- Email: [email protected]
A hydrodynamic analysis of thermocapillary convection in evaporating thin liquid films
International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A hydrodynamic analysis of thermocapillary convection in evaporating thin liquid films Elaine Lim a,b, Yew Mun Hung a,⇑, Boon Thong Tan a a b
School of Engineering, Monash University, 47500 Bandar Sunway, Malaysia Faculty of Engineering and Technology, Multimedia University, 75450 Melaka, Malaysia
a r t i c l e
i n f o
Article history: Received 15 July 2016 Received in revised form 26 October 2016 Accepted 31 December 2016 Available online 9 January 2017 Keywords: Critical turning point Evaporating thin film Recirculation flow pattern Thermocapillary convection
a b s t r a c t Evaporation in a thin film induces pronounced temperature gradient and surface tension gradient along the liquid-vapor interface and in turn engenders thermocapillary flow. This study aims to investigate the fluid flow characteristics attributed to the thermocapillarity in an evaporating thin liquid film of polar and nonpolar liquids. A numerical steady-flow model is derived based on the fundamental principles of fluid flow and heat transfer by applying the long-wave evolution technique. To scrutinize the underlying physical transport phenomena associated with the significance of thermocapillary effect in an evaporating thin liquid film, we investigate the hydrodynamic characteristics of thermocapillary convection which is typically characterized by the recirculation flow patterns. The two-dimensional recirculation flow patterns in different excess-temperature regimes are analyzed and a critical turning point at where the flow is reversed due to the thermocapillary action can be identified. Compared to other working fluids, water depicts a unique thermocapillary flow characteristic where its flow lines manifests in the form of swirls along the liquid-vapor interface. The normal and the shear stress distributions further provide a clearer picture on the strength of thermocapillarity to identify the manifestation of thermocapillary flow. The analysis of flow patterns and hydrodynamic behaviors of evaporating thin liquid films provide essential insights in discerning the occurrence of thermocapillary flow as well as the significance of thermocapillarity in polar and nonpolar liquids. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The intricate solid-liquid and liquid-vapor interfacial interactions of an evaporating thin liquid film intrinsically influence the heat and fluid flow characteristics. Fig. 1 illustrates a schematic diagram of an evaporating thin liquid film. The liquid film layer can be divided into three regions, namely adsorbed layer region, evaporating thin film region and intrinsic meniscus region [1–3]. The adsorbed layer region is also known as non-evaporating region. The adsorbed film which is suppressed by disjoining pressure is extremely thin and evaporation does not take place in this region [2,3]. As disjoining pressure diminishes in the evaporating thin film region, the low thermal resistance in this region leads to high evaporation rates [4,5]. The liquid is circulated from the intrinsic meniscus region, such that it is driven from a lowtemperature region to a high-temperature region to maintain the evaporation rates [6,7]. The temperature gradient leads to the surface tension gradient during the evaporation process [8,9]. The ⇑ Corresponding author. E-mail address: [email protected] (Y.M. Hung). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.111 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
effect of surface tension gradient induces thermocapillary flow which drives the liquid from a low-surface-tension region to a high-surface-tension region [10,11]. The high-surface-tension strength of the intrinsic meniscus region imposes an opposing force which prevents the liquid from flowing to the evaporating thin film region, as depicted in Fig. 1. The effect of surface tension gradient essentially affects the capillarity of a microscale phasechange heat transfer device when the ratio of surface area relative to volume is large [12]. Thermocapillary effect has been observed in an evaporating thin film of polar and nonpolar liquids [13]. The disjoining pressure is strongly associated with the polarity [14–16] and the thermophysical properties [7] of a working fluid which significantly influence the thermal and fluid characteristics. From the aspect of thermal behavior, our recent study [13] demonstrated that different liquid temperature gradient magnitudes attributed to the variations of thermo-physical properties lead to different degree of thermocapillary effect on the thermal characteristics. The liquid temperature gradient is intimately related to the excess temperature, which in turn affects the thermocapillary strength of an evaporating thin film. The thermocapillary effect is observed to be more
1104
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Nomenclature A A0 C E I J k K lc le L Mw M n p pðv Þ P q00e R R S t T DT
non-dimensional Hamaker constant Hamaker constant (J) capillary number evaporation number identity tensor mass flux (kg/s m2) thermal conductivity (W/m K) parameter of degree of non-equilibrium at the evaporating interface critical turning-point length (m) evaporating film length (m) latent heat of vaporization (J/kg) molecular weight (kg/mol) Marangoni number unit normal vector liquid pressure (Pa) vapor pressure (Pa) Prandtl number local evaporative heat flux (W/m2) universal gas constant (J/mol K) radius of curvature (m) non-dimensional surface tension unit tangent vector local surface temperature (K) excess temperature (K)
Ts Tw T Tðv Þ
v
x y
saturation temperature (K) wall temperature (K) liquid stress tensor (Pa) vapor stress tensor (Pa) liquid velocity (m/s) horizontal coordinate (m) vertical coordinate (m)
Greek symbols a accommodation coefficient adsorbed film thickness (m) d0 d liquid film thickness (m) c surface tension gradient (N/m K) j thermal diffusivity (m2/s) k wave number l dynamic viscosity (N s/m2) m kinematic viscosity (m2/s) q liquid density (kg/m3) ðv Þ q vapor density (kg/m3) r surface tension (N/m) r0 surface tension at saturation reference temperature (N/m) s rate of deformation tensor in liquid (s1) P disjoining pressure (Pa)
Fig. 1. A schematic illustration of an evaporating thin film depicting the transport phenomena in the adsorbed layer region, the evaporating thin film region and the intrinsic meniscus region.
significant at higher excess temperature [3,13]. When a low excess temperature of less than 10 K is applied, thermocapillary effect is deemed to be insignificant in the evaporating thin liquid film [4,17]. The Marangoni number, defined as the ratio of surface tension force to viscous force, is used as a manifestation of thermocapillary flow [18]. It can be shown that for a thermocapillary evaporation, the Marangoni number is proportional to the excess temperature [3,13]. When the Marangoni number is less than its critical value, the thermocapillary effect is considered to be absent. While thermocapillary convection prevails in most of the volatile liquids [19,20], its presence in water, a strong polar liquid, remains a subject of controversy [21–25]. A number of studies have shown that the thermocapillary convection is insubstantial in the water despite its Marangoni number is much greater than the critical value. The thermocapillary flow in water is assumed to be suppressed by the surface contaminants as the presence of impurities or surface active agents on the water surface counteracts the thermocapillary flow in water [21,26,27]. On the other hand, the prevalence of interfacial temperature gradient suggested the existence
of thermocapillary convection in pure water [28–32]. Therefore, the claim that the thermocapillary effect is significant in water remains ambiguous. To scrutinize the underlying physical transport phenomena associated with the significance of thermocapillary effect in an evaporating thin liquid film, we extend our study to numerically investigate the hydrodynamic characteristics of thermocapillary convection in this paper. Thermocapillary flow is typically characterized by the recirculation pattern [33–36]. The analysis of flow patterns and hydrodynamic behaviors of evaporating thin liquid films would be essential in discerning the occurrence of thermocapillary flow as well as the significance of thermocapillarity in polar and nonpolar liquids. From the streamline plots, the turning point which characterizes the occurrence of thermocapillary flow can be identified. By analyzing the normal and the shear stress distributions, a clearer picture on the strength of thermocapillarity can be obtained. The direction and velocity of the thermocapillary flow are dependent on the excess temperature and hence the recirculation flow patterns in different excess-temperature regimes are
1105
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
analyzed. Based on the present hydrodynamic analysis, we can ascertain the criteria of the materialization of thermocapillary flow in different types of working fluid. This study provides interesting insights on the thermocapillary flow patterns and hydrodynamic behaviors of evaporating thin liquid films which are useful for the design of microscale two-phase cooling devices such as micro heat pipes and two-phase flow microchannels.
1
T n n ¼ p þ 2lð1 þ d2x Þ ½ux ðd2x 1Þ ðuy þ v x Þdx
The capillary pressure on the right side of Eq. (7) is given by [39,40]
rr n ¼ 2rH ¼ rdxx ð1 þ d2x Þ
3=2
The evaporating liquid film characteristics are modeled based on the fundamental principles of fluid mechanics and heat transfer. To investigate the thermocapillary characteristics of the evaporating liquid film from the hydrodynamic aspect, we make use of our previous long-wave evolution model developed for investigating the impact of thermocapillary effect on the thermal behavior [3,13]. In what follows, a summary of the numerical model will be presented for the purpose of demonstrating a self-sufficient picture of the model. The working fluid is an incompressible Newtonian fluid with constant liquid properties. Here, four different working fluids are employed, i.e. acetone, ammonia, heptane, and water. Water is a strong polar liquid with a dielectric constant, e, exceeding a value of 15 [37]. Acetone and ammonia are considered as weak polar liquids while heptane is typically a nonpolar liquid. The film thickness of the adsorbed layer region, d0 , is extremely thin and constant due to the presence of disjoining pressure. Neglecting the gravitational effect, the equation of conservation of momentum can be expressed as
qv rv ¼ rp þ lr v 2
ð1Þ
where v is the liquid velocity vector whose components u and v varies in the x- and y-directions, respectively. The corresponding continuity equation is
rv ¼0
ð2Þ
and the energy equation is given by
v rT ¼ jr2 T
ð3Þ
where T is the temperature and j is the thermal diffusivity. At the liquid-vapor interface, y ¼ dðxÞ, the jump mass balance and jump energy balance in the outward normal direction are, respectively, given by
ð10Þ
where H is the mean curvature of the interface. The surface tension is a linear function of liquid temperature and expressed as [8]
r ¼ r0 cðT T s Þ
2. Mathematical formulation
ð9Þ
ð11Þ
where r0 is the surface tension at the reference saturated temperature and c ¼ dr=dT is the surface tension gradient. For nonpolar liquids, the disjoining pressure in Eq. (7) is expressed in terms of film thickness as [1,41]
P ¼ A00 d3 00
ð12Þ 0
0
where A ¼ A =6p and A is the Hamaker constant. For polar disjoining pressure, it is associated with the molecular and structural components as [16,42]
P ¼ A00 d3 þ K 0 ed=‘
ð13Þ
0
where K and ‘ are the empirical parameters for a combination of water and mica substrate, with values of 3 103 N=cm2 and 0:8 nm, respectively [42]. The shear stress balance is given by
T n t ¼ rr t
ð14Þ
where the left-side term can be expressed as 1
T n t ¼ lð1 þ d2x Þ ½ðuy þ v x Þð1 d2x Þ 4ux dx
ð15Þ
and the right-side term is
rr t ¼ cðT x þ T y dx Þð1 þ d2x Þ
1=2
ð16Þ
The steady-state liquid model of evaporation is derived by decoupling the dynamics of vapor phase from the dynamics of liquid phase [39]. By recasting the problem into non-dimensional variables, the length is scaled by the adsorbed film thickness, d0 , and the velocity and pressure are, respectively scaled by the viscous scales: v =d0 and qv 2 =d20 . The temperature and mass flux are scaled, respectively by ðT T s Þ=DT and kDT=d0 L [41], where the excess temperature is DT ¼ T w T s . Subsequently the nondimensional governing equations can be obtained as
J ¼ qv n
ð4Þ
^u ^ ^x þ v^ u ^ y^ ¼ p ^^x þ u ^^x^x þ u ^ y^y^ u
ð17Þ
JL ¼ krT n
ð5Þ
^ v^ ^x þ v^ v^ y^ ¼ p ^y^ þ v^ ^x^x þ v^ y^y^ u
ð18Þ
^^x þ v^ y^ ¼ 0 u
ð19Þ
^ T^ ^x þ v^ T^ y^ Þ ¼ T^ ^x^x þ T^ y^y^ Pðu
ð20Þ
The outward unit normal vector, n, and unit tangent vector, t, are
respectively,
given
by
1=2
n ¼ ð1 þ d2x Þ
hdx ; 1i
and
1=2
h1; dx i, where the subscript indicates the derivative t ¼ ð1 þ d2x Þ with respect to the independent variable. The mass flux, J is derived from the kinetic theory as [38]
!
J¼
aqðv Þ L T 3=2 s
1=2 Mw ðT T s Þ 2pR
ð6Þ
ð7Þ
where pðv Þ is the constant vapor pressure. The stress tensor of liquid, T, is expressed as
T ¼ pI þ 2ls
1=2 ^ ^d^x þ v^ Þð1 þ ^d2^x Þ E^J ¼ ðu
ð21Þ
where E ¼ kDT=qmL is the evaporation number. The nondimensional energy balance is expressed as
The normal stress balance is given by [39]
pðv Þ T n n ¼ rr n þ P
where P ¼ m=j is the Prandtl number. The non-dimensional jump mass balance is given by
ð8Þ
where I is the identity tensor and s is the rate of deformation tensor in the liquid. The second term of Eq. (7) is given by
^J ¼ ðT^ ^x ^d^x T^ y^ Þð1 þ ^d2 Þ1=2 ^x
ð22Þ
The non-dimensional normal stress balance is given by
1 h i ^ ðv Þ þ p ^ 2 1 þ ^d2^x ^ y^ þ v^ ^x Þ^d^x ^ ^x ð^d^2x 1Þ ðu p u 32 ^ ^ þ A^d3 þ Bed=‘ ¼ S 1 C T^ ^d^x^x 1 þ ^d^2x
ð23Þ
1106
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
For nonpolar liquids, the coefficient B vanishes. For polar liquids, B ¼ K 0 d20 =qm2 is the non-dimensional coefficient of the structural components of disjoining pressure where A ¼ A0 =ð6pd0 qm2 Þ is the non-dimensional Hamaker constant. The non-dimensional surface tension and capillary number are respectively, given by S ¼ r0 d0 =qm2 and C ¼ cDT=r0 . The non-dimensional shear stress balance is
12 ^ y^ þ v^ ^x Þ 1 ^d2^x 4u ^^x ^d^x ¼ 2MP1 ðT^ ^x þ T^ y^ ^d^x Þ 1 þ ^d2^x ðu
ð24Þ
where M ¼ cDTd0 =ð2qmjÞ is the Marangoni number. The mass flux ^ where in Eq. (6) can also be non-dimensionalized as K^J ¼ T, 1=2 3=2 2 wÞ K ¼ ðkT =ad0 qðv Þ L Þð2pR=M is the parameter to measure the s
degree of non-equilibrium at the evaporating interface. By applying the long-wave evolution theory (as detailed in Appendix A), the evolution equation incorporating thermocapillary effect is given by [3,13]
1 B ^^ EðK þ ^dÞ ^d2 ^d^x S^d^x^x^x þ 3A^d4 ^d^x þ ed=‘ ‘^ 1 B ^ ^ ^d3 S^d^x^x^x þ A^d4 ^d^x þ ed=‘ 3 3‘^ ^ n h x io 2 2 1 1 2 ^ ^ ^ ^ MP KðK þ dÞ d d^x^x 2d^x ðK þ ^dÞ ^d1 ¼ 0
ð25Þ
The fourth-order Runge-Kutta Fehlberg method is applied to solve Eq. (25) numerically with a sufficiently small relative error of 106. Four boundary conditions are required to solve the fourth-order, nonlinear ordinary differential equation. The first boundary condition is set at ^ dð0Þ where the initial film thickness of the evaporating thin liquid is equaled to the adsorbed film thickness, d0 , which is 10 nm [3,43]. The second boundary condition is the first-order derivative of film thickness which is set to be the initial slope for the wetting film with zero contact angle. The liquid film is almost flat due to the evaporation suppression by the disjoining pressure. Therefore, any value of ^ d^x ð0Þ which below 1011 does not alter the film profile [44]. The third boundary condition is a far field boundary condition where the second-order derivative, ^ d^x^x ð1Þ is obtained with a shooting method [16,44]. The far field boundary is set at the intersection of the evaporating thin film region and intrinsic meniscus region, and ^ d^x^x ð0Þ is evaluated itera-
tively until ^ d^x^x ð1Þ converges to the radius of curvature of the intrinsic meniscus region, which is a constant value of approximately half of the channel height for a completely wetting liquid. The fourth boundary condition, ^ d^x^x^x ð0Þ has its initial value sets to
Fig. 2. Comparison of liquid film thickness profiles between the experimental model of Hanchak et al. [46] and the present long-wave evolution model, under the same operating conditions. The significance of the thermocapillary effect is only discernible at a higher Marangoni number.
a value of 5:91 106 , the difference between both numerical models (with and without thermocapillarity) is insignificant. At a 2-order-of-magnitude higher Marangoni number of 5:91 104 , the difference between the two models becomes discernible. The thermocapillary effect is neglected when the Marangoni number is typically small [17]. The variation of thermo-physical properties of different working fluids gives rise to distinct temperature distributions and hence temperature gradients [13]. The evaporation-induced temperature gradient in turn affects the surface tension gradient. The thermocapillary action induced by the surface tension gradient acts in the opposing direction of liquid flow and impedes the evaporation rate. The total evaporative heat transfer rate can be expressed in a function of excess temperature over the evaporating film length as Rl qc ¼ 0e q00e dx, where q00e ¼ JL. The significance of thermocapillary effect among the working fluids is determined by defining a ratio w ¼ qc;1 =qc;2 , where qc;1 is the total evaporative heat transfer rate of Model 1 (with thermocapillary effect), and qc;2 is that of Model 2 (without thermocapillary effect). Fig. 3 plots the ratio w as a function of the excess temperature for different types of working fluids. The ratio which decreases with increasing excess temperature is always smaller than 1, indicating that Model 2 overestimates the total evaporative heat transfer when the thermocapillary effect is
be zero as its influence on the thin film profile is marginal according to the flat adsorbed layer assumption [45]. 3. Results and discussion The long-wave evolution model has been employed to investigate the thermal characteristics of an evaporating thin film and justified the significance of thermocapillary effect [3,13]. Before embarking the hydrodynamic analysis of thermocapillary effect on the evaporating liquid thin film, we proceed with the validation of the numerical results. Fig. 2 shows the comparison of the liquid film thickness profiles between the numerical results and the experimental results by Hanchak et al. [46]. With the same pertinent parameters and under the same operating conditions, the numerical data agrees well with the experimental data, validating the accuracy of the numerical model. This type of Marangoni flow is intrinsically associated with the Marangoni number which is defined as the ratio of surface tension to viscous force [13]. As the Marangoni number of the experiment is remarkably small with
Fig. 3. Relative total evaporative heat flux for Model 1 and Model 2 of all working fluids with different excess temperatures [13].
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
neglected. Among the four working fluids, ammonia has the highest discrepancy between the models, followed by acetone, heptane and water. The descending order of the discrepancy is in accordance with the liquid temperature gradient profiles where ammonia manifests the largest temperature gradient, followed by acetone, heptane and water [13]. Therefore, the temperature gradient can be a significance indicator of thermocapillary effect on an evaporating thin liquid film. Our previous investigations justified the significance of thermocapillary effect of an evaporating thin liquid film for different working fluids from the thermal aspect [3,13]. In this paper, we focus on the fluid flow characteristics of the evaporating thin liquid film for various types of working fluids as employed in Fig. 3. When a low excess temperature of less than 10 K is applied, thermocapillary effect is deemed to be insignificant in the evaporating thin liquid film [4,45]. To scrutinize the underlying physical transport phenomena associated with the significance of thermocapillary effect in an evaporating thin liquid film, we extend our study to investigate the hydrodynamic characteristics of thermocapillary convection. In what follows, we divide our analysis into two sub-sections based on the imposed excess temperature, i.e. low excess temperature and high excess temperature, to evaluate the thermocapillary effect for different working fluids from the hydrodynamic aspect. 3.1. Low excess temperature Fig. 4 plots the streamlines of different types of working fluids without considering thermocapillary effect (Model 2) at an excess temperature of 5 K. Similar fluid flow patterns are observed in all working fluids. The unidirectional liquid streamlines show that the liquid flow is streaming from the intrinsic meniscus region to the evaporating thin film region as a result of capillary action
1107
due to evaporation. High evaporation rate prevails at the leading edge of the evaporating thin film region because of a lower thermal resistance. As the excess temperature increases, the evaporative heat flux increases, and the thermal efficiency of working fluid is overestimated. The thermocapillary force induced by surface tension gradient obstructs the liquid flow circulation and impedes the evaporation rates. It is justified in the thermal characteristics of the evaporating thin liquid film that thermocapillary effect should not be neglected especially when high excess temperature is applied [3,13]. In order to visualize the thermocapillary flow, we then proceed to study the fluid flow behaviors of different working fluids by taking thermocapillary effect into consideration. Intrinsically, the liquid circulates from intrinsic meniscus region to evaporating thin film region to maintain the evaporation rates. During the evaporation process, the transverse temperature gradient induces the surface tension gradient which draws the liquid from the low-surface-tension region to the high-surface-tension region. Fig. 5 depicts the flow streamlines of evaporating thin films for different types of working fluids incorporating the thermocapillary effect at an excess temperature of 5 K. The thermocapillary flow lines induced by the surface tension gradient are evident in all working fluids except for water. The streamlines of ammonia, acetone and heptane are diverging into two opposite directions in respect of the capillary action and thermocapillary action. A turning point is identified at the location where the thermocapillary action overcomes the capillary action due to evaporation. Here, we define a critical turning point at where the flow is reversed due to the thermocapillary action at a location nearest to the adsorbed film. The critical turning point length, lc , is measured from the origin of the evaporating thin film ð^ x ¼ 0Þ. The shorter the lc , the stronger the thermocapillary action is, based on the fact that the flow reversal prevails in a larger portion of the film. The critical turning point of ammonia is located at the
Fig. 4. Streamlines plots for different working fluids of Model 2 at an excess temperature of 5 K.
1108
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Fig. 5. Streamlines plots for different working fluids of Model 1 at an excess temperature of 5 K.
nearest to the origin where the liquid flow lines are heading towards the intrinsic meniscus region. The critical turning point of acetone occurs at further downstream of the evaporating thin film region and that of heptane takes place at the vicinity of the intrinsic meniscus region. Therefore, the thermocapillary action in ammonia is the strongest. On the other hand, no turning point is detected in water. The streamlines of water are observed to be slightly distorted by the thermocapillary action, as compared to those in Fig. 4. Most of the previous research studies concluded that the absence of thermocapillary flow in water is attributed to the presence of surface contaminants in pure water [21,26,27]. However, based on the heat transfer analysis, Fig. 3 shows that the thermocapillary effect is present in water although it is less conspicuous than that in other working fluids. As the excess temperature increases, the effect of thermocapillary on the thermal performance of water becomes more significant. The presence of thermocapilary flow in water will be further examined with the application of higher excess temperature in the next section. Fig. 6 shows the liquid velocity distribution of different working fluids without thermocapillary effect. Heat is transferred across the thin liquid layer to the liquid-vapor interface and expended for evaporation which induces high liquid velocity at the liquidvapor interface. The highest liquid-vapor interface velocity occurs at the location where the liquid film is thin and the evaporative flux is intense. Ammonia has the highest liquid velocity while water manifests the lowest. The evaporation-driven liquid velocity is dependent on the thermo-physical properties of the working fluids. We can observe that the ascending order of the liquid velocity magnitude among the working fluids is the same as that of the liquid temperature gradient, as shown in Ref. [13]. This shows that
indicates the critical turning point of thermocapillary flow.
the flow of liquid is solely driven by evaporation when thermocapillary effect is not taken into account. By taking thermocapillary effect into account, the liquid velocity contour plots of various working fluids are plotted in Fig. 7. It has been justified in Fig. 5 that the liquid flow direction is affected by the thermocapillary action, thus the liquid velocity contours of Model 1 are different from those of Model 2. The velocity magnitude of Model 1 is higher than that of Model 2. The liquid velocity of Model 1 is attributed to both capillary thermocapillary actions, while in Model 2 the flow is solely driven by capillary action. The lowest liquid velocity of Model 2 occurs at the solid-liquid interface where the no-slip boundary condition is applied. For Model 1, the liquid velocity is retarded in the vicinity of the turning points. The flow is decelerated due to the reversal in direction induced by the thermocapillary action. Consistent with the trends of the streamline and velocity plots, the thermocapillary effect of water is the most insignificant among all the working fluids as evidenced by the relative total evaporative heat flux between Model 1 and Model 2 in Fig. 3. Figs. 8 and 9 show the distributions of normal stress acting within the evaporating thin film for Model 1 and Model 2 at an excess temperature of 5 K. According to Eq. (7), the normal stress exerting on the liquid-vapor interface is attributed to the vapor pressure, disjoining pressure and capillary pressure. As disjoining pressure diminishes downstream along the axial direction, the normal stress is dominated by the capillary pressure and vapor pressure. According to the coordinate system in Fig. 1, the negative sign of normal stress indicates that the pressure is applied onto the liquid-vapor interface. Ammonia has the highest normal stress, followed by acetone, heptane and water. The normal stress of
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
1109
Fig. 6. Liquid velocity contours for different working fluids of Model 2 at an excess temperature of 5 K.
Fig. 7. Liquid velocity contours for different working fluids of Model 1 at an excess temperature of 5 K.
Model 1 and Model 2 have the same order of magnitude. The difference of normal stress balance between both models are insignificant for the respective working fluids, implying that the thermocapillary effect on the normal stress of an evaporating thin liquid film is marginal [18]. Fig. 10 plots the shear stress distributions of Model 1 at an excess temperature of 5 K. Unlike normal stress, the shear stress is dependent on the surface tension gradient and the temperature gradient, as depicted in Eq. (16). Ammonia has the highest shear stress yielded by its high surface tension gradient and temperature gradient values [13]. The low shear stress of heptane is due to its low surface tension gradient while the low temperature gradient
causes a low shear stress in water [13]. The shear stress diminishes along the y-direction in line with the application of no-slip boundary condition at the wall. In comparison with Fig. 5, the shear stress magnitude decreases with the critical turning point length, lc , showing that the former is intimately related to the strength of thermocapillary action. The shearing stress increases when the flow direction is reversed due to the thermocapillary action. For water, the maximum shear stress takes place in close proximity to the intrinsic meniscus region, indicating that thermocapillary flow exists there albeit rather indiscernibly. On the other hand, the shear stresses of Model 2 are negligibly small due to the absence of thermocapillary effect. In conclusion, the shear stress
1110
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Fig. 8. Normal stress distributions for different working fluids of Model 1 at an excess temperature of 5 K.
Fig. 9. Normal stress distributions for different working fluids of Model 2 at an excess temperature of 5 K.
within an evaporating thin film is essentially associated with the thermocapillary action. In contrary, the thermocapillary effect on the normal stress is negligibly insignificant. 3.2. High excess temperature The excess temperature plays a key role in affecting the heat transfer rate of a working fluid. As discussed earlier in Fig. 3, the thermocapillary effect should not be neglected when a high excess temperature is applied. Fig. 11 depicts the streamlines of various working fluids incorporating thermocapillary effect (Model 1) at a large excess temperature of 30 K. The streamline plots are similar to those in Fig. 5, except that of water. As the thermocapillary
action becomes stronger at higher excess temperature, the critical turning point emerges at a shorter distance from the origin. When the excess temperature is low, thermocapillary recirculation flow lines are elusive in water as discussed earlier. However, we can observe discernible liquid swirls towards the liquid-vapor interface when the excess temperature increases. These clockwise swirls which are resulted from the thermocapillary action are in a counter direction compared to the thermocapillary flow lines of other working fluids. It has been pointed out that the thermocapillary flow direction of an evaporation droplet is dependent on the ratio of thermal conductivities of solid substrate and liquid [47]. The thermal conductivity of water is higher than that of other working fluids. It can be deduced that the counter-direction ther-
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
1111
Fig. 10. Shear stress distributions for different working fluids of Model 1 at an excess temperature of 5 K.
Fig. 11. Streamlines plots for different working fluids of Model 1 at an excess temperature of 30 K.
mocapillary flow of water is attributed to its high thermal conductivity. At a reference temperature of 333 K, the thermal conductivity values of acetone, heptane, ammonia and water are 0.1224, 0.1411, 0.3896 and 0.6544 W/m K, respectively. By setting the thermal conductivity of the solid substrate as a constant, the high thermal conductivity of water reverses the direction of the thermocapillary flow compared to that of other working fluids [47]. In this study, consistent with the experimental results [47], the numerical
indicates the critical turning point of thermocapillary flow.
results evidence the existence of thermocapillary flow in water albeit only discernible at a high excess temperature. Fig. 12 depicts the velocity contours of Model 1 at an excess temperature of 30 K. Ammonia has the highest liquid velocity, followed by acetone, heptane and water. The liquid velocity is higher at higher excess temperature. Except for water, other liquids achieve the highest velocity at the liquid-vapor interface, while the lowest liquid velocity takes place along the solid-liquid inter-
1112
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Fig. 12. Liquid velocity contours for different working fluids of Model 1 at an excess temperature of 30 K.
Fig. 13. Shear stress distributions for different working fluids of Model 1 at an excess temperature of 30 K.
face and in the vicinity of the critical turning point. The velocity contour of water shows that the minimum liquid velocity occurs in the vicinity of the liquid swirls near the liquid-vapor interface and along the solid-liquid interface. The flow in the vicinity of the liquid swirl is decelerated due to the reversal direction induced by the thermocapillary action. On the other hand, the minimum liquid velocity occurs along the solid-liquid interface due to the no-slip boundary condition. Therefore, thermocapillary effect is justified to be significant in liquids although the presence of thermocapillary flow is not easily to be distinguished in water due to its unique thermo-physical properties characteristics. Fig. 13 plots the shear stress distributions of Model 1 at an excess temperature of 30 K. Compared to Fig. 10, the trends of the shear stress distributions at a high excess temperature are similar with those at a low excess temperature. As discussed in Fig. 10 earlier, the shear stress magnitude increases with the decreasing critical turning point length, lc . The shearing stress increases when the flow direction is reversed due to the thermocapillary action. In
the case of high excess temperature, the magnitudes of the shear stress for all liquids are higher than those of low excess temperature. This implies that lc decreases and the coverage of the flow reversal becomes larger when the excess temperature increases. For acetone, ammonia and heptane, this is consistent with the streamline plots in Fig. 11 where a larger area is covered by the flow reversal due to the thermocapillary action. For water, the maximum shear stress occurs further upstream in the evaporating thin film and this explains that the existence of thermocapillary flow becomes more discernible. 4. Conclusions We delineate the thermocapillary effect on an evaporating thin liquid film for different types of working fluid from the hydrodynamic aspect. The long-wave evolution theory is employed to derive a two-dimensional mathematical model based on the basic principles of heat transfer and fluid mechanics applicable to polar
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
and nonpolar liquids in this study. This model could be expanded to model other types of liquid such as the non-wetting fluids with non-zero contact angle and the non-Newtonian fluids with sheardependent viscosity. The two-dimensional recirculation flow patterns in different excess-temperature regimes are analyzed and a critical turning point at where the flow is reversed due to the thermocapillary action can be identified. By neglecting the thermocapillary effect, the evaporative heat transfer rate is observed to be overrated and the overestimate increases with increasing excess temperature. The surface tension gradient eliciting thermocapillary flow is dominantly attributed to the temperature gradient induced by evaporation. Water exhibits a unique thermocapillary flow characteristics where the thermocapillary flow lines are inconspicuous compared to other working fluids at a low excess temperature. As the excess temperature increases, the thermocapillary flow in water manifests in the form of swirls along the liquidvapor interface. The thermocapillary flow direction of water is intrinsically dictated by the ratio of thermal conductivities of solid substrate and liquid. Water can be considered as the best working fluid in terms of thermocapillary suppression. The normal and shear stresses within the evaporating thin liquid film correlate well to the temperature gradient and surface tension gradient of the working fluids. Consistent with the heat transfer analysis, we conclude that the thermocapillary effect of an evaporating thin film should not be simply neglected in the hydrodynamic analysis as acute errors can be incurred in the thermal performance analysis. Acknowledgment The third author acknowledges the support of the Ministry of Higher Education, Malaysia, through the Fundamental Research Grant Scheme (FRGS/1/2014/TK01/MUSM/02/1). Appendix A As the characteristic film thickness is much smaller than the characteristic length scale of the film thickness variation, the long-wave theory can be employed to describe the film dynamics behaviors. The dependent variables of non-isothermal evaporating layer are expected to vary slowly along the axial direction. A small parameter denoted as the wave number, k, which is defined in line with lubrication theory, is used in rescaling the non-dimensional governing equations and boundary conditions. The following non-dimensional variables are defined as [39,48]
f ¼ k^x
g ¼ y^
ðA1Þ
The order of magnitude of dependent variables are outlined as ^ ; ^J; T^ ¼ Oð1Þ and p ^; P ¼ Oðk1 Þ. The film thickness ^ follows: u dðfÞ is expressed as an unspecified unit-order function. To retain continu^ ¼ OðkÞ is set. The dependent variables are expanded in the ity, v powers of k as [39]
^1 þ ^¼u ^ 0 þ ku u ^ ^ v ¼ kðv 0 þ kv^ 1 þ Þ ^J ¼ ^J 0 þ k^J 1 þ
T^ ¼ T^ 0 þ kT^ 1 þ ^0 þ kp ^1 þ Þ ^ ¼ k1 ðp p 1 ^ ^ ^ 1 þ Þ P ¼ k ðP0 þ kP
^ 0;f þ u ^ 0;gg ¼ 0 P
ðA3Þ
^ 0;g ¼ 0 P
ðA4Þ
^ 0;f þ v^ 0;g ¼ 0 u
ðA5Þ
T^ 0;gg ¼ 0
ðA6Þ
The non-dimensional groups are related to k as
1 ; Bk 3 Þ 1 ; Ek; Mk 1 ; Sk ðA; B; E; M; SÞ ¼ ðAk
Based on the asymptotic analysis with k ! 0, the expansions in (A2) are substituted into the governing equations where the coefficients of like powers of k in each equation and boundary condition are equated to zero. The governing equations can be simplified at the leading order in k as
ðA7Þ
where the order of magnitude of the non-dimensional groups are set as follows: A and B ¼ Oðk1 Þ, E ¼ OðkÞ to include the mass loss, M ¼ Oðk1 Þ to retain the thermocapillary effect in shear stress condition, S ¼ Oðk3 Þ to retain the surface tension effect in normal stress condition, as well as K ¼ Oð1Þ and P ¼ Oð1Þ. As k ! 0, the quantities with over-bar are of unit order of magnitude. The boundary conditions incorporating the non-dimensional groups at the interface can be expressed as
^0 ^df þ v^ 0 E^J 0 ¼ u
ðA8Þ
^J 0 ¼ T^ 0;g
ðA9Þ
^d3 þ Be ^dff þ A ^d=‘^ ^ ðv Þ þ P^ 0 ¼ S P 0
ðA10Þ
^ 0;g ¼ 2MP1 ðT^ 0;f þ T^ 0;g ^df Þ u
ðA11Þ
K^J 0 ¼ T^ 0
ðA12Þ
¼ 2k2 MP1 , Eq. (A10) is obtained By eliminating the term SC from Eq. (23). The Marangoni number, M, is absent at leading order in the normal stress balance while it exists in the shear stress balance in Eq. (A11). At the wall surface (g ¼ 0), the no-slip and notemperature-jump boundary conditions are specified as
^ 0 ¼ 0; u
v^ 0 ¼ 0;
T^ 0 ¼ 1
ðA13Þ
The governing equations (A3)–(A6) are solved together with the corresponding boundary conditions (A8)–(A13) to yield the longwave evolution equation as a function of the liquid film thickness distribution by incorporating the evaporative and thermocapillary effects. By applying the boundary conditions in Eqs. (A9), (A12) and (A13), Eq. (A6) is solved and the liquid temperature distribution is given by 1 T^ 0 ¼ 1 ðK þ ^dÞ g
ðA14Þ
The evaporative mass flux is obtained as
^J 0 ¼ ðK þ ^dÞ1
ðA15Þ
Subject to the boundary condition (A10), Eq. (A4) is solved to yield the liquid pressure as
^d3 þ Be ^dff þ A ^ 0 ¼ P^ ðv Þ S ^d=‘^ P 0 ðA2Þ
1113
ðA16Þ
By differentiating Eq. (A16) with respect to x and substituting it into Eq. (A3), together with the application of boundary conditions in (A11) and (A13), we obtain the x-component of liquid velocity as
^0 ¼ U u
2 1 2 ^ g dg þ 2MP1 KðK þ ^dÞ ^df g 2
ðA17Þ
^ ^ ^d=‘^=‘Þ. ^ By substituting Eq. (A17) df þ Be where U ¼ ðS dfff þ 3A d4 ^ into Eq. (A5) with the application of boundary condition in (A13), the y-component of liquid velocity is given by
1114
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
1 1 1 v^ 0 ¼ Uf g3 ^dg2 þ U^df g2 6 2 2 h i 2 1 1 ^ ^ MP KðK þ dÞ dff 2ðK þ ^dÞ ^d2f g2
ðA18Þ
References [1] M. Potash Jr., P.C. Wayner Jr., Evaporation from a two-dimensional extended meniscus, Int. J. Heat Mass Transf. 15 (1972) 1851–1863. [2] S.S. Panchamgam, J.L. Plawsky, P.C. Wayner Jr., Experimental evaluation of Marangoni shear in the contact line region of an evaporating 99+% pure octane meniscus, ASME J. Heat Transf. 129 (2007) 1476–1485. [3] E. Lim, Y.M. Hung, Thermocapillary flow in evaporating thin liquid films with long-wave evolution model, Int. J. Heat Mass Transf. 73 (2014) 849–858. [4] R. Ranjan, J.Y. Murthy, S.V. Garimella, A microscale model for thin-film evaporation in capillary wick structures, Int. J. Heat Mass Transf. 54 (2011) 169–179. [5] R. Raj, C. Kunkelmann, P. Stephan, J. Plawsky, J. Kim, Contact line behavior for a highly wetting fluid under superheated conditions, Int. J. Heat Mass Transf. 55 (2012) 2664–2675. [6] S.S. Panchamgam, A. Chatterjee, J.L. Plawsky, P.C. Wayner Jr., Comprehensive experimental and theoretical study of fluid flow and heat transfer in a microscopic evaporating meniscus in a miniature heat exchanger, Int. J. Heat Mass Transf. 51 (2008) 5368–5379. [7] J.-J. Zhao, Y.-Y. Duan, X.-D. Wang, B.-X. Wang, Effects of superheat and temperature-dependent thermophysical properties on evaporating thin liquid films in microchannels, Int. J. Heat Mass Transf. 54 (2011) 1259–1267. [8] M.J. Tan, S.G. Bankoff, S.H. Davis, Steady thermocapillary flows of thin liquid layers. I. Theory, Phys. Fluids A 2 (1990) 313–321. [9] C. Buffone, K. Sefiane, J.R.E. Christy, Experimental investigation of self-induced thermocapillary convection for an evaporating meniscus in capillary tubes using micro-particle image velocimetry, Phys. Fluids 17 (2005) 052104. [10] X. Xu, J. Luo, Marangoni flow in an evaporating water droplet, Appl. Phys. Lett. 91 (2007) 124102. [11] A.M. Benselama, S. Harmand, K. Sefiane, Thermocapillary effects on steadily evaporating contact line: a perturbative local analysis, Phys. Fluids 24 (2012) 072105. [12] F.L. Chang, Y.M. Hung, The coupled effects of working fluid and solid wall on thermal performance of micro heat pipes, Int. J. Heat Mass Transf. 73 (2014) 76–87. [13] E. Lim, Y.M. Hung, Thermophysical phenomena of working fluids of thermocapillary convection in evaporating thin liquid films, Int. Commun. Heat Mass Transfer 66 (2015) 203–211. [14] W. Qu, T. Ma, Effects of the polarity of working fluids on vapor-liquid flow and heat transfer characteristics in a capillary, Microscale Thermophys. Eng. 6 (2002) 175–190. [15] S.-K. Wee, K.D. Kihm, K.P. Hallinan, Effects of the liquid polarity and the wall slip on the heat and mass transport characteristics of the micro-scale evaporating transition film, Int. J. Heat Mass Transf. 48 (2005) 265–278. [16] S.-Y. Du, Y.-H. Zhao, New boundary conditions for the evaporating thin-film model in a rectangular micro channel, Int. J. Heat Mass Transf. 54 (2011) 3694– 3701. [17] V.S. Ajaev, G.M. Homsy, Steady vapor bubbles in rectangular microchannels, J. Colloid Interface Sci. 240 (2001) 259–271. [18] J.R.A. Pearson, On convection cells induced by surface tension, J. Fluid Mech. 4 (1958) 489–500. [19] A.-T. Chai, N. Zhang, Experimental study of Marangoni-Benard convection in a liquid layer induced by evaporation, Exp. Heat Transf. 11 (1998) 187–205. [20] T. Qin, Z˘. Tukovic´, R.O. Grigoriev, Buoyancy-thermocapillary convection of volatile fluids under atmospheric conditions, Int. J. Heat Mass Transf. 75 (2014) 284–301. [21] J.C. Berg, M. Boudart, A. Acrivos, Natural convection in pools of evaporating liquids, J. Fluid Mech. 24 (1966) 721–735.
[22] H.K. Cammenga, D. Schreiber, G.T. Barnes, D.S. Hunter, On Marangoni convection during the evaporation of water, J. Colloid Interface Sci. 98 (1984) 585–586. [23] C.A. Ward, D. Stanga, Interfacial conditions during evaporation or condensation of water, Phys. Rev. E 64 (2001) 051509. [24] K. Sefiane, C.A. Ward, Recent advances on thermocapillary flows and interfacial conditions during the evaporation of liquids, Adv. Colloid Interface Sci. 134–135 (2007) 201–223. [25] I. Thompson, F. Duan, C.A. Ward, Absence of Marangoni convection at Marangoni numbers above 27,000 during water evaporation, Phys. Rev. E 80 (2009) 056308. [26] R. Savino, D. Paterna, N. Favaloro, Buoyancy and Marangoni effects in an evaporating drop, J. Thermophys. Heat Transfer 16 (2002) 562–574. [27] H. Hu, R.G. Larson, Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet, Langmuir 21 (2005) 3972–3980. [28] C.A. Ward, F. Duan, Turbulent transition of thermocapillary flow induced by water evaporation, Phys. Rev. E 69 (2004) 056308. [29] F. Duan, C.A. Ward, Surface excess properties from energy transport measurements during water evaporation, Phys. Rev. E 72 (2005) 056302. [30] E.A. Chinnov, Wave – Thermocapillary effects in heated liquid films at high Reynolds numbers, Int. J. Heat Mass Transf. 71 (2014) 106–116. [31] L. Zhou, X. Du, Y. Yang, P. Jiang, B. Wang, Thermocapillary effect on bubble sweeping and circling during subcooled nucleate pool boiling of water over microwire, Int. J. Heat Mass Transf. 88 (2015) 276–283. [32] E.A. Chinnov, Enhancement of thermocapillary effect in heated liquid films for large waves at high Reynolds numbers, J. Heat Transfer 138 (2016) 092901. [33] M.L. Cordero, H.O. Rolfsnes, D.R. Burnham, P.A. Campbell, D. McGloin, C.N. Baroud, Mixing via thermocapillary generation of flow patterns inside a microfluidic drop, New J. Phys. 11 (2009) 075033. [34] H. Chraibi, J.-P. Delville, Thermocapillary flows and interface deformations produced by localized laser heating in confined environment, Phys. Fluids 24 (2012) 032102. [35] C.-M. Wu, Y.-R. Li, R.-J. Liao, Rotating and thermocapillary-buoyancy-driven flow in a cylindrical enclosure with a partly free surface, Phys. Fluids 26 (2014) 104105. [36] C.-M. Wu, D.-F. Ruan, Y.-R. Li, R.-J. Liao, Flow pattern transition driven by the combined Marangoni effect and rotation of crucible and crystal in a Czochralski configuration, Int. J. Therm. Sci. 86 (2014) 394–407. [37] T.H. Lowry, K.S. Richardson, Mechanism and Theory in Organic Chemistry, Harper & Row, New York, 1976. [38] V. Carey, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corporation, New York, 1992. [39] J.P. Burelbach, S.G. Bankoff, S.H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech. 195 (1988) 463–494. [40] R.W. Atherton, G.M. Homsy, On the derivation of evolution equations for interfacial waves, Chem. Eng. Commun. 2 (1976) 57–77. [41] J.A. Schonberg, S. DasGupta, P.C. Wayner Jr., An augmented Young-Laplace model of an evaporating meniscus in a microchannel with high hext flux, Exp. Thermal Fluid Sci. 10 (1995) 163–170. [42] N.V. Churaev, B.V. Derjaguin, Inclusion of structural forces in the theory of stability of colloids and films, J. Colloid Interface Sci. 103 (1985) 542–553. [43] J.A. Schonberg, P.C. Wayner Jr., Analytical solution for the integral contact line evaporative heat sink, J. Thermophys. Heat Transfer 6 (1992) 128–134. [44] H. Wang, S.V. Garimella, J.Y. Murthy, Characteristics of an evaporating thin film in a microchannel, Int. J. Heat Mass Transf. 50 (2007) 3933–3942. [45] V.S. Ajaev, G.M. Homsy, Steady vapor bubbles in rectangular microchannels, J. Colloid Interface Sci. 240 (2011) 259–271. [46] M.S. Hanchak, M.D. Vangsness, L.W. Byrd, J.S. Ervin, Thin film evaporation of noctane on silicon: experiments and theory, Int. J. Heat Mass Transf. 75 (2014) 196–206. [47] W.D. Ristenpart, P.G. Kim, C. Domingues, J. Wan, H.A. Stone, Influence of substrate conductivity on circulation reversal in evaporating drops, Phys. Rev. Lett. 99 (2007) 234502. [48] A. Oron, S.H. Davis, S.G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69 (1997) 931–980.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
A hydrodynamic analysis of thermocapillary convection in evaporating thin liquid films Elaine Lim a,b, Yew Mun Hung a,⇑, Boon Thong Tan a a b
School of Engineering, Monash University, 47500 Bandar Sunway, Malaysia Faculty of Engineering and Technology, Multimedia University, 75450 Melaka, Malaysia
a r t i c l e
i n f o
Article history: Received 15 July 2016 Received in revised form 26 October 2016 Accepted 31 December 2016 Available online 9 January 2017 Keywords: Critical turning point Evaporating thin film Recirculation flow pattern Thermocapillary convection
a b s t r a c t Evaporation in a thin film induces pronounced temperature gradient and surface tension gradient along the liquid-vapor interface and in turn engenders thermocapillary flow. This study aims to investigate the fluid flow characteristics attributed to the thermocapillarity in an evaporating thin liquid film of polar and nonpolar liquids. A numerical steady-flow model is derived based on the fundamental principles of fluid flow and heat transfer by applying the long-wave evolution technique. To scrutinize the underlying physical transport phenomena associated with the significance of thermocapillary effect in an evaporating thin liquid film, we investigate the hydrodynamic characteristics of thermocapillary convection which is typically characterized by the recirculation flow patterns. The two-dimensional recirculation flow patterns in different excess-temperature regimes are analyzed and a critical turning point at where the flow is reversed due to the thermocapillary action can be identified. Compared to other working fluids, water depicts a unique thermocapillary flow characteristic where its flow lines manifests in the form of swirls along the liquid-vapor interface. The normal and the shear stress distributions further provide a clearer picture on the strength of thermocapillarity to identify the manifestation of thermocapillary flow. The analysis of flow patterns and hydrodynamic behaviors of evaporating thin liquid films provide essential insights in discerning the occurrence of thermocapillary flow as well as the significance of thermocapillarity in polar and nonpolar liquids. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The intricate solid-liquid and liquid-vapor interfacial interactions of an evaporating thin liquid film intrinsically influence the heat and fluid flow characteristics. Fig. 1 illustrates a schematic diagram of an evaporating thin liquid film. The liquid film layer can be divided into three regions, namely adsorbed layer region, evaporating thin film region and intrinsic meniscus region [1–3]. The adsorbed layer region is also known as non-evaporating region. The adsorbed film which is suppressed by disjoining pressure is extremely thin and evaporation does not take place in this region [2,3]. As disjoining pressure diminishes in the evaporating thin film region, the low thermal resistance in this region leads to high evaporation rates [4,5]. The liquid is circulated from the intrinsic meniscus region, such that it is driven from a lowtemperature region to a high-temperature region to maintain the evaporation rates [6,7]. The temperature gradient leads to the surface tension gradient during the evaporation process [8,9]. The ⇑ Corresponding author. E-mail address: [email protected] (Y.M. Hung). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.12.111 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
effect of surface tension gradient induces thermocapillary flow which drives the liquid from a low-surface-tension region to a high-surface-tension region [10,11]. The high-surface-tension strength of the intrinsic meniscus region imposes an opposing force which prevents the liquid from flowing to the evaporating thin film region, as depicted in Fig. 1. The effect of surface tension gradient essentially affects the capillarity of a microscale phasechange heat transfer device when the ratio of surface area relative to volume is large [12]. Thermocapillary effect has been observed in an evaporating thin film of polar and nonpolar liquids [13]. The disjoining pressure is strongly associated with the polarity [14–16] and the thermophysical properties [7] of a working fluid which significantly influence the thermal and fluid characteristics. From the aspect of thermal behavior, our recent study [13] demonstrated that different liquid temperature gradient magnitudes attributed to the variations of thermo-physical properties lead to different degree of thermocapillary effect on the thermal characteristics. The liquid temperature gradient is intimately related to the excess temperature, which in turn affects the thermocapillary strength of an evaporating thin film. The thermocapillary effect is observed to be more
1104
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Nomenclature A A0 C E I J k K lc le L Mw M n p pðv Þ P q00e R R S t T DT
non-dimensional Hamaker constant Hamaker constant (J) capillary number evaporation number identity tensor mass flux (kg/s m2) thermal conductivity (W/m K) parameter of degree of non-equilibrium at the evaporating interface critical turning-point length (m) evaporating film length (m) latent heat of vaporization (J/kg) molecular weight (kg/mol) Marangoni number unit normal vector liquid pressure (Pa) vapor pressure (Pa) Prandtl number local evaporative heat flux (W/m2) universal gas constant (J/mol K) radius of curvature (m) non-dimensional surface tension unit tangent vector local surface temperature (K) excess temperature (K)
Ts Tw T Tðv Þ
v
x y
saturation temperature (K) wall temperature (K) liquid stress tensor (Pa) vapor stress tensor (Pa) liquid velocity (m/s) horizontal coordinate (m) vertical coordinate (m)
Greek symbols a accommodation coefficient adsorbed film thickness (m) d0 d liquid film thickness (m) c surface tension gradient (N/m K) j thermal diffusivity (m2/s) k wave number l dynamic viscosity (N s/m2) m kinematic viscosity (m2/s) q liquid density (kg/m3) ðv Þ q vapor density (kg/m3) r surface tension (N/m) r0 surface tension at saturation reference temperature (N/m) s rate of deformation tensor in liquid (s1) P disjoining pressure (Pa)
Fig. 1. A schematic illustration of an evaporating thin film depicting the transport phenomena in the adsorbed layer region, the evaporating thin film region and the intrinsic meniscus region.
significant at higher excess temperature [3,13]. When a low excess temperature of less than 10 K is applied, thermocapillary effect is deemed to be insignificant in the evaporating thin liquid film [4,17]. The Marangoni number, defined as the ratio of surface tension force to viscous force, is used as a manifestation of thermocapillary flow [18]. It can be shown that for a thermocapillary evaporation, the Marangoni number is proportional to the excess temperature [3,13]. When the Marangoni number is less than its critical value, the thermocapillary effect is considered to be absent. While thermocapillary convection prevails in most of the volatile liquids [19,20], its presence in water, a strong polar liquid, remains a subject of controversy [21–25]. A number of studies have shown that the thermocapillary convection is insubstantial in the water despite its Marangoni number is much greater than the critical value. The thermocapillary flow in water is assumed to be suppressed by the surface contaminants as the presence of impurities or surface active agents on the water surface counteracts the thermocapillary flow in water [21,26,27]. On the other hand, the prevalence of interfacial temperature gradient suggested the existence
of thermocapillary convection in pure water [28–32]. Therefore, the claim that the thermocapillary effect is significant in water remains ambiguous. To scrutinize the underlying physical transport phenomena associated with the significance of thermocapillary effect in an evaporating thin liquid film, we extend our study to numerically investigate the hydrodynamic characteristics of thermocapillary convection in this paper. Thermocapillary flow is typically characterized by the recirculation pattern [33–36]. The analysis of flow patterns and hydrodynamic behaviors of evaporating thin liquid films would be essential in discerning the occurrence of thermocapillary flow as well as the significance of thermocapillarity in polar and nonpolar liquids. From the streamline plots, the turning point which characterizes the occurrence of thermocapillary flow can be identified. By analyzing the normal and the shear stress distributions, a clearer picture on the strength of thermocapillarity can be obtained. The direction and velocity of the thermocapillary flow are dependent on the excess temperature and hence the recirculation flow patterns in different excess-temperature regimes are
1105
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
analyzed. Based on the present hydrodynamic analysis, we can ascertain the criteria of the materialization of thermocapillary flow in different types of working fluid. This study provides interesting insights on the thermocapillary flow patterns and hydrodynamic behaviors of evaporating thin liquid films which are useful for the design of microscale two-phase cooling devices such as micro heat pipes and two-phase flow microchannels.
1
T n n ¼ p þ 2lð1 þ d2x Þ ½ux ðd2x 1Þ ðuy þ v x Þdx
The capillary pressure on the right side of Eq. (7) is given by [39,40]
rr n ¼ 2rH ¼ rdxx ð1 þ d2x Þ
3=2
The evaporating liquid film characteristics are modeled based on the fundamental principles of fluid mechanics and heat transfer. To investigate the thermocapillary characteristics of the evaporating liquid film from the hydrodynamic aspect, we make use of our previous long-wave evolution model developed for investigating the impact of thermocapillary effect on the thermal behavior [3,13]. In what follows, a summary of the numerical model will be presented for the purpose of demonstrating a self-sufficient picture of the model. The working fluid is an incompressible Newtonian fluid with constant liquid properties. Here, four different working fluids are employed, i.e. acetone, ammonia, heptane, and water. Water is a strong polar liquid with a dielectric constant, e, exceeding a value of 15 [37]. Acetone and ammonia are considered as weak polar liquids while heptane is typically a nonpolar liquid. The film thickness of the adsorbed layer region, d0 , is extremely thin and constant due to the presence of disjoining pressure. Neglecting the gravitational effect, the equation of conservation of momentum can be expressed as
qv rv ¼ rp þ lr v 2
ð1Þ
where v is the liquid velocity vector whose components u and v varies in the x- and y-directions, respectively. The corresponding continuity equation is
rv ¼0
ð2Þ
and the energy equation is given by
v rT ¼ jr2 T
ð3Þ
where T is the temperature and j is the thermal diffusivity. At the liquid-vapor interface, y ¼ dðxÞ, the jump mass balance and jump energy balance in the outward normal direction are, respectively, given by
ð10Þ
where H is the mean curvature of the interface. The surface tension is a linear function of liquid temperature and expressed as [8]
r ¼ r0 cðT T s Þ
2. Mathematical formulation
ð9Þ
ð11Þ
where r0 is the surface tension at the reference saturated temperature and c ¼ dr=dT is the surface tension gradient. For nonpolar liquids, the disjoining pressure in Eq. (7) is expressed in terms of film thickness as [1,41]
P ¼ A00 d3 00
ð12Þ 0
0
where A ¼ A =6p and A is the Hamaker constant. For polar disjoining pressure, it is associated with the molecular and structural components as [16,42]
P ¼ A00 d3 þ K 0 ed=‘
ð13Þ
0
where K and ‘ are the empirical parameters for a combination of water and mica substrate, with values of 3 103 N=cm2 and 0:8 nm, respectively [42]. The shear stress balance is given by
T n t ¼ rr t
ð14Þ
where the left-side term can be expressed as 1
T n t ¼ lð1 þ d2x Þ ½ðuy þ v x Þð1 d2x Þ 4ux dx
ð15Þ
and the right-side term is
rr t ¼ cðT x þ T y dx Þð1 þ d2x Þ
1=2
ð16Þ
The steady-state liquid model of evaporation is derived by decoupling the dynamics of vapor phase from the dynamics of liquid phase [39]. By recasting the problem into non-dimensional variables, the length is scaled by the adsorbed film thickness, d0 , and the velocity and pressure are, respectively scaled by the viscous scales: v =d0 and qv 2 =d20 . The temperature and mass flux are scaled, respectively by ðT T s Þ=DT and kDT=d0 L [41], where the excess temperature is DT ¼ T w T s . Subsequently the nondimensional governing equations can be obtained as
J ¼ qv n
ð4Þ
^u ^ ^x þ v^ u ^ y^ ¼ p ^^x þ u ^^x^x þ u ^ y^y^ u
ð17Þ
JL ¼ krT n
ð5Þ
^ v^ ^x þ v^ v^ y^ ¼ p ^y^ þ v^ ^x^x þ v^ y^y^ u
ð18Þ
^^x þ v^ y^ ¼ 0 u
ð19Þ
^ T^ ^x þ v^ T^ y^ Þ ¼ T^ ^x^x þ T^ y^y^ Pðu
ð20Þ
The outward unit normal vector, n, and unit tangent vector, t, are
respectively,
given
by
1=2
n ¼ ð1 þ d2x Þ
hdx ; 1i
and
1=2
h1; dx i, where the subscript indicates the derivative t ¼ ð1 þ d2x Þ with respect to the independent variable. The mass flux, J is derived from the kinetic theory as [38]
!
J¼
aqðv Þ L T 3=2 s
1=2 Mw ðT T s Þ 2pR
ð6Þ
ð7Þ
where pðv Þ is the constant vapor pressure. The stress tensor of liquid, T, is expressed as
T ¼ pI þ 2ls
1=2 ^ ^d^x þ v^ Þð1 þ ^d2^x Þ E^J ¼ ðu
ð21Þ
where E ¼ kDT=qmL is the evaporation number. The nondimensional energy balance is expressed as
The normal stress balance is given by [39]
pðv Þ T n n ¼ rr n þ P
where P ¼ m=j is the Prandtl number. The non-dimensional jump mass balance is given by
ð8Þ
where I is the identity tensor and s is the rate of deformation tensor in the liquid. The second term of Eq. (7) is given by
^J ¼ ðT^ ^x ^d^x T^ y^ Þð1 þ ^d2 Þ1=2 ^x
ð22Þ
The non-dimensional normal stress balance is given by
1 h i ^ ðv Þ þ p ^ 2 1 þ ^d2^x ^ y^ þ v^ ^x Þ^d^x ^ ^x ð^d^2x 1Þ ðu p u 32 ^ ^ þ A^d3 þ Bed=‘ ¼ S 1 C T^ ^d^x^x 1 þ ^d^2x
ð23Þ
1106
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
For nonpolar liquids, the coefficient B vanishes. For polar liquids, B ¼ K 0 d20 =qm2 is the non-dimensional coefficient of the structural components of disjoining pressure where A ¼ A0 =ð6pd0 qm2 Þ is the non-dimensional Hamaker constant. The non-dimensional surface tension and capillary number are respectively, given by S ¼ r0 d0 =qm2 and C ¼ cDT=r0 . The non-dimensional shear stress balance is
12 ^ y^ þ v^ ^x Þ 1 ^d2^x 4u ^^x ^d^x ¼ 2MP1 ðT^ ^x þ T^ y^ ^d^x Þ 1 þ ^d2^x ðu
ð24Þ
where M ¼ cDTd0 =ð2qmjÞ is the Marangoni number. The mass flux ^ where in Eq. (6) can also be non-dimensionalized as K^J ¼ T, 1=2 3=2 2 wÞ K ¼ ðkT =ad0 qðv Þ L Þð2pR=M is the parameter to measure the s
degree of non-equilibrium at the evaporating interface. By applying the long-wave evolution theory (as detailed in Appendix A), the evolution equation incorporating thermocapillary effect is given by [3,13]
1 B ^^ EðK þ ^dÞ ^d2 ^d^x S^d^x^x^x þ 3A^d4 ^d^x þ ed=‘ ‘^ 1 B ^ ^ ^d3 S^d^x^x^x þ A^d4 ^d^x þ ed=‘ 3 3‘^ ^ n h x io 2 2 1 1 2 ^ ^ ^ ^ MP KðK þ dÞ d d^x^x 2d^x ðK þ ^dÞ ^d1 ¼ 0
ð25Þ
The fourth-order Runge-Kutta Fehlberg method is applied to solve Eq. (25) numerically with a sufficiently small relative error of 106. Four boundary conditions are required to solve the fourth-order, nonlinear ordinary differential equation. The first boundary condition is set at ^ dð0Þ where the initial film thickness of the evaporating thin liquid is equaled to the adsorbed film thickness, d0 , which is 10 nm [3,43]. The second boundary condition is the first-order derivative of film thickness which is set to be the initial slope for the wetting film with zero contact angle. The liquid film is almost flat due to the evaporation suppression by the disjoining pressure. Therefore, any value of ^ d^x ð0Þ which below 1011 does not alter the film profile [44]. The third boundary condition is a far field boundary condition where the second-order derivative, ^ d^x^x ð1Þ is obtained with a shooting method [16,44]. The far field boundary is set at the intersection of the evaporating thin film region and intrinsic meniscus region, and ^ d^x^x ð0Þ is evaluated itera-
tively until ^ d^x^x ð1Þ converges to the radius of curvature of the intrinsic meniscus region, which is a constant value of approximately half of the channel height for a completely wetting liquid. The fourth boundary condition, ^ d^x^x^x ð0Þ has its initial value sets to
Fig. 2. Comparison of liquid film thickness profiles between the experimental model of Hanchak et al. [46] and the present long-wave evolution model, under the same operating conditions. The significance of the thermocapillary effect is only discernible at a higher Marangoni number.
a value of 5:91 106 , the difference between both numerical models (with and without thermocapillarity) is insignificant. At a 2-order-of-magnitude higher Marangoni number of 5:91 104 , the difference between the two models becomes discernible. The thermocapillary effect is neglected when the Marangoni number is typically small [17]. The variation of thermo-physical properties of different working fluids gives rise to distinct temperature distributions and hence temperature gradients [13]. The evaporation-induced temperature gradient in turn affects the surface tension gradient. The thermocapillary action induced by the surface tension gradient acts in the opposing direction of liquid flow and impedes the evaporation rate. The total evaporative heat transfer rate can be expressed in a function of excess temperature over the evaporating film length as Rl qc ¼ 0e q00e dx, where q00e ¼ JL. The significance of thermocapillary effect among the working fluids is determined by defining a ratio w ¼ qc;1 =qc;2 , where qc;1 is the total evaporative heat transfer rate of Model 1 (with thermocapillary effect), and qc;2 is that of Model 2 (without thermocapillary effect). Fig. 3 plots the ratio w as a function of the excess temperature for different types of working fluids. The ratio which decreases with increasing excess temperature is always smaller than 1, indicating that Model 2 overestimates the total evaporative heat transfer when the thermocapillary effect is
be zero as its influence on the thin film profile is marginal according to the flat adsorbed layer assumption [45]. 3. Results and discussion The long-wave evolution model has been employed to investigate the thermal characteristics of an evaporating thin film and justified the significance of thermocapillary effect [3,13]. Before embarking the hydrodynamic analysis of thermocapillary effect on the evaporating liquid thin film, we proceed with the validation of the numerical results. Fig. 2 shows the comparison of the liquid film thickness profiles between the numerical results and the experimental results by Hanchak et al. [46]. With the same pertinent parameters and under the same operating conditions, the numerical data agrees well with the experimental data, validating the accuracy of the numerical model. This type of Marangoni flow is intrinsically associated with the Marangoni number which is defined as the ratio of surface tension to viscous force [13]. As the Marangoni number of the experiment is remarkably small with
Fig. 3. Relative total evaporative heat flux for Model 1 and Model 2 of all working fluids with different excess temperatures [13].
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
neglected. Among the four working fluids, ammonia has the highest discrepancy between the models, followed by acetone, heptane and water. The descending order of the discrepancy is in accordance with the liquid temperature gradient profiles where ammonia manifests the largest temperature gradient, followed by acetone, heptane and water [13]. Therefore, the temperature gradient can be a significance indicator of thermocapillary effect on an evaporating thin liquid film. Our previous investigations justified the significance of thermocapillary effect of an evaporating thin liquid film for different working fluids from the thermal aspect [3,13]. In this paper, we focus on the fluid flow characteristics of the evaporating thin liquid film for various types of working fluids as employed in Fig. 3. When a low excess temperature of less than 10 K is applied, thermocapillary effect is deemed to be insignificant in the evaporating thin liquid film [4,45]. To scrutinize the underlying physical transport phenomena associated with the significance of thermocapillary effect in an evaporating thin liquid film, we extend our study to investigate the hydrodynamic characteristics of thermocapillary convection. In what follows, we divide our analysis into two sub-sections based on the imposed excess temperature, i.e. low excess temperature and high excess temperature, to evaluate the thermocapillary effect for different working fluids from the hydrodynamic aspect. 3.1. Low excess temperature Fig. 4 plots the streamlines of different types of working fluids without considering thermocapillary effect (Model 2) at an excess temperature of 5 K. Similar fluid flow patterns are observed in all working fluids. The unidirectional liquid streamlines show that the liquid flow is streaming from the intrinsic meniscus region to the evaporating thin film region as a result of capillary action
1107
due to evaporation. High evaporation rate prevails at the leading edge of the evaporating thin film region because of a lower thermal resistance. As the excess temperature increases, the evaporative heat flux increases, and the thermal efficiency of working fluid is overestimated. The thermocapillary force induced by surface tension gradient obstructs the liquid flow circulation and impedes the evaporation rates. It is justified in the thermal characteristics of the evaporating thin liquid film that thermocapillary effect should not be neglected especially when high excess temperature is applied [3,13]. In order to visualize the thermocapillary flow, we then proceed to study the fluid flow behaviors of different working fluids by taking thermocapillary effect into consideration. Intrinsically, the liquid circulates from intrinsic meniscus region to evaporating thin film region to maintain the evaporation rates. During the evaporation process, the transverse temperature gradient induces the surface tension gradient which draws the liquid from the low-surface-tension region to the high-surface-tension region. Fig. 5 depicts the flow streamlines of evaporating thin films for different types of working fluids incorporating the thermocapillary effect at an excess temperature of 5 K. The thermocapillary flow lines induced by the surface tension gradient are evident in all working fluids except for water. The streamlines of ammonia, acetone and heptane are diverging into two opposite directions in respect of the capillary action and thermocapillary action. A turning point is identified at the location where the thermocapillary action overcomes the capillary action due to evaporation. Here, we define a critical turning point at where the flow is reversed due to the thermocapillary action at a location nearest to the adsorbed film. The critical turning point length, lc , is measured from the origin of the evaporating thin film ð^ x ¼ 0Þ. The shorter the lc , the stronger the thermocapillary action is, based on the fact that the flow reversal prevails in a larger portion of the film. The critical turning point of ammonia is located at the
Fig. 4. Streamlines plots for different working fluids of Model 2 at an excess temperature of 5 K.
1108
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Fig. 5. Streamlines plots for different working fluids of Model 1 at an excess temperature of 5 K.
nearest to the origin where the liquid flow lines are heading towards the intrinsic meniscus region. The critical turning point of acetone occurs at further downstream of the evaporating thin film region and that of heptane takes place at the vicinity of the intrinsic meniscus region. Therefore, the thermocapillary action in ammonia is the strongest. On the other hand, no turning point is detected in water. The streamlines of water are observed to be slightly distorted by the thermocapillary action, as compared to those in Fig. 4. Most of the previous research studies concluded that the absence of thermocapillary flow in water is attributed to the presence of surface contaminants in pure water [21,26,27]. However, based on the heat transfer analysis, Fig. 3 shows that the thermocapillary effect is present in water although it is less conspicuous than that in other working fluids. As the excess temperature increases, the effect of thermocapillary on the thermal performance of water becomes more significant. The presence of thermocapilary flow in water will be further examined with the application of higher excess temperature in the next section. Fig. 6 shows the liquid velocity distribution of different working fluids without thermocapillary effect. Heat is transferred across the thin liquid layer to the liquid-vapor interface and expended for evaporation which induces high liquid velocity at the liquidvapor interface. The highest liquid-vapor interface velocity occurs at the location where the liquid film is thin and the evaporative flux is intense. Ammonia has the highest liquid velocity while water manifests the lowest. The evaporation-driven liquid velocity is dependent on the thermo-physical properties of the working fluids. We can observe that the ascending order of the liquid velocity magnitude among the working fluids is the same as that of the liquid temperature gradient, as shown in Ref. [13]. This shows that
indicates the critical turning point of thermocapillary flow.
the flow of liquid is solely driven by evaporation when thermocapillary effect is not taken into account. By taking thermocapillary effect into account, the liquid velocity contour plots of various working fluids are plotted in Fig. 7. It has been justified in Fig. 5 that the liquid flow direction is affected by the thermocapillary action, thus the liquid velocity contours of Model 1 are different from those of Model 2. The velocity magnitude of Model 1 is higher than that of Model 2. The liquid velocity of Model 1 is attributed to both capillary thermocapillary actions, while in Model 2 the flow is solely driven by capillary action. The lowest liquid velocity of Model 2 occurs at the solid-liquid interface where the no-slip boundary condition is applied. For Model 1, the liquid velocity is retarded in the vicinity of the turning points. The flow is decelerated due to the reversal in direction induced by the thermocapillary action. Consistent with the trends of the streamline and velocity plots, the thermocapillary effect of water is the most insignificant among all the working fluids as evidenced by the relative total evaporative heat flux between Model 1 and Model 2 in Fig. 3. Figs. 8 and 9 show the distributions of normal stress acting within the evaporating thin film for Model 1 and Model 2 at an excess temperature of 5 K. According to Eq. (7), the normal stress exerting on the liquid-vapor interface is attributed to the vapor pressure, disjoining pressure and capillary pressure. As disjoining pressure diminishes downstream along the axial direction, the normal stress is dominated by the capillary pressure and vapor pressure. According to the coordinate system in Fig. 1, the negative sign of normal stress indicates that the pressure is applied onto the liquid-vapor interface. Ammonia has the highest normal stress, followed by acetone, heptane and water. The normal stress of
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
1109
Fig. 6. Liquid velocity contours for different working fluids of Model 2 at an excess temperature of 5 K.
Fig. 7. Liquid velocity contours for different working fluids of Model 1 at an excess temperature of 5 K.
Model 1 and Model 2 have the same order of magnitude. The difference of normal stress balance between both models are insignificant for the respective working fluids, implying that the thermocapillary effect on the normal stress of an evaporating thin liquid film is marginal [18]. Fig. 10 plots the shear stress distributions of Model 1 at an excess temperature of 5 K. Unlike normal stress, the shear stress is dependent on the surface tension gradient and the temperature gradient, as depicted in Eq. (16). Ammonia has the highest shear stress yielded by its high surface tension gradient and temperature gradient values [13]. The low shear stress of heptane is due to its low surface tension gradient while the low temperature gradient
causes a low shear stress in water [13]. The shear stress diminishes along the y-direction in line with the application of no-slip boundary condition at the wall. In comparison with Fig. 5, the shear stress magnitude decreases with the critical turning point length, lc , showing that the former is intimately related to the strength of thermocapillary action. The shearing stress increases when the flow direction is reversed due to the thermocapillary action. For water, the maximum shear stress takes place in close proximity to the intrinsic meniscus region, indicating that thermocapillary flow exists there albeit rather indiscernibly. On the other hand, the shear stresses of Model 2 are negligibly small due to the absence of thermocapillary effect. In conclusion, the shear stress
1110
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Fig. 8. Normal stress distributions for different working fluids of Model 1 at an excess temperature of 5 K.
Fig. 9. Normal stress distributions for different working fluids of Model 2 at an excess temperature of 5 K.
within an evaporating thin film is essentially associated with the thermocapillary action. In contrary, the thermocapillary effect on the normal stress is negligibly insignificant. 3.2. High excess temperature The excess temperature plays a key role in affecting the heat transfer rate of a working fluid. As discussed earlier in Fig. 3, the thermocapillary effect should not be neglected when a high excess temperature is applied. Fig. 11 depicts the streamlines of various working fluids incorporating thermocapillary effect (Model 1) at a large excess temperature of 30 K. The streamline plots are similar to those in Fig. 5, except that of water. As the thermocapillary
action becomes stronger at higher excess temperature, the critical turning point emerges at a shorter distance from the origin. When the excess temperature is low, thermocapillary recirculation flow lines are elusive in water as discussed earlier. However, we can observe discernible liquid swirls towards the liquid-vapor interface when the excess temperature increases. These clockwise swirls which are resulted from the thermocapillary action are in a counter direction compared to the thermocapillary flow lines of other working fluids. It has been pointed out that the thermocapillary flow direction of an evaporation droplet is dependent on the ratio of thermal conductivities of solid substrate and liquid [47]. The thermal conductivity of water is higher than that of other working fluids. It can be deduced that the counter-direction ther-
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
1111
Fig. 10. Shear stress distributions for different working fluids of Model 1 at an excess temperature of 5 K.
Fig. 11. Streamlines plots for different working fluids of Model 1 at an excess temperature of 30 K.
mocapillary flow of water is attributed to its high thermal conductivity. At a reference temperature of 333 K, the thermal conductivity values of acetone, heptane, ammonia and water are 0.1224, 0.1411, 0.3896 and 0.6544 W/m K, respectively. By setting the thermal conductivity of the solid substrate as a constant, the high thermal conductivity of water reverses the direction of the thermocapillary flow compared to that of other working fluids [47]. In this study, consistent with the experimental results [47], the numerical
indicates the critical turning point of thermocapillary flow.
results evidence the existence of thermocapillary flow in water albeit only discernible at a high excess temperature. Fig. 12 depicts the velocity contours of Model 1 at an excess temperature of 30 K. Ammonia has the highest liquid velocity, followed by acetone, heptane and water. The liquid velocity is higher at higher excess temperature. Except for water, other liquids achieve the highest velocity at the liquid-vapor interface, while the lowest liquid velocity takes place along the solid-liquid inter-
1112
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
Fig. 12. Liquid velocity contours for different working fluids of Model 1 at an excess temperature of 30 K.
Fig. 13. Shear stress distributions for different working fluids of Model 1 at an excess temperature of 30 K.
face and in the vicinity of the critical turning point. The velocity contour of water shows that the minimum liquid velocity occurs in the vicinity of the liquid swirls near the liquid-vapor interface and along the solid-liquid interface. The flow in the vicinity of the liquid swirl is decelerated due to the reversal direction induced by the thermocapillary action. On the other hand, the minimum liquid velocity occurs along the solid-liquid interface due to the no-slip boundary condition. Therefore, thermocapillary effect is justified to be significant in liquids although the presence of thermocapillary flow is not easily to be distinguished in water due to its unique thermo-physical properties characteristics. Fig. 13 plots the shear stress distributions of Model 1 at an excess temperature of 30 K. Compared to Fig. 10, the trends of the shear stress distributions at a high excess temperature are similar with those at a low excess temperature. As discussed in Fig. 10 earlier, the shear stress magnitude increases with the decreasing critical turning point length, lc . The shearing stress increases when the flow direction is reversed due to the thermocapillary action. In
the case of high excess temperature, the magnitudes of the shear stress for all liquids are higher than those of low excess temperature. This implies that lc decreases and the coverage of the flow reversal becomes larger when the excess temperature increases. For acetone, ammonia and heptane, this is consistent with the streamline plots in Fig. 11 where a larger area is covered by the flow reversal due to the thermocapillary action. For water, the maximum shear stress occurs further upstream in the evaporating thin film and this explains that the existence of thermocapillary flow becomes more discernible. 4. Conclusions We delineate the thermocapillary effect on an evaporating thin liquid film for different types of working fluid from the hydrodynamic aspect. The long-wave evolution theory is employed to derive a two-dimensional mathematical model based on the basic principles of heat transfer and fluid mechanics applicable to polar
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
and nonpolar liquids in this study. This model could be expanded to model other types of liquid such as the non-wetting fluids with non-zero contact angle and the non-Newtonian fluids with sheardependent viscosity. The two-dimensional recirculation flow patterns in different excess-temperature regimes are analyzed and a critical turning point at where the flow is reversed due to the thermocapillary action can be identified. By neglecting the thermocapillary effect, the evaporative heat transfer rate is observed to be overrated and the overestimate increases with increasing excess temperature. The surface tension gradient eliciting thermocapillary flow is dominantly attributed to the temperature gradient induced by evaporation. Water exhibits a unique thermocapillary flow characteristics where the thermocapillary flow lines are inconspicuous compared to other working fluids at a low excess temperature. As the excess temperature increases, the thermocapillary flow in water manifests in the form of swirls along the liquidvapor interface. The thermocapillary flow direction of water is intrinsically dictated by the ratio of thermal conductivities of solid substrate and liquid. Water can be considered as the best working fluid in terms of thermocapillary suppression. The normal and shear stresses within the evaporating thin liquid film correlate well to the temperature gradient and surface tension gradient of the working fluids. Consistent with the heat transfer analysis, we conclude that the thermocapillary effect of an evaporating thin film should not be simply neglected in the hydrodynamic analysis as acute errors can be incurred in the thermal performance analysis. Acknowledgment The third author acknowledges the support of the Ministry of Higher Education, Malaysia, through the Fundamental Research Grant Scheme (FRGS/1/2014/TK01/MUSM/02/1). Appendix A As the characteristic film thickness is much smaller than the characteristic length scale of the film thickness variation, the long-wave theory can be employed to describe the film dynamics behaviors. The dependent variables of non-isothermal evaporating layer are expected to vary slowly along the axial direction. A small parameter denoted as the wave number, k, which is defined in line with lubrication theory, is used in rescaling the non-dimensional governing equations and boundary conditions. The following non-dimensional variables are defined as [39,48]
f ¼ k^x
g ¼ y^
ðA1Þ
The order of magnitude of dependent variables are outlined as ^ ; ^J; T^ ¼ Oð1Þ and p ^; P ¼ Oðk1 Þ. The film thickness ^ follows: u dðfÞ is expressed as an unspecified unit-order function. To retain continu^ ¼ OðkÞ is set. The dependent variables are expanded in the ity, v powers of k as [39]
^1 þ ^¼u ^ 0 þ ku u ^ ^ v ¼ kðv 0 þ kv^ 1 þ Þ ^J ¼ ^J 0 þ k^J 1 þ
T^ ¼ T^ 0 þ kT^ 1 þ ^0 þ kp ^1 þ Þ ^ ¼ k1 ðp p 1 ^ ^ ^ 1 þ Þ P ¼ k ðP0 þ kP
^ 0;f þ u ^ 0;gg ¼ 0 P
ðA3Þ
^ 0;g ¼ 0 P
ðA4Þ
^ 0;f þ v^ 0;g ¼ 0 u
ðA5Þ
T^ 0;gg ¼ 0
ðA6Þ
The non-dimensional groups are related to k as
1 ; Bk 3 Þ 1 ; Ek; Mk 1 ; Sk ðA; B; E; M; SÞ ¼ ðAk
Based on the asymptotic analysis with k ! 0, the expansions in (A2) are substituted into the governing equations where the coefficients of like powers of k in each equation and boundary condition are equated to zero. The governing equations can be simplified at the leading order in k as
ðA7Þ
where the order of magnitude of the non-dimensional groups are set as follows: A and B ¼ Oðk1 Þ, E ¼ OðkÞ to include the mass loss, M ¼ Oðk1 Þ to retain the thermocapillary effect in shear stress condition, S ¼ Oðk3 Þ to retain the surface tension effect in normal stress condition, as well as K ¼ Oð1Þ and P ¼ Oð1Þ. As k ! 0, the quantities with over-bar are of unit order of magnitude. The boundary conditions incorporating the non-dimensional groups at the interface can be expressed as
^0 ^df þ v^ 0 E^J 0 ¼ u
ðA8Þ
^J 0 ¼ T^ 0;g
ðA9Þ
^d3 þ Be ^dff þ A ^d=‘^ ^ ðv Þ þ P^ 0 ¼ S P 0
ðA10Þ
^ 0;g ¼ 2MP1 ðT^ 0;f þ T^ 0;g ^df Þ u
ðA11Þ
K^J 0 ¼ T^ 0
ðA12Þ
¼ 2k2 MP1 , Eq. (A10) is obtained By eliminating the term SC from Eq. (23). The Marangoni number, M, is absent at leading order in the normal stress balance while it exists in the shear stress balance in Eq. (A11). At the wall surface (g ¼ 0), the no-slip and notemperature-jump boundary conditions are specified as
^ 0 ¼ 0; u
v^ 0 ¼ 0;
T^ 0 ¼ 1
ðA13Þ
The governing equations (A3)–(A6) are solved together with the corresponding boundary conditions (A8)–(A13) to yield the longwave evolution equation as a function of the liquid film thickness distribution by incorporating the evaporative and thermocapillary effects. By applying the boundary conditions in Eqs. (A9), (A12) and (A13), Eq. (A6) is solved and the liquid temperature distribution is given by 1 T^ 0 ¼ 1 ðK þ ^dÞ g
ðA14Þ
The evaporative mass flux is obtained as
^J 0 ¼ ðK þ ^dÞ1
ðA15Þ
Subject to the boundary condition (A10), Eq. (A4) is solved to yield the liquid pressure as
^d3 þ Be ^dff þ A ^ 0 ¼ P^ ðv Þ S ^d=‘^ P 0 ðA2Þ
1113
ðA16Þ
By differentiating Eq. (A16) with respect to x and substituting it into Eq. (A3), together with the application of boundary conditions in (A11) and (A13), we obtain the x-component of liquid velocity as
^0 ¼ U u
2 1 2 ^ g dg þ 2MP1 KðK þ ^dÞ ^df g 2
ðA17Þ
^ ^ ^d=‘^=‘Þ. ^ By substituting Eq. (A17) df þ Be where U ¼ ðS dfff þ 3A d4 ^ into Eq. (A5) with the application of boundary condition in (A13), the y-component of liquid velocity is given by
1114
E. Lim et al. / International Journal of Heat and Mass Transfer 108 (2017) 1103–1114
1 1 1 v^ 0 ¼ Uf g3 ^dg2 þ U^df g2 6 2 2 h i 2 1 1 ^ ^ MP KðK þ dÞ dff 2ðK þ ^dÞ ^d2f g2
ðA18Þ
References [1] M. Potash Jr., P.C. Wayner Jr., Evaporation from a two-dimensional extended meniscus, Int. J. Heat Mass Transf. 15 (1972) 1851–1863. [2] S.S. Panchamgam, J.L. Plawsky, P.C. Wayner Jr., Experimental evaluation of Marangoni shear in the contact line region of an evaporating 99+% pure octane meniscus, ASME J. Heat Transf. 129 (2007) 1476–1485. [3] E. Lim, Y.M. Hung, Thermocapillary flow in evaporating thin liquid films with long-wave evolution model, Int. J. Heat Mass Transf. 73 (2014) 849–858. [4] R. Ranjan, J.Y. Murthy, S.V. Garimella, A microscale model for thin-film evaporation in capillary wick structures, Int. J. Heat Mass Transf. 54 (2011) 169–179. [5] R. Raj, C. Kunkelmann, P. Stephan, J. Plawsky, J. Kim, Contact line behavior for a highly wetting fluid under superheated conditions, Int. J. Heat Mass Transf. 55 (2012) 2664–2675. [6] S.S. Panchamgam, A. Chatterjee, J.L. Plawsky, P.C. Wayner Jr., Comprehensive experimental and theoretical study of fluid flow and heat transfer in a microscopic evaporating meniscus in a miniature heat exchanger, Int. J. Heat Mass Transf. 51 (2008) 5368–5379. [7] J.-J. Zhao, Y.-Y. Duan, X.-D. Wang, B.-X. Wang, Effects of superheat and temperature-dependent thermophysical properties on evaporating thin liquid films in microchannels, Int. J. Heat Mass Transf. 54 (2011) 1259–1267. [8] M.J. Tan, S.G. Bankoff, S.H. Davis, Steady thermocapillary flows of thin liquid layers. I. Theory, Phys. Fluids A 2 (1990) 313–321. [9] C. Buffone, K. Sefiane, J.R.E. Christy, Experimental investigation of self-induced thermocapillary convection for an evaporating meniscus in capillary tubes using micro-particle image velocimetry, Phys. Fluids 17 (2005) 052104. [10] X. Xu, J. Luo, Marangoni flow in an evaporating water droplet, Appl. Phys. Lett. 91 (2007) 124102. [11] A.M. Benselama, S. Harmand, K. Sefiane, Thermocapillary effects on steadily evaporating contact line: a perturbative local analysis, Phys. Fluids 24 (2012) 072105. [12] F.L. Chang, Y.M. Hung, The coupled effects of working fluid and solid wall on thermal performance of micro heat pipes, Int. J. Heat Mass Transf. 73 (2014) 76–87. [13] E. Lim, Y.M. Hung, Thermophysical phenomena of working fluids of thermocapillary convection in evaporating thin liquid films, Int. Commun. Heat Mass Transfer 66 (2015) 203–211. [14] W. Qu, T. Ma, Effects of the polarity of working fluids on vapor-liquid flow and heat transfer characteristics in a capillary, Microscale Thermophys. Eng. 6 (2002) 175–190. [15] S.-K. Wee, K.D. Kihm, K.P. Hallinan, Effects of the liquid polarity and the wall slip on the heat and mass transport characteristics of the micro-scale evaporating transition film, Int. J. Heat Mass Transf. 48 (2005) 265–278. [16] S.-Y. Du, Y.-H. Zhao, New boundary conditions for the evaporating thin-film model in a rectangular micro channel, Int. J. Heat Mass Transf. 54 (2011) 3694– 3701. [17] V.S. Ajaev, G.M. Homsy, Steady vapor bubbles in rectangular microchannels, J. Colloid Interface Sci. 240 (2001) 259–271. [18] J.R.A. Pearson, On convection cells induced by surface tension, J. Fluid Mech. 4 (1958) 489–500. [19] A.-T. Chai, N. Zhang, Experimental study of Marangoni-Benard convection in a liquid layer induced by evaporation, Exp. Heat Transf. 11 (1998) 187–205. [20] T. Qin, Z˘. Tukovic´, R.O. Grigoriev, Buoyancy-thermocapillary convection of volatile fluids under atmospheric conditions, Int. J. Heat Mass Transf. 75 (2014) 284–301. [21] J.C. Berg, M. Boudart, A. Acrivos, Natural convection in pools of evaporating liquids, J. Fluid Mech. 24 (1966) 721–735.
[22] H.K. Cammenga, D. Schreiber, G.T. Barnes, D.S. Hunter, On Marangoni convection during the evaporation of water, J. Colloid Interface Sci. 98 (1984) 585–586. [23] C.A. Ward, D. Stanga, Interfacial conditions during evaporation or condensation of water, Phys. Rev. E 64 (2001) 051509. [24] K. Sefiane, C.A. Ward, Recent advances on thermocapillary flows and interfacial conditions during the evaporation of liquids, Adv. Colloid Interface Sci. 134–135 (2007) 201–223. [25] I. Thompson, F. Duan, C.A. Ward, Absence of Marangoni convection at Marangoni numbers above 27,000 during water evaporation, Phys. Rev. E 80 (2009) 056308. [26] R. Savino, D. Paterna, N. Favaloro, Buoyancy and Marangoni effects in an evaporating drop, J. Thermophys. Heat Transfer 16 (2002) 562–574. [27] H. Hu, R.G. Larson, Analysis of the effects of Marangoni stresses on the microflow in an evaporating sessile droplet, Langmuir 21 (2005) 3972–3980. [28] C.A. Ward, F. Duan, Turbulent transition of thermocapillary flow induced by water evaporation, Phys. Rev. E 69 (2004) 056308. [29] F. Duan, C.A. Ward, Surface excess properties from energy transport measurements during water evaporation, Phys. Rev. E 72 (2005) 056302. [30] E.A. Chinnov, Wave – Thermocapillary effects in heated liquid films at high Reynolds numbers, Int. J. Heat Mass Transf. 71 (2014) 106–116. [31] L. Zhou, X. Du, Y. Yang, P. Jiang, B. Wang, Thermocapillary effect on bubble sweeping and circling during subcooled nucleate pool boiling of water over microwire, Int. J. Heat Mass Transf. 88 (2015) 276–283. [32] E.A. Chinnov, Enhancement of thermocapillary effect in heated liquid films for large waves at high Reynolds numbers, J. Heat Transfer 138 (2016) 092901. [33] M.L. Cordero, H.O. Rolfsnes, D.R. Burnham, P.A. Campbell, D. McGloin, C.N. Baroud, Mixing via thermocapillary generation of flow patterns inside a microfluidic drop, New J. Phys. 11 (2009) 075033. [34] H. Chraibi, J.-P. Delville, Thermocapillary flows and interface deformations produced by localized laser heating in confined environment, Phys. Fluids 24 (2012) 032102. [35] C.-M. Wu, Y.-R. Li, R.-J. Liao, Rotating and thermocapillary-buoyancy-driven flow in a cylindrical enclosure with a partly free surface, Phys. Fluids 26 (2014) 104105. [36] C.-M. Wu, D.-F. Ruan, Y.-R. Li, R.-J. Liao, Flow pattern transition driven by the combined Marangoni effect and rotation of crucible and crystal in a Czochralski configuration, Int. J. Therm. Sci. 86 (2014) 394–407. [37] T.H. Lowry, K.S. Richardson, Mechanism and Theory in Organic Chemistry, Harper & Row, New York, 1976. [38] V. Carey, Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment, Hemisphere Publishing Corporation, New York, 1992. [39] J.P. Burelbach, S.G. Bankoff, S.H. Davis, Nonlinear stability of evaporating/condensing liquid films, J. Fluid Mech. 195 (1988) 463–494. [40] R.W. Atherton, G.M. Homsy, On the derivation of evolution equations for interfacial waves, Chem. Eng. Commun. 2 (1976) 57–77. [41] J.A. Schonberg, S. DasGupta, P.C. Wayner Jr., An augmented Young-Laplace model of an evaporating meniscus in a microchannel with high hext flux, Exp. Thermal Fluid Sci. 10 (1995) 163–170. [42] N.V. Churaev, B.V. Derjaguin, Inclusion of structural forces in the theory of stability of colloids and films, J. Colloid Interface Sci. 103 (1985) 542–553. [43] J.A. Schonberg, P.C. Wayner Jr., Analytical solution for the integral contact line evaporative heat sink, J. Thermophys. Heat Transfer 6 (1992) 128–134. [44] H. Wang, S.V. Garimella, J.Y. Murthy, Characteristics of an evaporating thin film in a microchannel, Int. J. Heat Mass Transf. 50 (2007) 3933–3942. [45] V.S. Ajaev, G.M. Homsy, Steady vapor bubbles in rectangular microchannels, J. Colloid Interface Sci. 240 (2011) 259–271. [46] M.S. Hanchak, M.D. Vangsness, L.W. Byrd, J.S. Ervin, Thin film evaporation of noctane on silicon: experiments and theory, Int. J. Heat Mass Transf. 75 (2014) 196–206. [47] W.D. Ristenpart, P.G. Kim, C. Domingues, J. Wan, H.A. Stone, Influence of substrate conductivity on circulation reversal in evaporating drops, Phys. Rev. Lett. 99 (2007) 234502. [48] A. Oron, S.H. Davis, S.G. Bankoff, Long-scale evolution of thin liquid films, Rev. Mod. Phys. 69 (1997) 931–980.