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Theoretical and experimental study of departure duration of condensate droplets from radiant cooling ceiling surfaces
Building and Environment 114 (2017) 445e454
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Building and Environment journal homepage: www.elsevier.com/locate/buildenv
Theoretical and experimental study of departure duration of condensate droplets from radiant cooling ceiling surfaces Haida Tang a, Tao Zhang a, b, Xiaohua Liu a, *, Yi Jiang a a b
Department of Building Science, Tsinghua University, Beijing 100084, China School of Aerospace, Tsinghua University, Beijing 100084, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 September 2016 Received in revised form 2 January 2017 Accepted 3 January 2017 Available online 4 January 2017
In this paper, a mathematical model for predicting the departure duration of the first condensate droplet from a radiant ceiling surface was proposed on the basis of the condensation water mass. The simulation results indicate a dependence of condensation water mass on the apparent contact angle of the substrate, but almost in no relation with the surface temperature. The condensation water mass firstly increases with the increase of the apparent contact angle. It reaches a maximum weight of 522 g/m2 at an apparent contact angle of 110 , and then decreases. A visualization experiment of condensation on a radiant ceiling panel with a conventional aluminum alloy surface was performed in a climate chamber to measure the departure duration of the droplet. The measured departure duration fluctuates due to the variance of apparent contact angle and the randomness of condensation process, but it decreases sharply with the sub-cooled degree (air dew point minus surface temperature). And the average departure duration is 10 h with a sub-cooled degree of 5 C. The theoretical model is validated as the average relative biases between the experimental and theoretical results are within 25%. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Radiant cooling ceiling Condensation Condensate droplet departure duration Condensation water mass Contact angle
1. Introduction Nowadays building energy consumption is still increasing with the development of economy and society, accounting for about 20% of the total energy in China [1]. The air-conditioning system including heating and cooling usually consumes about 30%e60% [1,2] of the total building energy. It is crucial to reduce the energy consumption of the air-conditioning system in buildings. Continuous efforts have been paid in proposing novel appropriate approaches for creating comfortable living conditions with a low energy cost. The radiant cooling ceiling system is such a new solution serving as a high temperature cooling system [3e5]. It has been used in multiple types of buildings, such as office buildings, residential buildings and airports, and its popularity is still increasing [6e8]. Compared with all-air systems, radiant cooling ceiling systems can achieve energy savings by increasing chiller's efficiency due to high temperature cooling and reducing energy consumption of fans [9,10]. Khan et al. [11] investigated the energy consumption of radiant cooling ceiling systems with a dedicated outside air system (DOAS) in a hot and humid climate. A
* Corresponding author. E-mail address: [email protected] (X. Liu). http://dx.doi.org/10.1016/j.buildenv.2017.01.001 0360-1323/© 2017 Elsevier Ltd. All rights reserved.
comparison of energy consumption calculated that the radiant systems was 17.5% more efficient than a conventional all-air system. Besides, a side-by-side comparison between radiant and variable air volume systems was conducted in two identical buildings in India. The monitoring data indicated that the radiant system used 34% less energy compared to the variable air volume systems during the two years of operation [12]. Furthermore, previous studies revealed that a radiant cooling ceiling system could achieve better thermal comfort than a conventional all-air system [13e15]. Radiant cooling ceiling system is also claimed as a promising indoor terminal unit for temperature and humidity independent control. As radiant cooling ceiling systems only handle indoor sensible load, indoor moisture load needs to be handled by auxiliary dehumidification systems [16,17]. However, there are condensation risks on the panel surfaces of radiant cooling ceiling systems when applied in a hot and humid climate. Research on avoiding condensation of radiant cooling surfaces has mainly focused on the control strategies and system configuration. It's suggested that the lowest surface temperature of radiant panel should be maintained above the indoor dew point temperature to guarantee no condensation. Yuan et al. [18] studied the inherent correlation between total heat flux capability and panel surface temperature, and proposed a condensation free control logic based on a simplified model of panel surface
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temperature. Ning et al. [19] improved a novel radiant ceiling panel with a uniform surface temperature distribution which is beneficial for condensation control. Xie et al. [20] investigated the influence of water temperature and tube spacing on the non-uniformity of surface temperature for a capillary radiant ceiling panel by a computational fluid dynamics (CFD) simulation. On the other hand, it is also essential to control the indoor moisture level carefully. Seo et al. [21] developed a control logic that a dehumidification ventilation system will be activated to remove indoor moisture facing potential condensation risks. Zhang et al. [22] and Ge et al. [23] studied the optimal pre-dehumidification time with neural network models to eliminate condensation risks during the startup process where the air conditioning system operated intermittently. However, the condensation issue on a radiant cooling ceiling close to external windows and doors, or wherever the indoor moisture load increases sharply, is sometimes inevitable. Natural convection flows arise from the combined buoyancy effects of thermal and mass diffusion in buildings. Condensation on a radiant cooling ceiling involving the natural convection is relatively a slow process [24,25]. The departure duration of droplet usually takes several hours from the beginning of the condensation to the first condensate droplet departure from a radiant ceiling. This provides a quantitative requirement for the responsiveness of control strategy due to thermal inertia of the radiant system and a limited dehumidifying ability of the air system. Yin et al. [26] experimentally examined the growth rate of average droplet radius on a plastic capillary panel varying with inlet temperature of chilled water. It's indicated that the departure duration of droplet is affected by the largest droplet radius instead of the average value. Mei et al. [27,28] and Sikarwar et al. [29] simulated the processes of nucleation, growth due to direct condensation, coalescence, and fall-off of droplets on the underside of horizontal substrate. The dropwise condensation of water vapor was studied correspondingly. Nevertheless, the condensation on radiant ceiling arises from natural convection between humid air and panel surface rather than condensation in pure water vapor. And the great amount of computation becomes an obstacle in application. Because the departure duration of droplet is closely correlated to the condensation water mass, the frost melting water retention on a vertical substrate was investigated theoretically and experimentally by Liang et al. [30]. The experimental results indicated that the retained water mass of the superhydrophobic fin decreases by 75.82% compared with that of the bare fins. However, there are seldom studies on the departure duration of condensate droplet from a radiant ceiling surface due to the complexity and randomness of the condensation process. In this paper, a theoretical model based on the condensation water mass, which took the surface characteristics of apparent contact angle into account, was developed for predicting the departure duration of the first condensate droplet from a radiant ceiling panel. Meanwhile, a series of condensation experiments with different sub-cooled degrees were conducted on a radiant ceiling with a conventional aluminum alloy surface. The experimental results helped to investigate the mass of condensation water and the departure duration of droplet. The present research would be beneficial for a better operation and control of radiant cooling systems. 2. Theoretical analysis 2.1. Critical size of the gravity-induced falling droplet As shown in Fig. 1, metal panels are usually adopted as the radiant cooling ceiling for a better heat transfer performance. When the panel surface temperature is below dew-point temperature, condensate droplets grow and coalesce underneath the radiant
Fig. 1. Schematic of a radiant cooling ceiling panel.
ceiling. And they fall off due to gravity upon approaching the critical size. The condensation phenomenon is significantly influenced by the wettability of a surface. It is always characterized by the contact angle (q), which indicates the degree of wetting when a solid and liquid interact. When an interface exists between a liquid and a solid, the angle between the surface of the liquid and the outline of the contact surface is described as the contact angle. In the case of complete wetting (spreading), the contact angle is 0 . The solid is partially wetting if q is between 0 and 90 , and it is partially non-wetting as q is above 90 . In the case of superhydrophobic materials with so-called lotus effect, the contact angle approaches the theoretical limit of 180 . Here we investigate the critical size of the gravity-induced falling droplet from a horizontal radiant ceiling surface. The capillary length is a characteristic length scale for an interface between water and air which is subject both to gravitational acceleration and to a surface force in the interface, which is expressed as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lcap ¼
g
ðrw ra Þg
(1)
where g represents the surface tension of water; rw and ra are the density of water and air, respectively; g is the gravitational acceleration. The shape of a pendent droplet depends on the equilibrium between the forces of surface tension, gravity, and pressure. It was described by the classical Young-Laplace equation of capillarity [31]. According to the hypothesis of rotational symmetry of pendent droplets underneath horizontal surfaces, solving the three-dimensional shape of the droplet is simplified into solving its profile. It can be written as the following ordinary differential equations as a function of arc length s, as shown in Fig. 2.
dx ¼ cos 4 ds
(2a)
dz ¼ sin 4 ds
(2b)
d4 2 z sin 4 ¼ þ ds b l2cap x
(2c)
dV ¼ px2 sin 4 ds
(2d)
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Fig. 2. Coordinate system used in simulation of a pendent droplet profile on a radiant ceiling.
xð0Þ ¼ zð0Þ ¼ 4ð0Þ ¼ Vð0Þ ¼ 0
(2e)
where b is the radius of curvature at the apex of the droplet (O point); 4 is the tangential angle that, for pendent droplets, becomes the apparent contact angle (q) at the triple-phase contact line. It should be noted that, (sin 4/x)s¼0 ¼ 1/b at the apex of the droplet. The profile of the pendant droplet is generated by numerically integrating the initial value problem via fourth-order Runge-Kutta Method. The integration is stopped when the tangential angle (4) reaches the apparent contact angle (q) or 0 . It is indicated by Pitts [32] and Fuchikami [33] that there is more than one equilibrium droplet shape for a fixed droplet volume and a given apparent contact angle and only the shortest one in z direction is the stable one. Furthermore, for a fixed contact angle q, the volume of the pendant droplet reaches the maximum equilibrium, and the droplet falls off the substrate at the critical condition. The critical condition is shown in Eq. (3):
dV ¼0 db q
(3)
Fig. 3 illustrates the profiles of the falling droplets on a substrate with various apparent contact angles via simulation, where rmax and Vmax represents the projection radius and volume of the critical falling droplet from the substrate, respectively. The shape of the
Fig. 4. Size of a critical gravity-induced falling droplet as a function of apparent contact angle (q).
pendent droplet tends towards a spherical segment with the rising apparent contact angle, especially on hydrophobic surfaces. Furthermore, the radius (rmax) and volume (Vmax) of a critical gravity-induced shedding droplet are plotted as a function of the apparent contact angle (q) in Fig. 4. It could be found that the radius and volume of a critical falling pendent droplet both decrease with the increase of apparent contact angle.
2.2. Volume of pendent droplets underneath radiant ceiling From condensation nucleuses, a condensate droplet on a radiant ceiling grows via condensation and coalescence until reaching the maximum size. The volumes and radii of condensate droplets on a substrate vary in a quite large range. And the droplet radius has a correlation to the droplet volume with a given apparent contact angle. On the basis of Young-Laplace differential equations and the criteria of the gravity-induced falling droplet, Fig. 5 illuminates the variance of the pendent droplet volume (V(r)) as a function of the droplet radius (r) on radiant ceiling surfaces with various contact angles. The droplet radius firstly increases with the increase of pendent droplet volume underneath a radiant ceiling surface with
Fig. 3. Profiles of critical gravity-induced falling droplets with various apparent contact angles.
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Fig. 5. Pendent droplet volume as a function of its projection radius on radiant ceiling surfaces with various contact angles.
a given apparent contact angle. Then it decreases after a transition volume. The transition results from the non-uniqueness of the pendent droplet profile with a given apparent contact angle and a given projection radius. Fig. 6 shows that the volumes of pendent droplets differ with the same projection radius of 5.9 mm and an apparent contact angle of 35 . The smaller one is 164.5 mm3 and the larger one is 269.0 mm3. 2.3. Condensation water mass and departure duration of condensate droplets The departure duration (Dt) of condensate droplet is defined as a period of time from the beginning of condensation on radiant ceiling surface to the departure of the first condensate droplet. The departure of droplet occurs at the moment when the largest condensate droplet on the radiant ceiling surface reaches the critical size, i.e. when the body forces exceed surface tension at the triple-phase contact line holding it to the solid surface. The departure duration could be obtained according to the growth rate of the largest pendent droplet on a radiant ceiling during the condensation process. Condensation on the ceiling begins with the droplet formation at preferred nucleation sites at the atomic scale. The growth of a condensate droplet occurs by the combination of condensation and coalescence among neighboring droplets. As a consequence, the largest condensate droplet on a radiant ceiling surface is difficult to trace unless the whole condensation process is simulated. However, the condensation water mass progressively increases during the condensation process from the perspective of the condensation water holdup on the radiant ceiling surface. And it reaches the maximum value (Mdew) when the first droplet falls off. Then the fall-off droplets expose substrate area to provide a continuous source of nucleation sites. When the condensation
Fig. 6. Non-unique profiles of pendent droplets on a substrate with an apparent contact angle of 35 and a projection radius of 5.9 mm.
comes into a balance stage, the growth of droplets is equal to the shedding. As a result, the mass of condensate water on the radiant ceiling surface reaches a relatively invariable value (Mdew,s). Watanabe et al. [34] investigated the time-series characteristic of drop-size distribution density by experiment. The experimental results show that the theoretical drop-size distribution density of Rose [35] located on the center of obtained drop-size distribution densities from the first departure of condensate droplet to a steady state, particularly in the equilibrium region of small drops. On the other hand, the correlation between the droplet volume and droplet radius with a fixed apparent contact angle maintains constant during the whole condensation process on the substrate. Therefore, the stable retained mass of condensation water (Mdew,s) is approximately equal to the condensation water holdup (Mdew) on the substrate at the moment of the departure of the first droplet. Fig. 7 illustrates the flowchart of the theoretical model to predict the departure duration. The droplet departure duration could be calculated as the ratio between Mdew,s and the condensation rate (md) on the radiant ceiling surface. The stable retained condensation mass can be predicted with the aid of the droplet size distribution (N(r)) and the pendent droplet volume (V(r)). The size distribution proposed by Rose referred to the stable condition is adopted here because of its simplicity and agreement with the experimental data [35],
NðrÞ ¼
1 3pr 2 rmax
r
2=3
rmax
(4)
where rmax represents the projection radius of a critical shedding droplet which has been revealed in Fig. 4. N(r) is the population density of the condensate droplets in number per unit area per unit droplet radius. The condensation water mass underneath a radiant ceiling is
Zrmax Mdew;s ¼
rw VðrÞNðrÞdr
(5)
rmin
where rmin represents the radius of the condensation nucleus [36]. The plot of condensation water mass as a function of the apparent contact angle is shown in Fig. 8. The non-uniqueness problem of the droplet profile with the same projection radius may exist on a hydrophilic surface. As a result, the condensation water mass ranges between the lower and upper limits. But the relative deviation of the upper and lower limits is no more than 5%. A conservative estimation, i.e., the lower limit can be used as the
Fig. 7. Flowchart of the theoretical model framework for the prediction of departure duration from radiant ceiling surfaces.
H. Tang et al. / Building and Environment 114 (2017) 445e454
Fig. 8. Plot of the simulated stable condensation water mass on radiant ceiling as a function of the apparent contact angle of the substrate.
condensation water mass on the radiant ceiling in the view of engineering application. The condensation water mass on substrates with various surface characteristics range from 80 g/m2 to 520 g/ m2. Both of the lower and upper limits of the condensation water mass firstly increase and then decrease with the increase of apparent contact angle. The condensation water mass reaches a maximum with an apparent contact angle of 110 . Large droplets spread on the substrate and occupy too much space on a hydrophilic surface with good wettability, resulting in a small mass of condensation water per area. The smaller size of each single droplet contributes to a smaller condensation water mass on a superhydrophobic surface with bad wettability. Similar to Newton cooling formula, the condensation rate (md) on a radiant cooling ceiling surface can be obtained by the mass transfer coefficient (hm), which is expressed as
md ¼ hm ra ðua us Þ
(6)
where ua and us represent the water vapor mass fractions of indoor humid air and at the radiant ceiling surface respectively. ra denotes the density of humid air. The mass transfer coefficient hm has been measured experimentally in our previous study [37]. And it could be expressed in a correlation of average Sherwood number (Sh) vs. Rayleigh number (Ra).
Sh ¼ 0:1745Ra0:3313 ;
2:5 108 Ra 4:5 109
(7)
Sh ¼ hm L=DAB
(8)
Ra ¼ Pr,gL3 ðrs ra Þ=rs y2
(9)
where L is the characteristic length of radiant ceiling panel, DAB is the mass diffusivity of humid air, and Pr is the Prandtl number. Therefore, the departure duration of condensate droplet could be calculated, as
Dt ¼ Mdew =md
(10)
Fig. 9 shows the lower limit of the departure duration of the first condensate droplet from radiant ceiling surfaces, where there are three different sub-cooled degrees as a function of the apparent contact angle. It could be found that the droplet departure duration firstly increases and then decreases with the rise of apparent contact angle at a given surface temperature (Ts) and ambient air state. Additionally, the departure duration decreases with the increase of sub-cooled degree. Specifically, the sub-cooled degree is 5 C as the surface temperature is 14.5 C in ambient air of 28 C and 60% RH.
449
Fig. 9. Lower limit of the departure duration of condensate droplets from radiant ceiling surfaces of different surface temperatures as a function of the apparent contact angle.
The departure duration of condensate droplets is 6.0 h for the radiant ceiling made of conventional aluminum alloy with an apparent contact angle of 60 correspondingly. 3. Experimental setup 3.1. Experimental platform and procedure Fig. 10 is the schematic diagram of the experimental platform. A climate chamber (3 m 5 m 3 m) with hole-board ventilation was used to maintain indoor air temperature and humidity in the experiments. Indoor parameters were controlled by a cooling coil, an electric heater and a humidifier. A square radiant panel with a length of 1.0 m acting as the radiant ceiling was installed horizontally in the climate chamber, as shown in Fig. 11(a). The crosssection of the radiant panel is shown in Fig. 11(b). The outermost layer of the radiant panel is a 0.5-mm-thick aluminum alloy plate with electrostatic spraying. The metal surface is attached to a 20mm-thick concrete slab embedded with copper pipes. The bottom surface of the concrete layer is attached to a 100-mm-thick block of extruded polystyrene board (thermal conductivity k ¼ 0.05 W/(m$K)). Cooling water is supplied to the copper pipes, which snake throughout the concrete layer at an interval of 80 mm to maintain a uniform temperature field on the aluminum alloy surface. In addition, the radiant panel was surrounded by wind deflectors. It's ensured that no external disturbances in the air interfered with the condensation experiments. The radiant panel was cooled by continuous cooling water provided by a thermostatic water tank. The temperature of the cooling water was regulated from 4 C to 30 C using a temperature controller. The growth, coalescence and departure of condensate droplets on the radiant panel were visualized by an upturned optical microscope (Canon EOS 60D camera installed with an EF 135 mm f/5.6 USM focus). The indoor air temperature was measured by T-type thermocouples, and the dew point temperature was measured by a lithium chloride dew point sensor. In the experiment, the radiant panel is divided into 10 square units, the size of which is 25 cm 25 cm. The temperature of each unit was measured by a platinum resistance (RTD), which was monitored by a data acquisition unit (HP-34970 A) connected to a personal computer. There are an advancing (maximal) contact angle and a receding (minimal) contact angle on a flat surface. It depends on whether a given droplet is formed by an advancing front (i.e., a droplet formed by increasing its volume) or a receding front (i.e., a droplet formed by decreasing its volume), respectively. Condensation occurred on the aluminum alloy surface of the radiant panel in the experiment.
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Fig. 10. Schematic of the experimental system.
Fig. 11. Structure of the radiant cooling panel: (a) Photograph from upward view; and (b) cross section.
The apparent (q), advancing (qad), and receding (qre) contact angles of the aluminum alloy surface were measured by a contact angle meter (JC2000CD1) at an air temperature of 28 ± 0.5 C and a relative humidity of 60 ± 2%, as shown in Table 1. The apparent contact angle was obtained via a sessile droplet method by dropping a water droplet with a volume of 3 mm3 on the aluminum alloy surface, as shown in Fig. 12. The measurement of the
Table 1 Contact angle measurement.
Contact angle ( ) Apparent (q)
Advancing (qad)
Receding (qre)
62.4
79.5
37.5
H. Tang et al. / Building and Environment 114 (2017) 445e454
advancing and receding contact angles was performed by the tilted plate method. A water droplet with volume of 10 mm3 was dropped on the sample platform of the contact angle meter, and then the sample platform was continuously tilted. When the water droplet just rolled, the contact angles obtained at the lowest point (qad) and highest point (qre) are considered as the advancing and receding contact angles, respectively. 3.2. Parameter measurement The aim of this investigation was to study the condensation water mass (Mdew) and the droplet departure duration (Dt) with different temperature differences between indoor air and radiant ceiling surface. In the experiment, the given supplied cooling water temperature ranged from 6 C to 14 C. The air temperature and dew point temperature in the chamber were controlled as 28 C and 20 C, respectively. Due to the thermal inertia of concrete layer in the radiant panel, the start of each condensation experiment was judged to be when the panel surface temperature remained invariable for 0.5 h. With the condensation process on the radiant panel monitored by the video camera, the departure duration of each unit on the panel was recorded when the first condensate droplet departed from the substrate. The radii (r) of condensate droplets on the panel were measured by a comparative method. The photographs of droplets were compared with the scale plate at the same magnification. And the size distribution of condensate droplet can be further obtained according to measuring the radius of single droplet. An electronic balance, with an accuracy of ±0.01 g and measuring range of 0e100 g, was used to weight the retained condensation water. A piece of absorbent paper was used to absorb the retained condensation water thoroughly as soon as the first condensate droplet departed from a unit of the radiant panel. And then the retained water was weighted. The experimental result of the condensation water mass per area was slightly underestimated because it was measured after the departure of the largest condensate droplet.
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falling droplets ranges between the lower and upper limits determined by the advancing and receding contact angles, respectively. Fig. 13 shows the photographs of the condensate droplets on the panel at different sub-cooled degrees when the first droplet departed from the substrate. For different sub-cooled degrees, droplet size distribution when the first droplet departed from the panel was measured experimentally via photograph, as shown in Fig. 14(a). The droplet size distributions with different sub-cooled degrees were close to each other and the fluctuation was induced by the randomness of the condensation process. It's indicated that there is almost no effect of sub-cooled degrees upon the droplet size distribution. And the size distribution is mainly determined by the wettability of panel surface. The population density of condensate droplets decreases exponentially with the rising droplet radius. Fig. 14(b) shows the comparison between statistical average value of the measured droplet population density and theoretical result of Rose. The experimental results demonstrate the same trend with the theoretical correlation. While the population density of large droplets, radii of which are over the capillary length (i.e., 2.72 mm for water) is average 2.2 times of Rose's results. The deviation of population density for large droplets results in the underestimation of condensation water mass on the radiant panel. 4.2. Departure duration of condensate droplet During the condensation experiment, the departure of condensate droplets from the radiant panel was monitored by an upturned video camera. The departure duration was plotted as a function of the sub-cooled degree of the panel surface, shown in Fig. 15. It is found that the droplet departure duration fluctuates at the same sub-cooled degree due to the variant apparent contact
4. Experimental results and discussion 4.1. Droplet size distribution when the first droplet departed The condensation water mass plays an important role in the departure duration of the first condensate droplet. It is mainly determined by the droplet size distribution. Condensation experiments were performed on the radiant panel with a surface temperature ranging from 9 C to 18 C, i.e., a sub-cooled degree of 1.5 C-10.0 C. In the experiment, the measured projection radii (rmax) of the gravity-induced falling droplets from the radiant panel were random and ranged from 4.3 mm to 6.5 mm. It is due to that the apparent contact angle is variable during the process of condensation, a number between the receding and advancing contact angles. As a result, the critical size of the gravity-induced
Fig. 12. Apparent contact angle measurement via the sessile droplet method.
Fig. 13. Photographs of condensate droplets on the panel at the moment of the departure of the first droplet: (a) Under a sub-cooled degree of 9.6 C; and (b) under a sub-cooled degree of 5.1 C.
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Fig. 14. Droplet size distribution when the first condensate droplet departed: (a) Under different sub-cooled degrees; and (b) statistical average values.
angles during the condensation process. But it decreases sharply with the sub-cooled degree. Fig. 15 also shows the comparison of droplet departure duration between the experiment and simulation. The measured results of droplet departure durations
demonstrate the same changing tendency as the simulated results. As a whole, the experimental duration is longer than the theoretical estimation. The average relative biases between experiment and simulation were within 25%, which validate the feasibility of the theoretical model. As for a sub-cooled degree of 5 C, the measured duration from the commencement of condensation to the first departure was 10 ± 1.0 h, while the theoretical result corresponding to an advancing contact angle of 79.5 is 6.8 h. The theoretical model provides a relatively conservative estimate of the droplet departure duration. According to the theoretical model described in section 2.3, the theoretical departure duration is the ratio between condensation water mass and condensation rate on the radiant ceiling. These two main influencing factors were also measured to validate the theoretical model. In the experiment, the condensation water mass (Mdew) on the panel when the first droplet departed was collected via absorbent paper and weighted by an electric balance. Fig. 16 shows the comparison of condensation water mass values through experiment and theoretical analysis. The condensation water mass per unit area with different sub-cooled degrees ranged from 450 g/m2 to 550 g/m2. It was almost not influenced by the panel surface temperature. By virtue of numerical simulation, theoretical condensation water masses are 450 g/m2, 404 g/m2 and 310 g/m2 corresponding to the advancing, apparent and receding contact angles, respectively. The average condensation water mass measured on the conventional aluminum alloy surface is 509 g/m2, which is 11.6% higher than the theoretical value corresponding to the advancing contact angle. On the other hand, condensation rate (md) on the panel surface was derived by the condensation water mass and time duration. Fig. 17 shows the comparison of condensation rates through experiment and theory with different sub-cooled degrees. The average condensation rate linearly increases with the sub-cooled degree. The experimental results are about 20% smaller than the theoretical values. The reason is that thermal resistance for the later stage of condensation process is higher than that for the early stage due to the accumulation and coverage of dew on the substrate. And the theoretical correlation formula of condensation rate was measured during the early stage of condensation process in the previous experiment [37]. Therefore, the underestimation of condensation water mass and overestimation of condensation rate both result in the theoretical departure duration shorter than the measured value.
Fig. 15. Plot of the departure duration as a function of the sub-cooled degree of the radiant panel surface.
H. Tang et al. / Building and Environment 114 (2017) 445e454
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Fig. 16. Comparison of condensation water mass when the first droplet departed through experiment and theoretical analysis under different sub-cooled degrees.
Fig. 17. Comparison of condensation rate between experiment and theory under various sub-cooled degrees.
5. Conclusions Radiant cooling ceiling system is an energy efficient approach to provide thermal comfort environment in buildings. Condensation risk is a main obstacle for its widespread application. The departure duration of condensate droplets reflects how long it takes condensation on the radiant ceiling to negatively affect indoor occupants. It is a basic index concerning with both condensation risk and control strategy of radiant cooling systems. In this paper, a theoretical model was developed for predicting the departure duration of the first condensate droplet from the radiant ceiling. An experimental system was performed further to measure the departure duration with different sub-cooled degrees. The main results of this study can be summarized as follows: (1) According to the theoretical analysis, the condensation water mass is mainly determined by the surface wettability. It is almost not affected by the surface temperature, which is also validated by the experiment. (2) The droplet departure duration increases firstly with the increase of apparent contact angle through simulation
results. And it reaches the maximum with a contact angle of 110 . Then it decreases if apparent contact angle still increases. The droplet departure duration fluctuates, due to the variance of apparent contact angle during the condensation process. (3) The experimental results indicate that the droplet departure duration for a conventional aluminum alloy surface decreases sharply with the increase of sub-cooled degree. The changing tendency also consists with the theoretical analysis. (4) The experimental droplet departure duration was 10 ± 1 h with a sub-cooled degree of 5 C. However, the measured droplet departure duration is averagely 25% longer than theoretical estimation. Thus, the theoretical model provides a relatively conservative estimation of condensate droplet departure duration. Acknowledgments The research described in this paper was supported by the National Natural Science Foundation of China (No.51422808,
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No.51521005), and Application and development program of Chongqing (cstc2014yykfB30003). Nomenclature b DAB g hm L lcap md Mdew Mdew,s N Pr r Ra rmax s Sh Dt V
Radius of curvature at the apex of the pendent droplet, (m) Mass diffusivity, (m2/s) Gravitational acceleration, (N/kg) Mass transfer coefficient, (mm/s) Characteristic length of radiant panel in Eq. (6), (m) Capillary length, (mm) Condensation rate on the radiant ceiling, (g/(h$m2)) Condensation water mass when the first droplet departed, (g/m2) Stable retained condensation water mass, (g/m2) Droplet size distribution, (mm3) Prandtl number, () Droplet radius, (mm) Rayleigh number, Eq. (5) Projection radius of critical falling droplet, (mm) Arc length, (m) Sherwood number, Eq. (5) Condensate droplet departure duration, (h) Volume of the pendent droplet, (mm3)
Greek symbols g Surface tension, (N/m) 4 Tangential angle of the pendent droplet profile, (rad) q Apparent contact angle, (rad) qad Advancing contact angle, (rad) qre Receding contact angle, (rad) r Density, (kg/m3) u Non-dimensional water vapor fraction Subscripts a Air s Radiant panel surface w Water References [1] Building Energy Research Center of Tsinghua University, Annual Report on China Building Energy Efficiency, China Architecture & Building Press, Beijing, 2015. rez-Lombard, J. Ortiz, C. Pout, A review on buildings energy consumption [2] L. Pe information, Energy Build. 40 (2008) 394e398. [3] K.N. Rhee, K.W. Kim, A 50 year review of basic and applied research in radiant heating and cooling systems for the built environment, Build. Environ. 91 (2015) 166e190. [4] E.M. Saber, K.W. Tham, H. Leibundgut, A review of high temperature cooling systems in tropical buildings, Build. Environ. 96 (2016) 237e249. [5] T. Mikeska, S. Svendsen, Dynamic behavior of radiant cooling system based on capillary tubes in walls made of high performance concrete, Energy Build. 108 (2015) 92e100. [6] R. Bean, B.W. Olesen, K.W. Kim, History of radiant heating & cooling systems: part 1, ASHRAE J. 52 (2010) 40e41. [7] K. Zhao, X.H. Liu, Y. Jiang, Application of radiant floor cooling in large space buildingseA review, Renew. Sustain. Energy Rev. 55 (2016) 1083e1096. [8] T. Yu, P. Heiselberg, B. Lei, et al., Experimental investigation of cooling performance of a novel HVAC system combining natural ventilation with diffuse ceiling inlet and TABS, Energy Build. 105 (2015) 165e177. [9] J.D. Feng, F. Chuang, F. Borrelli, F. Bauman, Model predictive control of radiant slab systems with evaporative cooling sources, Energy Build. 87 (2015) 199e210.
[10] X. Hao, G. Zhang, Y. Chen, S. Zou, D.J. Moschandreas, A combined system of chilled ceiling, displacement ventilation and desiccant dehumidification, Build. Environ. 42 (2007) 3298e3308. [11] Y. Khan, V.R. Khare, J. Mathur, et al., Performance evaluation of radiant cooling system integrated with air system under different operational strategies, Energy Build. 97 (2015) 118e128. [12] G. Sastry, VAV vs. radiant: side-by-side comparison, ASHRAE J. 56 (5) (2014) 16. [13] W.H. Chiang, C.Y. Wang, J.S. Huang, Evaluation of cooling ceiling and mechanical ventilation systems on thermal comfort using CFD study in an office for subtropical region, Build. Environ. 48 (2012) 113e127. [14] J. Le Dreau, P. Heiselberg, Sensitivity analysis of the thermal performance of radiant and convective terminals for cooling buildings, Energy Build. 82 (2014) 482e491. [15] Y. He, N. Li, Q. Huang, A field study on thermal environment and occupant local thermal sensation in offices with cooling ceiling in Zhuhai, China, Energy Build. 102 (2015) 277e283. [16] E.M. Saber, R. Iyengar, M. Mast, F. Meggers, K.W. Tham, H. Leibundgut, Thermal comfort and IAQ analysis of a decentralized DOAS system coupled with radiant cooling for the tropics, Build. Environ. 82 (2014) 361e370. [17] M.M. Hassan, Y. Beliveau, Modeling of an integrated solar system, Build. Environ. 43 (5) (2008) 804e810. [18] Y.L. Yuan, X. Zhang, X. Zhou, A study on inherent correlation of thermal performances and condensation free control of the radiant system, Energy Build. 129 (2016) 19e31. [19] B. Ning, Y. Chen, H. Liu, S. Zhang, Cooling capacity improvement for a radiant ceiling panel with uniform surface temperature distribution, Build. Environ. 102 (2016) 64e72. [20] D. Xie, Y. Wang, H. Wang, S. Mo, M. Liao, Numerical analysis of temperature non-uniformity and cooling capacity for capillary ceiling radiant cooling panel, Renew. Energy 87 (2016) 1154e1161. [21] J.M. Seo, D. Song, K.H. Lee, Possibility of coupling outdoor air cooling and radiant floor cooling under hot and humid climate conditions, Energy Build. 81 (2014) 219e226. [22] L.Z. Zhang, J.L. Niu, Indoor humidity behaviors associated with decoupled cooling in hot and humid climates, Build. Environ. 38 (2003) 99e107. [23] G.M. Ge, F. Xiao, S.W. Wang, Neural network based prediction method for preventing condensation in chilled ceiling systems, Energy Build. 45 (2012) 290e298. [24] S.A. Mumma, Chilled ceilings in parallel with dedicated outdoor air systems: Addressing the concerns of condensation, capacity, and cost, ASHRAE Trans. 108 (2002) 220. [25] H.D. Tang, X.H. Liu, Experimental study of dew formation on metal radiant panels, Energy Build. 85 (2014) 515e523. [26] Y.L. Yin, R.Z. Wang, X.Q. Zhai, T.F. Ishugah, Experimental investigation on the heat transfer performance and water condensation phenomenon of radiant cooling panels, Build. Environ. 71 (2014) 15e23. [27] M. Mei, B. Yu, M. Zou, L. Luo, A numerical study on growth mechanism of dropwise condensation, Int, J. Heat. Mass Transf. 54 (9) (2011) 2004e2013. [28] M. Mei, F. Hu, C. Han, Y. Cheng, Time-averaged droplet size distribution in steady-state dropwise condensation, Int. J. Heat. Mass Transf. 88 (2015) 338e345. [29] B.S. Sikarwar, N.K. Battoo, S. Khandekar, K. Muralidhar, Dropwise condensation underneath chemically textured surfaces: simulation and experiments, J. Heat. Transf. 133 (2) (2011) 021501. [30] C. Liang, F. Wang, Y. Lü, M. Yang, X. Zhang, Experimental and theoretical study of frost melting water retention on fin surfaces with different surface characteristics, Exp. Therm. Fluid Sci. 71 (2016) 70e76. [31] O.I. del Río, A.W. Neumann, Axisymmetric drop shape analysis: computational methods for the measurement of interfacial properties from the shape and dimensions of pendant and sessile drops, J. Colloid Interface Sci. 196 (2) (1997) 136e147. [32] E. Pitts, The stability of pendent liquid drops. Part 2. Axial symmetry, J. Fluid Mech. 63 (3) (1974) 487e508. [33] N. Fuchikami, S. Ishioka, K. Kiyono, Simulation of a dripping faucet, J. Phys. Soc. Jpn. 68 (4) (1999) 1185e1196. [34] N. Watanabe, M. Aritomi, A. Machida, Time-series characteristics and geometric structures of drop-size distribution density in dropwise condensation, Int. J. Heat. Mass Transf. 76 (2014) 467e483. [35] J.W. Rose, L.R. Glicksman, Dropwise condensationdthe distribution of drop sizes, Int. J. Heat. Mass Transf. 16 (2) (1973) 411e425. [36] E.J. Le Fevre, J.W. Rose, An experimental study of heat transfer by dropwise condensation, Int. J. Heat. Mass Transf. 8 (8) (1965) 1117e1133. [37] H.D. Tang, X.H. Liu, Y. Jiang, Theoretical and experimental study of condensation rates on radiant cooling surfaces in humid air, Build. Environ. 97 (2016) 1e10.
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Building and Environment journal homepage: www.elsevier.com/locate/buildenv
Theoretical and experimental study of departure duration of condensate droplets from radiant cooling ceiling surfaces Haida Tang a, Tao Zhang a, b, Xiaohua Liu a, *, Yi Jiang a a b
Department of Building Science, Tsinghua University, Beijing 100084, China School of Aerospace, Tsinghua University, Beijing 100084, China
a r t i c l e i n f o
a b s t r a c t
Article history: Received 6 September 2016 Received in revised form 2 January 2017 Accepted 3 January 2017 Available online 4 January 2017
In this paper, a mathematical model for predicting the departure duration of the first condensate droplet from a radiant ceiling surface was proposed on the basis of the condensation water mass. The simulation results indicate a dependence of condensation water mass on the apparent contact angle of the substrate, but almost in no relation with the surface temperature. The condensation water mass firstly increases with the increase of the apparent contact angle. It reaches a maximum weight of 522 g/m2 at an apparent contact angle of 110 , and then decreases. A visualization experiment of condensation on a radiant ceiling panel with a conventional aluminum alloy surface was performed in a climate chamber to measure the departure duration of the droplet. The measured departure duration fluctuates due to the variance of apparent contact angle and the randomness of condensation process, but it decreases sharply with the sub-cooled degree (air dew point minus surface temperature). And the average departure duration is 10 h with a sub-cooled degree of 5 C. The theoretical model is validated as the average relative biases between the experimental and theoretical results are within 25%. © 2017 Elsevier Ltd. All rights reserved.
Keywords: Radiant cooling ceiling Condensation Condensate droplet departure duration Condensation water mass Contact angle
1. Introduction Nowadays building energy consumption is still increasing with the development of economy and society, accounting for about 20% of the total energy in China [1]. The air-conditioning system including heating and cooling usually consumes about 30%e60% [1,2] of the total building energy. It is crucial to reduce the energy consumption of the air-conditioning system in buildings. Continuous efforts have been paid in proposing novel appropriate approaches for creating comfortable living conditions with a low energy cost. The radiant cooling ceiling system is such a new solution serving as a high temperature cooling system [3e5]. It has been used in multiple types of buildings, such as office buildings, residential buildings and airports, and its popularity is still increasing [6e8]. Compared with all-air systems, radiant cooling ceiling systems can achieve energy savings by increasing chiller's efficiency due to high temperature cooling and reducing energy consumption of fans [9,10]. Khan et al. [11] investigated the energy consumption of radiant cooling ceiling systems with a dedicated outside air system (DOAS) in a hot and humid climate. A
* Corresponding author. E-mail address: [email protected] (X. Liu). http://dx.doi.org/10.1016/j.buildenv.2017.01.001 0360-1323/© 2017 Elsevier Ltd. All rights reserved.
comparison of energy consumption calculated that the radiant systems was 17.5% more efficient than a conventional all-air system. Besides, a side-by-side comparison between radiant and variable air volume systems was conducted in two identical buildings in India. The monitoring data indicated that the radiant system used 34% less energy compared to the variable air volume systems during the two years of operation [12]. Furthermore, previous studies revealed that a radiant cooling ceiling system could achieve better thermal comfort than a conventional all-air system [13e15]. Radiant cooling ceiling system is also claimed as a promising indoor terminal unit for temperature and humidity independent control. As radiant cooling ceiling systems only handle indoor sensible load, indoor moisture load needs to be handled by auxiliary dehumidification systems [16,17]. However, there are condensation risks on the panel surfaces of radiant cooling ceiling systems when applied in a hot and humid climate. Research on avoiding condensation of radiant cooling surfaces has mainly focused on the control strategies and system configuration. It's suggested that the lowest surface temperature of radiant panel should be maintained above the indoor dew point temperature to guarantee no condensation. Yuan et al. [18] studied the inherent correlation between total heat flux capability and panel surface temperature, and proposed a condensation free control logic based on a simplified model of panel surface
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temperature. Ning et al. [19] improved a novel radiant ceiling panel with a uniform surface temperature distribution which is beneficial for condensation control. Xie et al. [20] investigated the influence of water temperature and tube spacing on the non-uniformity of surface temperature for a capillary radiant ceiling panel by a computational fluid dynamics (CFD) simulation. On the other hand, it is also essential to control the indoor moisture level carefully. Seo et al. [21] developed a control logic that a dehumidification ventilation system will be activated to remove indoor moisture facing potential condensation risks. Zhang et al. [22] and Ge et al. [23] studied the optimal pre-dehumidification time with neural network models to eliminate condensation risks during the startup process where the air conditioning system operated intermittently. However, the condensation issue on a radiant cooling ceiling close to external windows and doors, or wherever the indoor moisture load increases sharply, is sometimes inevitable. Natural convection flows arise from the combined buoyancy effects of thermal and mass diffusion in buildings. Condensation on a radiant cooling ceiling involving the natural convection is relatively a slow process [24,25]. The departure duration of droplet usually takes several hours from the beginning of the condensation to the first condensate droplet departure from a radiant ceiling. This provides a quantitative requirement for the responsiveness of control strategy due to thermal inertia of the radiant system and a limited dehumidifying ability of the air system. Yin et al. [26] experimentally examined the growth rate of average droplet radius on a plastic capillary panel varying with inlet temperature of chilled water. It's indicated that the departure duration of droplet is affected by the largest droplet radius instead of the average value. Mei et al. [27,28] and Sikarwar et al. [29] simulated the processes of nucleation, growth due to direct condensation, coalescence, and fall-off of droplets on the underside of horizontal substrate. The dropwise condensation of water vapor was studied correspondingly. Nevertheless, the condensation on radiant ceiling arises from natural convection between humid air and panel surface rather than condensation in pure water vapor. And the great amount of computation becomes an obstacle in application. Because the departure duration of droplet is closely correlated to the condensation water mass, the frost melting water retention on a vertical substrate was investigated theoretically and experimentally by Liang et al. [30]. The experimental results indicated that the retained water mass of the superhydrophobic fin decreases by 75.82% compared with that of the bare fins. However, there are seldom studies on the departure duration of condensate droplet from a radiant ceiling surface due to the complexity and randomness of the condensation process. In this paper, a theoretical model based on the condensation water mass, which took the surface characteristics of apparent contact angle into account, was developed for predicting the departure duration of the first condensate droplet from a radiant ceiling panel. Meanwhile, a series of condensation experiments with different sub-cooled degrees were conducted on a radiant ceiling with a conventional aluminum alloy surface. The experimental results helped to investigate the mass of condensation water and the departure duration of droplet. The present research would be beneficial for a better operation and control of radiant cooling systems. 2. Theoretical analysis 2.1. Critical size of the gravity-induced falling droplet As shown in Fig. 1, metal panels are usually adopted as the radiant cooling ceiling for a better heat transfer performance. When the panel surface temperature is below dew-point temperature, condensate droplets grow and coalesce underneath the radiant
Fig. 1. Schematic of a radiant cooling ceiling panel.
ceiling. And they fall off due to gravity upon approaching the critical size. The condensation phenomenon is significantly influenced by the wettability of a surface. It is always characterized by the contact angle (q), which indicates the degree of wetting when a solid and liquid interact. When an interface exists between a liquid and a solid, the angle between the surface of the liquid and the outline of the contact surface is described as the contact angle. In the case of complete wetting (spreading), the contact angle is 0 . The solid is partially wetting if q is between 0 and 90 , and it is partially non-wetting as q is above 90 . In the case of superhydrophobic materials with so-called lotus effect, the contact angle approaches the theoretical limit of 180 . Here we investigate the critical size of the gravity-induced falling droplet from a horizontal radiant ceiling surface. The capillary length is a characteristic length scale for an interface between water and air which is subject both to gravitational acceleration and to a surface force in the interface, which is expressed as:
rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi lcap ¼
g
ðrw ra Þg
(1)
where g represents the surface tension of water; rw and ra are the density of water and air, respectively; g is the gravitational acceleration. The shape of a pendent droplet depends on the equilibrium between the forces of surface tension, gravity, and pressure. It was described by the classical Young-Laplace equation of capillarity [31]. According to the hypothesis of rotational symmetry of pendent droplets underneath horizontal surfaces, solving the three-dimensional shape of the droplet is simplified into solving its profile. It can be written as the following ordinary differential equations as a function of arc length s, as shown in Fig. 2.
dx ¼ cos 4 ds
(2a)
dz ¼ sin 4 ds
(2b)
d4 2 z sin 4 ¼ þ ds b l2cap x
(2c)
dV ¼ px2 sin 4 ds
(2d)
H. Tang et al. / Building and Environment 114 (2017) 445e454
447
Fig. 2. Coordinate system used in simulation of a pendent droplet profile on a radiant ceiling.
xð0Þ ¼ zð0Þ ¼ 4ð0Þ ¼ Vð0Þ ¼ 0
(2e)
where b is the radius of curvature at the apex of the droplet (O point); 4 is the tangential angle that, for pendent droplets, becomes the apparent contact angle (q) at the triple-phase contact line. It should be noted that, (sin 4/x)s¼0 ¼ 1/b at the apex of the droplet. The profile of the pendant droplet is generated by numerically integrating the initial value problem via fourth-order Runge-Kutta Method. The integration is stopped when the tangential angle (4) reaches the apparent contact angle (q) or 0 . It is indicated by Pitts [32] and Fuchikami [33] that there is more than one equilibrium droplet shape for a fixed droplet volume and a given apparent contact angle and only the shortest one in z direction is the stable one. Furthermore, for a fixed contact angle q, the volume of the pendant droplet reaches the maximum equilibrium, and the droplet falls off the substrate at the critical condition. The critical condition is shown in Eq. (3):
dV ¼0 db q
(3)
Fig. 3 illustrates the profiles of the falling droplets on a substrate with various apparent contact angles via simulation, where rmax and Vmax represents the projection radius and volume of the critical falling droplet from the substrate, respectively. The shape of the
Fig. 4. Size of a critical gravity-induced falling droplet as a function of apparent contact angle (q).
pendent droplet tends towards a spherical segment with the rising apparent contact angle, especially on hydrophobic surfaces. Furthermore, the radius (rmax) and volume (Vmax) of a critical gravity-induced shedding droplet are plotted as a function of the apparent contact angle (q) in Fig. 4. It could be found that the radius and volume of a critical falling pendent droplet both decrease with the increase of apparent contact angle.
2.2. Volume of pendent droplets underneath radiant ceiling From condensation nucleuses, a condensate droplet on a radiant ceiling grows via condensation and coalescence until reaching the maximum size. The volumes and radii of condensate droplets on a substrate vary in a quite large range. And the droplet radius has a correlation to the droplet volume with a given apparent contact angle. On the basis of Young-Laplace differential equations and the criteria of the gravity-induced falling droplet, Fig. 5 illuminates the variance of the pendent droplet volume (V(r)) as a function of the droplet radius (r) on radiant ceiling surfaces with various contact angles. The droplet radius firstly increases with the increase of pendent droplet volume underneath a radiant ceiling surface with
Fig. 3. Profiles of critical gravity-induced falling droplets with various apparent contact angles.
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Fig. 5. Pendent droplet volume as a function of its projection radius on radiant ceiling surfaces with various contact angles.
a given apparent contact angle. Then it decreases after a transition volume. The transition results from the non-uniqueness of the pendent droplet profile with a given apparent contact angle and a given projection radius. Fig. 6 shows that the volumes of pendent droplets differ with the same projection radius of 5.9 mm and an apparent contact angle of 35 . The smaller one is 164.5 mm3 and the larger one is 269.0 mm3. 2.3. Condensation water mass and departure duration of condensate droplets The departure duration (Dt) of condensate droplet is defined as a period of time from the beginning of condensation on radiant ceiling surface to the departure of the first condensate droplet. The departure of droplet occurs at the moment when the largest condensate droplet on the radiant ceiling surface reaches the critical size, i.e. when the body forces exceed surface tension at the triple-phase contact line holding it to the solid surface. The departure duration could be obtained according to the growth rate of the largest pendent droplet on a radiant ceiling during the condensation process. Condensation on the ceiling begins with the droplet formation at preferred nucleation sites at the atomic scale. The growth of a condensate droplet occurs by the combination of condensation and coalescence among neighboring droplets. As a consequence, the largest condensate droplet on a radiant ceiling surface is difficult to trace unless the whole condensation process is simulated. However, the condensation water mass progressively increases during the condensation process from the perspective of the condensation water holdup on the radiant ceiling surface. And it reaches the maximum value (Mdew) when the first droplet falls off. Then the fall-off droplets expose substrate area to provide a continuous source of nucleation sites. When the condensation
Fig. 6. Non-unique profiles of pendent droplets on a substrate with an apparent contact angle of 35 and a projection radius of 5.9 mm.
comes into a balance stage, the growth of droplets is equal to the shedding. As a result, the mass of condensate water on the radiant ceiling surface reaches a relatively invariable value (Mdew,s). Watanabe et al. [34] investigated the time-series characteristic of drop-size distribution density by experiment. The experimental results show that the theoretical drop-size distribution density of Rose [35] located on the center of obtained drop-size distribution densities from the first departure of condensate droplet to a steady state, particularly in the equilibrium region of small drops. On the other hand, the correlation between the droplet volume and droplet radius with a fixed apparent contact angle maintains constant during the whole condensation process on the substrate. Therefore, the stable retained mass of condensation water (Mdew,s) is approximately equal to the condensation water holdup (Mdew) on the substrate at the moment of the departure of the first droplet. Fig. 7 illustrates the flowchart of the theoretical model to predict the departure duration. The droplet departure duration could be calculated as the ratio between Mdew,s and the condensation rate (md) on the radiant ceiling surface. The stable retained condensation mass can be predicted with the aid of the droplet size distribution (N(r)) and the pendent droplet volume (V(r)). The size distribution proposed by Rose referred to the stable condition is adopted here because of its simplicity and agreement with the experimental data [35],
NðrÞ ¼
1 3pr 2 rmax
r
2=3
rmax
(4)
where rmax represents the projection radius of a critical shedding droplet which has been revealed in Fig. 4. N(r) is the population density of the condensate droplets in number per unit area per unit droplet radius. The condensation water mass underneath a radiant ceiling is
Zrmax Mdew;s ¼
rw VðrÞNðrÞdr
(5)
rmin
where rmin represents the radius of the condensation nucleus [36]. The plot of condensation water mass as a function of the apparent contact angle is shown in Fig. 8. The non-uniqueness problem of the droplet profile with the same projection radius may exist on a hydrophilic surface. As a result, the condensation water mass ranges between the lower and upper limits. But the relative deviation of the upper and lower limits is no more than 5%. A conservative estimation, i.e., the lower limit can be used as the
Fig. 7. Flowchart of the theoretical model framework for the prediction of departure duration from radiant ceiling surfaces.
H. Tang et al. / Building and Environment 114 (2017) 445e454
Fig. 8. Plot of the simulated stable condensation water mass on radiant ceiling as a function of the apparent contact angle of the substrate.
condensation water mass on the radiant ceiling in the view of engineering application. The condensation water mass on substrates with various surface characteristics range from 80 g/m2 to 520 g/ m2. Both of the lower and upper limits of the condensation water mass firstly increase and then decrease with the increase of apparent contact angle. The condensation water mass reaches a maximum with an apparent contact angle of 110 . Large droplets spread on the substrate and occupy too much space on a hydrophilic surface with good wettability, resulting in a small mass of condensation water per area. The smaller size of each single droplet contributes to a smaller condensation water mass on a superhydrophobic surface with bad wettability. Similar to Newton cooling formula, the condensation rate (md) on a radiant cooling ceiling surface can be obtained by the mass transfer coefficient (hm), which is expressed as
md ¼ hm ra ðua us Þ
(6)
where ua and us represent the water vapor mass fractions of indoor humid air and at the radiant ceiling surface respectively. ra denotes the density of humid air. The mass transfer coefficient hm has been measured experimentally in our previous study [37]. And it could be expressed in a correlation of average Sherwood number (Sh) vs. Rayleigh number (Ra).
Sh ¼ 0:1745Ra0:3313 ;
2:5 108 Ra 4:5 109
(7)
Sh ¼ hm L=DAB
(8)
Ra ¼ Pr,gL3 ðrs ra Þ=rs y2
(9)
where L is the characteristic length of radiant ceiling panel, DAB is the mass diffusivity of humid air, and Pr is the Prandtl number. Therefore, the departure duration of condensate droplet could be calculated, as
Dt ¼ Mdew =md
(10)
Fig. 9 shows the lower limit of the departure duration of the first condensate droplet from radiant ceiling surfaces, where there are three different sub-cooled degrees as a function of the apparent contact angle. It could be found that the droplet departure duration firstly increases and then decreases with the rise of apparent contact angle at a given surface temperature (Ts) and ambient air state. Additionally, the departure duration decreases with the increase of sub-cooled degree. Specifically, the sub-cooled degree is 5 C as the surface temperature is 14.5 C in ambient air of 28 C and 60% RH.
449
Fig. 9. Lower limit of the departure duration of condensate droplets from radiant ceiling surfaces of different surface temperatures as a function of the apparent contact angle.
The departure duration of condensate droplets is 6.0 h for the radiant ceiling made of conventional aluminum alloy with an apparent contact angle of 60 correspondingly. 3. Experimental setup 3.1. Experimental platform and procedure Fig. 10 is the schematic diagram of the experimental platform. A climate chamber (3 m 5 m 3 m) with hole-board ventilation was used to maintain indoor air temperature and humidity in the experiments. Indoor parameters were controlled by a cooling coil, an electric heater and a humidifier. A square radiant panel with a length of 1.0 m acting as the radiant ceiling was installed horizontally in the climate chamber, as shown in Fig. 11(a). The crosssection of the radiant panel is shown in Fig. 11(b). The outermost layer of the radiant panel is a 0.5-mm-thick aluminum alloy plate with electrostatic spraying. The metal surface is attached to a 20mm-thick concrete slab embedded with copper pipes. The bottom surface of the concrete layer is attached to a 100-mm-thick block of extruded polystyrene board (thermal conductivity k ¼ 0.05 W/(m$K)). Cooling water is supplied to the copper pipes, which snake throughout the concrete layer at an interval of 80 mm to maintain a uniform temperature field on the aluminum alloy surface. In addition, the radiant panel was surrounded by wind deflectors. It's ensured that no external disturbances in the air interfered with the condensation experiments. The radiant panel was cooled by continuous cooling water provided by a thermostatic water tank. The temperature of the cooling water was regulated from 4 C to 30 C using a temperature controller. The growth, coalescence and departure of condensate droplets on the radiant panel were visualized by an upturned optical microscope (Canon EOS 60D camera installed with an EF 135 mm f/5.6 USM focus). The indoor air temperature was measured by T-type thermocouples, and the dew point temperature was measured by a lithium chloride dew point sensor. In the experiment, the radiant panel is divided into 10 square units, the size of which is 25 cm 25 cm. The temperature of each unit was measured by a platinum resistance (RTD), which was monitored by a data acquisition unit (HP-34970 A) connected to a personal computer. There are an advancing (maximal) contact angle and a receding (minimal) contact angle on a flat surface. It depends on whether a given droplet is formed by an advancing front (i.e., a droplet formed by increasing its volume) or a receding front (i.e., a droplet formed by decreasing its volume), respectively. Condensation occurred on the aluminum alloy surface of the radiant panel in the experiment.
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Fig. 10. Schematic of the experimental system.
Fig. 11. Structure of the radiant cooling panel: (a) Photograph from upward view; and (b) cross section.
The apparent (q), advancing (qad), and receding (qre) contact angles of the aluminum alloy surface were measured by a contact angle meter (JC2000CD1) at an air temperature of 28 ± 0.5 C and a relative humidity of 60 ± 2%, as shown in Table 1. The apparent contact angle was obtained via a sessile droplet method by dropping a water droplet with a volume of 3 mm3 on the aluminum alloy surface, as shown in Fig. 12. The measurement of the
Table 1 Contact angle measurement.
Contact angle ( ) Apparent (q)
Advancing (qad)
Receding (qre)
62.4
79.5
37.5
H. Tang et al. / Building and Environment 114 (2017) 445e454
advancing and receding contact angles was performed by the tilted plate method. A water droplet with volume of 10 mm3 was dropped on the sample platform of the contact angle meter, and then the sample platform was continuously tilted. When the water droplet just rolled, the contact angles obtained at the lowest point (qad) and highest point (qre) are considered as the advancing and receding contact angles, respectively. 3.2. Parameter measurement The aim of this investigation was to study the condensation water mass (Mdew) and the droplet departure duration (Dt) with different temperature differences between indoor air and radiant ceiling surface. In the experiment, the given supplied cooling water temperature ranged from 6 C to 14 C. The air temperature and dew point temperature in the chamber were controlled as 28 C and 20 C, respectively. Due to the thermal inertia of concrete layer in the radiant panel, the start of each condensation experiment was judged to be when the panel surface temperature remained invariable for 0.5 h. With the condensation process on the radiant panel monitored by the video camera, the departure duration of each unit on the panel was recorded when the first condensate droplet departed from the substrate. The radii (r) of condensate droplets on the panel were measured by a comparative method. The photographs of droplets were compared with the scale plate at the same magnification. And the size distribution of condensate droplet can be further obtained according to measuring the radius of single droplet. An electronic balance, with an accuracy of ±0.01 g and measuring range of 0e100 g, was used to weight the retained condensation water. A piece of absorbent paper was used to absorb the retained condensation water thoroughly as soon as the first condensate droplet departed from a unit of the radiant panel. And then the retained water was weighted. The experimental result of the condensation water mass per area was slightly underestimated because it was measured after the departure of the largest condensate droplet.
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falling droplets ranges between the lower and upper limits determined by the advancing and receding contact angles, respectively. Fig. 13 shows the photographs of the condensate droplets on the panel at different sub-cooled degrees when the first droplet departed from the substrate. For different sub-cooled degrees, droplet size distribution when the first droplet departed from the panel was measured experimentally via photograph, as shown in Fig. 14(a). The droplet size distributions with different sub-cooled degrees were close to each other and the fluctuation was induced by the randomness of the condensation process. It's indicated that there is almost no effect of sub-cooled degrees upon the droplet size distribution. And the size distribution is mainly determined by the wettability of panel surface. The population density of condensate droplets decreases exponentially with the rising droplet radius. Fig. 14(b) shows the comparison between statistical average value of the measured droplet population density and theoretical result of Rose. The experimental results demonstrate the same trend with the theoretical correlation. While the population density of large droplets, radii of which are over the capillary length (i.e., 2.72 mm for water) is average 2.2 times of Rose's results. The deviation of population density for large droplets results in the underestimation of condensation water mass on the radiant panel. 4.2. Departure duration of condensate droplet During the condensation experiment, the departure of condensate droplets from the radiant panel was monitored by an upturned video camera. The departure duration was plotted as a function of the sub-cooled degree of the panel surface, shown in Fig. 15. It is found that the droplet departure duration fluctuates at the same sub-cooled degree due to the variant apparent contact
4. Experimental results and discussion 4.1. Droplet size distribution when the first droplet departed The condensation water mass plays an important role in the departure duration of the first condensate droplet. It is mainly determined by the droplet size distribution. Condensation experiments were performed on the radiant panel with a surface temperature ranging from 9 C to 18 C, i.e., a sub-cooled degree of 1.5 C-10.0 C. In the experiment, the measured projection radii (rmax) of the gravity-induced falling droplets from the radiant panel were random and ranged from 4.3 mm to 6.5 mm. It is due to that the apparent contact angle is variable during the process of condensation, a number between the receding and advancing contact angles. As a result, the critical size of the gravity-induced
Fig. 12. Apparent contact angle measurement via the sessile droplet method.
Fig. 13. Photographs of condensate droplets on the panel at the moment of the departure of the first droplet: (a) Under a sub-cooled degree of 9.6 C; and (b) under a sub-cooled degree of 5.1 C.
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Fig. 14. Droplet size distribution when the first condensate droplet departed: (a) Under different sub-cooled degrees; and (b) statistical average values.
angles during the condensation process. But it decreases sharply with the sub-cooled degree. Fig. 15 also shows the comparison of droplet departure duration between the experiment and simulation. The measured results of droplet departure durations
demonstrate the same changing tendency as the simulated results. As a whole, the experimental duration is longer than the theoretical estimation. The average relative biases between experiment and simulation were within 25%, which validate the feasibility of the theoretical model. As for a sub-cooled degree of 5 C, the measured duration from the commencement of condensation to the first departure was 10 ± 1.0 h, while the theoretical result corresponding to an advancing contact angle of 79.5 is 6.8 h. The theoretical model provides a relatively conservative estimate of the droplet departure duration. According to the theoretical model described in section 2.3, the theoretical departure duration is the ratio between condensation water mass and condensation rate on the radiant ceiling. These two main influencing factors were also measured to validate the theoretical model. In the experiment, the condensation water mass (Mdew) on the panel when the first droplet departed was collected via absorbent paper and weighted by an electric balance. Fig. 16 shows the comparison of condensation water mass values through experiment and theoretical analysis. The condensation water mass per unit area with different sub-cooled degrees ranged from 450 g/m2 to 550 g/m2. It was almost not influenced by the panel surface temperature. By virtue of numerical simulation, theoretical condensation water masses are 450 g/m2, 404 g/m2 and 310 g/m2 corresponding to the advancing, apparent and receding contact angles, respectively. The average condensation water mass measured on the conventional aluminum alloy surface is 509 g/m2, which is 11.6% higher than the theoretical value corresponding to the advancing contact angle. On the other hand, condensation rate (md) on the panel surface was derived by the condensation water mass and time duration. Fig. 17 shows the comparison of condensation rates through experiment and theory with different sub-cooled degrees. The average condensation rate linearly increases with the sub-cooled degree. The experimental results are about 20% smaller than the theoretical values. The reason is that thermal resistance for the later stage of condensation process is higher than that for the early stage due to the accumulation and coverage of dew on the substrate. And the theoretical correlation formula of condensation rate was measured during the early stage of condensation process in the previous experiment [37]. Therefore, the underestimation of condensation water mass and overestimation of condensation rate both result in the theoretical departure duration shorter than the measured value.
Fig. 15. Plot of the departure duration as a function of the sub-cooled degree of the radiant panel surface.
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Fig. 16. Comparison of condensation water mass when the first droplet departed through experiment and theoretical analysis under different sub-cooled degrees.
Fig. 17. Comparison of condensation rate between experiment and theory under various sub-cooled degrees.
5. Conclusions Radiant cooling ceiling system is an energy efficient approach to provide thermal comfort environment in buildings. Condensation risk is a main obstacle for its widespread application. The departure duration of condensate droplets reflects how long it takes condensation on the radiant ceiling to negatively affect indoor occupants. It is a basic index concerning with both condensation risk and control strategy of radiant cooling systems. In this paper, a theoretical model was developed for predicting the departure duration of the first condensate droplet from the radiant ceiling. An experimental system was performed further to measure the departure duration with different sub-cooled degrees. The main results of this study can be summarized as follows: (1) According to the theoretical analysis, the condensation water mass is mainly determined by the surface wettability. It is almost not affected by the surface temperature, which is also validated by the experiment. (2) The droplet departure duration increases firstly with the increase of apparent contact angle through simulation
results. And it reaches the maximum with a contact angle of 110 . Then it decreases if apparent contact angle still increases. The droplet departure duration fluctuates, due to the variance of apparent contact angle during the condensation process. (3) The experimental results indicate that the droplet departure duration for a conventional aluminum alloy surface decreases sharply with the increase of sub-cooled degree. The changing tendency also consists with the theoretical analysis. (4) The experimental droplet departure duration was 10 ± 1 h with a sub-cooled degree of 5 C. However, the measured droplet departure duration is averagely 25% longer than theoretical estimation. Thus, the theoretical model provides a relatively conservative estimation of condensate droplet departure duration. Acknowledgments The research described in this paper was supported by the National Natural Science Foundation of China (No.51422808,
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No.51521005), and Application and development program of Chongqing (cstc2014yykfB30003). Nomenclature b DAB g hm L lcap md Mdew Mdew,s N Pr r Ra rmax s Sh Dt V
Radius of curvature at the apex of the pendent droplet, (m) Mass diffusivity, (m2/s) Gravitational acceleration, (N/kg) Mass transfer coefficient, (mm/s) Characteristic length of radiant panel in Eq. (6), (m) Capillary length, (mm) Condensation rate on the radiant ceiling, (g/(h$m2)) Condensation water mass when the first droplet departed, (g/m2) Stable retained condensation water mass, (g/m2) Droplet size distribution, (mm3) Prandtl number, () Droplet radius, (mm) Rayleigh number, Eq. (5) Projection radius of critical falling droplet, (mm) Arc length, (m) Sherwood number, Eq. (5) Condensate droplet departure duration, (h) Volume of the pendent droplet, (mm3)
Greek symbols g Surface tension, (N/m) 4 Tangential angle of the pendent droplet profile, (rad) q Apparent contact angle, (rad) qad Advancing contact angle, (rad) qre Receding contact angle, (rad) r Density, (kg/m3) u Non-dimensional water vapor fraction Subscripts a Air s Radiant panel surface w Water References [1] Building Energy Research Center of Tsinghua University, Annual Report on China Building Energy Efficiency, China Architecture & Building Press, Beijing, 2015. rez-Lombard, J. Ortiz, C. Pout, A review on buildings energy consumption [2] L. Pe information, Energy Build. 40 (2008) 394e398. [3] K.N. Rhee, K.W. Kim, A 50 year review of basic and applied research in radiant heating and cooling systems for the built environment, Build. Environ. 91 (2015) 166e190. [4] E.M. Saber, K.W. Tham, H. Leibundgut, A review of high temperature cooling systems in tropical buildings, Build. Environ. 96 (2016) 237e249. [5] T. Mikeska, S. Svendsen, Dynamic behavior of radiant cooling system based on capillary tubes in walls made of high performance concrete, Energy Build. 108 (2015) 92e100. [6] R. Bean, B.W. Olesen, K.W. Kim, History of radiant heating & cooling systems: part 1, ASHRAE J. 52 (2010) 40e41. [7] K. Zhao, X.H. Liu, Y. Jiang, Application of radiant floor cooling in large space buildingseA review, Renew. Sustain. Energy Rev. 55 (2016) 1083e1096. [8] T. Yu, P. Heiselberg, B. Lei, et al., Experimental investigation of cooling performance of a novel HVAC system combining natural ventilation with diffuse ceiling inlet and TABS, Energy Build. 105 (2015) 165e177. [9] J.D. Feng, F. Chuang, F. Borrelli, F. Bauman, Model predictive control of radiant slab systems with evaporative cooling sources, Energy Build. 87 (2015) 199e210.
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