Heat transfer model for dropwise condensation on hydrophobic and superhydrophobic interfaces

Abstract

Keywords: Dropwise Condensation; Biot Number; Heat Transfer; Hydrophobic Surface; Superhydrophobic Surface; Interface Temperature

Introduction

Condensation is a phase change mechanism widely used in many applications such as power generation [1-3], desalination [4], electronic thermal management [5], heat exchangers [6], building energy management [7-8], refrigeration, and air conditioning system [9]. Condensation on

non-wetting surfaces offers dropwise condensation due to low surface energy. The focus on dropwise condensation has increased widely since its discovery in 1930 [10]. This mechanism yields heat transfer rates up to an order of magnitude higher than filmwise condensation [11-12]. The low surface energy of a substrate offers increases in contact

*Corresponding author. *E-mail address: yuvarajr@sonatech.ac.in This paper was recommended for publication in revised form by Regional Editor Omid Mahian Published by Yıldız Technical University Press, İstanbul, Turkey Copyright 2021, Yıldız Technical University. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

angle (θ) from hydrophilic (θ < 90°) to hydrophobic (θ > 90°) surfaces. Several works have since fabricated superhydrophobic (θ ≥ 150°) θ ≥ 150° surfaces to achieve a variety of applications, including self-cleaning, defrosting, and condensation heat transfer enhancement. The initial droplet size of a diameter < 1 μm condensate on the dropwise condensation process at discrete nucleation sites distributed over a cooled surface [13]. The minimum thermodynamically viable radius (rmin) [14] of the condensed droplet grows due to condensation and coalescence with neighboring droplets until they become large enough (rmax) to be removed by gravity. The Rolling and merging of large droplets grow up to the maximum diameter and depart from the surface. This allows new primary droplets of radius rmin to form and grow from nucleation sites to the maximum radius rmax. In dropwise condensation, heat from the vapor transfers through the liquid-vapor interface and then transfers through the droplet by conduction to the cooled wall. Though many experimental studies [15-31] and numerical studies [32-35] of dropwise condensation on hydrophobic and superhydrophobic surfaces have been conducted in the past decades, accurate modeling of the heat transfer process has lagged. Kim and Kim proposed an analytical model to predict the heat transfer rate (Qd) through a single droplet by considering heat drop due to interfacial resistance (ΔTi ), curvature resistance (ΔTc), droplet resistance (ΔTdrop), and the coating resistance (ΔTcoat) [36]. Chavan et al. proposed an individual heat transfer model with the Nusselt number Nu defined as the ratio of convective resistance to conductive resistance and the Biot number Bi defined as the ratio of internal resistance to surface resistance. Adhikari et al. presented the average heat flux through the base of the droplet as [35] with the scaling factor f (Bi, θ)is a function of Biot number Bi and contact angle θ, for the conduction contribution to the overall thermal resistance. Most of the previous studies’ assumed f = 0 for an infinitely conductive droplet [14,39]. Several studies have been performed to determine the formulae for f (Bi, θ). Sadhal and Martin proposed a model to find the scaling factor for contact angle (θ < 90°) [41]. Yuvaraj et al. [42] reported dropwise condensation on the superhydrophobic surface with regular complete sphere-like droplets, increasing the contact surface area between the vapor and the plate surface. There is no room for the formation of films on the superhydrophobic surface. Instead, the water particles are formed as droplets, resulting decrease in resistance to the flow of heat and the fall-off diameter of the droplet from the superhydrophobic surface. This causes an increase in the heat transfer coefficient of dropwise condensation and the rate of condensation. Chavan et al. have reported numerical and experimental results for hydrophobic surfaces (θ > 90°) along with three different correlations to predict f (Bi, θ) [21]. Generally, this model would provide discontinuity in the simulation for dropwise condensation analyses. Accurate modeling of the continuous heat transfer process

from nano-size to the macro size of the condensate droplet on hydrophobic and superhydrophobic interfaces is currently unavailable. In the present study, a simple heat transfer model for dropwise condensation in terms of Biot number is developed to provide continuous results for heat transfer, temperature variation at the interfaces, and the growth rate of droplet condensate on the hydrophobic and superhydrophobic interfaces by using Archimedes’ hat-box theorem. The present model is validated with analytical results obtained using the Kim and Kim model [36] and experimental results given by Chavan et al. [21] of similar working parameters made excellent agreements. The continuous results from the nano to the macro level are obtained from the present model and compared with the experimental data presented by Chavan et al. [21]. The modified model is used for analyzing heat transfer, heat flux, the growth rate of the droplet, and heat drops due to various thermal resistances and interface temperatures by varying Biot numbers Bi = 0.001 → 1000 using Python 3.6.1 with an accuracy of 10-4. Python is an open-source high-level computer language and very simple compared to other high-level languages. The present dropwise condensation heat transfer model is developed and executed by creating code using Python version 3.6.1. Python contains many libraries, including NumPy, MatPlotLib, Xlsxwriter, etc., for solving linear algebraic equations, plotting the results, and storing the data. Python 3.6.1 version also supports the plotting of contours and heat maps.

Modeling Approach

The analytical heat transfer model presented by Kim and Kim is given as [36] (1)

where hi is the interfacial heat transfer coefficient, r is the radius of the droplet, θ is the contact angle, k is the thermal conductivity of the water droplet, δ is promoter coating layer thickness, kcoat is the thermal conductivity of the coating, and rmin is the minimum thermodynamic viable radius of the droplet. The heat transfer through a droplet also be calculated from the enthalpy dissipated by the condensing vapor [34] (6) where ρ is the density of water, hfg is the latent heat and is the growth rate of droplet condensate. The interfacial heat transfer coefficient is determined by using the kinetic theory [37-38]. (7) where α is the condensation coefficient, Rg is the gas constant, vg is the specific volume of vapor and Tsat is the saturation temperature of the vapor. Condensation coefficient α is defined as the ratio of vapor molecules captured by the liquid phase to the total number of vapor molecules reaching the liquid surface (α = 0 → 1). In the present work, a clean environment is assumed, and α is taken as 1. Chavan et al. proposed an individual heat transfer model as [21]

(8) where Nu is the Nusselt number defined as the ratio of convective resistance to conductive resistance, and Bi is the

Biot number defined as the ratio of internal resistance to surface resistance. Adhikari et al. presented the average heat flux through the base of the droplet as [35] (9) where, f (Bi, θ) is the scaling factor, a function of Biot number Bi and contact angle θ, for the conduction contribution to the overall thermal resistance. Most of the previous studies’ assumed f = 0 for an infinitely conductive droplet [14,39]. For f = 0.25 as assumed by Graham and Griffith [40], the conduction-front propagation velocity is greater than the growth velocity . Several studies have been performed to determine the formulae for f (Bi, θ). Figure 1 shows the water droplet underneath a flat surface of radius r, contact radius rc, droplet height h, contact angle θ with negligible hysteresis, equivalent height H, the volume of the droplet V, and its equivalent volume Ve is obtained from Archimedes’ hat-box theorem [43]. By equating V = Ve, the relation between h and H can be written as (10) Rose and Glicksman reported that most heat is transferred through tiny droplets of constant condensation rate with time [46]. Droplet diameter d < 150 μm transfer 90% of total heat for low-pressure condensation, which occupies 35% of surface area, and d < 40 μm transfer 90% of heat at atmospheric pressure with 23% of the surface area [40]. Droplets in this size range are sufficiently small such that Marangoni circulation and other convection effects can be neglected [45-46]. Hence, conduction is the primary

mode of heat transfer in dropwise condensation. As droplets grow, the internal temperature distribution varies, and conduction-front propagation velocity is much greater than the droplet growth rate. Therefore, the heat transfer process can be modeled as quasi-steady, and the in-droplet condensation heat transfer process reduces to conduction in a solid medium [35]. With this assumption, the conductive resistance of the droplet can be rewritten as (11) The Biot number Bi and the Nusselt number Nu are defined in terms of contact radius rc as (12)

(13) After substituting and rearranging the eq (10) and eq (11) in eq (5), the heat transfer model in terms of Biot number can be expressed as

(14) where c = 6πk2/hi is the condensation coefficient, a is the empirical value and b = Bicoat/Bi is the Biot number ratio between a coating to the droplet. The following parameters are assumed as similar to the previous works [34-36], interface heat transfer coefficient hi = 400 MW/ m2 K , the thermal conductivity of the water droplet k =

Results And Discussion

Validation of the Present Model with Analytical Model [36] and Numerical Model [21] The present model is validated by an analytical model proposed by Kim and Kim [36] and a numerical model presented by Chavan et al. [21] for the growth rate of the droplet as a function of the radius (r) of the droplet given in eq (16). The analytical model produces continuous results for different contact angles (θ = 90°, 120°, 150° & 170°) as shown in Figure 2. Whereas, in the numerical model, the discontinuities are noted in Figure 2. due to three different correlations, i) Bi < 0.5 ii) 0.5 < Bi < 2, and iii) Bi > 2, for predicting the Nu are used by Chavan et al. [21] Chavan et al. [21] compared the experimental growth of the droplet with numerical results and Kim’s analytical results and reported analytical model underestimates the growth rate due to the lack of taking into account local heat transfer effects at the three-phase contact line. Also, it was observed in the experiments that the large droplet diameters

(15) Eq (15) is the modified heat transfer model used in the present work without noncondensable gases during condensation. The present model can simulate heat transfer

Figure 2. Validation of the present model with Kim’s analytical model [36] and Chavan’s numerical model [21] regarding growth rate.

growing on the superhydrophobic surface lower the growth rates due to added conduction thermal resistance and low droplet diameters growing faster due to interfacial thermal resistance dominated growth [21]. When the contact angle θ = 90°, the present model deviates from the numerical model at a radius of the droplet r ≈ 50 μm and it produces an identical result at r = 1 mm. The present model made good agreements with the numerical results when the contact angle increases from θ = 90° to θ = 170° as shown in Figure 2. Hence, the present model is best suitable for dropwise condensation on hydrophobic and superhydrophobic interfaces. Validation of the Present Model with the Experimental works [21,24] The present model is also validated by experimental results obtained by Chavan et al. [21] and Enright et al. [24] for the radius of the droplet r as a function of time t. The experimental results are shown in Figures 3 (a) and

(b) obtained by Chavan et al. for the subcooling temperature ΔTexperimental = 5°C and corresponding contact angles θ = 140° and θ = 170° respectively. In the present work, the ΔTnumerical = 0.0545°C for θ = 140° and ΔTnumerical = 0.092°C for θ = 170° are taken by mapping the experimental results and made an excellent agreement for the case θ = 140° and good agreement made for the θ = 170°. The difference between the present model and the experiments can be recognized as the presence of noncondensable gases (NCG). Rose reported that the condensation of water vapor in the presence of NGCs (air) acts as a diffusion barrier for water vapor [15]. The present model is also validated by another experimental result obtained by Enright et al. [24] best fits with ΔTnumerical = 0.064°C as shown in Figure 3 (c). To obtain the complete simulation of the present model by varying contact angles from θ = 90° → 180°, the ΔTnumerical = 0.0545°C is used due to its best fit with experimental results.

Figure 3. Validation of present work with Kim’s analytical [36] and Chavan’s experimental work [21] for (a) contact angle θ = 140°, (b) contact angle θ = 170° and Enright’s experimental work [24] for (c) contact angle θ = 120°.

Simulation of Heat Transfer Through a Single Droplet In the present work, the heat transfer, heat drop, interface temperature variation, and growth rates are analyzed by simulating the eq (14) for different values of Biot number and contact angles. Since the Biot number is a function of the contact angle and radius of the droplet, Bi = f (θ,r) the variation of droplet radius is shown in Figure 4. The droplet radius measures the macro scale for the Biot numbers in the range of 10 > Bi > 1000, microscale for the Biot numbers in the range of 0.01 > Bi > 1, and nanoscale for the Biot numbers 0.00001 > Bi > 0.001. Since the minimum thermodynamically viable radius rmin ≈ 10 nm size, the range of Biot numbers in the present work is chosen from Bi = 0.0001 → 1000. Because of the spherical nature of the droplet, the radius of the droplet varies in a similar pattern against the contact angles θ = 90° to θ ≈ 180° for different Biot numbers, as shown in Figure 4.

Figure 4. Variation of droplet radius with contact angle for different Biot numbers (Bi = 0.001 → 1000).

The heat transfer through a single droplet Qd is shown in Figure 5 for different Biot numbers Bi = 0.01 → 1 in Figure (a) and Bi = 4 → 1000 in Figure 5 (b). The thermodynamically viable radius of a droplet can be expressed in the form as (17) Where σ is the surface tension of the water, in the literature, the subcooling temperature has been taken in the range ΔT = 1K - 10K. The subcooling temperature ΔT = 10K and the corresponding minimum viable radius of the droplet rmin ≈ 10 nm are used to solve the present model [21,34,36]. In the present work, the numerical subcooling temperature is taken as ΔTnumerical = 0.0545 K for the best fit of experimental data and the respective viable radius rmin ≈ 400 nm is used for simulation. The curvature effect is used in eq (14) becomes negative when the radius of the droplet r is smaller than the rmin value i.e. 400 nm. From this point forward, whatever the values are in the negative sign means that r < rmin. The heat transfer through a single droplet Qd is plotted as a function of contact angles varying from θ = 90° → 180° as shown in Figure 5. When the Bi < 0.3, the heat transfer through the droplet is very small due to high interface resistance and curvature resistance. The heat transfer rate increases with an increase in contact angle for Bi > 0.3 as shown in Figure 5 (a). Further, with an increase in Biot number towards Bi = 1 the variation of heat transfer follows linearly and approaches a constant line when Bi = 4 as shown in Figure 5 (b). The heat transfer remains constant for all the values of contact angles in both hydrophobic and superhydrophobic sides at Bi = 4. The heat transfer rate decreases with an increase in contact angle when the Biot number is increased from Bi = 4 → 1000 as shown in Figure 5 (b). It is essential to study the variations in the region of Bi < 0.3, to 1000.

(a) (b) Figure 5. Heat transfer rate for different contact angles (a) Bi = 0.01 → 1 and (b) Bi = 4 → 1000.

Figure 6. Heat transfer contours for different contact angles (a) Bi = 0.001 → 1, (b) Bi = 1 → 1000 , (c) Bi = 0.01 → 1 , (d) Bi = 0.1 → 0.3, (e) Bi = 1 → 4, and (f) Bi = 4 → 10. The heat transfer variations from Bi = 0.001 → 1000 are predicted by using the modified model eq (14), and contours are plotted as shown in Figure 6 (a-b). The maximum heat transfer rate in the superhydrophobic region for lower Biot numbers shown in Figure 6 (a) is shifted to the hydrophobic region for higher Biot numbers as shown in Figure 6 (b). Figure 6 (c-f) shows the higher

heat transfer shifting behavior for different ranges of Biot numbers Bi = 0.01 → 0.1, Bi = 0.1 → 0.3, Bi = 1 → 4, and Bi = 4 → 10 respectively. The cold spot (lower heat transfer rate) in the hydrophobic region and the hot spot (higher heat transfer rate) in the superhydrophobic region for Bi = 0.01 → 0.1 are shown in Figure 6 (c). The cold spot propagates hydrophobic to the superhydrophobic region,

(a) (b) Figure 7. Heat flux with different contact angles for (a) Bi = 0.01 → 1 and (b) Bi = 1 → 1000. and Bi = 0.1 → 0.3 both regions become cold spots, as shown in Figure 6 (d). An important finding is observed in Figure 6 (e) that the hot spot is almost the same for all contact angles at Bi = 4. In Figure 6 (f), the hot spot is shifted from the superhydrophobic to the hydrophobic region at Bi = 10 and the heat transfer rate decreases as the contact angle increases. The heat flux qd is the ratio between the heat transfer rate of a single droplet Qd and the contact area of the droplet between the solid and liquid interface. Figure 7 (a) shows the heat flux varies nearly similar to the heat transfer rate up to Bi = 4. In Figure 7 (b), the heat flux is constant throughout the hydrophobic and superhydrophobic region at Bi = 4, and afterward, increase in Biot number decreases the heat flux due to the high conductive resistance offered by the solid-liquid interface area. When the Bi > 4, the heat flux decreases for increases in contact angle.

The heat drop due to various resistances, interface resistance ΔTInt, curvature resistance ΔTCurv, droplet resistance ΔTDrop, and coating resistance ΔTCoat, are analyzed and shown in Figures 8-11 for different Biot numbers. In the condensation process, the first and foremost resistance is interfacial resistance. Eq (1) is used to predict the heat drop due to interfacial resistance for Bi = 0.01 → 1 as shown in Figure 8 (a), the maximum heat drop is attained at Bi = 1, and for Bi = 1 → 1000 as shown in Figure 8 (b), the heat drop decreases when the Biot number increases. For all the Biot numbers, when Bi > 0.3, minimum heat drop in the superhydrophobic region is occurred due to the large interface area offered by the superhydrophobic surface. When the contact angle exceeds 170°, Figures 8 (a) and (b) show a sudden heat drop. The increase in the contact angle decreases the heat drop due to the greater interfacial resistance offered by the liquid-vapor

(a) (b) Figure 8. Heat drop due to interfacial resistance between droplet and vapor with contact angles (θ = 90° → 179°) for different Biot numbers (a) Bi = 0.01 → 1 and (b) Bi = 1 → 1000.

Figure 9. Heat drop due to curvature effect with contact angles (θ = 90° → 179°) for different Biot numbers (Bi = 0.0001 → 1000).

Figure 10. Heat drop due to conductive resistance inside the droplet with contact angles (θ = 90° → 179°) for different Biot numbers (Bi = 0.01 → 1000).

interface. For a lower Biot number, the liquid-vapor surface area approaches zero, resulting in heat drop due to interfacial resistance, as shown in Figure 8 (a). The maximum heat drop of ΔTInt = 0.008 K is obtained for Bi = 1 and θ = 90°, which is 15% of the total heat drop ΔTInt = 0.0545 K. The heat drop due to curvature resistance ΔTCurv is predicted using the eq (2) for different Biot numbers shown in Figure 9. The curvature resistance only depends on the droplet’s radius, and the variation shows an identical pattern for all the Biot numbers. The maximum heat drop in Figure 9 is noted for the lowest Biot number Bi = 0.0001 and decreases with an increase in Biot numbers. For larger values of Biot numbers, the curvature effect can be neglected. The curvature resistance decreases for increases in contact angles, and the maximum heat drop for the Bi = 1 is obtained as ΔTCurv = 0.02, which is 37% of the total heat drop.

Eq (11) is the modified heat drop ΔTDrop due to droplet resistance and predicts the heat drops for different Biot numbers, as shown in Figure 10. The conductive resistance offered by the droplet is the primary source of heat drop for Bi > 0.1. The heat drop due to conductive resistance increases with an increase in contact angles for the Biot numbers Bi = 0.1 → 10 and closely follows linear variation. When the Biot number is greater than 100, the heat drop decreases in the hydrophobic region from contact angle θ = 90° → 140° and increases in the superhydrophobic region. The maximum heat drop occurs at a contact angle θ ≈ 180° for Bi = 1 is ΔTDrop = 0.02 K and at θ = 90° ΔTDrop = 0.005 K which is 9% of the total heat drop. The coating resistance is predicted by using Eq (4) for different Biot numbers shown in Figures 11 (a) and (b). The

(a) (b) Figure 11. Heat drop due to coating resistance with contact angles (θ = 90° → 179°) for different Biot numbers (a) Bi =

16. The maximum interface temperature Tsl for the contact

angle, θ = 140° and θ = 170°are obtained at Bi ≈ 0.5 and Bi ≈

Conclusion

An unique heat transfer model for dropwise condensation on the hydrophobic and superhydrophobic interface is developed by correlating the Biot number and contact angle. The developed model is validated with analytical, numerical, and experimental work, offering excellent agreement. The present heat transfer model is modified for the best fit of observed values. The modified heat transfer model is used to predict the heat transfer through the single droplet, heat flux, heat drops due to various resistances such as interfacial resistance, curvature resistance, droplet resistance, coating resistance, the growth rate of the droplet, and interfacial temperatures for different Biot numbers varies from Bi = 0.0001 → 1000 and different contact angles in the range θ = 90° → 180°. Besides, heat transfer contours are plotted for other Biot numbers as a function of contact angles. From the simulation of the present modified model, the following are the findings for dropwise condensation on hydrophobic and superhydrophobic interfaces, • A continuous single correlation model predicts the heat transfer for a vast range of Biot numbers Bi = 0.0001 → 1000 • For the best fit, the numerical value for subcooling temperature is found as ΔTnumerical = 0.0545 K against the experimental value ΔTexperimental = 5 K. • At Bi = 4, the heat transfer rate remains unchanged for all the contact angles in the hydrophobic and superhydrophobic regions. • A wide variation of the growth rate for all the Biot numbers in the range Bi = 0.0001 → 1000 is presented. • The heat drop due to interfacial resistance reaches a maximum value at Bi = 1. • For Bi = 1, the total heat drops are almost constant for all the contact angles in the hydrophobic and superhydrophobic regions. • The Nusselt number shift occurs at Bi ≈ 5. • The interfacial resistance and curvature effect, and coating resistance are significant for Bi < 0.01, and droplet resistance is substantial for Bi > 0.05. • The liquid-vapor interface temperature reaches its maximum value at Bi ≈ 5 and remains constant afterward. • The solid-liquid interface temperature reaches a maximum at Bi ≈ 0.4 → 0.5 and decreases afterward.

Nomenclature

Bi Biot number Nu Nusselt number θ Contact angle, degree r Droplet radius, m rmin Minimum viable radius, m rmax Maximum radius, m Q Heat transfer rate, W ΔT Temperature difference, K hi Interface heat transfer coefficient, W/m2K δ Coating thickness, m k Thermal conductivity, W/mK ρ Density, kg/m2 hfg Latent heat, J/kg dr/dt Growth rate, m/s rc Contact radius, m h Droplet height, m V Volume, m3 H Effective height, m a & b Empirical value c Condensation coefficient

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