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Condensation heat transfer in a sessile droplet at varying Biot number and contact angle
International Journal of Heat and Mass Transfer 115 (2017) 926–931
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Condensation heat transfer in a sessile droplet at varying Biot number and contact angle Sanjay Adhikari, Mahdi Nabil, Alexander S. Rattner ⇑ Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA 16802, United States
a r t i c l e
i n f o
Article history: Received 17 February 2017 Received in revised form 13 July 2017 Accepted 14 July 2017
Keywords: Dropwise condensation Conduction heat transfer
a b s t r a c t Dropwise condensation has been identified as a promising heat transfer mechanism because it can yield heat fluxes up to an order of magnitude higher than typically found in filmwise condensation. Models for dropwise condensation generally assume a statistical distribution of droplet sizes and integrate heat transfer over the droplet size spectrum, considering droplet curvature effects on saturation temperature, conduction thermal resistance, and interfacial resistance. Most earlier studies have assumed a constant heat transfer factor (f = O(1)) to account for the conduction contribution to total thermal resistance. However, f varies with droplet Biot number (Bi) and contact angle (h). Formulations for f with broad ranges of applicability are not currently available. In this study, finite element simulations are performed to determine f and corresponding numerical uncertainties for 0.0001 6 Bi 6 1000 and 10° 6 h 6 170°. This spans the active droplet size range considered in most droplet condensation studies (e.g., for water condensing at Patm on a surface 10 K below the ambient temperature, active droplets have 0.0005 < Bi < 300). An explicit correlation is proposed for f and is validated with published results. The proposed correlation can facilitate modeling and analysis of dropwise condensation. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Dropwise condensation has been identified as a promising highflux heat transfer mechanism for applications in power generation [1], desalination [2], and electronics thermal management [3]. Dropwise condensation often yields heat transfer rates up to an order of magnitude higher than that filmwise condensation [4,5]. The dropwise condensation process initiates from minute primary droplets (<1 mm diameter) at discrete nucleation sites distributed over a cooled surface [6,7]. Droplets with a radius greater than the minimum thermodynamically viable radius [4,8] grow due to condensation and coalescence with neighboring droplets until they become large enough to be removed by body forces (e.g., gravity). Sliding and merging of large droplets clear portions of the cooled surface, allowing new primary droplets to form and grow from nucleation sites. In dropwise condensation, heat from the condensation process transfers through the liquid-vapor interface (interfacial resistance R00i ¼ 1hi ) and then conducts through the droplet to the cooled wall (Rcond) (Fig. 1). Internal circulation and convection within droplets
⇑ Corresponding author at: 236A Reber Building, University Park, PA 16802, United States. E-mail address: [email protected] (A.S. Rattner). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.07.077 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
is generally assumed negligible [9]. Interfacial heat transfer resistance is usually significant in dropwise condensation because the conduction length from the interface to the wall approaches zero at the contact line. Therefore, interfacial thermal resistance is dominant in the near-wall region. The relative overall contributions of interfacial and conduction resistances in droplet condensation heat transfer depend on the Biot number Bi ¼ hki r and contact angle h l
[10]. Average heat flux through the base of the droplet is given by the following relation:
DT q00 ¼ f ðBi;hÞr þ 2h1 i k
ð1Þ
l
Here, q00 is the average heat flux through the base of the droplet, DT is the temperature difference between ambient vapor temperature and the base, r is the radius of curvature for the droplet, and kl is the liquid-phase thermal conductivity. f ðBi;hÞ is a scaling factor for the conduction contribution to overall droplet thermal resistance. f can be considered as an inverse fin efficiency for the droplet on the cooled wall, and should approach 0 for an infinitely conductive droplet ðlimBi!0 f ¼ 0Þ. Most prior studies assumed a constant factor f = O(1) [8,11,12]. This assumption may be suitable for steady-state analyses where the distribution of drop sizes is constant. However, an effective average f value would still be specific
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arguments, the in-droplet condensation heat transfer process reduces to conduction in a solid medium, with the following equation:
r2 T ¼ 0
ð2Þ
For small droplets, as is the focus here, surface tension forces dominate gravity. Therefore, the droplets can be modeled as truncated spheres [16], and evaluated with 2-D axisymmetric domains (Fig. 2). A fixed temperature boundary condition is applied on the base of the axisymmetric droplet domain (1 K below the saturation temperature), representing the cooled condenser surface temperature. A convection boundary condition is applied to the liquid-vapor interface to account for interfacial resistance, with the following heat transfer coefficient [21]:
! 1=2 2 2r hlv M hi ¼ 2pRT v 2r Tv vl
Fig. 1. (a) Schematic of the Droplet with radius r, contact angle h, conduction resistance Rcond and interfacial resistance Rint. (b) Magnified view of the interfacial resistance.
to the operating conditions. For cases where the droplet size distribution evolves over time (e.g., dropwise condensation startup), the variation of f as the droplet size distribution develops must be considered. Several studies have been performed to determine formulae for f(Bi, h). Sadhal and Martin [13] and Kim and Kim [14] have proposed analytical models for hydrophilic surfaces (h < 90°). Recently, Miljkovic et al. [15] and Chavan et al. [16] have reported numerical and experimental results for hydrophobic surfaces (h > 90°). Generally valid models for f(Bi, h) would facilitate dropwise condensation analyses, but are not currently available. In this study, finite element heat transfer analyses are performed for 2958 cases to develop a single broadly applicable droplet condensation heat transfer formulation for 0.0001 6 Bi 6 1000 and 10° 6 h 6 170°. Discretization errors are evaluated to determine uncertainty bounds for Richardsonextrapolated f factors. Results are validated against prior findings from Sadhal and Martin [13] and Chavan et al. [16]. Summarized data and simulation results from this study are publicly available at [17].
2. Modeling approach In this study, finite element analyses are performed for heat transfer in sessile droplets using COMSOL MultiPhysicsÒ (v. 5.2 [18]). In typical dropwise condensation conditions, >90% of the heat is transferred through droplets less than 200 lm in diameter [19]. Droplets in this size range are sufficiently small, such that Marangoni circulation and other convection effects can be neglected [20]. Therefore, conduction is the predominant mode of heat transfer. As droplets grow, the internal temperature distribution varies. However, the conduction-front propagation velocity al is generally much greater than the droplet growth rate dr ; dt r therefore, the heat transfer process can be modeled as quasisteady. For example, for a water droplet with 1 lm diameter and a 90° contact angle on a surface at 99.15 °C ðT sat ¼ 100:15 CÞ the conduction front velocity is arl 1:6 101 ms1 . Applying a standard droplet condensation heat transfer model (e.g., from [11] with constant f = 0.25), the predicted growth velocity would only be
dr dt
¼ 8:4 105 ms1 For similar conditions a 1 mm droplet will
have
al r
¼ 1:6 104 ms1 and
dr dt
¼ 6 107 ms1 . Based on these
ð3Þ
Here, hi is the interfacial heat transfer coefficient, r is the accommodation constant, hlv is the enthalpy of vaporization, T v is the ambient temperature, M is the molecular mass of the working fluid, and R is the specific gas constant. The saturation temperature of the vapor at the ambient pressure ðT sat Þ is substituted for T v in Eq. (3). The domain was meshed with nearly uniform size triangular elements (Fig. 2). The governing equation was discretized with second order accurate elements and solved to a relative residual of 1014 . The total droplet heat transfer rates obtained from the FEA were used to determine f ðBi; hÞ, using the relation given in Eq. (1), for varying Biot number and contact angle values. 3. Simulation studies A parametric study for droplet condensation heat transfer was performed. The contact angle ðhÞ was varied to span both the hydrophilic and hydrophobic regimes (10°–170°) in 10° increments. Droplet conductivity ðkl Þ was varied with fixed radius (r = 1 mm) and interfacial resistance (hi = 10,000 W m2 K1) to evaluate 174 Bi values from 0.0001 to 1000 for each contact angle. For water at atmospheric pressure, this Bi range spans droplet radii from 310 nm to 3:1 mm. In total, 2958 cases were studied. Grid sensitivity studies were performed for each case with at least three meshes to extrapolate converged values of f ðBi;hÞ and corresponding uncertainties following the method of Celik et al. [22]. The technique is briefly described below. Meshes (Fig. 2) with average element dimensions:
D3 ¼ 2:2 102 mm, D2 ¼ 1:2 102 mm and D1 ¼ 6:0 103 mm are employed. Subscript 3 refers to the coarsest mesh and subscript 1 refers to the finest mesh. The steps are as follows: e ð4Þ e32 ¼ f 3 f 2 ; e21 ¼ f 2 f 1 ; s ¼ sgn 32
e21
R32 ¼
D3 ; D2
R21 ¼
D2 D1
p R s ; qðpÞ ¼ ln 21 p R32 s
ð5Þ
pðqÞ ¼
þ q ln ee32 21
f ext ¼ ðRp21 f 1 f 2 Þ=ðRp21 1Þ e21 a
f f 2 ; ¼ 1 f1
unc ¼ f ext GCI
GCI ¼
1:25e21 a Rp21 1
lnðR21 Þ
ð6Þ ð7Þ ð8Þ ð9Þ
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Fig. 2. Representative mesh sizes: (a) coarse D3 ¼ 2:2 102 mm; (b) fine D2 ¼ 1:2 102 mm; (c) finest D1 ¼ 6:0 103 mm.
The e values are the changes in f (the heat transfer scaling factor) between meshes. R values are the mesh refinement ratios. p is the empirical convergence rate, defined implicitly in terms of parameter q. f ext is the extrapolated value of f for D ! 0 (infinitely fine is the relative error between the two finest meshes. mesh). e21 a GCI is the grid convergence index, a relative uncertainty estimate for f ext . The absolute uncertainty in f ext (unc) is determined from Eq. (9). All studied cases were found to be monotonically converging ðs ¼ 1Þ and therefore the grid convergence index (GCI) uncertainty estimate for f ext was reported (<3% uncertainty in all cases). Summarized data and simulation results from this study are publicly available at Ref. [17]. Fig. 3 shows the variation of f with (Bi1) for different contact angles (50°–130°). Linear trends of f with Bi1 were found to continue for Bi = 101–104, and are not included in Fig. 3 so that sharper variations for large Bi are apparent. Similar trends of increasing f with Bi1 for h < 50 and decreasing f with Bi1 h > 130 , were found. Corresponding very low and high contact angle curves are therefore omitted from Fig. 3 for presentation clarity. Negative f values were found for hydrophobic droplets with low Bi values. Such droplets have relatively low internal conduction resistances and greater surface area for condensation than for the baseline h = 90° droplet assumed in Eq. (1). Thus, the negative f value yields a lower overall thermal resistance than for interfacial resistance alone on a h = 90° droplet. 4. Results and discussion 4.1. Proposed correlation Simulation data (0.0001 Bi 1000, h ¼ 50 130 ) for f ðBi;hÞ were fit to the following analytic correlation using non-linear
regression. The range of f is spread over 4 orders of magnitude. 2
Therefore, data points were weighted by f in the fitting procedure to offset skew toward high magnitude values. The correlation is given in Eq. (10) and the coefficients are provided in Table 1. In Eq. (10), the contact angle h is in degrees. 1
1
1
f ðBi; hÞ ¼ ðA þ B Bi Þ þ ðC þ D Bi Þ h þ ½E log10 ðBi Þ 1
2
1
3
þ F flog10 ðBi Þg þ G flog10 ðBi Þg
ð10Þ
The absolute average deviation (AAD) for the data range (0.0001 Bi 1000, 50° h 130°) is 7.4%. Fig. 4 compares the absolute values of f obtained from the correlation with conduction simulation results. The correlation predicts 91% of cases with |f| < 1 within 25% of simulation values. All cases with |f| > 1 are predicted within 25% of simulation values. Because of the wide range of f values, it was difficult to fit a simple analytic correlation over 10° h 170°. Therefore, a MatlabÒ [23] function for f ðBi;hÞ is provided [17] that directly interpolates simulation results to provide precise values overall the full parametric space (0.0001 Bi 1000, 10° h 170°). 4.2. Validation of results For hydrophilic conditions the proposed correlation for f (Eq. (10)) was validated with data from Sadhall and Martin [13]. q Droplet Nusselt number Nu ¼ DTrk was compared. q is the total heat transferred through the base of a droplet. The AAD from the study, for the whole range of Bi and h, is 3.4%. Fig. 5 compares the variation of Nu with Bi. Numerical uncertainties are not presented with the simulation data points in the figure because they are too small to be visible ð< 104 %Þ. The only available data for f in hydrophobic droplets was reported by Chavan et al. [16]. They have provided three correlations for 101 < Bi < 105. This span of Bi does not cover the typical full size range of active droplets condensing on a surface. For example, for water condensing at P atm on a surface 10 K below the ambient temperature, active droplets have 0.0005 < Bi < 300 (0.0001 6 Bi 6 1000 evaluated in the present study). Good agreement between the proposed correlation (Eq. (10)) and the correlations of Chavan et al. [16] are found in the overlapping data region
Table 1 Coefficients for the correlation given in Eq. (10).
Fig. 3. f ðBi; hÞ vs. Bi1 for different contact angles.
Coefficient
Value (Uncertainty, 95% confidence interval)
A B C D E F G
0.2160 (±0.0015) 0.7278 (±0.0015) 0.001465 (±0.000013) 0.008086 (±0.000017) 0.1012 (±0.0007) 0.01378 (±0.00036) 0.007361 (±0.00013)
S. Adhikari et al. / International Journal of Heat and Mass Transfer 115 (2017) 926–931
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Volume generation is disabled to model the quasi-steady conduction problem.
@ ðkTÞ ¼ q_ pc @xi @xi
ð13Þ
The VOF domain is axisymmetric and 4 mm 4 mm in height and width. Meshes with average element dimensions:
Fig. 4. Comparison of absolute values of f predicted with proposed correlation and conduction simulations.
D3 ¼ 2:5 102 mm, D2 ¼ 1:25 102 mm and D1 ¼ 6:25 103 mm are employed. The base of the droplet is maintained at a temperature 2 K below the saturation temperature ðT sat ¼ 373:15 KÞ. The heat transfer in the vapor region is not included for comparison, and only the heat transfer through the base of the droplet is employed for comparison with the COMSOLÒ (droplet-only) data. The results from interThermalPhaseChangeFoam are also Richardson-extrapolated following the procedure from Eqs. (4)–(9). Table 2 compares the average heat flux q00 , obtained using interThermalPhaseChangeFoam (VOF), COMSOL conduction simulations, and the proposed correlation (Eq. (10)). Maximum error is less than 1.4%. This close agreement between results from two different software packages with different simulation methodologies further confirms the accuracy of reported hydrophobic droplet heat transfer data. 4.3. Internal temperature distribution in droplets
Fig. 5. Nu vs. Bi for different contact angles (50°–80°).
ð0:1 < Bi < 1000; 90 < h < 130 Þ, with AAD = 10.93%. The comparison is provided in table format [17]. As there are no other published data to validate hydrophobic droplet condensation heat transfer, five representative hydrophobic cases were also evaluated in interThermalPhaseChangeFoam [24], a volume-of-fluid (VOF) based CFD solver for phase-change heat transfer processes. VOF is an interface capturing technique, which tracks the volume fraction of the liquid phase in each mesh cell with a scalar / 2 ½0; 1. Eq. (11) is the advection equation for /, where ui is the velocity vector and /_ is the volumetric phase fraction source due to condensation phase change. A single set of governing equations is solved for both the liquid and the vapor phases. Fluid material properties are weighted by Eq. (12).
@/ @ þ ðui /Þ ¼ /_ @t @xi b ¼ /bL þ ð1 /ÞbV
ð11Þ b 2 ½q; l; k
ð12Þ
To reduce the VOF computation time for the sessile, quasisteady droplets considered here, the momentum and phase fraction advection (Eq. (11)) equations are not evaluated (velocity field u ¼ 0, phase-fraction field / constant). The steady state VOF thermal transport equation (Eq. (13)) is solved for a water droplet ðr ¼ 1 mmÞ with different contact angles (90°–130°). q_ pc is the volumetric condensation phase-change heat source term applied on the interface. Interfacial resistance is applied as a distributed source term on the diffuse interface region (4 mesh cells thick).
Temperature distribution trends inside droplets were found to depend more strongly on Bi than h. As Bi reduces, the conduction resistance also reduces relative to the interfacial resistance. Thus, as Bi reduces, the internal droplet temperature distribution becomes more uniform. For higher Bi, internal conduction resistance is significant and condensation heat transfer primarily occurs in a thin region near the contact line. Fig. 5 presents the temperature distribution inside a droplet with varying Bi (h ¼ 90 , droplet geometry and conditions described in Section 3). It is evident from Fig. 6 that for Bi < 0.2, the droplet is effectively isothermal. This effect can also be observed in Fig. 7, which presents the local heat flux at a given angle ranging from b = 0° at the contact line to b = 90° at the top of the droplet. For Bi = 0.2, condensation heat flux only varies by 19% over the surface of the droplet. In contrast, for Bi = 1000 (high Bi), 90% of the heat transfer occurs in the 20° of the droplet surface closest to the contact line. Trends of temperature distribution inside droplets were not found to depend strongly on the contact angle. As can be observed in Fig. 8, qualitatively similar temperature distributions are found for h = 40°–140° for a droplet at Bi = 1000. 4.4. Implications of varying conduction resistance factor in dropwise condensation Many prior dropwise condensation studies have assumed a constant value of conduction factor f. However, this approximation can yield incorrect heat transfer predictions for cases with timevarying droplet size distributions. As an example, consider the growth of a single water droplet ðh ¼ 90 ; r ¼ 100 lmÞ, on a surface maintained 10 K below the saturation temperature ðT sat ¼ 373:15 KÞ. The quasi-steady heat transfer through a growing droplet is given as [25]:
q ¼ qhlv 2pr2
dr dt
ð14Þ
Equating Eq. (14) with Eq. (1), the growth rate of a droplet is:
dr DT 1 ¼ dt 2qhlv frk þ 2h1
ð15Þ i
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Table 2 Average heat flux through the base of droplets with representative hydrophobic Bi and h values, computed with VOF solver, COMSOL conduction studies, and proposed correlation. Biot no. (Bi)
q00 (W m2) VOF
Contact angle (h)
90
8.87 10 ± 4300
8.75 10 ± 3.15
8.75 105
100
5.43 103 ± 0.50
5.40 103 ± 700
5.57 103
110 120 130
3.75E 105 ± 6.0 1.50 106 ± 7.0 1.32 106 ± 3200
3.72 105 ± 8600 1.47 106 ± 27000 1.30 106 ± 38000
3.82 105 1.43 106 1.28 106
5:88 10 2:40 0:10
3:68 101
Fig. 6. Non-dimensional temperature
TT wall T sat T wall
5
q00 (W m2) correlation
2
1:98 10
5
q00 (W m2) COMSOL
3
distribution inside a droplet with varying Bi. (a) Bi = 1000, (b) Bi = 2 and (c) Bi = 0.2.
Fig. 7. Heat flux along the condensing surface of the droplet with h ¼ 90 for different Bi. b is the angle subtended by any point on the droplet surface and the horizontal (inset).
Eq. (15) is integrated for 1.5 s to predict the growth of the droplet. This analysis is performed assuming both a constant value of f = 0.25 [12] and the proposed correlation for f (Eq. (10)), which varies with the instantaneous droplet radius. This droplet growth case is also evaluated using the interfacecapturing volume-of-fluid (VOF) based solver, interThermalPhaseChangeFoam to provide a high-resolution reference case. The domain is axisymmetric and 400 lm 400 lm in height and width. A fine mesh with an average element dimension D ¼ 0:625 lm is employed. The momentum and phase fraction
Fig. 9. Comparison of droplet growth between OpenFoam VOF simulation, theoretical relation with f obtained using the correlation and theoretical relation with f = 0.25.
advection equations were solved here, unlike in the quasi-steady cases described in the previous section. Assuming a constant value of f yields a 23% relative error in the droplet growth with respect to the VOF simulation (Drf = 0.25 = 71 lm, DrVOF = 92 lm). Based on the trend at t = 1.5 s (Fig. 9), this deviation would continue to grow if the analysis were continued to a later time. However, the proposed correlation for f results in only a 3% error. This analysis also indicates the accuracy of the proposed correlation, and its applicability to unsteady droplet condensation cases.
Fig. 8. Non-dimensional temperature distribution inside droplets with different contact angles for Bi = 1000. (a) h ¼ 40 , (b) h ¼ 90 and (c) h ¼ 140 .
S. Adhikari et al. / International Journal of Heat and Mass Transfer 115 (2017) 926–931
5. Conclusion In this study, finite element analyses were performed for condensation heat transfer in sessile droplets accounting for internal conduction and interfacial resistance. Parametric studies were performed over a wide range of Biot numbers (0.0001 6 Bi 6 1000) and contact angles (10° 6 h 6 170°) to determine conduction thermal resistance factors (f) and corresponding uncertainties. Summarized data and simulation results from this study are publicly available at [17]. A new correlation was proposed for f ðBi; hÞ that is valid over the studied range of conditions. This model was validated against an earlier hydrophilic model, only applicable for part of this range. Independent VOF simulations were performed to validate hydrophobic predictions. An analysis was performed to highlight the importance of accounting for the variation of f with different size droplets. As shown in Fig. 9, assuming a fixed value of f can lead to large errors in droplet growth predictions. The proposed, broadly applicable, correlation for f can thus facilitate future dropwise condensation analyses and research.
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[7] S.A. Stylianou, J.W. Rose, Drop-to-filmwise condensation transition: heat transfer measurements for ethanediol, Int. J. Heat Mass Transfer 26 (5) (1983) 747–760. [8] B.M. Burnside, H.A. Hadi, Digital computer simulation of dropwise condensation from equilibrium droplet to detectable size, Int. J. Heat Mass Transfer 42 (16) (1999) 3137–3146. [9] A. Umur, P. Griffith, Mechanism of dropwise condensation, J. Heat Transfer 87 (2) (1965) 275–282. [10] N. Miljkovic, R. Enright, E.N. Wang, Effect of droplet morphology on growth dynamics and heat transfer during condensation on superhydrophobic nanostructured surfaces, ACS Nano 6 (2) (2012) 1776–1785. [11] C. Graham, P. Griffith, Drop size distributions and heat transfer in dropwise condensation, Int. J. Heat Mass Transfer 16 (2) (1973) 337–346. [12] L.R. Glicksman, A.W. Hunt, Numerical simulation of dropwise condensation, Int. J. Heat Mass Transfer 15 (11) (1972) 2251–2269. [13] S.S. SadhalL, W.W. Martin, Heat transfer through using differential drop condensate inequalities, Int. J. Heat Mass Transfer 20 (12) (1977) 1401–1407. [14] S. Kim, K.J. Kim, Dropwise condensation modeling suitable for superhydrophobic surfaces, J. Heat Transfer 133 (8) (2011) 81502. [15] N. Miljkovic, R. Enright, E.N. Wang, Growth dynamics during dropwise condensation on nanostructured superhydrophobic surfaces, in: ASME 2012 3rd Int. Conf. Micro/Nanoscale Heat Mass Transf. MNHMT 2012, 2012, pp. 427–436. [16] S. Chavan, H. Cha, D. Orejon, K. Nawaz, N. Singla, Y.F. Yeung, D. Park, D.H. Kang, Y. Chang, Y. Takata, N. Miljkovic, Heat transfer through a condensate droplet on hydrophobic and nanostructured superhydrophobic surfaces, Langmuir 32 (31) (2016) 7774–7787. [17] S. Adhikari, A.S. Rattner, Single Droplet Condensation Heat Transfer Simulation Data [Online], 2017. Available: https://scholarsphere.psu. edu/concern/generic_works/z316q1768. [18] COMSOL, v. 5.2, 2016, COMSOL Inc., Stockholm, Sweden. [19] J.W. Rose, L.R. Glicksman, Dropwise condensation-the distribution of drop sizes, Int. J. Heat Mass Transfer 16 (2) (1973) 411–425. [20] H. Tanaka, T. Tsuruta, A microscopic study of dropwise condensation, Int. J. Heat Mass Transfer 27 (3) (1984) 327–335. [21] K.C. Hayes, H. Hatze, Passive visco-elastic properties of the structures spanning the human elbow joint, Eur. J. Appl. Physiol. Occup. Physiol. 37 (4) (1977) 265– 274. [22] I.B. Celik, U. Ghia, P.J. Roache, C.J. Freitas, H. Coleman, P.E. Raad, Procedure for estimation and reporting of uncertainty due to discretization in CFD applications, J. Fluids Eng. 130 (7) (2008) 78001. [23] MATLAB R2013, 2013, Natick, MA, Mathworks Inc. [24] M. Nabil, A.S. Rattner, InterThermalPhaseChangeFoam-a framework for twophase flow simulations with thermally driven phase change, SoftwareX 5 (2016) 216–226. [25] S. Vemuri, K.J. Kim, An experimental and theoretical study on the concept of dropwise condensation, Int. J. Heat Mass Transfer 49 (3–4) (2006) 649–657.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Condensation heat transfer in a sessile droplet at varying Biot number and contact angle Sanjay Adhikari, Mahdi Nabil, Alexander S. Rattner ⇑ Department of Mechanical and Nuclear Engineering, The Pennsylvania State University, University Park, PA 16802, United States
a r t i c l e
i n f o
Article history: Received 17 February 2017 Received in revised form 13 July 2017 Accepted 14 July 2017
Keywords: Dropwise condensation Conduction heat transfer
a b s t r a c t Dropwise condensation has been identified as a promising heat transfer mechanism because it can yield heat fluxes up to an order of magnitude higher than typically found in filmwise condensation. Models for dropwise condensation generally assume a statistical distribution of droplet sizes and integrate heat transfer over the droplet size spectrum, considering droplet curvature effects on saturation temperature, conduction thermal resistance, and interfacial resistance. Most earlier studies have assumed a constant heat transfer factor (f = O(1)) to account for the conduction contribution to total thermal resistance. However, f varies with droplet Biot number (Bi) and contact angle (h). Formulations for f with broad ranges of applicability are not currently available. In this study, finite element simulations are performed to determine f and corresponding numerical uncertainties for 0.0001 6 Bi 6 1000 and 10° 6 h 6 170°. This spans the active droplet size range considered in most droplet condensation studies (e.g., for water condensing at Patm on a surface 10 K below the ambient temperature, active droplets have 0.0005 < Bi < 300). An explicit correlation is proposed for f and is validated with published results. The proposed correlation can facilitate modeling and analysis of dropwise condensation. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction Dropwise condensation has been identified as a promising highflux heat transfer mechanism for applications in power generation [1], desalination [2], and electronics thermal management [3]. Dropwise condensation often yields heat transfer rates up to an order of magnitude higher than that filmwise condensation [4,5]. The dropwise condensation process initiates from minute primary droplets (<1 mm diameter) at discrete nucleation sites distributed over a cooled surface [6,7]. Droplets with a radius greater than the minimum thermodynamically viable radius [4,8] grow due to condensation and coalescence with neighboring droplets until they become large enough to be removed by body forces (e.g., gravity). Sliding and merging of large droplets clear portions of the cooled surface, allowing new primary droplets to form and grow from nucleation sites. In dropwise condensation, heat from the condensation process transfers through the liquid-vapor interface (interfacial resistance R00i ¼ 1hi ) and then conducts through the droplet to the cooled wall (Rcond) (Fig. 1). Internal circulation and convection within droplets
⇑ Corresponding author at: 236A Reber Building, University Park, PA 16802, United States. E-mail address: [email protected] (A.S. Rattner). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.07.077 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
is generally assumed negligible [9]. Interfacial heat transfer resistance is usually significant in dropwise condensation because the conduction length from the interface to the wall approaches zero at the contact line. Therefore, interfacial thermal resistance is dominant in the near-wall region. The relative overall contributions of interfacial and conduction resistances in droplet condensation heat transfer depend on the Biot number Bi ¼ hki r and contact angle h l
[10]. Average heat flux through the base of the droplet is given by the following relation:
DT q00 ¼ f ðBi;hÞr þ 2h1 i k
ð1Þ
l
Here, q00 is the average heat flux through the base of the droplet, DT is the temperature difference between ambient vapor temperature and the base, r is the radius of curvature for the droplet, and kl is the liquid-phase thermal conductivity. f ðBi;hÞ is a scaling factor for the conduction contribution to overall droplet thermal resistance. f can be considered as an inverse fin efficiency for the droplet on the cooled wall, and should approach 0 for an infinitely conductive droplet ðlimBi!0 f ¼ 0Þ. Most prior studies assumed a constant factor f = O(1) [8,11,12]. This assumption may be suitable for steady-state analyses where the distribution of drop sizes is constant. However, an effective average f value would still be specific
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arguments, the in-droplet condensation heat transfer process reduces to conduction in a solid medium, with the following equation:
r2 T ¼ 0
ð2Þ
For small droplets, as is the focus here, surface tension forces dominate gravity. Therefore, the droplets can be modeled as truncated spheres [16], and evaluated with 2-D axisymmetric domains (Fig. 2). A fixed temperature boundary condition is applied on the base of the axisymmetric droplet domain (1 K below the saturation temperature), representing the cooled condenser surface temperature. A convection boundary condition is applied to the liquid-vapor interface to account for interfacial resistance, with the following heat transfer coefficient [21]:
! 1=2 2 2r hlv M hi ¼ 2pRT v 2r Tv vl
Fig. 1. (a) Schematic of the Droplet with radius r, contact angle h, conduction resistance Rcond and interfacial resistance Rint. (b) Magnified view of the interfacial resistance.
to the operating conditions. For cases where the droplet size distribution evolves over time (e.g., dropwise condensation startup), the variation of f as the droplet size distribution develops must be considered. Several studies have been performed to determine formulae for f(Bi, h). Sadhal and Martin [13] and Kim and Kim [14] have proposed analytical models for hydrophilic surfaces (h < 90°). Recently, Miljkovic et al. [15] and Chavan et al. [16] have reported numerical and experimental results for hydrophobic surfaces (h > 90°). Generally valid models for f(Bi, h) would facilitate dropwise condensation analyses, but are not currently available. In this study, finite element heat transfer analyses are performed for 2958 cases to develop a single broadly applicable droplet condensation heat transfer formulation for 0.0001 6 Bi 6 1000 and 10° 6 h 6 170°. Discretization errors are evaluated to determine uncertainty bounds for Richardsonextrapolated f factors. Results are validated against prior findings from Sadhal and Martin [13] and Chavan et al. [16]. Summarized data and simulation results from this study are publicly available at [17].
2. Modeling approach In this study, finite element analyses are performed for heat transfer in sessile droplets using COMSOL MultiPhysicsÒ (v. 5.2 [18]). In typical dropwise condensation conditions, >90% of the heat is transferred through droplets less than 200 lm in diameter [19]. Droplets in this size range are sufficiently small, such that Marangoni circulation and other convection effects can be neglected [20]. Therefore, conduction is the predominant mode of heat transfer. As droplets grow, the internal temperature distribution varies. However, the conduction-front propagation velocity al is generally much greater than the droplet growth rate dr ; dt r therefore, the heat transfer process can be modeled as quasisteady. For example, for a water droplet with 1 lm diameter and a 90° contact angle on a surface at 99.15 °C ðT sat ¼ 100:15 CÞ the conduction front velocity is arl 1:6 101 ms1 . Applying a standard droplet condensation heat transfer model (e.g., from [11] with constant f = 0.25), the predicted growth velocity would only be
dr dt
¼ 8:4 105 ms1 For similar conditions a 1 mm droplet will
have
al r
¼ 1:6 104 ms1 and
dr dt
¼ 6 107 ms1 . Based on these
ð3Þ
Here, hi is the interfacial heat transfer coefficient, r is the accommodation constant, hlv is the enthalpy of vaporization, T v is the ambient temperature, M is the molecular mass of the working fluid, and R is the specific gas constant. The saturation temperature of the vapor at the ambient pressure ðT sat Þ is substituted for T v in Eq. (3). The domain was meshed with nearly uniform size triangular elements (Fig. 2). The governing equation was discretized with second order accurate elements and solved to a relative residual of 1014 . The total droplet heat transfer rates obtained from the FEA were used to determine f ðBi; hÞ, using the relation given in Eq. (1), for varying Biot number and contact angle values. 3. Simulation studies A parametric study for droplet condensation heat transfer was performed. The contact angle ðhÞ was varied to span both the hydrophilic and hydrophobic regimes (10°–170°) in 10° increments. Droplet conductivity ðkl Þ was varied with fixed radius (r = 1 mm) and interfacial resistance (hi = 10,000 W m2 K1) to evaluate 174 Bi values from 0.0001 to 1000 for each contact angle. For water at atmospheric pressure, this Bi range spans droplet radii from 310 nm to 3:1 mm. In total, 2958 cases were studied. Grid sensitivity studies were performed for each case with at least three meshes to extrapolate converged values of f ðBi;hÞ and corresponding uncertainties following the method of Celik et al. [22]. The technique is briefly described below. Meshes (Fig. 2) with average element dimensions:
D3 ¼ 2:2 102 mm, D2 ¼ 1:2 102 mm and D1 ¼ 6:0 103 mm are employed. Subscript 3 refers to the coarsest mesh and subscript 1 refers to the finest mesh. The steps are as follows: e ð4Þ e32 ¼ f 3 f 2 ; e21 ¼ f 2 f 1 ; s ¼ sgn 32
e21
R32 ¼
D3 ; D2
R21 ¼
D2 D1
p R s ; qðpÞ ¼ ln 21 p R32 s
ð5Þ
pðqÞ ¼
þ q ln ee32 21
f ext ¼ ðRp21 f 1 f 2 Þ=ðRp21 1Þ e21 a
f f 2 ; ¼ 1 f1
unc ¼ f ext GCI
GCI ¼
1:25e21 a Rp21 1
lnðR21 Þ
ð6Þ ð7Þ ð8Þ ð9Þ
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Fig. 2. Representative mesh sizes: (a) coarse D3 ¼ 2:2 102 mm; (b) fine D2 ¼ 1:2 102 mm; (c) finest D1 ¼ 6:0 103 mm.
The e values are the changes in f (the heat transfer scaling factor) between meshes. R values are the mesh refinement ratios. p is the empirical convergence rate, defined implicitly in terms of parameter q. f ext is the extrapolated value of f for D ! 0 (infinitely fine is the relative error between the two finest meshes. mesh). e21 a GCI is the grid convergence index, a relative uncertainty estimate for f ext . The absolute uncertainty in f ext (unc) is determined from Eq. (9). All studied cases were found to be monotonically converging ðs ¼ 1Þ and therefore the grid convergence index (GCI) uncertainty estimate for f ext was reported (<3% uncertainty in all cases). Summarized data and simulation results from this study are publicly available at Ref. [17]. Fig. 3 shows the variation of f with (Bi1) for different contact angles (50°–130°). Linear trends of f with Bi1 were found to continue for Bi = 101–104, and are not included in Fig. 3 so that sharper variations for large Bi are apparent. Similar trends of increasing f with Bi1 for h < 50 and decreasing f with Bi1 h > 130 , were found. Corresponding very low and high contact angle curves are therefore omitted from Fig. 3 for presentation clarity. Negative f values were found for hydrophobic droplets with low Bi values. Such droplets have relatively low internal conduction resistances and greater surface area for condensation than for the baseline h = 90° droplet assumed in Eq. (1). Thus, the negative f value yields a lower overall thermal resistance than for interfacial resistance alone on a h = 90° droplet. 4. Results and discussion 4.1. Proposed correlation Simulation data (0.0001 Bi 1000, h ¼ 50 130 ) for f ðBi;hÞ were fit to the following analytic correlation using non-linear
regression. The range of f is spread over 4 orders of magnitude. 2
Therefore, data points were weighted by f in the fitting procedure to offset skew toward high magnitude values. The correlation is given in Eq. (10) and the coefficients are provided in Table 1. In Eq. (10), the contact angle h is in degrees. 1
1
1
f ðBi; hÞ ¼ ðA þ B Bi Þ þ ðC þ D Bi Þ h þ ½E log10 ðBi Þ 1
2
1
3
þ F flog10 ðBi Þg þ G flog10 ðBi Þg
ð10Þ
The absolute average deviation (AAD) for the data range (0.0001 Bi 1000, 50° h 130°) is 7.4%. Fig. 4 compares the absolute values of f obtained from the correlation with conduction simulation results. The correlation predicts 91% of cases with |f| < 1 within 25% of simulation values. All cases with |f| > 1 are predicted within 25% of simulation values. Because of the wide range of f values, it was difficult to fit a simple analytic correlation over 10° h 170°. Therefore, a MatlabÒ [23] function for f ðBi;hÞ is provided [17] that directly interpolates simulation results to provide precise values overall the full parametric space (0.0001 Bi 1000, 10° h 170°). 4.2. Validation of results For hydrophilic conditions the proposed correlation for f (Eq. (10)) was validated with data from Sadhall and Martin [13]. q Droplet Nusselt number Nu ¼ DTrk was compared. q is the total heat transferred through the base of a droplet. The AAD from the study, for the whole range of Bi and h, is 3.4%. Fig. 5 compares the variation of Nu with Bi. Numerical uncertainties are not presented with the simulation data points in the figure because they are too small to be visible ð< 104 %Þ. The only available data for f in hydrophobic droplets was reported by Chavan et al. [16]. They have provided three correlations for 101 < Bi < 105. This span of Bi does not cover the typical full size range of active droplets condensing on a surface. For example, for water condensing at P atm on a surface 10 K below the ambient temperature, active droplets have 0.0005 < Bi < 300 (0.0001 6 Bi 6 1000 evaluated in the present study). Good agreement between the proposed correlation (Eq. (10)) and the correlations of Chavan et al. [16] are found in the overlapping data region
Table 1 Coefficients for the correlation given in Eq. (10).
Fig. 3. f ðBi; hÞ vs. Bi1 for different contact angles.
Coefficient
Value (Uncertainty, 95% confidence interval)
A B C D E F G
0.2160 (±0.0015) 0.7278 (±0.0015) 0.001465 (±0.000013) 0.008086 (±0.000017) 0.1012 (±0.0007) 0.01378 (±0.00036) 0.007361 (±0.00013)
S. Adhikari et al. / International Journal of Heat and Mass Transfer 115 (2017) 926–931
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Volume generation is disabled to model the quasi-steady conduction problem.
@ ðkTÞ ¼ q_ pc @xi @xi
ð13Þ
The VOF domain is axisymmetric and 4 mm 4 mm in height and width. Meshes with average element dimensions:
Fig. 4. Comparison of absolute values of f predicted with proposed correlation and conduction simulations.
D3 ¼ 2:5 102 mm, D2 ¼ 1:25 102 mm and D1 ¼ 6:25 103 mm are employed. The base of the droplet is maintained at a temperature 2 K below the saturation temperature ðT sat ¼ 373:15 KÞ. The heat transfer in the vapor region is not included for comparison, and only the heat transfer through the base of the droplet is employed for comparison with the COMSOLÒ (droplet-only) data. The results from interThermalPhaseChangeFoam are also Richardson-extrapolated following the procedure from Eqs. (4)–(9). Table 2 compares the average heat flux q00 , obtained using interThermalPhaseChangeFoam (VOF), COMSOL conduction simulations, and the proposed correlation (Eq. (10)). Maximum error is less than 1.4%. This close agreement between results from two different software packages with different simulation methodologies further confirms the accuracy of reported hydrophobic droplet heat transfer data. 4.3. Internal temperature distribution in droplets
Fig. 5. Nu vs. Bi for different contact angles (50°–80°).
ð0:1 < Bi < 1000; 90 < h < 130 Þ, with AAD = 10.93%. The comparison is provided in table format [17]. As there are no other published data to validate hydrophobic droplet condensation heat transfer, five representative hydrophobic cases were also evaluated in interThermalPhaseChangeFoam [24], a volume-of-fluid (VOF) based CFD solver for phase-change heat transfer processes. VOF is an interface capturing technique, which tracks the volume fraction of the liquid phase in each mesh cell with a scalar / 2 ½0; 1. Eq. (11) is the advection equation for /, where ui is the velocity vector and /_ is the volumetric phase fraction source due to condensation phase change. A single set of governing equations is solved for both the liquid and the vapor phases. Fluid material properties are weighted by Eq. (12).
@/ @ þ ðui /Þ ¼ /_ @t @xi b ¼ /bL þ ð1 /ÞbV
ð11Þ b 2 ½q; l; k
ð12Þ
To reduce the VOF computation time for the sessile, quasisteady droplets considered here, the momentum and phase fraction advection (Eq. (11)) equations are not evaluated (velocity field u ¼ 0, phase-fraction field / constant). The steady state VOF thermal transport equation (Eq. (13)) is solved for a water droplet ðr ¼ 1 mmÞ with different contact angles (90°–130°). q_ pc is the volumetric condensation phase-change heat source term applied on the interface. Interfacial resistance is applied as a distributed source term on the diffuse interface region (4 mesh cells thick).
Temperature distribution trends inside droplets were found to depend more strongly on Bi than h. As Bi reduces, the conduction resistance also reduces relative to the interfacial resistance. Thus, as Bi reduces, the internal droplet temperature distribution becomes more uniform. For higher Bi, internal conduction resistance is significant and condensation heat transfer primarily occurs in a thin region near the contact line. Fig. 5 presents the temperature distribution inside a droplet with varying Bi (h ¼ 90 , droplet geometry and conditions described in Section 3). It is evident from Fig. 6 that for Bi < 0.2, the droplet is effectively isothermal. This effect can also be observed in Fig. 7, which presents the local heat flux at a given angle ranging from b = 0° at the contact line to b = 90° at the top of the droplet. For Bi = 0.2, condensation heat flux only varies by 19% over the surface of the droplet. In contrast, for Bi = 1000 (high Bi), 90% of the heat transfer occurs in the 20° of the droplet surface closest to the contact line. Trends of temperature distribution inside droplets were not found to depend strongly on the contact angle. As can be observed in Fig. 8, qualitatively similar temperature distributions are found for h = 40°–140° for a droplet at Bi = 1000. 4.4. Implications of varying conduction resistance factor in dropwise condensation Many prior dropwise condensation studies have assumed a constant value of conduction factor f. However, this approximation can yield incorrect heat transfer predictions for cases with timevarying droplet size distributions. As an example, consider the growth of a single water droplet ðh ¼ 90 ; r ¼ 100 lmÞ, on a surface maintained 10 K below the saturation temperature ðT sat ¼ 373:15 KÞ. The quasi-steady heat transfer through a growing droplet is given as [25]:
q ¼ qhlv 2pr2
dr dt
ð14Þ
Equating Eq. (14) with Eq. (1), the growth rate of a droplet is:
dr DT 1 ¼ dt 2qhlv frk þ 2h1
ð15Þ i
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Table 2 Average heat flux through the base of droplets with representative hydrophobic Bi and h values, computed with VOF solver, COMSOL conduction studies, and proposed correlation. Biot no. (Bi)
q00 (W m2) VOF
Contact angle (h)
90
8.87 10 ± 4300
8.75 10 ± 3.15
8.75 105
100
5.43 103 ± 0.50
5.40 103 ± 700
5.57 103
110 120 130
3.75E 105 ± 6.0 1.50 106 ± 7.0 1.32 106 ± 3200
3.72 105 ± 8600 1.47 106 ± 27000 1.30 106 ± 38000
3.82 105 1.43 106 1.28 106
5:88 10 2:40 0:10
3:68 101
Fig. 6. Non-dimensional temperature
TT wall T sat T wall
5
q00 (W m2) correlation
2
1:98 10
5
q00 (W m2) COMSOL
3
distribution inside a droplet with varying Bi. (a) Bi = 1000, (b) Bi = 2 and (c) Bi = 0.2.
Fig. 7. Heat flux along the condensing surface of the droplet with h ¼ 90 for different Bi. b is the angle subtended by any point on the droplet surface and the horizontal (inset).
Eq. (15) is integrated for 1.5 s to predict the growth of the droplet. This analysis is performed assuming both a constant value of f = 0.25 [12] and the proposed correlation for f (Eq. (10)), which varies with the instantaneous droplet radius. This droplet growth case is also evaluated using the interfacecapturing volume-of-fluid (VOF) based solver, interThermalPhaseChangeFoam to provide a high-resolution reference case. The domain is axisymmetric and 400 lm 400 lm in height and width. A fine mesh with an average element dimension D ¼ 0:625 lm is employed. The momentum and phase fraction
Fig. 9. Comparison of droplet growth between OpenFoam VOF simulation, theoretical relation with f obtained using the correlation and theoretical relation with f = 0.25.
advection equations were solved here, unlike in the quasi-steady cases described in the previous section. Assuming a constant value of f yields a 23% relative error in the droplet growth with respect to the VOF simulation (Drf = 0.25 = 71 lm, DrVOF = 92 lm). Based on the trend at t = 1.5 s (Fig. 9), this deviation would continue to grow if the analysis were continued to a later time. However, the proposed correlation for f results in only a 3% error. This analysis also indicates the accuracy of the proposed correlation, and its applicability to unsteady droplet condensation cases.
Fig. 8. Non-dimensional temperature distribution inside droplets with different contact angles for Bi = 1000. (a) h ¼ 40 , (b) h ¼ 90 and (c) h ¼ 140 .
S. Adhikari et al. / International Journal of Heat and Mass Transfer 115 (2017) 926–931
5. Conclusion In this study, finite element analyses were performed for condensation heat transfer in sessile droplets accounting for internal conduction and interfacial resistance. Parametric studies were performed over a wide range of Biot numbers (0.0001 6 Bi 6 1000) and contact angles (10° 6 h 6 170°) to determine conduction thermal resistance factors (f) and corresponding uncertainties. Summarized data and simulation results from this study are publicly available at [17]. A new correlation was proposed for f ðBi; hÞ that is valid over the studied range of conditions. This model was validated against an earlier hydrophilic model, only applicable for part of this range. Independent VOF simulations were performed to validate hydrophobic predictions. An analysis was performed to highlight the importance of accounting for the variation of f with different size droplets. As shown in Fig. 9, assuming a fixed value of f can lead to large errors in droplet growth predictions. The proposed, broadly applicable, correlation for f can thus facilitate future dropwise condensation analyses and research.
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