resistance D. Niu a, L. Guo a, H.W. Hu b, G.H. Tang a,⇑ a b

MOE Key Laboratory of Thermo-Fluid Science and Engineering, School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, PR China School of Environment and Energy Engineering, Anhui Jianzhu University, Hefei 230601, PR China

a r t i c l e

i n f o

Article history: Received 28 December 2016 Received in revised form 10 April 2017 Accepted 12 April 2017

Keywords: Interfacial thermal resistance Molecular dynamics simulation Droplet nucleation Dropwise condensation heat transfer Experiment

a b s t r a c t Dropwise condensation is a multiscale process including droplet nucleation, growth, coalescence and departure stages. As the initial stage, droplet nucleation plays a significant role in condensation heat transfer by determining the droplet nucleation radius and nucleation density. In this work, the molecular dynamics simulation is employed to determine the liquid-solid interfacial thermal resistance (ITR) for different surface wettability. To derive the modified model, the liquid-solid ITR is incorporated into the existing heat transfer model for a single droplet, on the basis of which, the effect of liquid-solid ITR is introduced to modify the critical nucleation radius and nucleation density. The results show that the introduction of liquid-solid ITR leads to not only an increasing of critical nucleation radius but also a reduction of nucleation density. Dropwise condensation experiments are conducted and the present model predicts the experimental data we performed and those in the literature, more accurately than the existing model which neglects the liquid-solid ITR, particularly for large contact angles or large sub-cooled degrees. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Dropwise condensation has attracted a large amount of attention due to its higher heat transfer coefficient than filmwise condensation and its potentials in many fields such as energy conversion [1–3], water desalination [4], electronic device cooling [5] and water recovery [6,7]. The mechanism of dropwise condensation is explored extensively and a number of models are proposed to predict the heat flux under different subcooled degrees. LeFevre and Rose [8] are the first to propose the heat transfer model by combing the heat transfer model of a single droplet and the distribution function of droplet sizes. Then, Tanaka [9] modified the droplet size distribution for small non-coalescing droplets based on a population balance theory. Two different mechanisms of droplet growth including direct vapor condensation and coalescence with adjacent droplets were introduced. A similar theory was also adopted by Abu-Orabi [10] to estimate the droplet size distribution and develop a model considering the effect of thermal resistance of coating materials. Then the dependence of droplet size distribution on contact angles were evaluated by Kim and Kim [11] and the

⇑ Corresponding author. E-mail address: [email protected] (G.H. Tang). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.04.061 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.

results showed that small droplets were demonstrated to bear higher heat flux. With superhydrophobic surfaces based on micro/nanostructures applied to dropwise condensation, a model was developed by Miljkovic et al. [12] to consider the effects of surface geometry, nucleation density and promoter coating on heat transfer performance. In our previous investigation [13], the existing heat transfer model is modified for dropwise condensation on a horizontal tube and the predictions agree with experimental data well. Liu and Cheng [14,15] proposed an improved model in which the critical nucleation radius is determined based on a thermodynamic analysis. In their model, the thermal resistances of coating layer, liquid-vapor interface and curvature were taken into account to calculate the change of Gibbs free energy accurately. A formation of a droplet begins from a nucleation process of vapor molecules and the critical nucleation radius is determined by the nucleation theory and can be regarded as the size of minimum droplets in heat transfer models. Therefore, it is essential to develop a reasonable heat transfer model to evaluate the nucleation process. In previous studies, nucleation process has been extensively investigated with experimental, theoretical and numerical methods [16–21]. Yamada et al. [16] investigated water condensation on a hybrid hydrophilic-hydrophobic surface to reveal nucleation mechanisms at the microscale. Condensation experiments revealed that droplets could nucleate on the hydrophilic areas under unsaturated conditions. An extended nucleation

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Nomenclature G Gibbs free energy (J) h interfacial heat transfer coefficient (W m
2 K
1) specific latent heat (kJ kg
1) Hfg q heat flux (W m
2) k thermal conductivity (W m
1 K
1) N(r), n(r) drop size distribution (m
3) nucleation density (m
2) Ns P pressure (Pa) r radius (m) rmin minimum droplet radius (m) coalescence radius (m) re rmax maximum droplet radius (m) r0 minimum droplet radius (m) rc critical nucleation radius (m) T temperature (°C) Greek symbol U heat transfer rate (W) q density (kg m
3)

model taking into account the attracted water molecules on the hydrophilic surface was used to explain the measured droplet intervals. Mu et al. [17] experimentally studied the relationship between nucleation density and surface topography on surfaces of magnesium in nanoscale. The results showed that the surface topography had a great influence on nucleation density and larger fractal dimensions would lead to more nucleation sites. Zeng and Xu [18] developed a thermodynamic model to capture the formation of a droplet embryo on fractal surfaces. The results showed that the differences between the critical size of the embryos on the fractal surfaces and those on the flat surfaces were negligible for the hydrophobic nucleation. Molecular dynamics simulation was also used to investigate the droplet nucleation on a solid surface [19]. The nucleation rate, critical nucleation size and the required free energy obtained from the simulation agreed with classical heterogeneous theory for the smaller cooling rate or less wettable solid surface. Sheng et al. [20] investigated the onset of surface condensation for different surface wettability using molecular dynamics simulation. Different condensation modes including the no-condensation, dropwise condensation and filmwise condensation were quantitatively analyzed in the simulation by temporal profiles of surface clusters. Sun and Wang [21] used molecular dynamics simulation to investigate the early and developed stages of surface condensation and they revealed the competition mechanism of thermal resistances that the interfacial thermal resistance dominates at the onset of condensation while the condensate bulk thermal resistance gradually takes over with condensate thickness growing. The nanoscale droplet embryo forms during the nucleation process. As we know, the interface effect plays a significant role in nanoscale heat transfer [22]. However, the effect of liquid-solid ITR has not been considered during the droplet nucleation process. Actually, the liquid-solid ITR on phase-change processes has been investigated extensively [23–26]. Ghasemi and Ward [23] showed that a large liquid-solid ITR would impede the heat transfer normal to the liquid-solid interface effectively in their investigation of the evaporation for water droplet on Au substrates. Hu and Sun [24] also showed that the reduced liquid-solid ITR caused by the effect of nanopattern could enhance the heat transfer during the boiling of water on a gold surface using molecular dynamics simulation. A temperature jump is generated at the liquid-solid interface during

rlv h

eij u d

rij a

liquid vapor surface tension (N m
1) contact angle (°) characteristic surface energy (eV) angle (°) coating thickness (m) van der Waals radius (nm) inclination angle of the plate

Subscripts c condensate cur curvature d droplet l liquid s solid sub subcooled sat saturated v vapor

a rapid solidification process and would lead to a reduced solidification rate [25]. In our previous work [26], we employed the molecular dynamics simulation in examining the role of liquidsolid ITR in nanoscale condensation of water vapor onto surfaces with different wettability. The results showed that, at the onset of condensation, the filmwise has a higher heat transfer efficiency than the dropwise condensation, while the dropwise is higher at macroscale. This is because the liquid-solid ITR dominates in the total thermal resistance when other ones are extremely small at nanoscale. Although the theoretical and experimental investigations for dropwise condensation heat transfer have been widely carried out, and the effects of interfacial thermal resistance on the evaporation [23], boiling [24] and solidification [25] processes have also been investigated, the effect of liquid-solid ITR on droplet nucleation and dropwise condensation heat transfer performance has not been reported. Therefore, the liquid-solid ITR is worth considering for predicting heat transfer of dropwise condensation. In this study, we will propose a modified heat transfer model considering the effect of liquid-solid ITR. Furthermore, the experimental measurements are performed to validate the model. This paper is organized as follows. In Section 2, the modified heat transfer is derived and in Section 3, the dependence of liquid-solid ITR on surface wettability is determined using molecular dynamics simulation. Section 4 presents the experiment system while in Section 5, the experimental data are employed to validate the present model with a detailed discussion. Finally, Section 6 gives a conclusion. 2. Heat transfer model A droplet experiences the nucleation, growth, coalescence and departure processes in an existence cycle. Formation of droplets in dropwise condensation is based on two different mechanisms: by direct vapor condensation and by coalescence with adjacent droplets. Therefore, droplets on surfaces can be divided into two groups: one is the droplets of small sizes with rmin < r < re, and the other is the droplets of large sizes with re < r < rmax, where r represents the radius of droplets: rmin, rmax and re denote the minimum droplet radius, the maximum droplet radius and the coalescence radius, respectively. The maximum droplet radius is also

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known as the departure radius and is decided by the inclination angle of the plate, a. For the dropwise condensation on a vertical plate with an inclination angle a as 90°, the heat flux on the solid surface q could be given based on both the heat transfer through a single droplet and the droplet size distribution functions as

Z

q¼

re

Z

/d nðrÞdr þ

r min

r max

ð1Þ

/d NðrÞdr re

where n(r) and N(r) represent the size distribution functions of small droplets and large droplets, respectively. /d is the heat transfer rate through a droplet and is derived by building a heat transfer model through a single droplet as shown later. For the dropwise condensation on a horizontal tube, as our previous investigation shown [13], the horizontal tube surface is assumed to consist of numerous inclined plates with different inclination angles, varying from 0 to 180°. The departure radius depends on local inclination angles, as r max ðaÞ. The heat transfer rate for each inclined plate at a specific inclination angle could be given as Eq. (1) and the overall heat flux is derived as the integration of heat transfer rate for all inclined plates. The heat flux on a horizontal tube should be given as

q¼

Z pZ 0

re r min

/d nðrÞdrda þ

Z pZ 0

r max ðaÞ r min

/d NðrÞdrda

ð2Þ

The heat released from saturated vapor firstly flows through the liquid-vapor interface, leading to the liquid-vapor interfacial temperature drop. The liquid-vapor interfacial temperature drop (DTlv) is

DT l v ¼

DT sub ¼ DT lv þ DT drop þ DT coat þ DT curv þ DT ls

ð3Þ

ð4Þ

where h is the contact angle of droplets on the subcooled surface and hlv is the heat transfer coefficient of liquid-vapor interface, varying from 0.383 to 15.7 MW/m2 K with a vapor pressure ranging from 0.01 to 1.0 atm in [11]. As shown in Fig. 1a, the heat conduction through the bulk droplet is considered to transfer perpendicular to the isothermal surfaces. The temperature drop from one isothermal surface at an angle of u to the neighboring isothermal surface at an angle of u + du, where u is a shape-fitted coordinate, can be determined based on Fourier’s Law. The temperature drop due to the heat conduction through bulk liquid (DTdrop) is derived by introducing an integration of the temperature difference between two neighboring isothermal surfaces from 0 to h, as shown in Eq. (5), by Kim and Kim [11],

Z

h

/d de /d ¼ k A 2 p dkc s 0 c h/d ¼ 4pr sin hkc

DT drop ¼

2.1. Heat transfer through a single droplet or a droplet embryo Each droplet contributes to the dropwise condensation heat transfer by transferring heat through itself. So it is essential to build the single droplet heat transfer model firstly to predict the overall dropwise condensation heat transfer performance. As shown in Fig. 1a, the heat transfer through a single droplet is resisted by the thermal resistances of five parts, from the saturated vapor to the subcooled solid surface, containing the thermal resistances due to the liquid-vapor interface (Rlv), the conduction of the droplet (Rdrop), the conduction of coating layer (Rcoat), the curvature of the droplet (Rcurv) and the liquid-solid interface (Rls). These thermal resistances cause corresponding temperature drops, marked as DTlv, DTdrop, DTcoat, DTcurv, and DTls. The total subcooled degree DTsub is determined by DTsub = Tsat-Tw with Tsat and Tw as the temperatures of vapor and solid surface, respectively. It can also be expressed as the summation of the above temperature drops

/d 2pr 2 ð1
cos hÞhlv

Z

2

ðcsc2 u
cot u csc uÞsin u du 1
cos u 0 h

ð5Þ

where As is the area of an arbitrary isothermal surface, de means the distance between the two isothermal surfaces, and kc is the thermal conductivity of liquid. The temperature drop (DTcoat) due to the coating layer is given by Eq. (6), with kcoat as the thermal conductivity of coating material.

DT coat ¼

d/d

ð6Þ

2

kcoat pr 2 sin h

The temperature drop (DTcurv) caused by the droplet curvature is given as [11,14]

DT curv ¼

r0 DT sub r

ð7Þ

In the classical homogeneous nucleation theory, r0 is the expression of the minimum droplet radius, and r 0 ¼ H2TqsatDrTlv fg

c

sub

is obtained by

assuming that the total subcooled degree determines the minimum drop radius [11]; qc is the condensate density, rlv is the liquid-vapor surface tension, and Hfg is the latent heat of water vapor. To evaluate the temperature drop caused by the effect of liquidsolid ITR, hls is introduced as the heat transfer coefficient of liquidsolid interface, thus, the temperature drop (DTls) is derived as

DT ls ¼

p

/d 2 2 r sin hh

ð8Þ ls

The dependence of hls on contact angle is determined as hls = A (1 + cosh) using molecular dynamics simulation, which will be detailed in Section 3. Substituting Eqs. (4)–(8) into Eq. (3), we can derive the expression of /d as

/d ¼

Fig. 1a. Schematic of a condensate droplet or a droplet embryo on a subcooled surface.

d sin2 hkcoat

DT sub pr 2 ð1
r0 =rÞ 1 þ 4 sinhrhkc þ 2ð1
coshÞh þ sin21hh lv

ð9Þ

ls

Compared with the existing model, the underlined term in Eq. (9) is introduced to evaluate the effect of liquid-solid ITR on heat transfer through a single droplet with a radius of r.

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2.2. Determination of critical nucleation radius and nucleation density The size distribution functions of small droplets n(r) in Eqs. (1) and (2) are given as

nðrÞ ¼

 
2=3 1 re rðr e
r min Þ A2 r þ A3 expðB1 þ B2 Þ 3 3pre r max r max r
rmin A2 r e þ A3

The same method is adopted in the present determination of rc, however, the effect of liquid-solid ITR on the temperature distribution in the bulk droplet is considered to modify the critical radius (rc). The expression of nucleation free energy barrier derived by Liu and Cheng [14] is given as

Z ð10Þ

where the coalescence radius re is shown in Eq. (11) and other parameters can be referred in [11,15].

DGr ¼ qc

Hfg ðT l ðr; uÞ
T s Þ dv þ rlv ð2
3 cos h þ cos3 hÞpr2 T sat ð16Þ

dv ¼ pr 3 sin h 3

r e ¼ ð4Ns Þ


1=2

ð11Þ

where Ns is the theoretical expression of droplet nucleation density given by Rose [27] as

Ns ¼

0:037 r 2min

ð12Þ

The size distribution functions of large droplets N(r) is

 
2=3 1 r NðrÞ ¼ 3pr 2 rmax r max



3rlv sin hðcos hr
cos ha Þ 2qg sin að2
3 cos h þ cos3 hÞ

ð13Þ

1=2 ð14Þ

where rlv is the liquid-vapor surface tension, and hr and ha refer to the receding and advancing contact angle, respectively. A condensate droplet on an inclined surface is schematic in Fig. 1b. A detailed description for the dropwise condensation heat transfer model on horizontal tube is provided in our previous work [13]. The minimum radius rmin can be assumed as the expression of r0 by assuming that the total subcooled degree determines the minimum drop radius. However, with the effects of thermal resistance of coating, liquid-vapor interface and curvature depression taken into consideration, Liu and Cheng [14] proposed an improved thermodynamic model to obtain the critical radius rc as the minimum radius, thus,

r min ¼ rc

4

sin u

ð17Þ

du

in which T l ðr; uÞ is the local temperature of the isothermal surface at an angle of u. To determine T l ðr; uÞ, the temperature drop between the isothermal surface of u and the solid surface is marked as DT l ðr; uÞ ¼ T l ðr; uÞ
T w . Similar to Eq. (3), DT l ðr; uÞ can be expressed as

DT l ðr; uÞ ¼ DT coat þ DT drop ðr; uÞ þ DT ls

where the departure radius rmax depends on the contact angle of a plate with a certain inclination angle. In our previous study for dropwise condensation on horizontal tubes [13], the dependence of the departure radius (rmax) on inclination angle (a) based on the force balance is given by

r max ¼

ð1
cos uÞ2

ð15Þ

ð18Þ

Then, we can derive T l ðr; uÞ as

T l ðr; uÞ ¼ T w þ

d/d

pr2 sin2 hkcoat

þ

u/d /d þ 4pr sin hkc pr 2 sin2 hhls

ð19Þ

Compared with the existing model, the underlined term in Eq. (19) is introduced to evaluate the effect of liquid-solid ITR on the temperature distribution in bulk droplets. We obtain the nucleation free energy for a droplet with a radius of r, by submitting Eqs. (17) and (19) into Eq. (16), as 3

Rh

DGr ¼ rlv ð2
3cosh þ cos3 hÞpr 2 þ qc pr3 sin h

Hfg 0 T sat

d/d ð
DT sub þ pr2 sin 2 hk

2

coat

/d /d uÞ þ 4prusin þ pr2 sin Þ ð1
cos du 2 hkc hh sin4 u ls

ð20Þ Therefore, the role of liquid-solid ITR is introduced by the underlined term in Eq. (20), compared with the existing model. The critical nucleation radius rc could be obtained numerically by finding the location of the extreme point of the nucleation free energy, as shown in Eq. (21).

 @ DGr  ¼0 @r r¼rc

ð21Þ

With rc obtained, the minimum radius of droplet is obtained and the droplet nucleation density in Eq. (12) is determined as well. By substituting Eqs. (9)–(14) into Eqs. (1) and (2), we can obtain the heat transfer model of dropwise condensation on a flat surface and a horizontal tube, respectively. The above modification can be summarized as follows. With the liquid-solid ITR considered, the heat transfer rate though a single droplet is modified in Eq. (9) and the temperature distribution in bulk droplets is further determined in Eq. (19). The critical nucleation radius is reappraised in Eq. (21), which further helps improve the prediction of the droplet nucleation density in Eq. (12), and the size distribution functions of small droplets and large droplets in Eqs. (10) and (13), respectively. 3. Determination of liquid-solid ITR using molecular dynamics simulation

Fig. 1b. Schematic of a condensate droplet on an inclined surface.

Molecular dynamics simulation is performed based on the LAMMPS package [28] to demonstrate the relationship between the liquid-solid ITR and the wettability. A symmetrical liquidsolid system is built with a size of 136  35.28  35.28 Å in the x, y and z directions, respectively, as schematic in Fig. 2. Monatomic

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Fig. 2. Schematic of the liquid-solid simulation system.

water model [29,30] described by a Stillinger-Weber potential [31] is adopted. The EAM potential [32] is employed for the interatomic forces of Pt-like solid wall. The 12-6 LJ potential, E = 4eij[(rij/rij)12
(rij/rij)6], where eij is the characteristic surface energy, rij is the van der Waals radius, and rij is the inter-molecular distance, is used to describe the interaction between the water and the Pt-like surface with different energy parameters to cover the wettability from hydrophobic to hydrophilic. To confirm the dependence of surface wettability on energy parameter of liquid-solid interaction e used in the LJ potential, the contact angle of a water droplet on a solid surface with differ-

ent e are determined using molecular dynamics simulation, as shown in Fig. 3a. The water droplet is set with a thin-slab-like cylindrical droplet in the z direction, to avoid size-dependent contact angle caused by the line tension [33] and reduce computation time [34]. The contact angle is obtained by defining the droplet boundary as the contour line with 0.5 g/cm3 in the density field. As shown in Fig. 3b, the dependence of wettability on the surface energy parameter is established, from hydrophobic to hydrophilic, by changing e from 0.005 eV to 0.0175 eV. For preparing the system of thermal transport, firstly the NPT (constant number of atoms, pressure and temperature) ensemble runs at 1 atm and 300 K for 1 ns with a time step of 1 fs. Berendsen thermostat and barostat are used to maintain the temperature and pressure, respectively. Then the NVT ensemble is used to introduce a heat source and sink for inducing a steady heat flux as [35]. In this stage, the five layers of the Pt-like atoms in the middle are treated as the heat source with a higher temperature of 363 K and the symmetrical regions on both sides with a thickness of 15 Å in x direction are treated as the heat sink with a lower temperature of 293 K. Finally, the steady temperature distribution for Pt-like wall and water along the x direction is obtained by time averaging for 1 ns.

350

(a) 300

Extracted energy Injected energy

250

Energy (eV)

(a) 200 150 100 50 0 0

200

400

600

800

1000

Time (ps) 380 150

(b)

(b) 360

ε =0.01 eV

Temperature (K)

Contact angle (degree)

135 120 105

ε =0.005 eV ε =0.0125 eV

90

340

Solid region

Liquid region

Liquid region

320

300

ε =0.0075 eV

75 60 0.0025

280 -60 0.0050

0.0075

0.0100

ε (eV)

0.0125

0.0150

0.0175

Fig. 3. (a) Water droplet on the Pt-like surface in the simulation; (b) dependence of contact angles of droplets on the different surface wettability, with typical density contours inserted.

-40

-20

0

20

40

60

x (Å ) Fig. 4. Additional energy introduced by thermostat and temperature distribution in the x direction for e = 0.01 eV. (a) Extracted and injected energy from two different thermostats, (b) temperature profiles in the x direction and temperature jump at the interface.

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As shown in Fig. 4, the dependence of liquid-solid interfacial thermal conductance hls on surface wettability is determined according to q = hls DTjump, where q is the heat flux normal to the liquid-solid interface and DTjump is the corresponding temperature jump. To obtain the heat flux q, the accumulated energy, extracted and injected from the heat source and heat sink, respectively, are shown in Fig. 4a. Due to the symmetry of the simulation system, the heat flux q is calculated as half slope of the energy change with respective to the time and heat transfer area in the thermostats. To determine the temperature jump DTjump, water and solid are divided into thin bins with a thickness of 1 Å along the x direction and the time-averaged temperature distribution in each bin is obtained. Fig. 4b shows the case of e = 0.01 eV as an example. DTjump is calculated from the liquid temperature and solid temperature at the interface, both of which are predicted by linear fitting of the obtained values in bins. With q and DTjump obtained, the interfacial thermal conductance is given as, hls = q/DTjump. Combined with the results in Fig. 3b, a linear relationship of hls = A(1 + cosh) is obtained for surfaces with different contact angles, as shown in Fig. 5, where A is a proportionality coefficient of A = 98 MWm
2K
1, which is close to the results in [36].

A 5

P T

2

Conductance (MW/m K)

100

80

60

40 0.4

0.6

0.8

1.0

1.2

1+cos Fig. 5. Interfacial thermal conductance as a function of 1 + cosh.

1.4

3

1

T 8

Fig. 6a. Schematic of the custom-designed condensation heat transfer experiment setup. (1) Copper test specimen, (2) teflon block, (3) condensation chamber, (4) thermostatic bath, (5) water vapor reservoir, (6) heating power controller, (7) water vapor inflow valve, (8) check valve, and (9) vacuum pump. A–B: Water vapor channel, C–D: Cooling water channel, P: Pressure measurement, T: Temperature measurement.

Temperature(°C)

90

Fitting curve

2

6

B

The dropwise condensation on both horizontal tubes and vertical plates are experimentally studied. The experiment system for a horizontal tube was detailed in our previous work [37]. The experiment system for the condensation on a vertical plate is introduced as follows. As schematic in Fig. 6a, the experimental system consists of two separate circuits of water vapor flow and cooling water flow. The water vapor is generated by a vapor reservoir at controlled mass flow rates and then flows through the entire condensation chamber, marked as Circuit A-B while the cooling water is recirculated through the cooling chamber, marked as Circuit C-D. The condensation of pure water vapor on the vertical copper plate is carried out at specified surface subcooled degrees and a vapor pressure of 105 Pa. The temperature and pressure in the condensate chamber and the water vapor reservoir are monitored by Ttype thermocouples (TCV-TG-0300-10-M12, Omega) and pressure sensors (Tecisis P3276, Germany), respectively. Specified surface subcooled degree is obtained by changing the temperature of the cooling water, with a thermostatic bath (ThermoFlex 2500) employed.

Simulation results

C

9

93

120

4

P

D

4. Experimental measurement

140

7

87 84 81 78 75 72 0

5

10

15

20

25

Position of the hole (mm) Fig. 6b. The measured temperature distribution in the copper plate for the superhydrophobic surface, DTsub = 4.5 K.

To investigate the dropwise condensation on surfaces with different wettability, the hydrophobic and superhydrophobic surfaces of copper plate are prepared by the coatings of the self-assembled monolayer (n-octadecyl mercaptan with oxidation) and an etching treatment [37]. The characterizations of surface wettability of untreated and treated cooper surfaces are shown in Fig. 7, showing hydrophilic, hydrophobic and superhydrophobic properties, respectively. Prior to the experiments, the pressure in the condensation chamber is reduced to 600 Pa by evacuating the existing air. Then the water vapor continues injecting into the experiment system for over an hour to remove the remained non-condensable gas. The water-saturated vapor flow is supplied to the condensation chamber until the pressure reaches the target value, and then the copper plate is cooled to target temperatures by circulating cooling flow. Then, the condensation process is measured. The heating power controller is employed to maintain the temperature and pressure of the condensation chamber at the target values. The vertically-oriented copper plate is inserted into the Teflon insulator to construct a one-dimensional steady heat conduction model. Four equidistant (5 mm spacing) holes (U 0.6 mm), perpendicular to the axis, are drilled into the sidewall of the cooper plate

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Fig. 7. Surface wettability characterization of (a) untreated cooper surface, (b) hydrophobic surface, and (c) superhydrophobic surface.

1800

80

1600 1400 1200

Contact angle=120°

60

-2

Nucleation radius considering the IRT Nucleation radius without the IRT Nucleation density considering the IRT Nucleation density without the IRT

12

(a)

1000 800

40

600 400

20

200 0

0 0

5. Results and discussion

Droplet nucleation density (10 m )

100

Critical nucleation radius (nm)

for inserting thermocouples (TJC36, Omega). By regulating the pressure (P) of saturated vapor, corresponding to the saturation temperature of Tsat, and the temperature of coolant, with a fixed flow rate, we can measure the steady temperature gradient (rT) within copper plate, to calculate the surface temperature (TW). Further, the condensation heat transfer coefficient h can be obtained by h = kcopperrT/DTsub, where DTsub is the subcooled degree and kcopper is the thermal conductivity of copper plate. All signals are acquired using a data acquisition system (Keithley 3706A). The uncertainties of obtained data are evaluated from the precision of the thermocouple (±0.1 °C) and that of pressure sensor (±0.5%). As shown in Fig. 6b, the temperature gradient in the copper plate is measured. DTsub could be obtained by extrapolating the temperature at the position of x = 0.

1

2

3

4

5

Tsub (K)

5.1. Modified critical nucleation radius and nucleation density 240

5.2. Validation of modified heat transfer model of dropwise condensation Firstly, a filmwise condensation experiment is performed on the vertical copper plate cleaned thoroughly with hydrochloric acid solution and de-ionized water. The experiment results are compared with the predictions of Nusselt theory and the deviations between them are within 15%, as shown in Fig. 10a, which con-

1200

12

200

1000

Contact angle=150°

160

-2

Nucleation radius considering the IRT Nucleation radius without the IRT Nucleation density considering the IRT Nucleation density without the IRT

Droplet nucleation density (10 m )

1400

(b) Critical nucleation radius (nm)

The critical nucleation radius and nucleation density are modified based on Eqs. (21) and (12), respectively. The modified droplet nucleation radius and nucleation density predicted with and without the ITR considered, are compared under different subcooled degrees on a hydrophobic surface and a superhydrophobic surface, as shown in Fig. 8. We find that the incorporation of liquid-solid ITR can increase the critical nucleation radius and correspondingly reduce the nucleation density. For the superhydrophobic surface as shown in Fig. 8b, the nucleation density with the liquid-solid ITR considered could be one order of magnitude lower than the results neglecting the liquid-solid ITR. Therefore the liquid-solid ITR has a significant effect on the critical nucleation radius and nucleation density, especially for the superhydrophobic case. The effect of contact angle on the critical nucleation radius and nucleation density is investigated at a constant subcooled degree of DTsub = 4 K and the results are shown in Fig. 9. An increased contact angle leads to an increased critical radius and a reduced nucleation density, due to the smaller base area of the droplet and the increased interfacial thermal conductance. According to the previous model, the critical radius presents a slight increase from about 5.5 nm to 6.8 nm when the contact angle increases from 90° to 150°. However, considering the liquid-solid ITR, the critical radius exhibits a dramatic increase from 6.2 nm to 32.6 nm. Thus, the effect of liquid-solid ITR on the droplet nucleation is enhanced obviously when the contact angle becomes larger.

800 120 600 80 400 40

200 0

0 0

1

2

3

4

5

Tsub (K) Fig. 8. The effect of subcooled degree on droplet nucleation radius and density at (a) contact angle = 120°, (b) contact angle = 150°.

firms the reliability and accuracy of the experimental system. The slightly larger experimental results than the theory solutions could be attributed to the effect of water vapor velocity. Then, dropwise condensation experiments are carried out on two vertical copper plates, with contact angles of 120.5° and 155.4°, at 100 °C saturated vapor. The comparison between the experiment data and the predictions by the present model is shown in Fig. 10. For the dropwise condensation on the hydrophobic surface, as shown in Fig. 10a, the predictions of the models considering the liquidsolid ITR or not are both in good agreement with the experimental data. This feature can be explained by the findings in Fig. 8a that the nucleation parameters do not exhibit a particularly obvious difference for the hydrophobic surface at contact angle of 120°. However, for the superhydrophobic surface, as shown in Fig. 10b, the

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Nucleation radius considering the IRT without the IRT

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Contact angle (degree) Fig. 9. The effect of contact angle on droplet nucleation radius and density at DTsub = 4 K.

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(a) Hydrophobic surface Model considering the ITR Model without the IT Exp.

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demonstrates the importance of liquid-solid ITR for heat transfer prediction, the reason of which should be attributed to the obviously increased critical nucleation radius and the reduced nucleation density, as shown in Fig. 8b. In addition, as shown in Fig. 11, the prediction of present model is compared with the experiment data from [38] for the dropwise condensation on vertical titanium plates modified with contact angles of 127.45° and 156.85°. It is not surprising that the liquid-solid ITR almost has no effect on the condensation heat transfer performance for the hydrophobic surface, as shown in Fig. 11a. However, for the superhydrophobic case, a significant deviation between experimental data and the predictions is observed if the liquid-solid ITR is neglected, as shown in Fig. 11b. We also examine the present model for dropwise condensation on horizontal tubes, as shown in Fig. 12, with the experimental data from our previous study [37], where heat transfer was measured at 65 °C on the copper tubes modified with contact angles of 113.5° and 158.3°. As shown in Fig. 12a, similar to the case of vertical plate, the prediction fails to show an obvious difference no matter considering the liquid-solid ITR or not, for the hydrophobic case. When the hydrophobicity is further enhanced, the prediction without the incorporation of the liquid-solid ITR is much higher than the experimental data. Whereas, by taking into account the liquid-solid ITR, the present model exhibits a much closer prediction, despite suffering from a slight overestimation.

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Hydrophilic surface Exp. Nusselt theory

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Fig. 10. Comparison between the model and present experimental data on the vertical plate at (a) contact angle = 120.5°, (b) contact angle = 155.4°.

(b)

Superhydrophobic surface Model considering the ITR Model without the ITR Exp. [38]

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Contact angle=156.85° 60 45 30 15 0

present model agrees well with the experimental data, especially when the subcooled degree is enhanced; on the contrary, a missing of the liquid-solid ITR leads to a larger nucleation density and causes an overestimated prediction of heat flux. The finding

0

1

2

3

4

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Tsub(K) Fig. 11. Comparison between the model and experimental data [38] on the vertical plate at (a) contact angle = 127.45°, (b) contact angle = 156.85°.

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(b) Superhydrophobic surface

This work was supported by the National Natural Science Foundation of China under grant number of 51576156 and the 111 Project under grant number of B16038.

2

q (10 W/m )

120

dropwise condensation and a modified model is obtained. Moreover, the predictions of critical nucleation radius and density by the previous model and the present one are compared for different surface wettability and different subcooled degrees, exhibiting the effect of liquid-solid ITR on the nucleation clearly. The predictions of the present model show good agreement with the present experiment measurements and the experimental data in the literature. Furthermore, the present model could estimate heat flux more accurately in an extended range of subcooled degree, for dropwise condensations on not only vertical plates but also horizontal tubes. The advantages of the present model are highlighted particularly for superhydrophobic cases or under large subcooled degrees, compared with the model neglecting the liquid-solid ITR. The advantages of the present model can be explained by the modification of the prediction of critical nucleation radius and nucleation density with the liquid-solid interfacial thermal resistance considered.

Acknowledgments

Model considering the ITR Model without the ITR Exp. [37]

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References

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Contact angle=158.3°

80 60 40 20 18

20

22

24

26

28

30

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Tsub(K) Fig. 12. Comparison between the model predictions and experimental data [37] on the horizontal tubes at (a) contact angle = 113.5°, (b) contact angle = 158.3°.

The deviation of the present model could be attributed to two possible reasons. The first is that the large subcooled degree could produce more condensate, possibly leading to a complete wetting at some certain sites or generate a local liquid film, especially on the bottom of the horizontal tube due to the accumulation of condensate, because dropwise condensation can transform into filmwise condensation in certain conditions. The other is attributed to the constriction resistance [39], which is ignored in the present study but could have a significant effect on the condensation heat transfer at low pressure. Nevertheless, compared with the model without the effect of liquid-solid ITR considered, the present model introducing liquid-solid ITR is able to predict the dropwise condensation heat transfer performance more accurately within an extended range of subcooled degree on the superhydrophobic surface. The advantages of the present model for superhydrophobic under a large subcooled degree could be explained by the modified predictions of critical nucleation radius and nucleation density, as shown in Fig. 8. 6. Conclusions In this work, we have examined the dependence of thermal resistance of liquid-solid interface on wettability using the molecular dynamics simulation. The obtained quantitative expression of liquid-solid ITR is incorporated into existing heat transfer model of

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