pubs.acs.org/Langmuir © 2009 American Chemical Society
Investigation of Local Evaporation Flux and Vapor-Phase Pressure at an Evaporative Droplet Interface Fei Duan*,† and C. A. Ward‡ †
Division of Thermal and Fluids Engineering, School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore 639798, and ‡Thermodynamics and Kinetics Laboratory, Department of Mechanical and Industrial Engineering, University of Toronto, Toronto, Canada M5S 3G8 Received January 27, 2009. Revised Manuscript Received March 2, 2009 In the steady-state experiments of water droplet evaporation, when the throat was heating at a stainless steel conical funnel, the interfacial liquid temperature was found to increase parabolically from the center line to the rim of the funnel with the global vapor-phase pressure at around 600 Pa. The energy conservation analysis at the interface indicates that the energy required for evaporation is maintained by thermal conduction to the interface from the liquid and vapor phases, thermocapillary convection at interface, and the viscous dissipation globally and locally. The local evaporation flux increases from the center line to the periphery as a result of multiple effects of energy transport at the interface. The local vapor-phase pressure predicted from statistical rate theory (SRT) is also found to increase monotonically toward the interface edge from the center line. However, the average value of the local vapor-phase pressures is in agreement with the measured global vapor-phase pressure within the measured error bar.
Introduction The droplet evaporation is important in many applications such as coating, printing, electronic cooling, and medical applications. However, the process has not been well-understood. The reason might be that the evaporation is affected by the complex phenomena including hydrodynamic effects, contact line motion and wetting dynamics, buoyancy convection, and thermocapillary convection. The experiments of sessile water droplet evaporation on the glass substance1 visualized thermocapillary convection induced by evaporation even though thermocapillary convection was reported to be difficult to observe due to surfactant contaminants at the interface.2-4 Although the effects of contact line, buoyancy-driven convection, and hydrodynamic instability cannot be exactly eliminated in the experiments by Xu et al.,1 their experiments supported our previous studies on the evaporation of water and heavy water.5-8 In those steady-state experiments, the droplets were confined within stainless steel funnels; there was not contact line motion, buoyancy-driven convection, and hydrodynamic effects. The transition of thermocapillary flow was experimentally investigated on the basis of energy analysis at the evaporating interface and indicated with the Marangoni number, Ma, defined by Pearson.9 As Ma was less than about 100, the interface region was quiescent, thermal conduction in vapor phases provided the energy required to evaporate the liquid at the observed rate; as 100 < Ma < 22 000, thermocapillary convection was present, and thermal *
[email protected]. (1) Xu, X.; Luo, J. Appl. Phys. Lett. 2007, 91, 124102. (2) Deegan, R. D.; Bakajin, O.; Dupont, T. F.; Huber, G.; Nagel, S. R.; Witten, T. A. Nature (London) 1997, 329, 827. (3) Hu, H.; Larson, R. G. J. Phys. Chem. B 2006, 110, 7090. (4) Cammenga, H. K.; Schreiber, D.; Barnes, G. T.; Hunter, D. S. J. Colloid Interface Sci. 1984, 98, 585. (5) Duan, F.; Ward, C. A. Phys. Rev. E 2005, 72, 056302. (6) Duan, F.; Ward, C. A. Phys. Rev. E 2005, 72, 056304. (7) Ward, C. A.; Duan, F. Phys. Rev. E 2004, 69, 056308. (8) Duan, F.; Badam, V. K.; Durst, F.; Ward, C. A. Phys. Rev. E 2005, 72, 056303. (9) Pearson, J. R. A. J. Fluid Mech. 1958, 4, 489.
7424 DOI: 10.1021/la900337j
conduction through the liquid and vapor phases no longer provided sufficient energy to evaporate the liquid. Up to 40% energy was transported by thermocapillary convection from the funnel rim where the liquid was heated; as Ma > 22 000, the vapor-water interface became turbulent, and the interfacial viscous dissipation became important.5,7 By applying the Gibbs dividing-surface approximation,5,10 a new interfacial property of water, the surface-thermal capacity, cσ, was experimentally determined. It was found that the value is 30.6 ( 0.8 kJ/m2 K for water5 and 32.5 ( 0.8 kJ/m2 K for heavy water.6 The surfacethermal capacity played an important role in transferring energy by the thermocapillary flow during evaporation. However, the recent experiments in a poly(methyl methacrylate) funnel suggested that the Pearson’s Marangoni number cannot reflect the thermocapillary convection transition and the indicator of transition in our previous experimental studies5-8 might not be used in the circumstance. However, the transition was evaluated with the crucial energy balance analysis at the interface in the previous studies. To enhance the understanding of the energy conservation criterion, in this paper, a series of steady-state evaporation experiments are introduced by controlling the global vapor-phase pressure at about 600 Pa but at different throat temperature. The data are compared with the experiments with the same throat temperatures (about 3.6 C) but different vaporphase pressure.5,7 The energy transport analysis is conducted based on the three-dimensional temperature measurements. With the local evaporation flux calculated at the water-vapor interface, statistical rate theory (SRT)11,12 is applied to predict the local vapor-phase pressure along the evaporative droplet interface. The mean of predicted vapor-phase pressures is then compared with the globe vapor-phase pressure measured about 20 cm above the interface. (10) Willard Gibbs, J. Trans. Conn. Acad. Arts Sci. 1876, 3, 108. republished as The Scientific Papers of J. Willard Gibbs, edited by H. A. Bumstead and R. G. Van Name, Dover, NY, 1961, Vol. 1, p. 219. (11) Ward, C. A.; Fang, G. Phys. Rev. E 1999, 59, 429. (12) Ward, C. A. J. Non-Equilib. Thermodyn. 2002, 27, 289.
Published on Web 04/16/2009
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Experimental Methods The experimental equipment is the same as that in refs 5,7. Water was deionized, distilled, nanofiltered, and degassed at first. Simultaneously, the experimental chamber and syringe were evacuated to a pressure of 10-5 Pa (UHV reading) by turbo and backing mechanical pumps before the water was introduced into the chamber. A degassed water droplet, supplied with a pumped syringe, was formed on the conical stainless steel funnel mouth (Figure 1). To prevent subsequent vapor bubble formation, the chamber was pressurized with nitrogen to 142.0 kPa after the funnel and syringe were filled with water. Afterward, one-fifth the amount of water in the syringe was flushed out in order to remove the possibly dissolved nitrogen. A mechanical pump was then used to evaporate the chamber at a scheduled pressure. The spherical interface formed at a maximum height above the funnel mouth of approximately 1.0 mm. As shown in Figure 2, the curve of the droplet image matches a spherical cap with a interfacial radius R0
R0 ¼
rm sin½2arctanðrm =hÞ
ð1Þ
where h is the maximum height of the droplet above the funnel mouth, and rm is the maximum radius of conical funnel with a value of 3.5 mm. Steady-state conditions were adjusted by regulating the pumping rate of syringe to keep the droplet height at around 1.0 mm and maintaining the global vapor-phase pressure to a predetermined value. Under steady-state conditions, the global vapor-phase pressure was kept at about 600 Pa, but the temperature at the throat of the funnel was controlled by a cooling bath at different values from around 3.6 to 21.8 C as listed in Table 1. Then, the temperatures in the vapor and liquid phases were measured with a calibrated thermocouple (type K) that had been formed into a U shape. The thermocouple (25.4 μm in diameter) was mounted on a three-dimensional positioner. Two cathetometers, separated with an angle of 90, were used to determine the location of the thermocouple and the maximum height of the droplet. The temperatures were measured at 0.0 mm, 0.7 mm, 1.4 mm, 2.1 mm, and 2.8 mm from the center line of the funnel in one horizontal direction (Figure 1). At each vertical position, the distance between two measurements was 10 μm in the vapor phase near the interface and 20 μm in the liquid phase. At each point, the thermocouple readings were recorded each second for a period of 60 s by a LabView program using a 34970A HP data acquisition/ switch unit, and the mean and standard deviation of the temperatures were computed. Then, without changing the experimental conditions, the U-shaped thermocouple was rotated 90. The temperature distribution was measured at the center line and
Figure 1. The sketch of apparatus. The temperatures were measured across the water-vapor spherical interface. Langmuir 2009, 25(13), 7424–7431
Figure 2. The interface is assumed to be spherical. The calculated curve is compared with the measurement position with the U-shaped moveable thermocouple. There is not obvious variation found. 0.7 mm, 1.4 mm, 2.1 mm, and 2.8 mm away from the center line. The pressure in the vapor phase was measured approximately 20 cm above the funnel mouth with a Hg manometer. The interfacial liquid temperature at the center line (intf. liq. temp.), the interfacial vapor temperature at the center line (intf. vap. temp.), the temperature at the throat of funnel (throat temp.), the interfacial radius (intf. rad.), and global vapor-phase pressure (vap.-ph. press.) were listed in Table1 for each experiment. The average evaporation flux (avg. evap. flux) is determined by dividing the total evaporation rate recorded by the syringe pump by the surface area of the spherical water droplet above the funnel mouth.
Experimental Results and Discussions Temperature Profiles near the Interface. Six steady-state water evaporation experiments were conducted with the vapor-phase pressures at about 600 Pa when the throat temperature of conical funnel was maintained from 3.59 to 21.81 C, listed in Table 1. The temperature in the liquid and vapor phases was measured by moving down the moveable thermocouple (the diameter of wire is 25.4 μm and the bead is about 50 μm). The interfacial vapor temperature was measured within 10 μm before the thermocouple bead touched the liquid, while the interfacial liquid temperature was measured when the bead was just completely submerged in the liquid phase. With an increase in the throat temperature, the average evaporation flux dramatically increases from 1.371 g/m2 s in ES1 to 7.656 g/m2 s in ES6. However, the interfacial temperature discontinuity, which is the difference between the interfacial vapor and liquid temperatures, does not demonstrate a clearly increasing trend from ES1 to ES6. The maximum temperature discontinuity is 2.76 C in ES3 with the throat temperature at 6.35 C, while the minimum temperature discontinuity is 2.40 C in ES1. The temperature discontinuity is not of the same significant as in those experiments heated in the vapor phase,13,14 where as large as about 25 C temperature discontinuity was found. The possible reason might be that the liquid molecules at the interface cannot be easily activated to a higher energy level to escape to the vapor phase from liquid if it is heated from the liquid below. The temperature distributions in ES1, ES3, and ES6 are illustrated in Figure 3. The temperature profiles are shown at 0.0 mm, 1.4 mm, and 2.8 mm away from the center line. At each vertical position, a temperature discontinuity is found; the (13) Badam, V. K.; Kumar, V.; Durst, F.; Danov, K. Exp. Thermal Fluid Sci. 2007, 32, 276–292. (14) Duan, F.; Ward, C. A.; Badam, V. K.; Durst, F. Phys. Rev. E 2008, 78, 041130.
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Table 1. Thermal Conditions in Liquid and Vapor Phases at the Funnel Center Line Measured during Steady-State Evaporation When the Global Vapor-Phase Pressure Is about 600 Pa exp
vap.-ph. press. (Pa)
avg. evap. flux (g/m2s)
intf. vap. temp. (C)
intf. liq. temp. (C)
throat temp. (C)
intf. rad. R0 (mm)
Ra
ft
1.88 2.00 2.10 2.01 2.48 2.18
-0.52 -0.51 -0.66 -0.50 -0.21 -0.57
3.59 4.66 6.35 8.85 17.29 21.81
6.569 6.569 6.259 6.409 5.943 6.515
-244 -240 -209 32 2165 3783
0 0 0 0 0.76 0.84
ES1a 1.371 591.9 ES2 1.666 590.6 ES3 2.187 583.9 ES4 3.340 595.9 ES5 4.329 640.0 ES6 7.656 625.3 a ES1 is the experiment EV13 in ref 5.
interfacial vapor-phase temperature is greater than that in the liquid, the same as the previous experimental finding.5-8,13-16 A uniform temperature region is found just below the water-vapor interface. Below the uniform layer in the liquid phase, the temperature increases as a function of depth at each vertical position. It is suggested that the thermal conduction dominates the energy transport below the uniform temperature layer in the liquid phase. As shown in Figure 3, there is a temperature gradient in the vapor phase at each measured position in each experiment, but the gradients in the vapor phase do not illustrate more distinguishable changes than those in the liquid phase at a certain measured position when the throat temperature increases from ES1 to ES6. Such as, at the position of 1.4 mm, the vapor-phase temperature gradients are 9930, 11 941, and 14 952 C/m for ES1, ES3, and ES6, respectively; however, the liquid-phase temperature gradients are -1461, -2490, and -7085 C/m. The temperature gradient increased only 25% in the vapor phase from ES3 to ES6, but by 1.85 times in the liquid phase. Note that the interfacial temperature is the lowest at this measured vertical position at liquid and vapor phases, respectively; the thermal conduction is toward the interface. The thermal conductivity in the liquid phase is over 32 times that in the vapor phase at the interfacial temperatures. The thermal conduction energy transport to the interface from the liquid phase is more than that from the vapor phase in the six experiments. The contribution from the vapor phase becomes progressively smaller than that from the liquid phase with an increase of the throat temperature. In the liquid phase, the temperature gradients have a distinct increase from the center line to the periphery for each experiment; e.g., for ES6, the temperature gradients are 13 409, 14 952, and 19 218 C/m in the vapor phase and is 6217, 7085, and 23 187 C/m in the liquid phase at the position of 0.0 mm, 1.4 mm, and 2.8 mm, respectively. Although the vertical temperature gradients show a trend to increase along the interface in the vapor phase, the increase of the temperature gradient is much larger in the liquid phase. The higher thermal conduction at the stainless steel funnel wall compared to that in water might be the reason for the higher temperature gradient close to the rim of the funnel in the liquid phase. A higher throat temperature would cause a higher temperature gradient at the periphery of the droplet. At the position of 2.8 mm, ES6 has the highest temperature gradient, 23 187 C/m, in the liquid when the throat temperature is 21.81 C, but ES1 has the smallest value, 3696 C/m, with a throat temperature of 3.59 C. The differences between interfacial liquid temperature (TIL) at each measured position and the interfacial liquid temperature at the center line (TLI0) are fitted as a function of the distance from the center line toward the periphery of the droplet, as expressed (15) Fang, G.; Ward, C. A. Phys. Rev. E 1999, 59, 417. (16) Ward, C. A.; Stanga, D. Phys. Rev. E 2001, 64, 051509.
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Figure 3. Temperature measurements were demonstrated across the phase boundary at three horizontal positions (0.0 mm, 1.4 mm, and 2.8 mm) as water evaporated for ES1, ES3, and ES6.
in eq 2 and illustrated in Figure 4a. The interfacial liquid temperature parabolically increases from the center line to the edge of the droplet in each steady-state experiment. As the throat temperature increases, the temperature difference from the center line rises, reaching 3.07 C in ES6 at the periphery of the droplet but only 0.22 C in ES1. It is seen that a higher throat temperature resulted in both a higher interfacial liquid temperature gradient and a vertical liquid temperature gradient at the evaporative droplets above the stainless steel funnel. The interfacial liquid gradient would induce a surface tension gradient along the interface. Thermocapillary flows would be generated from the low surface tension periphery to the high surface tension center line during steady-state evaporation.5-8 If the flow arrives at the center line, it has to penetrate the liquid phase to form a mix process, which results in a uniform temperature layer. At each measured vertical position, a uniform temperature region with a certain thickness was measured immediately below the interface in the liquid phase. The uniform temperature layer had a maximum depth at the center line but minimum at the periphery. The fitting relation of the uniform temperature layer thickness is expressed in eq 3 and shown in Figure 4b. A thicker uniform temperature layer is present at the lower throat temperature. ES1 has the thickest uniform temperature layer on the center line, about 0.14 mm. TIL -TI0L ¼ a0 þ a1 sin2 θ
ð2Þ
δu ¼ b0 þ b1 sin2 θ þ b2 sin4 θ
ð3Þ
where TLI0 is the interfacial liquid temperature at the center line. ai and bi (i = 0, 1, 2) are the fitting coefficients, θ = arcsin(r/R0), r is the distance from 0 to rm, 3.5 mm. Once the interfacial liquid temperature difference (TLI - TLI0) and uniform temperature Langmuir 2009, 25(13), 7424–7431
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water-vapor interface; the local energy transport rate required for evaporation, qev, is qev ¼ jev ðhV - hL ÞI
ð6Þ
where jev is the local evaporation rate at interface, h is the intensive enthalpy, superscripts L and V indicate the liquid phase and vapor phase, and subscript I indicates interface. The local thermal conduction transport rate from both liquid and vapor phases, qcond, is expressed as qcond ¼ ðKV rT V - KL rT L ÞI 3 ir
ð7Þ
where ir is normal vector at interface, and κ is the thermal conductivity. The measured thermal conduction could be projected to the normal direction and fitted in a three-order polynomial along the water-vapor interface.5 The local energy rate, qconv, transported by thermocapillary convection at interface is
Figure 4. The thermocapillary convection velocities (c) were calculated from the interfacial liquid temperature gradients (a) and the uniform temperature layers (b) along the water-vapor interface.
thickness, δu, are known, the thermocapillary convection velocity be calculated in eq 4 based on equalizing the viscous shear stress and the surface tension gradient at the interface7 ! ! 1 dγLV dðTIL -TI0L Þ δu LV ln 1 ð4Þ νθ ¼ ηL dTIL dθ R0 where ηL is the dynamic viscosity in the liquid phase. Figure 4c demonstrates the velocity of the thermocapillary flows at the interface as a function of the distance from the center line. The stagnation point for the thermocapillary flows is on the center line at interface. The negative sign of the thermocapillary convection velocity suggests that the interfacial flow direction is from the periphery of the droplet to the center line. As shown in Figure 4c, ES6 has a maximum tangential velocity along the interface. The velocity curves in ES1, ES2, and ES3 have not shown a clear variation, resulting from a smaller difference in temperature gradients (Figure 4a) but a larger difference in uniform temperature layers (Figure 4b). Energy Balance at the Evaporative Interface. In the experiments listed in Table 1, the interface is assumed to be spherical if the liquid-vapor interface has a maximum height above the circular funnel mouth of about 1.00 mm. It is verified by our measurements in support of the moveable thermocouple. The points at the liquid-vapor interface touched by the moveable thermocouple are plotted in Figure 2. The deviation of touched points to the spherical interface curve is less than 1% in each experiment. Thus, we used spherical coordinates (r, θ, φ) to describe the interface. Since the droplet is axisymmetric, the azimuthal angle φ is avoided.5 The Gibbs dividing-surface approximation is adopted to describe the interfacial region.10 After the conservations of mass and energy were applied at an evaporation interface (see in detail in ref 5; we do not circumstantiate the derivation in this paper), the equation can be expressed as qev ¼ qcond þ qconv -ΦI
ð5Þ
where ΦI is the local viscous dissipation energy rate at the Langmuir 2009, 25(13), 7424–7431
qconv
cσ νLV DTIL ¼ - θ R0 Dθ
! ð8Þ
and cσ is the surface thermal capacity with a value of 30.6 kJ/m2 K for water, νLV θ is the interfacial tangential velocity. Note that the energy rate required for evaporation on the left side of eq 5 is expressed by those on the right: the thermal conduction energy transport rate into the interface from the liquid and vapor phases, the transport energy rate by the interfacial tangential flow driven by the surface tension gradient, and the viscous dissipation energy rate at the interface. The negative tangential velocity at the interface always enhances the evaporation in these evaporation experiments, as seen in eqs 5 and 8. The energy rates transported by the local thermal conduction and thermocapillary convection are illustrated in Figure 5. The local thermal conduction energy monotonically increases to the periphery of the droplet from the center line. ES1 has a value at 0.9 kW/m2 at the center line. With an increase of the throat temperature, ES6 reaches 3.7 kW/m2 at the center line. At the rim of funnel, the value reaches the maximum due to the contribution of the higher temperature gradient at the droplet periphery in the liquid phase, as caused by the higher-temperature thermal conduction in the stainless steel funnel wall. The energy transport rates by thermal conduction at the periphery of the droplet are 8.4 kW/m2 in ES1 and 53.4 kW/m2 in ES6. The maximum thermocapillary energy
Figure 5. Local energy transport rate by thermal conduction (left) and thermocapillary convection (right) as a function of the distance from the center line to the rim of the evaporative droplet. DOI: 10.1021/la900337j
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transport rate is in position with the maximum tangential velocity at interface. The quantity is less than that from thermal conduction in each experiment. ES6 has a maximum value at 32.3 kW/m2, but that for ES1 is 1.1 kW/m2. At the center line and the rim of the evaporative droplet, the energy transport rates by thermocapillary convection are zero due to the tangential velocity vanishing there. If eq 5 is integrated over the spherical interface of the droplet, we have : : : : Eev ¼ QN þ Eσ -Edissp
ð9Þ
: where Eev is the total evaporation energy change rate with an expression of J(hV - hL), J: is the total evaporation rate measured from the syringe pump, QN is the total energy transport rate by thermal conduction, which is integrated from the fitting curve : shown in Figure 5, Eσ is the total energy transport rate by thermocapillary convection, which is integrated from eq 8, and : Edissp is the total interfacial viscous dissipation energy rate. So, we could compare the global energy change rate required for evaporation with the sum of total energy rates transported by thermal conduction and thermocapillary convection. The comparison is shown in Figure 6. As seen there, the energy rate from thermal conduction in both phases is insufficient evaporation from ES1 to ES6. When the energy rate transported by thermocapillary convection is considered, the sum satisfies the energy change rate required for evaporation from ES1 to ES4 within the measured error, about 5%. However, the extra energy transport is found in the experiments of ES5 and ES6. The extra energy transport rate is the total interfacial viscous dissipation energy, : Edissp,5 which is the variation of the sum energy transport rate from thermocapillary convection and thermal conduction and the energy change rate required by evaporation mathematically. However, a question is yet to be answered here. In the vapor phase, only water vapor exists at the system; the interfacial vapor temperature has the lowest value. It indicates that the denser vapor is below the lighter vapor; buoyancy convection in the vapor phase could be eliminated. In the liquid phase, we notice that the throat temperature for the experiments from ES2 to ES6 is from 4.7 to 21.8 C; the interfacial temperature is less than 0 C. It suggests that a high-density layer should exist between the water-vapor interface and the throat, since water has maximum density at 4 C. The liquid might have buoyancy convection. The Rayleigh number, Ra, is calculated based on the data at the center line and listed in Table 1. Ra ¼
gβðTI0L -Tt Þd 3 υR
ð10Þ
where g is the acceleration due to gravity, d is the distance from the interface to the throat on the center line, Tt is the throat temperature, υ is the kinematic viscosity, R is the thermal diffusivity, and β is the thermal expansion coefficient. The Rayleigh number is negative in experiments ES1-ES3. For ES4, it has a value of 32, far less than the critical number for the onset of buoyancy-driven convection (about 1100). In experiments ES1-ES4, buoyancy-driven convection might be absent, but the Rayleigh numbers for ES5 and ES6 are larger than the critical number. There are two possibilities: (a) buoyancy-driven convection takes place, but cannot penetrate through the maximum density layer at 4 C; thermocapillary convection at interface does not be affected under this condition; (b) buoyancy-driven convection takes place, penetrates through the maximum density layer, and reaches the interface. Although 7428 DOI: 10.1021/la900337j
Figure 6. The comparison of the energy transport rate to the interface and the energy change rate required for evaporation. The abscissa is the energy change rate required by evaporation. The ordinate is the energy transport rate to the interface, : which includes energy rates transported by thermal conduction, : : QN, thermocapillary convection, Eσ, and viscous dissipation, Edissp.
it is possible, the likelihood is small in our options: once buoyancy-driven convection occurs, the liquid temperature below the uniform temperature layer should be parabolic instead of linear; the convection would generate a large-scale mixing process, but in the measurement of liquid-phase temperature, the standard derivations of thermocouple readings are within ( 0.03 C, similar to those in ES1-ES4. The uniform temperature layer has smaller values in ES5 and ES6. Thus, we can assume that the possible buoyancy-driven convection does not influence the evaporation flux by acting on the tangential velocity or the viscous dissipation at interface. Thereby, the global energy transport equation can be expressed in eq 9. In the experiments of ES5 and ES6, the additional energy transport shown in Figure 6 would be dissipated by the interfacial turbulent flows. The assumption was confirmed in the latter vapor-phase pressure prediction. Similar phenomena were found in the experiments of EV17-EV19 in Table 2 in which buoyancy-driven convection was absent.5 In those experiments, the interfacial turbulent transition was validated by the probe visualization experiments.7 Now, we could say that the energy balance at the interface could be applied to evaluate the interfacial turbulence transition. If we combine the data of EV1EV6 listed in Table 2, we could justify the transition from quiescent state to thermocapillary convection. The interfacial liquid temperatures are uniform in the experiments of EV1EV6, and thermal conduction maintains the evaporation while the throat temperature is reduced to close to the interfacial liquid temperature, as shown in the inset in Figure 6. The state of the liquid is quiescent. Once the thermal conduction energy transport rate is insufficient for evaporating the liquid, a thermocapillary flow would appear and make up the energy for the evaporation in the experiments of ES1-ES4 in Figure 6; the energy required for evaporation is supplied from both thermal conduction from the liquid and vapor phases and thermocapillary convection at the interface. For example, thermal conduction supplies about 80% and thermocapillary convection supplies about 20% in ES3. The extra energy in ES5 and ES6 could be dissipated by the interfacial turbulent flow, since buoyancy convection in liquid and vapor phases might be absent. Local Evaporation Flux along the Interface. In eq 9, the total energy transport rate for evaporation equals that the interfacial viscous dissipation rate is subtracted from the sum of Langmuir 2009, 25(13), 7424–7431
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Table 2. Thermal Conditions in Liquid and Vapor Phases on the Droplet Center Line Measured during Steady-State Evaporation When the Funnel Throat Is at 3.6 C5,7 exp
vap.-ph. press. (Pa)
avg. evap. flux (g/m2s)
intf. vap. temp. (C)
intf. liq. temp. (C)
throat temp. (C)
intf. rad. R0 (mm)
ft
EV1 EV2 EV3 EV4 EV5 EV6 EV17 EV18 EV19
797.2 795.9 786.8 791.9 787.9 786.6 285.3 264.0 258.6
0.027 0.034 0.044 0.050 0.057 0.063 3.026 3.421 4.242
4.95 4.88 4.62 4.65 4.68 4.62 -7.19 -7.31 -7.59
3.70 3.76 3.52 3.57 3.56 3.56 -11.61 -11.86 -12.06
3.52 3.60 3.54 3.56 3.56 3.57 3.59 3.57 3.59
6.740 6.625 6.740 6.682 6.569 6.462 6.625 6.682 6.625
0 0 0 0 0 0 0.91 0.75 0.57
thermal conduction energy transport rate to the interface and thermocapillary energy transport rate at the interface. Only ES5 and ES6 have measurable interfacial viscous dissipation. In the previous studies, the interfacial viscous dissipation was found to increase almost linearly with the maximum interfacial thermocapillary velocity.5 Thus, we might assume that the energy transferred at the interface by thermocapillary convection either enhances evaporation or is dissipated by the interfacial turbulent flow once the : through the turbulent transition. The : interface goes fraction of Eσ taken by Edissp increases as the thermocapillary flow velocity increases. It reaches 84% in ES6. If ft is defined as : Edissp ft ¼ : Eσ
ð11Þ Figure 7. The local evaporation flux as a function of position from the center line in EV2, EV19, and ES1-ES6. The local evaporation flux over a spherical droplet interface is shown in the inset.
Equation 12 can be obtained from eq 9 : : : Eev ¼ QN -ð1 -ft ÞEσ
ð12Þ
As a result, if we take the integrand in eq 12 out of the integration range, we have c νLV DTIL ðKV rT V -KL rT L ÞI 3 ir -ð1 -ft Þ σR0θ Dθ ð13Þ jev ¼ V L ðh -h ÞI Figure 7 demonstrates the local evaporation flux at the vapor-liquid interface as a function of the distance from the center line of the droplet in ES1-ES6, EV2, and EV19. Before the transition of thermocapillary convection, in EV2 for example, the local evaporation flux is uniform along the interface. After the transition, the local evaporation flux is nonuniform anymore along the interface as a result of thermal conduction to the interface in the liquid and vapor phases, thermocapillary convection, which transports and redistributes energy at the evaporating interface form the hot rim of the funnel, and the interfacial viscous dissipation when ft 6¼ 0, ES5 and ES6 in Table 1 and EV17-EV19 in Table 2. The local flux has a minimum value at the center line and a maximum value at the periphery of funnel. In ES6, 84% thermocapillary convection energy goes to the interfacial viscous dissipation; only 16% enhances the evaporation. Although the highest local energy transported by thermocapillary convection is close to the middle from the center line to the edge of droplet, the local evaporation flux is dominated by thermal conduction to the interface. The maximum local evaporation at the periphery is more than 14 times the minimum local evaporation flux at the center line in ES6. Note that the local interfacial turbulent dissipation is assumed to be proportional to the local tangential thermocapillary convection velocity as expressed in eq 13; the highest local interfacial viscous dissipation occurs at a Langmuir 2009, 25(13), 7424–7431
position with the highest thermocapillary velocity. This assumption will be evaluated in the next section. Predicting Local Vapor-Phase Pressures. The local evaporation flux, jev, can be obtained at each measured position at 0.0 mm, 0.7 mm, 1.4 mm, 2.1 mm, and 2.8 mm away from the center line, respectively, (see eq 13 and Figure 7). The local evaporation flux is applied in the expression of SRT at a spherical interface with an interfacial radius, R0. The SRT approach depends on the local evaporation fluxes, the interfacial temperature measurements, the material properties, and the molecular properties of the substrate in the liquid and vapor phases at steady-state water evaporation. The expression for the local evaporative flux could be denoted in terms of two thermodynamic functions: the entropy change at the interface, ΔsLV, and the molecular exchange rate, Ke11,12
jev
ΔsLV ¼ 2Ke sinh kb
ð14Þ
where kb is the Boltzmann constant. If the chemical potential at interface in phase i (L and V) is denoted as μi and the temperature by Ti, then the function ΔsLV and Ke could be written as ΔsLV
Ke ¼
μL μV ¼ TIL TIV
Psat ðTIL Þ
! þh
V
1 1 TIV TIL
!
vLsat L L exp kb T L Pe -Psat ðTI Þ I qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2πmw kb TIL DOI: 10.1021/la900337j
ð15Þ
ð16Þ
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Article
Duan and Ward
where Psat(TLI ) is the saturated pressure,17 νLsat, is the specific volume at the interfacial liquid temperature, mw, is the molecular weight of water, and the local liquid-phase pressure, PLe at a spherical interface, is determined by "
PLe
¼
Psat ðTIL Þ
#
vLsat ðTIL Þ L 2γLV L exp P -P ðT Þ þ ð17Þ sat e I R0 kb TIL
Then, an approximate expression is expressed at a spherical interface ΔsLV TV ¼ 4 1 - IL kb TI
!
1 1 þ TIV TIL
!
0 1 pωl 3 X pω B l kb
C þ @ Aþ 2kb exp pωl -1 l ¼1 kb
! LV vLsat 2γ þ PVI -Psat ðTIL Þ þ R0 kb TIL 2 2 !3 !4 !3 V L V T P ðT Þ q ðT Þ sat vib I I 5 ð18Þ 5 þ ln4 ln4 IL TI qvib ðTIL Þ PVI
Figure 8. Local predicted and average vapor-phase pressures are different from the measured pressure as a function of the measured vapor-phase pressure.
where qvib is the vibrational partition function:11,12 3
qvib ðTÞ ¼ P
expð -pωl =2kb TÞ expð -pωl =kb TÞ
l ¼1 1 -
ð19Þ
The vibration frequencies (ωl) of the covalent bonds in the water molecule involve the symmetric stretch frequency (3651 cm-1), asymmetric stretch frequency (3756 cm-1), and bending frequency (1590 cm-1).18 At each measured position, if the local evaporation flux, the interfacial liquid- and vapor-phase temperatures, the radius of interface, and the thermal properties are substituted in eqs 14 and 16-19, there is only a single unknown variable, PV I . The local vapor-phase pressure can be predicted at each measured position. The difference between the predicted local vapor-phase pressures and the measured global vapor-phase pressure at 20 mm above the funnel is illustrated as a function of the measured value in Figure 8. Before thermocapillary convection, the interface has a constant predicted vapor-phase pressure in the experiments EV1-EV6. Once thermocapillary convection is at the interface, the local predicted vapor-phase pressures demonstrate an increase from the center line to the edge of the droplet. The variation in the local predicted vapor-phase pressure for ES1-ES4 is in the measured error range ((13.3 Pa). However, in EV17-EV19, ES5, and ES6, the maximum deviation is above the measured error range. The local vapor-phase pressures are found to be less than the measured vapor-phase pressure at the center line and 0.7 mm away from the center line, but higher than the measured value at the position of 2.8 mm. If we assume the measured vapor-phase pressure as a universal vapor-phase pressure along the interface, the negative entropy change, ΔsLV, shown in Figure 9 is found at 0.0 mm, 0.7 mm, and 1.4 mm away from the center line in the experiment of ES6. If the negative entropy change is input in eq 14, negative evaporation fluxes result and are illustrated in Figure 9. Condensation would be expected to take place at these positions. It is contrary to what we obtained in Figure 7 from the energy balance at the interface. (17) Duan, F.; Thompson, I.; Ward, C. A. J. Phys. Chem. B 2008, 112, 8805– 8813. (18) G. Herzberg Molecular Spectra and Molecular Structure; Van Nostrand: New Jersey, 1964; Vol. 2, p 281.
7430 DOI: 10.1021/la900337j
Figure 9. Local entropy change and local evaporation flux as a function of position when the measured vapor-phase pressure was assumed as the local vapor-phase pressure at each measured position in ES6.
Hence, the measured vapor-phase pressure cannot be considered universally along the vapor-water interface. The mean of the predicted vapor-phase pressures is expressed in eq 20 P PVI ¼
PVI m
ð20Þ
where m is the number of measurements for each experiment. The difference between the mean of predicted vapor-phase pressures and measured global vapor-phase pressure is also shown in Figure 8, the difference of the pressures is less than the measured error bar ((13.3 Pa). Although the deviation is over 50 Pa in ES6, the fact that the average predicted vaporphase pressure agrees with the measured pressure provides us some support on the assumption that the local viscous dissipation is related to thermocapillary velocity at an evaporation interface after the interfacial turbulent transition.
Conclusions The steady-state evaporation of water over a conical stainless steel funnel was investigated when the global vapor-phase pressure is maintained at about 600 Pa with an increasing throat temperature from 3.59 to 21.81 C. The temperatures were measured in the liquid and vapor phases and at the interface. The average evaporation flux increases 4.6 times from ES1 to ES6 even though the vapor-phase conditions are similar for the measured vapor-phase pressures and the temperature distributions. In the liquid phase, the temperature gradient increases with an Langmuir 2009, 25(13), 7424–7431
Duan and Ward
increase of the throat temperature at the same measured vertical position in the experiments (Figure 3). The interfacial liquid temperatures become more parabolic with a higher throat temperature (Figure 4). The energy transport rate is balanced at the evaporation interface. The energy required for evaporation is satisfied with thermal conduction and thermocapillary convection in ES1-ES4. But in ES5 and ES6, the additional energy is found and explained with the interfacial turbulence dissipation induced by the thermocapillary flow at the interface (Figures 5 and 6). The thermocapillary flow transports the energy from the rim of the funnel and distributes it along the interface (Figure 4), thereby enhancing the local evaporation flux. The local evaporation flux progressively increases from the periphery
Langmuir 2009, 25(13), 7424–7431
Article
of the droplet to the center line as a result of thermal conduction, thermocapillary convection, and viscous dissipation, which is only for ES5 and ES6 in the series of experiments (Figure 7). The predicted local vapor-phase pressures illustrate a gradient along the interface of the evaporative droplet. The measured vaporphase pressure cannot be treated as the local vapor-phase pressure after the turbulent transition (Figure 9). However, the average predicted vapor-phase pressure agrees with the measured value for each experiment within the measured error bar (Figure 8). Acknowledgment. We wish to acknowledge the support from the Canadian Space Agency and SUG in Nanyang Technological University, Singapore.
DOI: 10.1021/la900337j
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