A fractal dropwise condensation heat transfer model including the effects of contact angle and drop size distribution

International Journal of Heat and Mass Transfer 83 (2015) 259–272

Contents lists available at ScienceDirect

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

A fractal dropwise condensation heat transfer model including the effects of contact angle and drop size distribution Baojin Qi a,b,⇑, Jinjia Wei b, Li Zhang c, Hong Xu c a

School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China c School of Mechanical and Power Engineering, East China University of Science and Technology, Shanghai 200237, China b

a r t i c l e

i n f o

Article history: Received 18 September 2013 Received in revised form 10 November 2014 Accepted 25 November 2014

Keywords: Dropwise condensation Fractal Drop size distribution function Contact angle Heat transfer model

a b s t r a c t In this paper, a new dropwise condensation heat transfer model is developed on the basis of Rose model for a drop heat transfer and random fractal functions for the drop size distributions of direct condensation drops and coalescence drops. Compared to the well-established Rose’s model, the new model considers the contact angle and its hysteresis, the nucleation site density, the fractal dimension for drop sizes and maximum and minimum drop radii. Expressions of the distributions functions for both the direct condensing drops and the coalescence drops are derived and calculated by introducing fractal theory, respectively. The total heat flux of dropwise condensation on the entire condensing surface is calculated as the sum of the contributions from the condensations of above mentioned two kinds of drops. The simulation results indicate that the fractal dimension for drop sizes distribution decreases with increase in contact angle, resulting in smaller departure diameters and even shorter life cycle, which plays an active role in higher heat transfer coefficients. But the decline of fractal dimension leads to the decrease in the number density of droplets, which has negative effects on heat transfer process. Attributing to the combined influences mentioned above, the maximum heat flux value is obtained at h = 135°. The model proposed is applicable to predict the heat flux of the condensing drops with a wide range of contact angles, and the prediction results can agree well with different experimental dropwise condensation heat transfer data obtained from experiments and majority of literature results by choosing the proper contact angle parameters. The study can also reasonably explain the cause of the diversities of the dropwise condensation heat transfer data from different references. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Filmwise condensation and dropwise condensation heat transfer are two important heat transfer processes in many industrial applications such as in the power generation industry and chemical engineering. Dropwise condensation may occur when condensation takes place on a surface that is not wetted by the condensate. Surface heat transfer coefficients for dropwise condensation are much higher than for the filmwise mode that occurs when the surface is wetted. Dropwise condensation was first recognized by Schmidt et al. [1], and much interest was stimulated by their report that heat transfer coefficients were between 5 and 7 times higher than those found with film condensation. Over the years, much work has been performed on the mechanism of dropwise condensation. Considerable progress was made by Le ⇑ Corresponding author at: School of Chemical Engineering and Technology, Xi’an Jiaotong University, Xi’an 710049, China. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2014.11.083 0017-9310/Ó 2014 Elsevier Ltd. All rights reserved.

Fevre and Rose [2] to calculate the heat flux through drops of the given size, and their revised model has been widely accepted. In recent years, the outstanding research on multiple scales for drop formation process and atomistic modeling of dropwise condensation were reported by Sikarwar and Khandekar [3,4]. In their research, an atomistic viewpoint of the drop formation involving clusters of adatoms that eventually lead to thermodynamically stable nuclei was presented, and a population balance model that captures the role of major engineering process parameters was highlighted. Moreover, Three-dimensional numerical analysis of flow and heat transfer of an individual droplet including fields of velocity, pressure, temperature, and transport fluxes was also discussed by Sikarwar and Khandekar [5,6]. Kim [7] modeled heat transfer through a single droplet by adapting population balance model to develop a drop distribution function for the small drops that grow by direct condensation. However, compared with the heat transfer of single droplets, the size and spatial distribution of drops is not clearly reported, despite the theoretical significance

260

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

Nomenclature A A Ak Asl df dB F(r) f(r) Hfg h hi K M m Mg n Nco Ndc Nk Nk Nd Nrk Ns P Q q qtot R

area covered by drops (m2) total area covered by drops (m2) area covered by drops of generation k (m2) interface are of a drop (m2) fractal dimension box dimension distribution function of coalescence drops (cm2 mm1) distribution function of direct condensation drops (cm2 mm1) latent heat of vaporization (J kg1) heat transfer coefficient (J m2 K1) interfacial heat transfer coefficient (J m2 K1) coefficient molar mass (g mol1) grid scale (m1) gravity moment (N m) drop size distributions exponent cumulative number of coalescence drops (m2) cumulative number of direct condensation drops (m2) amount of drops of generation k (m2) total number of drops with rk 6 r 6 rmax (m2) number of boxes used to cover a certain geometric figure number of drops with radius of rk (m2) number of nucleation sites on the condensing surface (m2) the fraction of available area heat transfer rate (W) heat flux though a drop (W m2) total heat flux (W m2) Molar gas constant (J K1 mol1)

of calculating the average heat flux through the condensing surface. At the early stage of the study, Fatica and Katz [8] assumed that, within a given area, all drops have the same size, and they uniformly distribute and grow via condensation on the solid surface. Wenzel [9] suggested that drops grow in uniform square array and that coalescences occur between four neighboring drops to form a larger drop in a new uniform square array. Gose et al. [10], Rose and Glickman [11], and Tanasawa and Tachibana [12] conducted numerical simulations to partially model the drop growth and coalescence process. More recently, a new random fractal model of dropwise condensation was presented by Wu et al. [13] to investigate the shape and size distributions of drops, which has drawn broad attention. Then the fractal dimension was put forward by Sun et al. [14], based on the random fractal model. In this model, expression for the fractal dimension was derived, and the parameters influencing the fractal dimension were also analyzed. However, the fractal dimension expression they presented was related to the drop-size, which might be contradicted with self-similarity and scale invariance of fractal. Moreover, another fractal model for dropwise condensation heat transfer was developed by Mei et al. [15], based on the fractal characteristics of the drop size distribution on the condensing surfaces. In this model, the condensation heat transfer was found to be a function of the fractal dimension for drop sizes, maximum and minimum drop radii, the temperature difference, and physical properties of fluid. As research continues, Mei and Yu [16] constructed a lifecycle model of droplet growth in dropwise condensation. In the model, the stage in dynamic processes of

r rc rk rmax rmin DT

DTc DTd DTi Tsat t tco tdc ttot V W Wa Z

ac c d

gk h kl

m q r s

drop radius (m) critical radius (m) drop radius of generation k (m) maximum radius of drop (m) minimum radius of drop (m) total temperature difference/subcooling temperature (K) temperature difference due to interface curvature (K) temperature difference due to conduction through the drop (K) temperature drop due to interfacial resistance (K) temperature of saturated vapor (K) time period (s) time period for drop to coalesce from rc to rmax (s) time period for drop to grow from rmin to rc (s) drop growth cycle time (s) volume of drop (m3) total work (J) adhesion work (N m1) basal diameter of drop (m) condensation coefficient the radio of the radius between any two neighboring generations of drops length of a single box’s side (m) basal radius of drop of generation k (m) contact angle degree liquid thermal conductivity (W m1 K1) specific volume of vapor (m3 kg1) liquid density (kg m3) liquid–vapor interfacial tension (N m1) value of drop generations

nucleation, growth, renucleation and sweeping of droplets and the static stage of drop size distributions were studied respectively. Sikarwar and Khandekar [5,6] presented a holistic framework of modeling and studying droplet distribution at multiple scales, and developed computer simulation on droplet distribution from atomistic sizes (form stable clusters) to macroscopic sizes (including Growth by Direct Condensation and Growth by Coalescence). The simulation results reported in the literatures mentioned above were in good agreement with their experimental results respectively. However, significant difference between experimental results can be found presented by different researchers. This may due mainly to the discrepancy in the contact angle and surface energy of the testing surface employed in their experiments, causing the difference in both the condensation pattern and drop size distribution. The concept of liquid–solid surface free energy difference was introduced by Ma et al. [17] and Lan et al. [18] to elucidate the influence of the contact angle for individual droplet, but its effect on the drop size distribution was not accounted for. In the current study, considering the effect of the contact angle hysteresis, the random fractal models, proposed to describe the drop size distributions of direct condensation drops and coalescence drops on the condensing surface, are derived in two different ways, respectively. Based on these fractal models, a dropwise condensation heat transfer model is developed, and expressions for the fractal dimension, drop size distribution exponent and drop size distribution functions of the dropwise condensation are also derived. The predicted heat flux based on the model is compared to the available experimental data. Moreover, the relationships between the contact angle and distribution function, and the

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

contact angle and fractal dimension are analyzed, and the effects of the fractal dimension on distribution function are also studied in this paper. 2. Heat transfer model The analytical model presented in this paper is based on the following assumptions [13]: (1) The drop-size distribution over the entire condensing surface during the dropwise condensation process is steady with randomly located nucleation sites on the surface. (2) The vapor phase heat transfer is via the natural convection with the condensation being continuous at random locations. The heat transfer characteristics are same and the surface temperature is uniform. (3) The droplet is a section of a sphere. Compared with condensation heat transfer, the effects of convection heat transfer on the surface not covered by drops is much little and can even be ignored in heat transfer process. That is, there is no heat transfer from vapor to the surface not covered by drops but only through the base area of the drops. (4) The thermal-physical properties of the condensate are estimated at the average temperature of the vapor and condensing surface. The general procedures for calculating the dropwise condensation heat transfer is: first, calculate the heat flux from a condensing surface through a single drop of a given size; secondly, find the averaged heat transfer from the product of heat flux and the drop number density. 2.1. Heat transfer through a drop underneath a horizontal substrate Heat transfer to a single drop is affected by many factors. Consider a drop with radius r on a plain surface that is coated with a hydrophobic material, as shown in Fig. 1. The contact angle h on the coated surface is assumed to be fixed regardless of the drop size r and the vapor and surface temperatures. Knowing the vapor saturation temperature Tsat and the plate surface temperature Tsurf, the heat transfer rate Q can be derived by considering all resistances. Four main resistances are taken into consideration in this paper: the resistance of surface curvature, Rc, the interfacial resistance between the liquid and vapor phases, Ri, the heat conduction resistance through the drop Rd, and the resistance of the coating material on the contact surface Rcoat. For the purposes of this

Fig. 1. A droplet mode on the condensing surface.

261

model, all resistances will be presented as the temperature drop for the region in question. The difference between the equilibrium temperatures of saturated vapor at a planar interface and at a curved interface is given by [19]

DT c ¼

2T sat r Hfg r q

ð1Þ

This resistance includes the loss of driving temperature potential due to the droplet interface curvature. Vapor–liquid interfacial resistance for a hemispherical drop may be expressed as a temperature difference as shown below [20,21]

DT i ¼

Q 2pr 2 ð1  cos hÞhi

ð2Þ

where the interfacial heat transfer coefficient can be obtained from

hi ¼

 1=2 2 Hfg 2ac M 2  ac 2pRT sat T sat m

ð3Þ

The droplet itself acts as the resistance to heat conduction through the droplet. In order to analyze heat conduction, a method of integration of the temperature difference between two neighboring isothermal surfaces is introduced. The heat is thought to conduct from one isothermal surface having an angle of / (see Fig. 2) to another isothermal surface of angle / + d/. The surface area of the lower isothermal surface As is obtained by integration with respect to the angle c [7]

dAs ¼ 2p

As ¼



2 d cos cdc 2 sin /

 2 d pd2 ð1  cos /Þ 2p cos cdc ¼ 2 2 sin / 2 sin / p=2/

Z p=2

ð4Þ

ð5Þ

The distance de between the two isothermal surfaces is the largest at c = 90°,

d f½cscð/ þ D/Þ  csc /  ½cotð/ þ D/Þ  cot /g 2 d ¼ ðcsc2 /  cot / csc /Þd/ 2

demax ¼

ð6Þ

Fig. 2. Heat transfer by conduction between two neighboring isothermal surfaces.

262

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

The mean distance is assumed to be half of the maximum distance demax,

 d 2 csc /  cot / csc / d/ 4

demax ¼

ð7Þ

The temperature difference DTd drop between the drop surface and the coating surface is calculated by integration with respect to the angle /,

Z

h

Q Q de ¼ kl As 2pdkl Qh ¼ 2pdkl sin h

DT d ¼

0

Z 0

h

2

2

ðcsc /  cot / csc /Þ sin / d/ ð1  cos /Þ ð8Þ

The temperature drop due to the resistance of the coating material on the contact surface is given by [3]

DT coat ¼

Qd 2

kcoat pr 2 sin h

ð9Þ

The total temperature difference between the vapor and the substrate surface is

DT ¼ DT c þ DT i þ DT d þ DT coat

rc ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1=4NS

ð11Þ

The influence of the contact angle is considered in Eq. (11), which gives a better approximation to the drop heat transfer rate, especially with small drop sizes. Therefore, it will be used to calculate the dropwise condensation heat flux in this work. 2.2. The fractal characteristics of drop size distributions on condensing surfaces

ð13Þ

The drop size distribution function of direct condensation drops, f(r), can be expressed as the number of drops whose radius range from r  dr to r + dr, and can be described by the following form

f ðrÞ ¼ 

dN dc ðrÞ df r c df þ1 ¼  dr rc r

ð14Þ

where df is fractal dimension. The value of df is calculated by the box-counting method applied to the drop size distribution

lg Nd ðAÞ  lg d

df ¼ dB ¼ lim d!0

ð15Þ

where d is the length of a single box’s side, and Nd is the number of boxes used to cover a certain geometric figure. For condensing drops, Eq. (15) can be written as

ð10Þ

combining Eqs. (1)–(10) we obtain the heat transfer rate through a single drop

  2T sat r Q ¼ pr 2 DT  Hfg r q  1 1 þ cos h d rh  þ þ 2 2hi kcoat sin h 4kl sin h

minimum radius to those with the critical radius. Abu-Orabi [23] assumed that the drop radius was equal to half the mean spacing between the active nucleation sites on the condensing surface. Assuming that the nucleation sites form a square array [10], the critical radius can be calculated as

df ¼ lim

r k !rmin

lg Nrk ðAÞ  lgð2  r k Þ

ð16Þ

where N rk is the number of drops with rk used to cover the area of A. The minimum drop radius is taken to be that of the smallest viable drop [19]

rmin ¼

2T sat r qHfg DT

ð17Þ

Fractal dimension, as stated in literatures, is independent of the drop size. The direct condensation drop size distribution function and the coalescence drop size distribution function are actually proposed to describe the different stages of a continuous process of condensing life cycle. Therefore, the fractal dimensions of the two random fractal functions can be thought equal to each other. (2) The drop size distribution function of coalescence drops

It is well known that the description of the drop size and spatial distribution is the key to simulating dropwise condensation heat transfer. Recent studies indicate that the drop size distributions are self-similar and follow the fractal scaling law during dropwise condensation. That is to say, the photographs of dropwise condensation taken at different instant or in different scale are similar, and a whole photograph can be obtained by enlarging properly a local photograph. In addition, dropwise condensation is a kind of transient growing process, and drop spatial distribution possesses randomly. Therefore, a random fractal model is presented here to describe drop size and spatial distribution. (1) The drop size distribution function of direct condensation drops

The total area covered by drops per unit area, having radius in the interval (r, rmax) can be written as



AðrÞ ¼ 1 

r

r max

n

with r c 6 r 6 r max

ð18Þ

If N co ðrÞ is defined as the cumulative number of the coalescence drops which have radius larger than r, then Nco ðrÞ is the total number of coalescence drops cover the area of A(r). Thus, the distribution function of coalescence drops, F(r), larger than r, can be described by

FðrÞ ¼ 

 n3 dNco ðrÞ 1 dA n r ¼ 2 ¼ 3 dr pr dr prmax rmax

ð19Þ

(3) The solution process of the drop size distribution functions A fractal characteristic of drop size distribution has been shown in literatures [13,14]. We can apply the fractal geometry theory to the model of dropwise condensation. It has been shown that the cumulative number of the direct condensation drops with the radius equal to or larger than a particular value, r, follows the fractal scaling law as [22]

Ndc ðrÞ ¼

r df c

r

with r min 6 r 6 r c

ð12Þ

where rc is the critical radius. Droplets with the radii smaller than rc are termed as small droplets, and they grow mainly by direct condensation; droplets with the radius larger than rc are termed as large droplets, and they grow mainly by coalescence. The equation above also gives the total number of drops from the drops with the

In order to obtain drop size distribution functions, f(r) and F(r), in Eqs. (16) and (19), it should analyze and solve three important parameters: the value of the fractal dimension, df, the distribution function parameter, n, and the maximum radius of drops, rmax. The drop size distribution based on the random fractal model presented by Wu [13] is employed in this paper. Based on the random fractal model, replacing small squares with inscribed circles and letting Z1 = 2g1 = 2rmax sin h = 1/m, Zk = 2gk = 2rk sin h (k P 2), the area covered by the drops for generation k can be written as

N k Ak ¼

! k1 X 1 N i Ai P i¼1

ð20Þ

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

where Nk and Ak are amount and area of condensing drops of generation k, respectively. From Eq. (20), we can obtain

Nk Ak ¼ ð1  PÞNk1 Ak1

ð21Þ

and the area covered by the drops of generation k is then given by

Nk Ak ¼ N1 A1 ð1  PÞk1 ¼ Pð1  PÞk1

ð22Þ

then, the total area covered by the drops for all generations can be calculated as k X A¼ Ni Ai ¼ 1  ð1  PÞk

ð23Þ

i¼1

According to the model’s property, the side length of any generation of drops to the next generation, c, can be expressed as

c ¼ rk1 =rk

ð24Þ

and the drop radius of generation k can be written as

r k ¼ cðk1Þ  r max

ð25Þ

the value of generations from the critical radius to maximum radius can be obtained as

s¼

lgðr max =r c Þ þ1 lg c

ð26Þ

the amount of droplets covering the area A with radius rk can be presented as

Nrk ðAÞ ¼

A ð2r k Þ2

¼

1  ð1  pÞk

ð27Þ

n h io 2ðk1Þ lg r 2  1  ð1  PÞk max  c  lgð2  cðk1Þ  r max Þ

k!s

lg½1  ð1  PÞk  k!s ðk  1Þ lg c  lgð2r max Þ

¼ 2 þ lim

rmax

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1=2 2 6 sin h r ¼ 2  3 cos h þ cos3 h ðql  qv Þg

ð28Þ

Plug df into Eq. (14), the drop size distribution function of direct condensation drops can be obtained.

ð29-iÞ

The maximum radius of drops on a vertical cold substrate, on the basis of dimensional analysis, is taken as [25]

rmax ¼ K



r ðql  qv Þg

1=2 ð29-iiÞ

Inclining the substrate causes an imbalance in the forces and results in drop deformation, to achieve necessary static balance. The critical radius of a deformed pendant droplet on a vertical substrate is shown in Fig. 3. Accordingly, the retention force arising from contact angle hysteresis, the difference in the advancing angle and receding angle, is equal to the component of weight parallel to the substrate. Thermodynamically, when the adhesion work of retention force is equal to or less than the total gravity moment the droplet began to slide-off on the substrate. The contact angle hysteresis, namely, the variation in the advancing to receding contact angle, is taken to vary linearly along the contact line with respect to azimuthal angle. The base of the droplet is taken to be circular as discussed earlier. The variation of contact angle, with respect to azimuthal angle along the drop contact line is formulated as [3]

  cos hrcd  cos hadv n

2

substituting Eqs. (25) and (27) into Eq. (16) we obtain the fractal dimension of the coalescence drop size distribution which is equal to that of the direct condensation drop size distribution

df ¼ lim

The maximum drop diameter underneath a horizontal substrate is calculated from balancing the surface tension with the weight of the drop and is derived as [24]

cos h ¼ cos hadv þ

½2c  r max  1 2 2ðk1Þ ¼ r max  c  ½1  ð1  PÞk  4 ðk1Þ

263

p

ð30Þ

Inclining the substrate causes an imbalance in the forces and results in drop deformation, to achieve necessary static balance, as shown in Fig. 3. For the convenience of calculation, suppose that placing a drop of liquid on a vertical plate leads to the formation of a segment of a sphere macroscopically, characterized by the average contact angle havg. The leading side contact angle havg is equal to the average of advancing angle and the receding angle of the liquid substrate combination (that is, havg = (hrcd ± hadv)/2) to minimize the errors caused by shape changing of droplets, as shown in Fig. 3. This defines surfaces the areas of which are functions of the value of the contact angle havg and the curvature radius of the liquid–vapor interface, r. The area of the plane surface corresponding to the interface on vertical plate is

Fig. 3. Force analysis of a drop on a vertical surface.

264

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

 2

Asl ¼ pr 1  cos2 hav g



ð31Þ

Adhesion work can be written as

Wa ¼

Z p

Dgk ¼ gk  gk1 ¼ ð1  cÞgk

rð1 þ cos hÞ cos ndn

ð32Þ

0

Total work required to overcome for departing drops can be given by

W ¼ W a Asl ¼

Z p 0

the following expressions, which relate the size and the amount of drops on condensing surface

pr2 rð1 þ cos hÞð1  cos2 hav g Þ cos ndn

ð33Þ

denoting

Nk ¼

k X Ni

 W ¼ 2pr2 r 1  cos2 hav g

then

   cos hrcd  cos hadv

ð34Þ

p

h

Nk ¼

N1 pg21 ð1  PÞk1

pg2k

ðql  qv Þg pr sin aðcos a  cos hÞ cos ndadn Z p   1 ¼ ðql  qv Þg pr 4 3  8 cos h þ 6 cos2 h  cos4 h cos ndn 12 0 " 1 4 ðq  qv Þg pr ð8 cos3 hadv  24 cos hadv þ 16Þ ¼ 12 l

Mg ¼

0

0



cos hrcd  cos hadv

p

2

þ ð12 cos2 hadv  12Þ 

þð12p cos hadv  24Þ  þ4

ðcos hrcd  cos hadv Þ

p

dNk DN k ¼ lim dgk Dgk !0 Dgk

ðcos hrcd  cos hadv Þ4

ð35Þ

p

Pð1  PÞk1

ð42Þ

pg2k

! ð43Þ

Eq. (25) and Eqs. (39)–(43) can be combined and rearranged to give the following equation as

dNk lg g dgk 3 1

p2

When the adhesion work equals to the total gravity moment, the departure radius was obtained as

r max



ðcos hrcd  cos hadv Þ3 #

¼

in addition

i

3

4

ð41Þ

From Eq. (22), the amount of drops of generation k can be obtained as

Total gravity moment of the drop can be obtained as h

ð40Þ

i¼1

DNk ¼ Nk  Nk1 ¼ Nk

Substituting Eq. (30) and havg = (hrcd ± hadv)/2 into Eq. (33) and integrating, one obtains

Z pZ

ð39Þ

!

" ¼ lg ðr max sin hÞ

3

Pð1  PÞk1

#

3

pðrk sin hÞ   lgð1  PÞ rk P lg þ lg ¼ 3þ lg c r max pðc  1Þ

ð44Þ

Comparing Eqs. (38) and (44), distribution function exponent of coalescence drops can be given as

n¼

lgð1  PÞ lg c

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi v u r 6  ð1  cos2 hav g Þ u" # ¼ u ðql  qv Þg u ð2 cos3 hadv  6 cos hadv þ 4Þ  þð3 cos2 hadv  3Þ  ðcos hrcd  cos hadv Þ t hadv Þ2 þ ðcos hrcd  cos hadv Þ3 þð3p cos hadv  8Þ  ðcos hrcd cos p

ð45Þ

ð36Þ

Comparing Eqs. (29-ii) and (36), we find that the coefficient K can be written as

vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u 6  ð1  cos2 hav g Þ u # K ¼ u" u ð2 cos3 hadv  6 cos hadv þ 4Þ  þð3 cos2 hadv  3Þ  ðcos hrcd  cos hadv Þ t

ð37Þ

2

hadv Þ þ ðcos hrcd  cos hadv Þ3 þð3p cos hadv  8Þ  ðcos hrcd cos p

The dimensionless size distribution of drops can be obtained from Eq. (19) as

Plug rmax, and n into Eq. (19), the drop size distribution function of coalescence drops, F(r), can be obtained.

lg r 3max

2.3. The total heat transfer through a horizontal condensing surface

"

#   n dNco ðrÞ r ¼ ð3  nÞ lg þ lg dr r max p

ð38Þ

According to the characteristics of the fractal model and the construction method and process of drops, Wu et al. [13] derived

The total heat transfer rate per unit area of dropwise condensation underneath a horizontal surface is calculated by integrating

265

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

the heat transfer rates of direct condensation droplets and the coalescence drops from the minimum to the maximum of drop radiuses.

qtot ¼

Z

rc

Q  f ðrÞ  dr þ

r min

Z

rmax

Q  FðrÞ  dr

ð46Þ

rc

the maximum drop diameter can be calculated from Eq. (29-i). 2.4. The total heat transfer on the vertical cold surface condensing surface Eq. (46) provides the heat flux for dropwise condensation underneath a horizontal surface, but excludes the effect of sweeping of the surface by the departing droplets. On an Inclined Substrate, the distribution of droplets will be affected by the drop sliding motion, and the heat transfer therefore need to recalculated base on fractal theory by considering the effect of the sweeping.

The drop growth rate can be found from the equality of the rate of increase of the droplet mass and the rate of condensation on its surface

q

2

dV pr sin hav g ¼ q dt Hfg

ð47Þ

A geometrical expression is available that gives the drop volume as a function of the drop radius r and the contact angle havg, as

V¼

pr3 h 3

2

2ð1  cos hav g Þ  sin hav g cos hav g

i

ð48Þ

Substituting Eq. (48) into Eq. (47) and taking the derivative

h dt ¼ qHfg

i 2 2ð1  cos hav g Þ  sin hav g cos hav g dr 2 q sin hav g

ð49Þ

Integrating Eq. (49) from the minimum radius, rmin, to a peculiar radius, r, and the time for growth can be given as

h t dc ¼ qHfg

i 2 2ð1  cos hav g Þ  sin hav g cos hav g Z 2

sin hav g

Ndep ðyÞ ¼

f s ðyÞ ¼ Ddep w=Ddep

Z 0

y

f d ðeÞde

ð52Þ

Since any droplet remaining in the strip above y sweeps the middle point, the average sweeping period is given by.

(1) Growth period of droplets

2

where fs (y) is the number of departing drops crossing elevation y, per unit time, for a condenser surface of width w. It is necessary to determine the frequency at which a point on the condenser surface is swept, since this frequency affects the rate of heat transfer at that point. The time interval between two successive sweepings at a particular point on the condenser surface can be defined as the sweeping period. The condenser surface is divided into vertical strips of width Ddep, which is taken as the departure diameter of a drop. Departing droplets above line y are randomly distributed over the strip such that on the average each strip has the same number of departing droplets, because of the random nature of dropwise condensation. The rate at which the number of departing droplets is generated within a strip above line y is then [26]:

r

r min

dr q

ð50Þ

When r = rc, the above equation calculates the entire growing period of an individual droplet for the direct condensation droplets. (2) The effect of sweeping motion

pðyÞ ¼

Ddep

1 Ry f ðeÞde 0 d

ð53Þ

In equations above, the time average sweeping frequency, fd (y), is obtained as an integral equation.

" # pD2dep Z y qtot ðgÞ f d ðyÞ ¼ 1 f d ðeÞde  ½1  ðt dc þ t co ÞDdep md hfg 4v 0 Z y  f d ðeÞde

where md is the mass of a departing droplet, v is the velocity of the falling droplets. The time period that elapse for the droplets to coalesce from critical radius, rc, to maximum radius, rmax, called tco can be found from the instantaneous mass balance of the condensate forming on the swept region and from the growth rate of the entire drop population, over the period from the onset of coalescence up to the time where the maximum drop size attains the departure size [26].

h iZ 2 tco ¼ 2ð1  cos hav g Þ  sin hav g cos hav g



 ðnþ1Þ pnqHfg 1 þ nrnþ1 rmax dr max c

g

rc

ð55Þ

3qtot

Therefore, the time elapsed between the sweeping and the next appearance of a departing droplet can be defined as the total growth period, given as.

ttot ¼ tdc þ t co The sweeping of the surface is affected by the height of the condenser surface; no sweeping takes place at the top of the condenser surface, while the sweeping rate can be expected to increase at the lower parts of the surface, since more and more departing droplets join the falling droplets. Therefore, the frequency of sweeping at any height can be calculated by summing the number of departing droplets generated at any point on the surface over the height of the condenser surface to that point. A new parameter fd (y) was introduced by Yamali [26], which can be described as the number of drops attaining the departure size, per unit time, per unit area, at a height y. The total number of departing droplets that cross a horizontal line at an altitude of y per unit time can be obtained by integrating the local number over the condenser area above this line [26].

f s ðyÞ ¼ w

Z 0

y

f d ðeÞde

ð51Þ

ð54Þ

0

ð56Þ

The total heat flux for dropwise condensation on the vertical cold surface is the sum of the sequential contributions from the drops growing by direct condensation and from coalescing drops.

qtot ¼

1 ttot

"Z

rc

r min

Q  tdc  f ðrÞ  dr þ

Z

#

r max

Q  tco  FðrÞ  dr

ð57Þ

rc

The contact angle h, used to calculate the heat transfer through a drop in Eq. (11), is replaced by average contact angle havg to limit effects of the change in geometric shape. Moreover, the maximum radius of drops on a vertical cold substrate should be obtained from Eq. (29-ii). The average heat transfer coefficient of the condensing surface can be calculated as.

h¼

qtot DT

ð58Þ

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

-4

10

r = 0.1

(a)

m

2

Resistance / Km /W

266

-5

10

Rd

RTot

Rcoat

-6

10

-7

10

-8

10

0.8

Rc

Ri βT =

Δ TSubscript

r = 0.1

ΔT

0.6

m

ΔTd ΔT

β

T

0.4 ΔTcoat ΔT

0.2 ΔTi ΔT

0.0 70

80

90

100

110

120

-4

10

r = 10

130

140

150

160

degree

(b)

m

RTot

2

Resistance / Km /W

Contact angle

ΔTc ΔT

-5

10

-6

10

Rcoat

Rd

-7

10

Ri

-8

10

0.8

βT =

β

T

0.6

Rc

ΔTSubscript ΔT

r = 10

m ΔTd ΔT

ΔTcoat ΔT

0.4

ΔTi ΔT

0.2

ΔTc ΔT

0.0 70

80

90

100

110

120

-4

10

130

140

150

160

degree

(c)

r = 1mm

2

Resistance / Km /W

Contact angle

-5

10

-6

10

Ri -8

10

βT =

0.6 T

Rc

-7

10

0.8

β

RTot

Rcoat

Rd

ΔTSubscript ΔT

r = 1mm ΔTd ΔT

ΔTcoat ΔT

0.4 ΔTi ΔT

0.2

ΔTc ΔT

0.0 70

80

90

100

110

Contact angle

120

130

140

150

160

degree

Fig. 4. Effect of contact angles on temperature differences and thermal resistances for single droplets with different sizes; (a) r = 0.1 lm, (b) r = 10 lm, and (c) r = 1 mm.

3. Analysis of influencing factors of heat transfer model 3.1. The effect of contact angle on a single drop and the drop size distribution functions The effect of the static contact angle on heat transfer model is mainly reflected in two aspects: the thermal resistance for a single drop and the drop size distribution functions.

Fig. 5. Distribution functions at different contact angles.

Fig. 4(a)–(c) show the temperature differences ratio bT (where bT = DTsubscript/DT) and thermal resistances of single droplets with sizes of r = 0.1 lm, r = 10 lm, and r = 1 mm on a condensing surface and contact angle ranges between 65° and 160°. For r = 0.1 lm drop, all thermal resistances illustrated are basically in the same order of magnitude. However, the temperature drops for r = 10 lm, and r = 1 mm sizes, the thermal resistance of conduction in condensate drops become dominant, which is in accordance with the result reported by Lee [27]. As shown in Fig. 4(a)–(c), the temperature differences due to the thermal conduction resistance in the condensate (DTd) generally increases and becomes dominant, as the contact angle and the drop size increase. It can be considered that the large size condensate drops with higher contact angles make lower thermal resistances at the liquid–vapor interfaces due to larger interfacial areas, however these large drops have a higher conduction thermal resistance in the condensate, because the conduction thermal resistance dramatically increases as the average thickness of the condensate drop which has very low thermal conductivity, increases. Therefore, it is natural that the conduction thermal resistance of a single condensate drop becomes dominant as the size of a condensate drop increases. In addition, for r = 0.1 lm, and r = 10 lm droplets, the temperature difference due to the thermal conduction (DTd) is below the temperature differences due to the droplet curvature (DTc) and interfacial heat transfer (DTi) first and then exceeding with increasing h. While for r = 1 mm, DTd is significantly higher than other temperature differences, and gaps is become more and more visible with increase in contact angle h. In this study, assume that the droplets larger than 102 lm can grow up from direct condensation stably, and the tiny droplets smaller than 102 lm may disappear due to the evaporation or diffusion process. The distribution functions, f(r) and F(r), of drop size larger than 102 lm at the same subcooling temperature (i.e. temperature difference DT between the vapor and the substrate surface) for various contact angles are illustrated in Fig. 5, along with experimental data from Tanasawa and Ochiai [28]. The values of f(r) and F(r) are evaluated from Eqs. (14) and (19) at DT = 1.0 K, c = 1.3, and P = 0.1063 [29]. The consistency between the fractal theory and the experimental data shows that the proposed fractal characteristic of drop size distribution is reasonable. As shown in Fig. 5, the number density of drop decreases with increase in contact angle and droplet radius obviously. When the droplet radius reaches the critical radius, rc = 1.18 lm, the distribution functions change suddenly, which shows that the distribution function for direct condensation drops decreases faster than that

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

Hydrophilic

Hydrophobic

Fractal dimension df

2.0

Superhydrophobic

1.9 1.8 1.7

T = 0.1K T = 1.0K T = 5.0K T = 10.0K

1.6 1.5

γ = 1.3; P = 0.1063

0

20

40

60

80

100 120 140 160 180

contact angle

degree

Fig. 6. The relationship between the contact angle and the fractal dimension under various temperature differences.

for coalescence drops. Moreover, the change rate of droplet number density presents the zonal distribution for contact angles from 75° to 150°, which can cover the most of experimental data in literatures. Besides, the value of f (r)/F (r) at h = 75° is about 3 orders of magnitude higher than that at h = 150° for various radii. This means that the effect the contact angle cannot be neglected in determination of the distribution function in dropwise condensation heat transfer. 3.2. The effect of contact angle on fractal dimension for direct condensation drops The relationship between the contact angle and fractal dimension is demonstrated in Fig. 6. The fractal dimension decreases with increasing in contact angle under various temperature differences, and the decline range increases gradually with the increase of contact angle. The theoretical curves of present model show that the contact angle has little impact on fractal dimension on hydrophilic surface. Taking DT = 1.0 K as an example, the value of the fractal dimension reduces from 2 to 1.96 as the contact angle increases from 0° to 90°, only a 0.022% decline per degree. Similarly, the contact angle shows minor effect on fractal dimension

267

on hydrophobic surface, and the declining rate increases slightly, to 0.067%, when the contact angle increases from 90° to 150°. On the contrary, the fractal dimension decreases sharply with increasing h when the contact angle is larger than 150° (super-hydrophobic surface). The declining rate is approximately 1.55% per degree when the contact angle increases from 150° to 170°, and a greater rate will appear with further increasing contact angle. For the same contact angle, the fractal dimension increases drastically with temperature difference when DT < 5.0 K, and then increases slowly with temperature difference. This is mainly due to the fact that the condensing surface is rapidly occupied by drops under small temperature difference, and the area fraction of drops increases drastically with temperature difference. In addition, when the contact angle is reduced to 0°, complete filmwise condensation will occur on the condensing surface, and the fractal dimension equals to 2, which is in good agreement with universal understanding. While the contact angle approaches 180°, droplets will depart from the condensing surface in no time after nucleation and the fractal dimension approximately equals to 1, as shown in Fig. 6. 3.3. The influence of drop size distribution function on fractal dimension for direct condensation drops Fig. 7 shows the distribution function versus the fractal dimension for direct condensation droplets, f(r), for r = 106 m, r = 107 m, and r = 108 m at DT = 1.0 K, c = 1.3, and P = 0.1063, respectively. As shown in this figure, the fractal dimension is affected obviously by the value of distribution function when df < 1.96, and when df > 1.96 the value of f (r) has minor effect on the fractal dimension. That is, the value of df increases sharply at low number density of drops, and for large number density, the reverse is true. For a certain radius droplets, the larger of the number of drops on condensing surface, the greater the area fraction would be expected, and so does the fractal dimension. Moreover, for the same f(r), the fractal dimension increases with the increase in drop sizes. This is because that the area fraction is also getting greater with the growing sizes of droplets at a given number density. In addition, when the number density of drops approach to the infinity, the value of df reaches 2, the condensing surface is completely covered by tiny droplets adhering to each other. That is to say, dropwise condensation has been transformed into filmwise condensation on condensing surface.

Fractal dimension df

2.0 1.9 1.8 1.7 -8

r = 10 m ΔT = 1.0K -7 r = 10 m γ =1.3 -6 r = 10 m P = 0.1063

1.6 1.5

1

10

3

10

5

10

7

10

9

10

-2

11

10

13

10

15

10

17

10

-1

f (r) / drops·cm ·mm

Fig. 7. The distribution function for direct condensation drops versus fractal dimension for different drop size.

Fig. 8. Distribution functions of condensate drops with respect to sizes for various surface densities.

268

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

6 4 7

9

2

5

3 12 11

10

1

8

15

14

13

Fig. 9. Schematic diagram of the experimental apparatus. 1 – Deionized water tank; 2,3 – Steam generator; 4 – Superheater; 5 – Test section; 6 – Heat transfer plate; 7 – Glass window; 8 – Diaphragm pump; 9,10 – Rotameter; 11 – Auxiliary condenser; 12,13 – Measuring tube; 14 – Pump; 15 – Cooling-water tank. d Temperature measuring position; water line; - - - - - Steam line.

3.4. The influence of the nucleation site density for the drop size distribution functions In this paper, assuming the dropwise condensation as a nucleation phenomenon, and the number of nucleate sites of drops Ns is obtained from Rose [19] in a range of 4.80  1012 m2–2.40  1014 m2. For the purpose of comparison, Fig. 8 shows the same distribution functions, both F(r) and f(r), as reported in the literature [27], based on the nucleation site density around 4.80  1012 m2, 5.0  1013 m2, and 2.40  1014 m2 with a surface subcooling of 5 K, respectively. As shown in the figure, the distribution functions decrease with the number of nucleation site of drops decreasing. Although the distribution functions of small droplets (r 6 1 lm) are significantly affect by the nucleation site density, the number of nucleation site of drops would have little or no effect on the distribution functions of large drops(r P 1 lm). By comparing, one can find that the calculation results of Ns = 2.40  1014 m2 and 5.0  1013 m2 are in agreement with the literature [27] for equivalent parameters. When Ns = 4.80  1012 m2, the level of consistency between the present model and the data from Lee [27] become worse, especially if the droplets tend to in small size.

4. Experimental study on dropwise condensation heat transfer 4.1. Experimental apparatus The experimental apparatus, shown schematically in Fig. 9, consists of four parts: the steam generating system, cooling system, steam condensing system, and data acquisition/control system.

Steam at about 100 °C was supplied from a steam generator, and then passed through a superheater to remove mist. The dry steam was led into the condensing chamber and condensed on the surface of the test condensing plate. The steam velocity in the duct was obtained by dividing total flow rate of steam through the duct by the cross-sectional area and was maintained at approximately 5 ± 0.5 m/s. The steam temperature was measured by a platinum resistance thermometer (Pt 100), with an accuracy of ±0.1 °C. The steam pressure was monitored by a manometer, and maintained at atmospheric pressure by adjusting the valve installed at the outlet of the condensing chamber. The condensing chamber is shown in Fig. 10. This condensing chamber can freely adjust angle (from horizontal to vertical) by rotating device to satisfy experiment requirement. A baffle was set at top of the condensing plate to eliminate the sweep effect of steam flow to the condensation droplets. The condensate on the test surface was collected by a funnel and then flowed into a measuring tube. A window was mounted at the chamber for visual observation of the condensation process. The chamber was insulated by rock wool with thickness of 50 mm to diminish the condensation that occurred at other place. By doing this, the influence of the unwanted condensation was very limited and can be neglected in the experiments. The excessive steam and the condensate condensed on the other surfaces of the condensing chamber flowed into an auxiliary condenser and were collected by another measuring tube. During the warm-up period, the steam rapidly flowed through the condensing chamber for 60 min to prevent accumulation of non-condensing gases. Water was used to cool the test surface. It was kept at a constant temperature in a cooling-water tank and was sprayed on the backside of the test condensing plate by means of a pump

269

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272 Table 1 Contact angles of water for all the test surfaces.

Fig. 10. Photograph of the condensing chamber.

Specimen

Contact angle h/°

Advancing angle hadv/°

Receding angle hrcd/°

Average contact angle havg/°

1# 2# 3#

159.3 128.6 100.4

161.6 132.2 113.5

152.1 122.7 75.2

156.85 127.45 94.35

As shown in Fig. 11, two holes of 1 mm in diameter were drilled at the condensing plate. Their depths were 15 mm and 25 mm, respectively. Two sheathed copper-constantan thermocouples (type T, Omega Engineering Inc.) were inserted into the two holes to measure the wall temperature. The uncertainty of temperature measurement was estimated as ±0.1 K. Experimental data are collected using Agilent 34970A data acquisition unit, and all the data including the pressure, thermocouple readings, surface subcooling temperature, heat flux, condensation heat transfer coefficients can be calculated by date reduction software, and the real-time data profiles can be displayed on computer monitor. 4.3. Experimental data reduction The heat transfer coefficient h is defined as.

h¼

q q ¼ DT ðT s  T w Þ

ð59Þ

In order to calculate the heat transfer coefficient with Eq. (59), it is necessary to obtain the heat flux q and surface temperature Tw. In the current research, the heat produced by condensation on the condensing surface was transferred to the cooling water through the heat transfer plate.The heat transfer rate calculated from the amount of the condensate per unit time can be expressed as.

Q1 ¼ Fig. 11. Schematic diagram of the test heat transfer plate.

M  hfg t

ð60Þ

and the heat transfer rate calculated from the flow rate and temperature increment of the cooling water was obtained from the following equation.

and a nozzle. Heat flux through the heat transfer plate was regulated continuously by adjusting the pressure and flux of cooling water. The inlet and outlet water temperature were also measured by two platinum resistance thermometers (Pt 100). The uncertainty of temperature measurement was estimated as ±0.1 °C.

The difference between Q1 and Q2 was less than 5%, and therefore it is reasonable to calculate the heat flux using the following equation.

4.2. Heat transfer plate

q¼

Details of the test heat transfer plate are shown in Fig. 11. Three 3-mm-thick square condensing plates, with a test area of 50 mm  50 mm are made of commercially pure titanium, TA2. These test surfaces were polished, resulting in a mean surface roughness of approximately 0.15 lm. All titanium plates are chemical etched with 2 M hydrofluoric acid solution for 5 min at 25 °C, then immersing the titanium plates in a 25% hydrogen peroxide solution at 80 °C for 72 h, an oxide layer is formed on each titanium surface. The oxide layer formed on the substrate surface tends to increase the bonding between the substrate and the monolayers which eventually improves the longevity of the coating. After pre-oxidizing, one plate (1#) is immersed in a 2.5 mM solution of n-octadecyl mercaptan in ethanol for 10 h, and another plate (2#) is placed in a 2.5 mM solution of stearic acid in hexane for 6 h. By comparison, there is no any further treatment for the third plate (3#) after pre-oxidizing. Then, the two plates are washed with absolute ethanol and left to air dry. The contact angles (including advancing and receding contact angle) of water to these heat transfer plates were measured using a DSA-100 (KRUSS Instruments Ltd., Germany) type contact angle measurement apparatus with an accuracy of ±0.5°. The test results are listed in Table 1.

The temperature of the condensing surface Tw was calculated from Eq. (63) by using the heat flux q obtained in Eq. (62).

Q 2 ¼ G  C pl  ðT out  T in Þ

Q1 þ Q2 2A

Tw ¼ Ti þ

q  Dl k

ð61Þ

ð62Þ

ð63Þ

where Dl is the distance between the condensing surface and the measuring points of the sheathed thermocouples inserted into the heat transfer plate. 5. Simulation results and comparison with experimental data The relationship between heat flux and contact angle at various surface subcoolings are plotted in Fig. 12 in the range of 55° < h < 170°. While the subcooling temperature remained unchanged over this period, the heat flux increase firstly, and then decrease with the increment of contact angle, and the maximum heat flux value is obtained at about h = 135°. On one hand, larger contact angles contribute to smaller departure diameters and even shorter life cycle, which plays an active role in higher heat transfer coefficients. On the other hand, larger contact angles may also result in the decrease in the number density of droplets, which

270

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272 4

×10

Heat flux qtot / W m

-2

25 20

×104

20

Horizontal Surface T = 1K T = 3K T = 5K T = 10K

(a) Horizontal Surfaces

18

γ = 1.3 P = 0.1063 NS = 5.0×109cm-2

16

Heat flux qtot / W·m-2

30

15 10

14 12 10 8 6

Contact angle, θ

60

80

100

contact angle

120

140

160

0

/ degree

Fig. 12. The relationship between heat flux and contact angle at various surface subcoolings.

1

2

3

4

5

6

7

Subcooled temperature T / K

8

×104 80

(b) Vertical Surfaces 1#

Specimen

Heat flux qtot / W·m-2

70

2#

3#

Average Contact 156.85° 127.45° 94.35° angle, θavg Present model

60

Experimental data

50 40 30 20

γ = 1.3 P = 0.1063 NS = 5.0×109cm-2

10 0

0

1

2

3

4

5

6

7

Subcooled temperature T / K

8

Fig. 13. Comparison of heat flux between the present model and the experimental data (a) underneath horizontal surfaces, (b) on vertical surfaces.

×105 12 Present model ᧹75e ©᧹170e 10 ©᧹135e NS =5.0×109cm-2 8 P = 0.1063

Heat flux qtot / W·m-2

have negative effects on heat transfer process. The maximum value of heat flux increases with increasing subcooling temperature. Furthermore, the contact angle to reach such maximum value remains constant for various subcooling temperatures. In addition, under greater subcooling temperature, contact angle effect on heat flux and heat transfer coefficient will be more evident in the present model. Consequently, the contact angle plays an important role in the condensation heat transfer process, and thus is used in present fractal model for an improved prediction. The comparisons of heat transfer between the experimental data and the simulation results, both underneath horizontal surfaces and on vertical surfaces, are shown in Fig. 13. Firstly, the experimental data for horizontal surfaces are in good agreement with the simulation results obtained from Eq. (46), with the largest difference less than 10%. The experimental data of vertical surfaces also show good accordance with the simulation results calculated using Eq. (57), and the maximum error is less than 15%. Secondly, under the experimental condition, the experimental data show that the maximum heat flux is obtained from 2# plate (h = 128.6°, the closest value to the theoretical optimum value h = 135°), which is obviously higher than that of 1# (h = 159.3°), and 3# plate (h = 100.4°) under various subcooling temperature. Thirdly, both the experimental data and the simulation results indicate that an optimum contact angle exists between 100.4° and 159.3°, which is consistent with the previous analyses about the trends between heat flux and contact. The solid curves in Fig. 14 summarize most of the data reported since 1930 for dropwise condensation of steam at atmospheric pressure. The shaded region shows the results from Rose and the detail for all these curves are included in reference [25]. The heat flux curves for h = 75°, h = 135° and h = 170° obtained from Eq. (46) are also shown in this figure. According to the data in Fig. 14, significant differences of experimental data or calculated values from different researchers have been shown, and some data are located outside the region of Rose’s results. However, almost all literature data are located within the region marked by values from the present model for h from 75° to 170°, and the present models can accurately simulate heat fluxes for various contact angles. The present model can also show that for dropwise condensation, the heat transfer coefficient increases along with increasing contact angle up to a critical angle, while the contact angle exceeds the critical angle, heat transfer performance is reduced evidently. Fig. 15 shows the comparison between the experimental data from literature [30] and the simulation results based on the

3#

Experimental data

0

180

2#

159.3° 128.6° 100.4°

Present model

2 0 40

1#

Specimen

4

5

γ = 1.3

6 4 2 0

0

1

2

3

4

5

6

7

Subcooled temperatureT / K Fig. 14. Comparison between the present model and the experimental data from literatures.

present model with contact angle h = 117.3° on vertical surfaces. In this literature the contact angle was measured at room temperature and pressure, by the sessile drop method using an OCA20

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

Heat flux qtot / W·m-2

8

×5

literature [32] at 33.86 kPa and the present simulation results for h = 148.5° and h = 111.2° calculating from Eq. (58) are illustrated in Fig. 16, and the parameters for steam and condensate at 33.86 kPa are employed for calculation. The contact angles were measured using a CAM-100 (KSV Instruments Ltd., Finland) type contact angle measurement apparatus with an accuracy of ±0.5°. As shown in this figure, the simulation results are also in accordance with the experimental data. This has been further proved that the present heat transfer model is accurate and practicable.

γ = 1.3

P = 0.1063 9 -2 NS= 5.0×10 cm

6

©᧹117.3e

4

271

©᧹87.7e 6. Conclusions

2

0

Lan et al. [30] Qi et al. [31] Present model©᧹87.7e Present model©᧹117.3e

0

5

10

15

20

Subcooled temperatureT / K

Fig. 15. Comparison of heat flux between the present model and experimental data at atmospheric pressure.

(Dataphysics Co., Germany) type contact angle measurement apparatus with an accuracy of ±0.1°. The calculated values from Eq. (57) show good agreement with the experimental data of the literature [30] and the maximum difference between them is less than 20%. Moreover, another experimental results on vertical surfaces reported by the authors of this paper in literature [31] at h = 87.7° are also compared with the theoretical predictions in this figure, and the contact angle of H2O to the heat transfer plate was also measured at room temperature and pressure using a DSA-100 (KRUSS Instruments Ltd., Germany) type contact angle measurement apparatus with an accuracy of ±0.1°. It is seen that when the subcooling temperature is lower than 10 K, the theoretical predictions from Eq. (57) is highly consistent with the experimental data, and the maximum difference between experimental results and calculated values is less than 10% with a good repeatability. But when DT > 10 K, the reliability of the present model decreases gradually with the increase in the subcooling temperature, and the deviation is greater than 30% when the subcooling temperature is larger than 15 K. This is because that filmwise condensation occurs on the condensing surface in the form of rivulet, hence reducing the total heat transfer performance during condensing process. The third comparison between the experimental data from

7

Conflict of interest

Vemuri and Kim [32] Vemuri and Kim [32] Present model = 148.5 Present model = 111.2

6

None declared. Acknowledgments

-2

h / W· m · K

-1

5

A fractal model for dropwise condensation heat transfer is developed based on Rose’s model for heat transfer through a drop and the fractal characteristics of drop size distribution on condensing surface. In this model, two different random fractal models are employed to respectively describe the direct condensation drops and the coalescence drops drop size distributions on the condensing surface. A new model of dropwise condensation heat transfer is proposed considering the contact angle and its hysteresis, the nucleation site density, the fractal dimension for drop size and maximum and minimum drop radii. The simulation results show that the static contact angle may have great influence on the thermal resistance for a single drop and the drop size distribution functions. Because of this, the heat flux firstly increases and then decreases with increasing static contact angle, and the maximum heat transfer performance is observed when the static contact angle is about 135°, at the calculation conditions of this paper. The simulation results also indicated that the fractal dimension for drop size distribution decreases with increase in contact angle. The contact angle has little or minor impact on fractal dimension on hydrophilic surface and hydrophobic surface, but the fractal dimension decreases sharply with the values of h increasing on super-hydrophobic surface, especially in small temperature difference. On the contrary, the value of df increases sharply at low number density of drops, and for large number density, the reverse is true. The model proposed is applicable to predict the heat flux of the condensing drops with a wide range of contact angles, and the prediction results can agree well with the different experimental dropwise condensation heat transfer data obtained from experiments and majority of literature results by choosing the proper contact angle parameters. The study can also reasonably explain the cause of the diversities of the dropwise condensation heat transfer data from different references.

4

The authors are grateful to the financial supports by the National Natural Science Foundation of China under the Grants of 51306141 and 51225601, the Specialized Research Fund for the Doctoral Program of Higher Education under the Grants of 20120201120068.

= 148.5

3 γ = 1.3 P = 0.1063

2

9

0

= 111.2

-2

NS = 5.0×10 cm

1

References

Vacuum pressure = 33.86 kPa 0

1

2

3

Subcooling temperature

4

T/K

5

6

Fig. 16. Comparison of heat transfer coefficient between the present model and experimental data at 33.86 kPa.

[1] E. Schmidt, W. Schurig, Sellschopp, Versuche über die Kondensation in Filmund Tropfenform, Tech. Mech. Thermodyn. 1 (1930) 53–63. [2] E.J. Le Fevre, J.W. Rose. A theory of heat transfer by dropwise condensation, in: Proceedings of Third International Heat Transfer Conference, Chicago, 1966, pp. 362–375. [3] S. Khandekar, K. Muralidhar, Dropwise Condensation on Inclined Textured Surfaces, Springer, 2013.

272

B. Qi et al. / International Journal of Heat and Mass Transfer 83 (2015) 259–272

[4] B.S. Sikarwar, S. Khandekar, S. Agrawal, et al., Dropwise condensation studies on multiple scales, Heat Transfer Eng. 33 (2012) (2012) 301–341. [5] B.S. Sikarwar, S. Khandekar, K. Muralidhar, Simulation of flow and heat transfer in a liquid drop sliding underneath a hydrophobic surface, Int. J. Heat Mass Transfer 57 (2013) 786–811. [6] B.S. Sikarwar, S. Muralidhar, K. Khandekar, Effect of drop shape on heat transfer during dropwise condensation underneath inclined surfaces, in: Proc. Fourth Asian Symposium on Computational Heat Transfer and Fluid Flow, Hong Kong, 2013. [7] S. Kim, K. Kim, Dropwise condensation modeling suitable for superhydrophobic surfaces, J. Heat Transfer 133 (2011) 081502(1)–081502(8). [8] N. Fatica, D.L. Katz, Dropwise condensation, Chem. Eng. Prog. 45 (1949) 661– 674. [9] H. Wenzel, Der wrmeübergang bei der tropfenkondensation, Linde-Ber. Tech. Wiss. 18 (1964) 44–48. [10] E.E. Gose, A.N. Mucciardi, E. Baer, Model for dropwise condensation on randomly distributed sites, Int. J. Heat Mass Transfer 10 (1967) 15–22. [11] J.W. Rose, L.R. Glickman, Dropwise condensation – the distribution of drop sizes, Int. J. Heat Mass Transfer 16 (1973) 411–425. [12] I. Tanasawa, F. Tachibana. A synthesis of the total process of dropwise condensation using the method of computer simulation, in: Proc. Fourth Int. Heat Transfer Conference, Paris Versailles, 1970, pp. 1–11. [13] Y.T. Wu, C.Y. Yang, X.G. Yuan, et al., Drop distributions and numerical simulation of dropwise condensation heat transfer, Int. J. Heat Mass Transfer 44 (2001) 4455–4464. [14] F.Z. Sun, M. Gao, Y.B. Lei, et al., The fractal dimension of the fractal model of dropwise condensation and its experimental study, Int. J. Nonlinear Sci. Numer. Simul. 8 (2007) 211–222. [15] M.F. Mei, B.M. Yu, J.C. Cai, et al., A fractal analysis of dropwise condensation heat transfer, Int. J. Heat Mass Transfer 52 (2009) 4823–4828. [16] M.F. Mei, B.M. Yu, M.Q. Zou, et al., A numerical study on growth mechanism of dropwise condensation, Int. J. Heat Mass Transfer 54 (2011) 2004–2013. [17] X.H. Ma, Z. Lan, Y. Zhang, et al., Dropwise condensation heat transfer model: effect of the liquid–solid surface free energy difference, in: Proceedings of the ASME Fluids Engineering Division Summer Conference, Miami, 2006, pp. 655–662.

[18] Z. Lan, X.H. Ma, Theoretical study of dropwise condensation heat transfer: effect of the liquid–solid surface free energy difference, J. Enhanced Heat Transfer 16 (2009) 61–71. [19] J.W. Rose, Dropwise condensation theory, Int. J. Heat Mass Transfer 24 (1981) 191–194. [20] J.W. Rose, E.J. LeFevre, An experimental study of heat transfer by dropwise condensation, Int. J. Heat Mass Transfer 8 (1965) 1117–1133. [21] H.W. Wu, J.R. Maa, On the heat transfer on dropwise condensation, Chem. Eng. J. Biochem. Eng. J. 12 (1976) 225–231. [22] B.M. Yu, P. Cheng, Fractal models for the effective thermal conductivity of bidispersed porous media, AIAA J. Thermophys. Heat Transfer 16 (2002) 22– 29. [23] M. Abu-Orabi, Modeling of heat transfer in dropwise condensation, Int. J. Heat Mass Transfer 41 (1998) 81–87. [24] B.S. Sikarwar, N.K. Battoo, S. Khandekar, K. Muralidhar, Dropwise condensation underneath chemically textured surfaces: simulation and experiments, ASME J. Heat Transfer 133 (2011) 021501(1)–021501(15). [25] J.W. Rose, Dropwise condensation theory and experiment: a review, Proc. Inst. Mech. Eng. Part A, J. Power Energy 216 (2002) 115–128. [26] C. Yamali, H. Merte, A theory of dropwise condensation at large subcooling including the effect of the sweeping, Heat Mass Transfer 38 (2002) 191–202. [27] S. Lee, H.K. Yoon, K.J. Kim, et al., A dropwise condensation model using a nanoscale, pin structured surface, Int. J. Heat Mass Transfer 60 (2013) 664–671. [28] I. Tanasawa, J. Ochiai, Experimental study on dropwise condensation, Bull. Jpn. Soc. Mech. Eng. 16 (1973) 1184–1197. [29] K. Wan, J.C. Min, Characteristics of stochastic drop-size distributions during dropwise condensation, in: Proceedings of the CSHMT Conference, Beijing, 2003, pp. 341–344. [30] Z. Lan, X.H. Ma, S.F. Wang, et al., Effects of surface free energy and nanostructures on dropwise condensation, Chem. Eng. J. 156 (2010) 546–552. [31] B.J. Qi, L. Zhang, H. Xu, et al., Experimental study on condensation heat transfer of steam on vertical titanium plates with different surface energies, Exp. Therm. Fluid Sci. 35 (2011) 211–218. [32] S. Vemuri, K.J. Kim, B.D. Wood, et al., Long term testing for dropwise condensation using s elf-assembled monolayer coatings of n-octadecyl mercaptan, Appl. Therm. Eng. 26 (2006) 421–429.