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Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure
Applied Thermal Engineering xxx (2014) 1e9
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure Rongfu Wen, Zhong Lan, Benli Peng, Wei Xu, Xuehu Ma* Liaoning Provincial Key Laboratory of Clean Utilization of Chemical Resources, Institute of Chemical Engineering, Dalian University of Technology, Dalian 116024, China
h i g h l i g h t s Transients of initial droplet size distribution were experimentally studied. Surface coverage of steady condensation was strongly dependent on steam pressure. Effective heat transfer area reduced and resistance increased at low pressure. Heat transfer resistance distribution was sensitive to the steam pressure.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2014 Received in revised form 1 August 2014 Accepted 20 September 2014 Available online xxx
To investigate the transient characteristics of initial droplet size distribution, steady droplet size distribution and thermal resistance distribution at lower and ultra-lower steam pressure, dropwise condensation at the pressure range from atmospheric to 1.5 kPa has been studied. During the transient process, the initial nucleated droplets satisfied lognormal distribution, and then a bimodal distribution formed, finally revealed an exponential distribution. The peak value was smaller and the evolution was slower with the reduction of steam pressure. The corresponding surface coverage increased to 0.7e0.8 at the steady condensation which was strongly dependent on the pressure. Introducing a dimensionless time, the surface coverage evolution indicated that the time consumed by direct growth increased as the pressure decreased. The effect of steam pressure on droplet size distribution revealed a more scattered distribution, larger departure size, and denser large droplets at low pressure, resulting in the reduction of the effective heat transfer area. By comparing the thermal resistance distribution at various pressures, it showed that large droplets induced a greater proportion of resistance at low pressure. The findings help clarifying the limitations of droplet growth mechanism and offer guidelines for the optimization of surface morphology to enhance the steam condensation at low and ultra-low pressure. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Dropwise condensation Low pressure steam Droplet size distribution Surface coverage Dynamic evolution Thermal resistance distribution
1. Introduction The mechanisms governing steam condensation at low or ultralow pressure are crucial to a wide range of applications which have significant economic and environmental impacts, such as power generation [1], low temperature multi-effect seawater desalination [2], water harvesting in space station [3], thermal management systems [4,5] and environmental control [6]. Water vapor preferentially condenses on solid surfaces rather than directly from the vapor due to the reduced activation energy of heterogeneous nucleation as compared to homogeneous nucleation [7]. When * Corresponding author. Tel.: þ86 411 84707892; fax: þ86 411 84707700. E-mail addresses: [email protected] (R. Wen), [email protected] (X. Ma).
water vapor condenses on a surface, the condensate can form either a liquid film or distinct droplets, depending on the surface wettability. The latter, termed dropwise condensation is more desirable since droplets can efficiently remove from the surface in comparison to filmwise condensation. Droplet growth mechanism in dropwise condensation affects the heat transfer rate notably, for some cases with lower heat transfer rate, such as condensation at very low pressure [8] or with large amount non-condensable gas [9], droplet growth rate becomes very low. Meanwhile, condensed droplets stay on condensing surface longer so droplet dynamics and distribution will not be similar any more. Based on the population balance theory, Wu and Maa [10] deduced the distribution of small droplets less than the critical radius. For larger droplets, the droplet size distribution function proposed by Le Fevre and Rose [11] was still widely used.
http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069 1359-4311/© 2014 Elsevier Ltd. All rights reserved.
Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069
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Nomenclature
rvmin Rd
Variables Ad Ade Atotal c g h hi
Rdrop Rtotal t t' Tw DT
Hfg kcoat kw n N Nre Ns Pv r rd re rmax rmin
weighted base area of the same size droplets (m2) base area of individual droplet (m2) base area of all the condensed droplets (m2) constant dependent on the geometry of droplet (m/s2) gravitational acceleration (m/s2) droplet height (m) the condensation interfacial heat transfer coefficient (W/m2K) the latent heat of vaporization (J/kg) thermal conductivity of the coating material (W/m K) thermal conductivity of water (W/m K) density of small droplets (m3) density of large droplets (m3) population of droplets in the rectangular area density of nucleation site (m2) vapor saturation pressure (Pa) droplet radius (m) contact radius (m) effective droplet radius (m) maximum droplet radius (m) minimum droplet radius (m)
Subsequently, Wu [12] put forward a random fractal model to simulate the droplet size distribution from the primary to the departing droplet in dropwise condensation. Mei [13] derived expressions for fractal dimension and area fraction of droplet sizes correlated with surface subcooling. In addition, Lan [14] rebuilt the spatial conformation of droplet distribution into the temporal conformation based on the random fractal model and reflected the dynamic characteristics of dropwise condensation. According the dynamic nature of droplet growth on structured superhydrophobic surfaces, Miljkovic et al. [15] derived the droplet size distribution dependent on droplet morphology on structured superhydrophobic surfaces. In the investigations mentioned above, the droplet dynamics and distribution characteristics were mainly studied for the steady state dropwise condensation of pure steam at atmospheric or higher pressure. Owing to the complex coalescing mechanism among droplets, it is very hard to display dynamic evolution of the droplet size distribution theoretically. With the rapid improvement of operational performance of computer, the entire process of dropwise condensation was reappeared and the droplet size distribution was investigated by simulation [12,16e18]. In experiment, Chen, et al. [19] focused on the effect of microscopic topography on the cumulative departure volume and droplet number density. Ucar et al. [20] investigated the effect of surface roughness and contact angle hysteresis of polymeric substrates on the initial droplet density and surface coverage. Transient characteristics of droplet size distribution in the initial dropwise condensation were attributed to the growth and coalescence of the first generation droplets and the evolutions of transient stages on low thermal conductivity surface were affected by steam pressure obviously for the pressure range from atmospheric to 29.4 kPa [21]. As the steam pressure decreases, the droplet dynamics and distribution changes obviously, resulting in a great impact on heat transfer performance. Thus, the droplet dynamics and distribution at low or ultra-lower pressure are very desirable and significant to get insights in the mechanism of dropwise condensation heat transfer at low pressure. The purpose of this paper is to investigate the transient characteristics of initial droplet size distribution, steady droplet size
minimum discernible droplet radius (m) weighted thermal resistance of the same size droplets (m2 K/W) thermal resistance of individual droplet (m2 K/W) thermal resistance of all condensed droplets (m2 K/W) time (s) dimensionless time () surface temperature (K) surface subcooling temperature (K)
Greek letters contact angle ( ) advancing contact angle ( ) receding contact angle ( ) thickness of coating layer (m) density of the condensate (kg/m3) surface tension (N/m) the sweeping period (s) time from re to rmax (s) time from critical size to re (s) 4A weighted coverage ratio occupied by the same size () 4R weighted thermal resistance ratio of the same size () jA coverage ratio occupied by droplets () jR thermal resistance ratio of droplets ()
q qa qr d r s t tco tdi
distribution and heat transfer resistance distribution among various sizes droplets to clarify the limitations of droplet growth mechanism at low and ultra-low pressure. Droplet size distribution evolution and surface coverage evolution are studied in the circle of droplet growth from nucleation to the steady condensation. During the steady condensation, the droplet size distribution features, departure size and the density of large droplets are analyzed and compared with the simulation results. Furthermore, the heat transfer resistance distributions based on droplet size distribution at various steam pressures are compared.
2. Experiments 2.1. Apparatus The closed system mainly consists of a boiler, cooling water, condensing chamber, and data acquisition and control unit. A cylindrical condensing block, 13 mm in diameter and 22 mm long made of high purity copper was thermally insulated with PTFE to ensure the one dimensional steady-state conduction, as shown in Fig. 1. The condensing surface was oriented vertically. Five thermocouple holes, 0.8 mm in diameter, were drilled into the block in parallel along the stream. The width of space from condensing surface to window was approximately five mm. The pressure was measured by a manometer with the accuracy of 0.1 kPa combined with a McLeod vacuum gauge with the range from 0.1 Pa to 10 kPa. Experimental data were measured and collected with the Agilent34970A data acquisition system. An installed window on the test section facilitated the observation of the condensation process. The camera system (PHOTRON, FASTCAM APX-RS) mounted with a set of microscope lenses (HIROX, CX10C) was used to record the droplet behaviors. The highest shooting speed is 3000 fps at 1024 1024 pixels, maximum 250,000 fps in lower resolution. With various objective lenses, the magnification can achieve 7000. The smallest size of a droplet which can be observed by microscopy is 1 mm in diameter.
Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069
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Fig. 1. Schematic of experimental setup and condensing chamber.
2.2. Experimental procedure
2.3. Surfaces preparation and characterization
Great care was taken to degas the system as well as to make it gastight. Deionized water was supplied to the boiler. Setting the auxiliary vacuum pump in operation, the main system was evacuated and the pressure decreased to the saturation pressure corresponding to the room temperature. Then, the test loop was kept running at a high evaporation rate for an hour. The accumulated non-condensable gas was removed by the auxiliary vacuum pump at intervals. The degassed water was restored to the boiler. Setting the auxiliary vacuum pump in operation, the main system was evacuated and became dry. Further, the system was evacuated to a pressure Pv < 0.5 ± 0.1 Pa by the main vacuum pump. The chamber pressure and temperature were continuously monitored to ensure saturated condensations through the experiments. The steam generated by the boiler condensed partly in the chamber and the excess steam condensed in an auxiliary condenser. When the steam pressure in the chamber came to the expected pressure, experiments were conducted from large surface subcooling to small surface subcooling. As the surface subcooling decreased, the heat transfer performance of auxiliary condenser was improved by increasing the flow rate of cooling water. In this way, the increased excess steam condensed and the pressure of the chamber was adjusted. The wall temperature Tw was calculated by extrapolating the temperature gradient to the condensing surface. Additional data reduction was well described elsewhere [22]. The uncertainties of the measured and calculated parameters were estimated by the procedure described by Kline and McClintock [23,24]. The experimental uncertainties associated with the sensors and calculated parameters are listed in Table 1.
The n-octadecyl mercaptan was used as the surface promoter. The condensing side end of the copper block was finely polished. Then, the copper block was rinsed by deionized water and acetone, dried in the air and immersed in the 2.5 mmol/L solution of noctadecyl mercaptan in ethanol for 1 h at 343 K. Finally, a selfassembly monolayer formed. The scanning electron microscope (SEM) image shown in Fig. 2(a) illustrated the morphology of the hydrophobic surface. Contact angles (q), advancing contact angles (qa) and receding contact angles (qr) were measured to determine the surface wettability. The mean contact angle was 120.0 (DataPhysics OCAH200, Germany, ±0.1 ), as shown in Fig. 2(b).
Table 1 Experimental uncertainties containing sensors and key parameters. Sensors Thermocouple Pressure transducer Parameters Pressure (Pv) 1.5 kPa 31 kPa 101 kPa
±0.03 K ±10 Pa ±100 Pa
(0e10 kPa) (10e101 kPa)
Wall temperature (Tw) ±(0.035 Ke0.084 K) ±(0.061 Ke0.261 K) ±(0.064 Ke0.355 K)
Surface subcooling (DT) ±(1.3%e4.8%) ±(3.8%e7.8%) ±(5.2%e8.2%)
3. Results and discussion 3.1. Transient characteristics of initial droplet size distribution On a hydrophobic surface, droplet nucleation and growth proceed through two mechanisms: direct growth where droplets nucleate and grow on spatially random sites by direct condensation and coalescence-dominated growth where the droplet size becomes large enough to coalescence with neighboring droplets [25]. Therefore, a wide range of droplet sizes exist on the condensing surface from nucleation to steady condensation and the droplet size distribution reflects the instantaneity in the process of droplet growth [26]. Image analysis package Image-Pro Plus (Cold Spring, USA) was used to measure the droplet number with different sizes. The droplet zones were identified first and then the area and radius of each droplet were calibrated, the radius range in the image and corresponding droplet number were recorded, and finally the droplet size distribution was generated. The transient characteristics of droplet size distribution in the initial condensation at the steam pressure of 1.5 kPa and 60 kPa were investigated and the corresponding spatial droplet distribution images were shown in Fig. 3. At the beginning of nucleation, all the droplet sizes were uniform and droplet size distribution had a peak value, showing the lognormal distribution (t ¼ 0.4 s in Fig. 3(a) and t ¼ 0.08 s in Fig. 3(b)). With droplets growing up, the average droplet
Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069
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Fig. 2. Surface morphology and wettability of condensing surface. (a) SEM image; (b) Contact angle measurement.
size increased (t ¼ 0.8 s in Fig. 3(a)). Then coalescences among droplets led to wider span of droplet sizes and decline of right peak (t ¼ 1.8 s in Fig. 3(a) and t ¼ 0.12 s in Fig. 3(b)). With the increased space among the first generation droplets, new droplets nucleated
and grew at the exposed substrate where bimodal distribution formed (t ¼ 2.8 s in Fig. 3(a) and t ¼ 0.18 s in Fig. 3(b)). With further coalescences and nucleation, the number of larger droplets decreased and the number of smaller droplets increased, and droplet
Fig. 3. Transient steam condensation images and corresponding droplet size distribution. (a) Pv¼1.5kPa,
▵T¼1K; (b) Pv¼60kPa, ▵T¼1K.
Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069
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size range became wider. Finally, the right peak vanished and the distribution became unimodal (t ¼ 6.8 s in Fig. 3(a) and t ¼ 0.68 s in Fig. 3(b)). Note that the transients of droplet size distribution at different pressures showed the same tendency while the droplet density was strongly dependent on the steam pressure in accordance. In order to further indentify the effect of steam pressure on the droplet growth, surface coverage at different pressures were compared, as shown in Fig. 4(a)e(c). The surface coverage went through three typical periods: rapid growth period tdi where droplets nucleated and grew by direct condensation, slow growth period tco where droplets coalesced with neighboring droplets and new droplets nucleated at the exposed position, and steady period where the condensation process tended to be stable. The surface coverage at the beginning of coalescence increased from 0.5 to 0.58 with the pressure increasing from 1.5 kPa to 60 kPa, which was due to the larger density of active nucleation site at the higher pressure. In addition, a high pressure improved the surface coverage of steady condensation from 0.7 at 1.5 kPa to 0.79 at 60 kPa. Fig. 4(d) compares the proportion of time consumed by direct growth stage tdi/ (tdi þ tco) to indentify the droplet growth mechanism at low and ultra-low pressure. It revealed that the time consumed by direct condensation stage accounted for larger proportion at lower pressure. This was because the gas liquid interface transfer resistance rapidly increased and the growth rate of individual droplet reduced.
5
size distribution [28]. The steady state droplet size distribution of small droplets is given by,
1 re 2=3 rðre rmin Þ A2 r þ A3 nðrÞ ¼ expðB1 þ B2 Þ r rmin A2 re þ A3 3pre3 rmax rmax ðrmin < r < re Þ (1) where
A1 ¼
DT 2rHfg
(2)
A2 ¼
qð1 cos qÞ 4kw sin q
(3)
A3 ¼
1 dð1 cos qÞ þ 2hi kcoat sin2 q
(4)
B1 ¼
A2 re2 r 2 r rmin 2 þ rmin ðre rÞ rmin ln tA1 2 re rmin
(5)
3.2. Droplet size distribution features in the steady condensation
B2 ¼
A3 r rmin re r rmin ln tA1 re rmin
(6)
Dropwise condensation consists of the transient process occurring repeatedly. New tiny droplets nucleate on the space after coalescence and departure. These droplets form a new group and exhibit all the features of the first generation droplets. A steady state can thus be reached, with continuous formation of new droplets and flow of the large droplets [27]. The photographs at different instant or in different scales are similar. These features indicate that dropwise condensation appears with self-similarity [12].
where rmin is the droplet nucleation radius (rmin ¼ 2Tss/HfgrDT), re is the radius when droplets begin to merge, rmax is the maximum droplet radius, q is the apparent equilibrium contact angle, r is the water density, kw is the water thermal conductivity, Hfg is the latent heat of vaporization, hi is the condensation interfacial heat transfer coefficient, and t is the sweeping period. For large droplets, the droplet size distribution N(r) was established by Le Fevre and Rose [29],
3.2.1. Droplet size distribution model For small droplets that grow mainly by direct condensation, the population balance theory can be employed to determine droplet
NðrÞ ¼
Fig. 4. Surface coverage during transient steam condensation process. (a) Pv¼1.5kPa, dimensionless time t'¼t/(tdiþtco).
1 3pr 2 r
max
r rmax
2=3
ðre < r < rmax Þ
(7)
▵T¼1K; (b) Pv¼31kPa, ▵T¼1K; (c) Pv¼60kPa, ▵T¼1K; (d) Surface coverage with respect to the
Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069
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The maximum droplet radius rmax can be estimated by the force balance between the surface tension and gravity on a vertical surface, as follows [30],
rmax ¼
6cðcos qr cos qa Þsin q s 2 3 cos q þ cos3 q prg
1=2 (8)
where qr and qa are the advancing and receding contact angles, respectively. The effective radius re is defined as the boundary between the small droplets n(r) and large droplets N(r) by assuming the nucleation forms a square array [31],
1 re ¼ pffiffiffiffiffiffiffiffiffi 4NS
(9)
where NS is a number of nucleation site on a unit area of condensing surface.
3.2.2. Comparison of droplet size distribution Due to the self-similarity feature of droplet size distribution, a representative large enough rectangular area containing various growth stages was determined to investigate the droplet size distribution at different pressures. Droplets down to 10 mm in size were measured and counted on account of the limit of resolution. Fig. 5 compares the calculated results and experimental data at different pressures at a surface subcooling of 1 K. The development tendency of droplet size distribution was very consistent qualitatively with that from experiments at both steam pressures. It is also important to point out that the error between the calculated result and experimental data increased with the reduction of steam pressure. At low steam pressure, the number of small droplet was smaller than the model prediction results (Fig. 5(b)) while the number of large droplet was larger than the model prediction results (Fig. 5(c)). In addition, the frequency of large droplets with different sizes also increased at lower pressure, as shown in Fig. 5(c). In the droplet radius range from 180 mm to 1280 mm, there
Fig. 6. Schematic of heat transfer resistance distribution based on droplet size distribution.
were three sizes of large droplets at atmospheric while seven sizes of large droplets appeared at low pressure which led to the rapid reduction of effective heat transfer area. 3.3. Heat transfer resistance distribution The condensing surface covered by condensed droplets with various sizes has significantly different heat transfer performance. It is necessary to investigate thermal resistance distribution by condensed droplets with respect to the steam pressure. 3.3.1. Heat transfer resistance distribution model Since the area ratio of the liquid/solid interface and surface coverage can be given by the droplet size distribution, the heat
Fig. 5. Comparison of droplet size distribution between model prediction and experimental data. (A) Pv¼101kPa, (B) Pv¼1.5kPa.
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transfer resistance distribution through condensed droplets can also be analyzed by calculating the contact area of liquid/solid interface, liquid/gas interfaces and volume of every individual droplet combining with the droplet size distribution, as shown in Fig. 6. The spherical cap geometry of a droplet can be characterized by four different parameters: droplet height h, contact radius rd, droplet radius r, and contact angle q. Due to the large size of counted droplets in the study, more than 10 mm, the conduction through droplets is the dominant thermal resistance in the dropwise condensation [33]. Thus, the heat transfer resistance through the condensed droplet in the analysis is given as
Rdrop ðrÞ ¼
q 4prkw sin q
(10)
Atotal ¼
rmax X rvmin
4R ðrÞ ¼ Rd ðrÞ=Rtotal
(14)
where
Rd ðrÞ ¼ Rdrop ðrÞ$Nre ðrÞ
Rtotal ¼
rmax X
Rd ðrÞ
(15)
(16)
(11)
where Rdrop(r) refers to the thermal resistance through the droplet with radius r. The ratio of condensing surface covered by droplet with radius below r is expressed as
(12)
jA ðrÞ ¼
where
Ad ðrÞ ¼ Ade ðrÞ$Nre ðrÞ
where Ade(r) and Nre(r) refer to the base area and the number of droplet with radius r, respectively, and rvmin is the radius of minimum discernible droplets in the experiment. The weighted thermal resistance ratio of droplet with radius r can be written as
rvmin
The weighted coverage ratio occupied by droplets with radius r can be written as
4A ðrÞ ¼ Ad ðrÞ=Atotal
7
rmax X
4A ðrÞ
(17)
rvmin
Ad ðrÞ
(13)
The thermal resistance ratio of droplet with radius below r is expressed as
Fig. 7. Spatial droplet distribution images, droplet size distribution, and droplet population histograms. (a) Pv¼1.5kPa; (b) Pv¼31kPa; (c) Pv¼101kPa;(d) Pv¼1.5kPa; (e) Pv¼31kPa; (f) Pv¼101kPa;(g) Pv¼1.5kPa; (h) Pv¼31kPa; (i) Pv¼101kPa.
Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069
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Fig. 8. Droplet coverage ratio and thermal resistance ratio with respect to steam pressure.(a) Accumulated droplet coverage ratio vs. droplet radius; (b) Accumulated of thermal resistance ratio vs. droplet radius.
jR ðrÞ ¼
rmax X
4R ðrÞ
(18)
rvmin
3.3.2. Droplet size distribution dependant on steam pressure Image analysis package Image-Pro Plus was used to measure the number of condensed droplet with different sizes. The droplet zones were identified first and then the area and radius of each droplet were calibrated, the radius range in the image and corresponding droplet number were recorded, and finally the droplet size distribution generated. Fig. 7 shows the spatial droplet distribution images and droplet size distribution by Image-Pro Plus, as well as the droplet population histograms at different pressures. The strong dependence of droplet size distribution on steam pressure was clearly visible. An increase in the number of small droplets (blew ~0.1 mm) was obvious with the increase of pressure, as shown in Fig. 7(i), while the frequency of large droplets (above ~0.3 mm) increased obviously at low pressure, as shown in Fig. 7(g). 3.3.3. Heat transfer resistance distribution dependant on steam pressure It is obvious that more condensing surface covered by small droplets is preferred due to the smaller thermal resistance. The ratio of condensing surface covered by droplets with radiuses below r at different steam pressures is shown in Fig. 8(a). Visual observations confirmed that the surface coverage distribution was significantly influenced by the steam pressure. More condensing surfaces were covered by large droplets at low pressure. As the pressure reduced from 101 kPa to 1.5 kPa the ratio of condensing surface covered by small droplets (below 0.15 mm) decreased from 0.92 to 0.54. The surface coverage ratio by departure droplet increased from 0.08 to 0.13 as the pressure reduced from atmospheric to 1.5 kPa. Note also that the droplet size distribution is more dispersed at low pressure while it is mainly concentrated in small size at atmospheric. Fig. 8(b) shows the thermal resistance ratio of droplets with radiuses below r at different pressures. The thermal resistance focused on the departure droplet and the resistance of the other droplets was almost negligible at atmospheric. As pressure decreased, the thermal resistance ratio of droplet with radius above 0.2 mm increased obviously while the thermal resistance ratio of departure droplet decreased. The thermal resistance ratio of small droplets (below 0.15 mm) was smaller at low pressure which indicated large droplets caused a greater thermal resistance. Therefore, reducing the departure size is an effective way to enhance the condensation of atmospheric steam [32,33] while the improvement of heat transfer at low pressure should focus on the
reduction of the density of large droplets in addition to the reduction of departure size. 4. Conclusions Effect of steam pressure on the transient process of initial droplet size distribution, steady droplet size distribution and heat transfer resistance distribution among condensed droplets was investigated in dropwise condensation at the pressure range from atmospheric to 1.5 kPa. The initial nucleated droplets satisfied lognormal distribution, and then a bimodal distribution formed with the droplets coalesce, finally revealed an exponential distribution at the steady state. The peak value was smaller and the evolution was slower at low steam pressure, which was due to the reduction of the nucleation site density and the droplet growth rate. The surface coverage of steady condensation was strongly dependent on the steam pressure. As the steam pressure decreased, the time period for the droplet direct growth increased and the gas liquid interface transfer resistance rapidly increased. The effective heat transfer area reduced and the heat transfer resistance increased due to the more scattered droplet distribution, larger departure droplets, and denser large droplets at low steam pressure. A comparison of heat transfer resistance distribution at various steam pressures showed that large droplets led to a greater proportion of heat transfer resistance with the reduction of steam pressure. Acknowledgements The work is supported by the National Natural Science Foundation of China (No. 51236002) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120041110018). References [1] J.M. Beer, High efficiency electric power generation: the environmental role, Prog. Energy Combust. Sci. 33 (2007) 107e134. [2] A.D. Khawaji, I.K. Kutubkhanah, J.M. Wie, Advances in sea-water desalination technologies, Desalination 221 (2008) 47e69. [3] H.G. Andrews, E.A. Eccles, W.C.E. Schofield, et al., Three-dimensional hierarchical structures for fog harvesting, Langmuir 27 (2011) 3798e3802. [4] X.H. Ma, J.B. Chen, S.P. Li, et al., Application of absorption heat transformer to recover waste heat from a synthetic rubber plant, Appl. Therm. Eng. 23 (2003) 797e806. [5] D. Milani, A. Abbas, A. Vassallo, et al., Evaluation of using thermoelectric coolers in a dehumidification system to generate freshwater from ambient air, Chem. Eng. Sci. 66 (2011) 2491e2501. [6] L. Perez-Lomavard, J. Ortiz, C. Pout, A review on buildings energy consumption information, Energy Build. 40 (2008) 394e398.
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[21] X.H. Ma, T.Y. Song, Z. Lan, et al., Transient characteristics of initial droplet size distribution and effect of pressure on evolution of transient condensation on low thermal conductivity surface, Int. J. Therm. Sci. 49 (2010) 1517e1526. [22] Z. Lan, X.H. Ma, S.F. Wang, et al., Effect of surface free energy and nanostructures on dropwise condensation, Chem. Eng. J. 156 (2010) 546e552. [23] S.J. Kline, F.A. McClintock, Describing uncertainties in single-sample experiments, Mech. Eng. 75 (1953) 3e8. [24] S.J. Wilcox, W.M. Rohsenow, Film condensation of potassium using copper condensing block for precise wall-temperature measurement, ASME J. Heat Transfer (1970) 359e371. [25] D. Beysens, Dew nucleation and growth, C. R. Phys. 7 (2006) 1082e1100. [26] R. Enright, N. Miljkovic, A. Al-Obeidi, et al., Condensation on superhydrophobic surfaces: the role of local energy barriers and structure length scale, Langmuir 28 (2012) 14424e14432. [27] R.N. Leach, F. Stevens, S.C. Langford, et al., Dropwise condensation: experiments and simulation of nucleation and growth of water drops in a cooling system, Langmuir 22 (2006) 8864e8872. [28] S. Vemuri, K.J. Kim, An experimental and theoretical study on the concept of dropwise condensation, Int. J. Heat Mass Transfer 49 (2006) 649e657. [29] E.J. Le Fevre, J.W. Rose, A theory of heat transfer by dropwise condensation, in: Proceedings of the Third International Heat Transfer Conference, Chicago, vol. 2, 1966, pp. 362e375. [30] C. Graham, P. Griffith, Drop size distributions and heat transfer in dropwise condensation, Int. J. Heat Mass Transfer 16 (1973) 337e346. [31] P. Dimitrakopoulos, J.J.L. Higdon, On the gravitational displacement of threedimensional fluid droplets from inclined solid surfaces, J. Fluid Mech. 395 (1999) 181e209. [32] S. Lee, H.K. Yoon, K.J. Kim, et al., A dropwise condensation model using a nano-scale, pin structured surface, Int. J. Heat Mass Transfer 60 (2013) 664e671. [33] S. Kim, K.J. Kim, Dropwise condensation modeling suitable for superhydrophobic surfaces, ASME J. Heat Transfer 133 (2011) 081502e081511.
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Applied Thermal Engineering journal homepage: www.elsevier.com/locate/apthermeng
Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure Rongfu Wen, Zhong Lan, Benli Peng, Wei Xu, Xuehu Ma* Liaoning Provincial Key Laboratory of Clean Utilization of Chemical Resources, Institute of Chemical Engineering, Dalian University of Technology, Dalian 116024, China
h i g h l i g h t s Transients of initial droplet size distribution were experimentally studied. Surface coverage of steady condensation was strongly dependent on steam pressure. Effective heat transfer area reduced and resistance increased at low pressure. Heat transfer resistance distribution was sensitive to the steam pressure.
a r t i c l e i n f o
a b s t r a c t
Article history: Received 18 June 2014 Received in revised form 1 August 2014 Accepted 20 September 2014 Available online xxx
To investigate the transient characteristics of initial droplet size distribution, steady droplet size distribution and thermal resistance distribution at lower and ultra-lower steam pressure, dropwise condensation at the pressure range from atmospheric to 1.5 kPa has been studied. During the transient process, the initial nucleated droplets satisfied lognormal distribution, and then a bimodal distribution formed, finally revealed an exponential distribution. The peak value was smaller and the evolution was slower with the reduction of steam pressure. The corresponding surface coverage increased to 0.7e0.8 at the steady condensation which was strongly dependent on the pressure. Introducing a dimensionless time, the surface coverage evolution indicated that the time consumed by direct growth increased as the pressure decreased. The effect of steam pressure on droplet size distribution revealed a more scattered distribution, larger departure size, and denser large droplets at low pressure, resulting in the reduction of the effective heat transfer area. By comparing the thermal resistance distribution at various pressures, it showed that large droplets induced a greater proportion of resistance at low pressure. The findings help clarifying the limitations of droplet growth mechanism and offer guidelines for the optimization of surface morphology to enhance the steam condensation at low and ultra-low pressure. © 2014 Elsevier Ltd. All rights reserved.
Keywords: Dropwise condensation Low pressure steam Droplet size distribution Surface coverage Dynamic evolution Thermal resistance distribution
1. Introduction The mechanisms governing steam condensation at low or ultralow pressure are crucial to a wide range of applications which have significant economic and environmental impacts, such as power generation [1], low temperature multi-effect seawater desalination [2], water harvesting in space station [3], thermal management systems [4,5] and environmental control [6]. Water vapor preferentially condenses on solid surfaces rather than directly from the vapor due to the reduced activation energy of heterogeneous nucleation as compared to homogeneous nucleation [7]. When * Corresponding author. Tel.: þ86 411 84707892; fax: þ86 411 84707700. E-mail addresses: [email protected] (R. Wen), [email protected] (X. Ma).
water vapor condenses on a surface, the condensate can form either a liquid film or distinct droplets, depending on the surface wettability. The latter, termed dropwise condensation is more desirable since droplets can efficiently remove from the surface in comparison to filmwise condensation. Droplet growth mechanism in dropwise condensation affects the heat transfer rate notably, for some cases with lower heat transfer rate, such as condensation at very low pressure [8] or with large amount non-condensable gas [9], droplet growth rate becomes very low. Meanwhile, condensed droplets stay on condensing surface longer so droplet dynamics and distribution will not be similar any more. Based on the population balance theory, Wu and Maa [10] deduced the distribution of small droplets less than the critical radius. For larger droplets, the droplet size distribution function proposed by Le Fevre and Rose [11] was still widely used.
http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069 1359-4311/© 2014 Elsevier Ltd. All rights reserved.
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R. Wen et al. / Applied Thermal Engineering xxx (2014) 1e9
Nomenclature
rvmin Rd
Variables Ad Ade Atotal c g h hi
Rdrop Rtotal t t' Tw DT
Hfg kcoat kw n N Nre Ns Pv r rd re rmax rmin
weighted base area of the same size droplets (m2) base area of individual droplet (m2) base area of all the condensed droplets (m2) constant dependent on the geometry of droplet (m/s2) gravitational acceleration (m/s2) droplet height (m) the condensation interfacial heat transfer coefficient (W/m2K) the latent heat of vaporization (J/kg) thermal conductivity of the coating material (W/m K) thermal conductivity of water (W/m K) density of small droplets (m3) density of large droplets (m3) population of droplets in the rectangular area density of nucleation site (m2) vapor saturation pressure (Pa) droplet radius (m) contact radius (m) effective droplet radius (m) maximum droplet radius (m) minimum droplet radius (m)
Subsequently, Wu [12] put forward a random fractal model to simulate the droplet size distribution from the primary to the departing droplet in dropwise condensation. Mei [13] derived expressions for fractal dimension and area fraction of droplet sizes correlated with surface subcooling. In addition, Lan [14] rebuilt the spatial conformation of droplet distribution into the temporal conformation based on the random fractal model and reflected the dynamic characteristics of dropwise condensation. According the dynamic nature of droplet growth on structured superhydrophobic surfaces, Miljkovic et al. [15] derived the droplet size distribution dependent on droplet morphology on structured superhydrophobic surfaces. In the investigations mentioned above, the droplet dynamics and distribution characteristics were mainly studied for the steady state dropwise condensation of pure steam at atmospheric or higher pressure. Owing to the complex coalescing mechanism among droplets, it is very hard to display dynamic evolution of the droplet size distribution theoretically. With the rapid improvement of operational performance of computer, the entire process of dropwise condensation was reappeared and the droplet size distribution was investigated by simulation [12,16e18]. In experiment, Chen, et al. [19] focused on the effect of microscopic topography on the cumulative departure volume and droplet number density. Ucar et al. [20] investigated the effect of surface roughness and contact angle hysteresis of polymeric substrates on the initial droplet density and surface coverage. Transient characteristics of droplet size distribution in the initial dropwise condensation were attributed to the growth and coalescence of the first generation droplets and the evolutions of transient stages on low thermal conductivity surface were affected by steam pressure obviously for the pressure range from atmospheric to 29.4 kPa [21]. As the steam pressure decreases, the droplet dynamics and distribution changes obviously, resulting in a great impact on heat transfer performance. Thus, the droplet dynamics and distribution at low or ultra-lower pressure are very desirable and significant to get insights in the mechanism of dropwise condensation heat transfer at low pressure. The purpose of this paper is to investigate the transient characteristics of initial droplet size distribution, steady droplet size
minimum discernible droplet radius (m) weighted thermal resistance of the same size droplets (m2 K/W) thermal resistance of individual droplet (m2 K/W) thermal resistance of all condensed droplets (m2 K/W) time (s) dimensionless time () surface temperature (K) surface subcooling temperature (K)
Greek letters contact angle ( ) advancing contact angle ( ) receding contact angle ( ) thickness of coating layer (m) density of the condensate (kg/m3) surface tension (N/m) the sweeping period (s) time from re to rmax (s) time from critical size to re (s) 4A weighted coverage ratio occupied by the same size () 4R weighted thermal resistance ratio of the same size () jA coverage ratio occupied by droplets () jR thermal resistance ratio of droplets ()
q qa qr d r s t tco tdi
distribution and heat transfer resistance distribution among various sizes droplets to clarify the limitations of droplet growth mechanism at low and ultra-low pressure. Droplet size distribution evolution and surface coverage evolution are studied in the circle of droplet growth from nucleation to the steady condensation. During the steady condensation, the droplet size distribution features, departure size and the density of large droplets are analyzed and compared with the simulation results. Furthermore, the heat transfer resistance distributions based on droplet size distribution at various steam pressures are compared.
2. Experiments 2.1. Apparatus The closed system mainly consists of a boiler, cooling water, condensing chamber, and data acquisition and control unit. A cylindrical condensing block, 13 mm in diameter and 22 mm long made of high purity copper was thermally insulated with PTFE to ensure the one dimensional steady-state conduction, as shown in Fig. 1. The condensing surface was oriented vertically. Five thermocouple holes, 0.8 mm in diameter, were drilled into the block in parallel along the stream. The width of space from condensing surface to window was approximately five mm. The pressure was measured by a manometer with the accuracy of 0.1 kPa combined with a McLeod vacuum gauge with the range from 0.1 Pa to 10 kPa. Experimental data were measured and collected with the Agilent34970A data acquisition system. An installed window on the test section facilitated the observation of the condensation process. The camera system (PHOTRON, FASTCAM APX-RS) mounted with a set of microscope lenses (HIROX, CX10C) was used to record the droplet behaviors. The highest shooting speed is 3000 fps at 1024 1024 pixels, maximum 250,000 fps in lower resolution. With various objective lenses, the magnification can achieve 7000. The smallest size of a droplet which can be observed by microscopy is 1 mm in diameter.
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3
Fig. 1. Schematic of experimental setup and condensing chamber.
2.2. Experimental procedure
2.3. Surfaces preparation and characterization
Great care was taken to degas the system as well as to make it gastight. Deionized water was supplied to the boiler. Setting the auxiliary vacuum pump in operation, the main system was evacuated and the pressure decreased to the saturation pressure corresponding to the room temperature. Then, the test loop was kept running at a high evaporation rate for an hour. The accumulated non-condensable gas was removed by the auxiliary vacuum pump at intervals. The degassed water was restored to the boiler. Setting the auxiliary vacuum pump in operation, the main system was evacuated and became dry. Further, the system was evacuated to a pressure Pv < 0.5 ± 0.1 Pa by the main vacuum pump. The chamber pressure and temperature were continuously monitored to ensure saturated condensations through the experiments. The steam generated by the boiler condensed partly in the chamber and the excess steam condensed in an auxiliary condenser. When the steam pressure in the chamber came to the expected pressure, experiments were conducted from large surface subcooling to small surface subcooling. As the surface subcooling decreased, the heat transfer performance of auxiliary condenser was improved by increasing the flow rate of cooling water. In this way, the increased excess steam condensed and the pressure of the chamber was adjusted. The wall temperature Tw was calculated by extrapolating the temperature gradient to the condensing surface. Additional data reduction was well described elsewhere [22]. The uncertainties of the measured and calculated parameters were estimated by the procedure described by Kline and McClintock [23,24]. The experimental uncertainties associated with the sensors and calculated parameters are listed in Table 1.
The n-octadecyl mercaptan was used as the surface promoter. The condensing side end of the copper block was finely polished. Then, the copper block was rinsed by deionized water and acetone, dried in the air and immersed in the 2.5 mmol/L solution of noctadecyl mercaptan in ethanol for 1 h at 343 K. Finally, a selfassembly monolayer formed. The scanning electron microscope (SEM) image shown in Fig. 2(a) illustrated the morphology of the hydrophobic surface. Contact angles (q), advancing contact angles (qa) and receding contact angles (qr) were measured to determine the surface wettability. The mean contact angle was 120.0 (DataPhysics OCAH200, Germany, ±0.1 ), as shown in Fig. 2(b).
Table 1 Experimental uncertainties containing sensors and key parameters. Sensors Thermocouple Pressure transducer Parameters Pressure (Pv) 1.5 kPa 31 kPa 101 kPa
±0.03 K ±10 Pa ±100 Pa
(0e10 kPa) (10e101 kPa)
Wall temperature (Tw) ±(0.035 Ke0.084 K) ±(0.061 Ke0.261 K) ±(0.064 Ke0.355 K)
Surface subcooling (DT) ±(1.3%e4.8%) ±(3.8%e7.8%) ±(5.2%e8.2%)
3. Results and discussion 3.1. Transient characteristics of initial droplet size distribution On a hydrophobic surface, droplet nucleation and growth proceed through two mechanisms: direct growth where droplets nucleate and grow on spatially random sites by direct condensation and coalescence-dominated growth where the droplet size becomes large enough to coalescence with neighboring droplets [25]. Therefore, a wide range of droplet sizes exist on the condensing surface from nucleation to steady condensation and the droplet size distribution reflects the instantaneity in the process of droplet growth [26]. Image analysis package Image-Pro Plus (Cold Spring, USA) was used to measure the droplet number with different sizes. The droplet zones were identified first and then the area and radius of each droplet were calibrated, the radius range in the image and corresponding droplet number were recorded, and finally the droplet size distribution was generated. The transient characteristics of droplet size distribution in the initial condensation at the steam pressure of 1.5 kPa and 60 kPa were investigated and the corresponding spatial droplet distribution images were shown in Fig. 3. At the beginning of nucleation, all the droplet sizes were uniform and droplet size distribution had a peak value, showing the lognormal distribution (t ¼ 0.4 s in Fig. 3(a) and t ¼ 0.08 s in Fig. 3(b)). With droplets growing up, the average droplet
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Fig. 2. Surface morphology and wettability of condensing surface. (a) SEM image; (b) Contact angle measurement.
size increased (t ¼ 0.8 s in Fig. 3(a)). Then coalescences among droplets led to wider span of droplet sizes and decline of right peak (t ¼ 1.8 s in Fig. 3(a) and t ¼ 0.12 s in Fig. 3(b)). With the increased space among the first generation droplets, new droplets nucleated
and grew at the exposed substrate where bimodal distribution formed (t ¼ 2.8 s in Fig. 3(a) and t ¼ 0.18 s in Fig. 3(b)). With further coalescences and nucleation, the number of larger droplets decreased and the number of smaller droplets increased, and droplet
Fig. 3. Transient steam condensation images and corresponding droplet size distribution. (a) Pv¼1.5kPa,
▵T¼1K; (b) Pv¼60kPa, ▵T¼1K.
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size range became wider. Finally, the right peak vanished and the distribution became unimodal (t ¼ 6.8 s in Fig. 3(a) and t ¼ 0.68 s in Fig. 3(b)). Note that the transients of droplet size distribution at different pressures showed the same tendency while the droplet density was strongly dependent on the steam pressure in accordance. In order to further indentify the effect of steam pressure on the droplet growth, surface coverage at different pressures were compared, as shown in Fig. 4(a)e(c). The surface coverage went through three typical periods: rapid growth period tdi where droplets nucleated and grew by direct condensation, slow growth period tco where droplets coalesced with neighboring droplets and new droplets nucleated at the exposed position, and steady period where the condensation process tended to be stable. The surface coverage at the beginning of coalescence increased from 0.5 to 0.58 with the pressure increasing from 1.5 kPa to 60 kPa, which was due to the larger density of active nucleation site at the higher pressure. In addition, a high pressure improved the surface coverage of steady condensation from 0.7 at 1.5 kPa to 0.79 at 60 kPa. Fig. 4(d) compares the proportion of time consumed by direct growth stage tdi/ (tdi þ tco) to indentify the droplet growth mechanism at low and ultra-low pressure. It revealed that the time consumed by direct condensation stage accounted for larger proportion at lower pressure. This was because the gas liquid interface transfer resistance rapidly increased and the growth rate of individual droplet reduced.
5
size distribution [28]. The steady state droplet size distribution of small droplets is given by,
1 re 2=3 rðre rmin Þ A2 r þ A3 nðrÞ ¼ expðB1 þ B2 Þ r rmin A2 re þ A3 3pre3 rmax rmax ðrmin < r < re Þ (1) where
A1 ¼
DT 2rHfg
(2)
A2 ¼
qð1 cos qÞ 4kw sin q
(3)
A3 ¼
1 dð1 cos qÞ þ 2hi kcoat sin2 q
(4)
B1 ¼
A2 re2 r 2 r rmin 2 þ rmin ðre rÞ rmin ln tA1 2 re rmin
(5)
3.2. Droplet size distribution features in the steady condensation
B2 ¼
A3 r rmin re r rmin ln tA1 re rmin
(6)
Dropwise condensation consists of the transient process occurring repeatedly. New tiny droplets nucleate on the space after coalescence and departure. These droplets form a new group and exhibit all the features of the first generation droplets. A steady state can thus be reached, with continuous formation of new droplets and flow of the large droplets [27]. The photographs at different instant or in different scales are similar. These features indicate that dropwise condensation appears with self-similarity [12].
where rmin is the droplet nucleation radius (rmin ¼ 2Tss/HfgrDT), re is the radius when droplets begin to merge, rmax is the maximum droplet radius, q is the apparent equilibrium contact angle, r is the water density, kw is the water thermal conductivity, Hfg is the latent heat of vaporization, hi is the condensation interfacial heat transfer coefficient, and t is the sweeping period. For large droplets, the droplet size distribution N(r) was established by Le Fevre and Rose [29],
3.2.1. Droplet size distribution model For small droplets that grow mainly by direct condensation, the population balance theory can be employed to determine droplet
NðrÞ ¼
Fig. 4. Surface coverage during transient steam condensation process. (a) Pv¼1.5kPa, dimensionless time t'¼t/(tdiþtco).
1 3pr 2 r
max
r rmax
2=3
ðre < r < rmax Þ
(7)
▵T¼1K; (b) Pv¼31kPa, ▵T¼1K; (c) Pv¼60kPa, ▵T¼1K; (d) Surface coverage with respect to the
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The maximum droplet radius rmax can be estimated by the force balance between the surface tension and gravity on a vertical surface, as follows [30],
rmax ¼
6cðcos qr cos qa Þsin q s 2 3 cos q þ cos3 q prg
1=2 (8)
where qr and qa are the advancing and receding contact angles, respectively. The effective radius re is defined as the boundary between the small droplets n(r) and large droplets N(r) by assuming the nucleation forms a square array [31],
1 re ¼ pffiffiffiffiffiffiffiffiffi 4NS
(9)
where NS is a number of nucleation site on a unit area of condensing surface.
3.2.2. Comparison of droplet size distribution Due to the self-similarity feature of droplet size distribution, a representative large enough rectangular area containing various growth stages was determined to investigate the droplet size distribution at different pressures. Droplets down to 10 mm in size were measured and counted on account of the limit of resolution. Fig. 5 compares the calculated results and experimental data at different pressures at a surface subcooling of 1 K. The development tendency of droplet size distribution was very consistent qualitatively with that from experiments at both steam pressures. It is also important to point out that the error between the calculated result and experimental data increased with the reduction of steam pressure. At low steam pressure, the number of small droplet was smaller than the model prediction results (Fig. 5(b)) while the number of large droplet was larger than the model prediction results (Fig. 5(c)). In addition, the frequency of large droplets with different sizes also increased at lower pressure, as shown in Fig. 5(c). In the droplet radius range from 180 mm to 1280 mm, there
Fig. 6. Schematic of heat transfer resistance distribution based on droplet size distribution.
were three sizes of large droplets at atmospheric while seven sizes of large droplets appeared at low pressure which led to the rapid reduction of effective heat transfer area. 3.3. Heat transfer resistance distribution The condensing surface covered by condensed droplets with various sizes has significantly different heat transfer performance. It is necessary to investigate thermal resistance distribution by condensed droplets with respect to the steam pressure. 3.3.1. Heat transfer resistance distribution model Since the area ratio of the liquid/solid interface and surface coverage can be given by the droplet size distribution, the heat
Fig. 5. Comparison of droplet size distribution between model prediction and experimental data. (A) Pv¼101kPa, (B) Pv¼1.5kPa.
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transfer resistance distribution through condensed droplets can also be analyzed by calculating the contact area of liquid/solid interface, liquid/gas interfaces and volume of every individual droplet combining with the droplet size distribution, as shown in Fig. 6. The spherical cap geometry of a droplet can be characterized by four different parameters: droplet height h, contact radius rd, droplet radius r, and contact angle q. Due to the large size of counted droplets in the study, more than 10 mm, the conduction through droplets is the dominant thermal resistance in the dropwise condensation [33]. Thus, the heat transfer resistance through the condensed droplet in the analysis is given as
Rdrop ðrÞ ¼
q 4prkw sin q
(10)
Atotal ¼
rmax X rvmin
4R ðrÞ ¼ Rd ðrÞ=Rtotal
(14)
where
Rd ðrÞ ¼ Rdrop ðrÞ$Nre ðrÞ
Rtotal ¼
rmax X
Rd ðrÞ
(15)
(16)
(11)
where Rdrop(r) refers to the thermal resistance through the droplet with radius r. The ratio of condensing surface covered by droplet with radius below r is expressed as
(12)
jA ðrÞ ¼
where
Ad ðrÞ ¼ Ade ðrÞ$Nre ðrÞ
where Ade(r) and Nre(r) refer to the base area and the number of droplet with radius r, respectively, and rvmin is the radius of minimum discernible droplets in the experiment. The weighted thermal resistance ratio of droplet with radius r can be written as
rvmin
The weighted coverage ratio occupied by droplets with radius r can be written as
4A ðrÞ ¼ Ad ðrÞ=Atotal
7
rmax X
4A ðrÞ
(17)
rvmin
Ad ðrÞ
(13)
The thermal resistance ratio of droplet with radius below r is expressed as
Fig. 7. Spatial droplet distribution images, droplet size distribution, and droplet population histograms. (a) Pv¼1.5kPa; (b) Pv¼31kPa; (c) Pv¼101kPa;(d) Pv¼1.5kPa; (e) Pv¼31kPa; (f) Pv¼101kPa;(g) Pv¼1.5kPa; (h) Pv¼31kPa; (i) Pv¼101kPa.
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Fig. 8. Droplet coverage ratio and thermal resistance ratio with respect to steam pressure.(a) Accumulated droplet coverage ratio vs. droplet radius; (b) Accumulated of thermal resistance ratio vs. droplet radius.
jR ðrÞ ¼
rmax X
4R ðrÞ
(18)
rvmin
3.3.2. Droplet size distribution dependant on steam pressure Image analysis package Image-Pro Plus was used to measure the number of condensed droplet with different sizes. The droplet zones were identified first and then the area and radius of each droplet were calibrated, the radius range in the image and corresponding droplet number were recorded, and finally the droplet size distribution generated. Fig. 7 shows the spatial droplet distribution images and droplet size distribution by Image-Pro Plus, as well as the droplet population histograms at different pressures. The strong dependence of droplet size distribution on steam pressure was clearly visible. An increase in the number of small droplets (blew ~0.1 mm) was obvious with the increase of pressure, as shown in Fig. 7(i), while the frequency of large droplets (above ~0.3 mm) increased obviously at low pressure, as shown in Fig. 7(g). 3.3.3. Heat transfer resistance distribution dependant on steam pressure It is obvious that more condensing surface covered by small droplets is preferred due to the smaller thermal resistance. The ratio of condensing surface covered by droplets with radiuses below r at different steam pressures is shown in Fig. 8(a). Visual observations confirmed that the surface coverage distribution was significantly influenced by the steam pressure. More condensing surfaces were covered by large droplets at low pressure. As the pressure reduced from 101 kPa to 1.5 kPa the ratio of condensing surface covered by small droplets (below 0.15 mm) decreased from 0.92 to 0.54. The surface coverage ratio by departure droplet increased from 0.08 to 0.13 as the pressure reduced from atmospheric to 1.5 kPa. Note also that the droplet size distribution is more dispersed at low pressure while it is mainly concentrated in small size at atmospheric. Fig. 8(b) shows the thermal resistance ratio of droplets with radiuses below r at different pressures. The thermal resistance focused on the departure droplet and the resistance of the other droplets was almost negligible at atmospheric. As pressure decreased, the thermal resistance ratio of droplet with radius above 0.2 mm increased obviously while the thermal resistance ratio of departure droplet decreased. The thermal resistance ratio of small droplets (below 0.15 mm) was smaller at low pressure which indicated large droplets caused a greater thermal resistance. Therefore, reducing the departure size is an effective way to enhance the condensation of atmospheric steam [32,33] while the improvement of heat transfer at low pressure should focus on the
reduction of the density of large droplets in addition to the reduction of departure size. 4. Conclusions Effect of steam pressure on the transient process of initial droplet size distribution, steady droplet size distribution and heat transfer resistance distribution among condensed droplets was investigated in dropwise condensation at the pressure range from atmospheric to 1.5 kPa. The initial nucleated droplets satisfied lognormal distribution, and then a bimodal distribution formed with the droplets coalesce, finally revealed an exponential distribution at the steady state. The peak value was smaller and the evolution was slower at low steam pressure, which was due to the reduction of the nucleation site density and the droplet growth rate. The surface coverage of steady condensation was strongly dependent on the steam pressure. As the steam pressure decreased, the time period for the droplet direct growth increased and the gas liquid interface transfer resistance rapidly increased. The effective heat transfer area reduced and the heat transfer resistance increased due to the more scattered droplet distribution, larger departure droplets, and denser large droplets at low steam pressure. A comparison of heat transfer resistance distribution at various steam pressures showed that large droplets led to a greater proportion of heat transfer resistance with the reduction of steam pressure. Acknowledgements The work is supported by the National Natural Science Foundation of China (No. 51236002) and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 20120041110018). References [1] J.M. Beer, High efficiency electric power generation: the environmental role, Prog. Energy Combust. Sci. 33 (2007) 107e134. [2] A.D. Khawaji, I.K. Kutubkhanah, J.M. Wie, Advances in sea-water desalination technologies, Desalination 221 (2008) 47e69. [3] H.G. Andrews, E.A. Eccles, W.C.E. Schofield, et al., Three-dimensional hierarchical structures for fog harvesting, Langmuir 27 (2011) 3798e3802. [4] X.H. Ma, J.B. Chen, S.P. Li, et al., Application of absorption heat transformer to recover waste heat from a synthetic rubber plant, Appl. Therm. Eng. 23 (2003) 797e806. [5] D. Milani, A. Abbas, A. Vassallo, et al., Evaluation of using thermoelectric coolers in a dehumidification system to generate freshwater from ambient air, Chem. Eng. Sci. 66 (2011) 2491e2501. [6] L. Perez-Lomavard, J. Ortiz, C. Pout, A review on buildings energy consumption information, Energy Build. 40 (2008) 394e398.
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Please cite this article in press as: R. Wen, et al., Droplet dynamics and heat transfer for dropwise condensation at lower and ultra-lower pressure, Applied Thermal Engineering (2014), http://dx.doi.org/10.1016/j.applthermaleng.2014.09.069