International Journal of Heat and Mass Transfer 109 (2017) 482–500
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Generalized formulation for evaporation rate and flow pattern prediction inside an evaporating pinned sessile drop Chafea Bouchenna a,b,⇑, Mebrouk Ait Saada a, Salah Chikh a, Lounès Tadrist b a b
USTHB, Faculty of Mechanical and Process Engineering, LTPMP, Alger 16111, Algeria Aix-Marseille Université, CNRS, Laboratoire IUSTI, UMR 7343, 13453 Marseille, France
a r t i c l e
i n f o
Article history: Received 26 June 2016 Received in revised form 26 January 2017 Accepted 30 January 2017
Keywords: Drop evaporation Sessile drop Thermo-capillarity Buoyancy Flow patterns Coupling heat and mass transfer Solid-liquid-gas interactions Evaporation rate
a b s t r a c t When a sessile drop is heated from below, it evaporates and it induces a cooling effect in a zone close to the drop surface. The important evaporation rate at the contact line, the surface tension gradient at the liquid-air interface and the buoyancy generate the liquid motion inside the drop. Several parameters affect the evaporation rate among which the substrate properties, the moisture of the surrounding air and the heating conditions. Therefore, different flow patterns could be observed during the evaporation and they are mainly influenced by the relative importance of the evaporation rate, the thermo-capillarity and the buoyancy. The present study uses a generalized formulation to predict the flow patterns at any time during evaporation taking into account all these effects. The contribution and the relative importance of each effect are analyzed under isothermal and non-isothermal heating and different values of the relative humidity of the surrounding air. The correlation proposed by Hu and Larson for assessment of the evaporation rate is extended to non-isothermal surfaces for any evaporation conditions. Flow pattern maps are elaborated based on the dimensionless height of the drop apex and the evaporation conditions. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The scientific community showed a great interest in the phenomenon of sessile drop evaporation since the last century because of its various applications in metallurgy, biochemical assays, thin film coating, spray cooling, microelectronics, nano-devices and many others. At first, the authors focused on the study of global parameters such as the drop volume, the contact angle and the contact area [1,2]. The failure of some assumptions used in numerical and theoretical predictions, when compared with experimental data, confirms that the sessile drop evaporation is a complex phenomenon. Therefore, to predict reliably this phenomenon, one should include in the analysis all the coupled phenomena, i.e. the heat and mass transfer, the interactions between the solid, liquid and gas phases through different interfaces for a pinned or receding contact line, the evaporation cooling and the fluid flow. The literature review shows that a tremendous research work was carried out on many aspects of sessile drop evaporation. ⇑ Corresponding author at: Aix-Marseille Université, CNRS, Laboratoire IUSTI, UMR 7343, 13453 Marseille, France. E-mail addresses:
[email protected] (C. Bouchenna), m_aitsaada@ yahoo.fr (M. Ait Saada),
[email protected] (S. Chikh),
[email protected] (L. Tadrist). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2017.01.114 0017-9310/Ó 2017 Elsevier Ltd. All rights reserved.
Recently, Erbil [3] published a review paper about sessile droplets and nearly spherical suspended droplets of micrometer and millimeter size. He reported on research carried out on the topic for about 120 years. He concluded that the imposed parameters like the relative humidity of the surrounding air, the initial contact angle and the imposed temperature have a strong effect on the evaporation rate, the flow field and the temperature distribution. Therefore, it is essential to describe more precisely the operating conditions of the different experiments in order to classify more accurately these works. Cazabat and Guéna [4] presented a general review and some simple cases of analytical solutions existing in the literature to serve as reference for solving problems that are more complex. Larson [5] summarized analytical, numerical and experimental literature works on drying sessile droplets and deposition of suspended materials. He presented a list of useful dimensionless groups governing mass, momentum, and heat transfer effects in the droplet, the surrounding gas and the substrate. The wetting characteristic has been extensively investigated. The published literature shows that the sessile drop evaporation can occur in three different modes: (i) pinned mode with constant contact area and decreasing contact angle, (ii) de-pinned mode where the contact angle remains constant and the contact area decreases, or (iii) stick–slip mode in which both contact angle and contact area decrease with time [6–11]. Birdi and Vu [11]
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Nomenclature C cp Cv D h0 ha,b,/ h‘g Ha J Ja k Le _ M Ma Ra Ri Rk RaT R r, z r0 , u T
concentration (kg/m3) specific heat (J/kg K) saturated vapor concentration (kg/m3) vapor diffusion coefficient (m2/s) height of drop apex (m) metric coefficient (m) latent heat of vaporization (J/kg) air relative humidity (–) evaporation flux (kg/m2 s) Jacob number (–) thermal conductivity (W/m K) Lewis number (–) evaporation rate (kg/s) Marangoni number (–) Rayleigh number (–) Richman number (–) thermal conductivity ratio (–) thermal diffusivity ratio (–) contact radius (m) cylindrical coordinates (m) spherical coordinates (m, rd) temperature (°C)
studied experimentally the evaporation of water drop deposited on glass and found that the pinned mode dominates during most of the drop lifetime. Two possible cases were distinguished depending on initial wetting angle for water and n-octane drops deposited on glass and Teflon [9–11]. In the case of initial contact angles less than 90°, drops evaporate with a pinned contact line and the evaporation rate varies linearly with time, but in the case of initial contact angle larger than 90°, the evaporation rate is found to be non-linear. Several correlations were proposed to estimate the evaporation rate of sessile drops. In general, they are given as a product of two terms: the first one is a vapor mass diffusion term, and the second one, which has different expressions in the literature, is a function of the wetting angle [1,12–15]. The semiempirical correlation of Hu and Larson [14] is developed based on the assumption of isothermal drop surface. This correlation does not take into account the substrate nature, the fluid flow and the cooling effect at the drop surface induced by evaporation. Gatapova et al. [16] studied experimentally the evaporation of a water drop on a heated surface with controlled wettability and they developed a simple model of heat and mass diffusion to estimate the specific evaporation rate (evaporation rate per unit surface area). The calculated specific evaporation rates are found in good agreement with experimental data and in qualitative agreement with the correlation of Hu and Larson when considering the vapor concentration on the drop surface. Sefiane and Bennacer [17] proposed a correlation for the evaporation rate including a dimensionless number taking into account the effect of substrate thermal conductivity and thickness. Their results are in good agreement with experimental data. Nevertheless, the expression of the dimensionless number is very complicated, containing parameters not simple to determine accurately. Another aspect that has been explored by many researchers is the coffee ring phenomenon and the drop stain after evaporation. Undoubtedly, this involves fluid flow inside the drop and the interaction with the surrounding gas for either evaporation mode with a pinned or a receding contact line. Deegan et al. [18] explained the pinned evaporation mode by the liquid motion within the drop assimilated to an outward radial flow to compensate the strong mass loss near the contact line. This outward flow, which was studied experimentally, theoretically and numerically, has a driv-
VD u, v t
drop volume (m3) velocity component (m/s) time (s)
Greek symbols a, b, / toroidal coordinates (rd) aT thermal diffusivity (m2/s) thermal expansion coefficient (K1) bT r surface tension (N/m) q density (kg/m3) l dynamic viscosity (kg/m s) m kinetic viscosity (m2/s) h contact angle (°) Subscripts s; ‘; g solid, liquid, gas 1 conditions at infinity in surrounding air Superscripts ⁄ dimensionless variable
ing effect on the particles contained in drying drops. These particles are carried inside the drop and deposited near the contact line as in the coffee ring phenomenon [19–23]. It is well known that the flow inside the drop is strongly coupled to heat and mass transfer occurring during the evaporation of the drop. The early studies made the assumption of isothermal drop surface until demonstrated that this assumption does not always hold because of the high influence of evaporation cooling at the liquid-gas interface both on the liquid flow and heat transfer inside the drop and on the fluid flow and heat and mass transfer in the surrounding gas [17,24–27]. When the effect of evaporation cooling is considered, a temperature gradient is set at liquid-gas interface and imposes a surface tension gradient inducing thermo-capillary convection [28–33]. This later has a significant effect on drop evaporation and temperature distribution [34,35]. Although, thermo-capillary convection may be induced by heating the substrate or by hot surrounding air or by a very volatile liquid, the present study considers only the case of sessile water droplet in a surrounding air at room temperature. Therefore, thermo-capillary convection may enhance substantially the global evaporation rate in the case of heated substrate, but with negligible effect in the case of nonheated substrate. Moreover, the dependence of fluid density on temperature induces buoyancy convection in the liquid and surrounding gas [35–39], but its importance is still questionable. Some authors demonstrated that buoyancy effect in liquid phase can be neglected in the case where the substrate is at room temperature [40], whereas numerical results of Ait Saada et al. [38] revealed that neglecting buoyancy effect in the gas phase underestimates the evaporation rate especially for heated cases. Carle et al. [41,42] confirmed experimentally that when the substrate is heated, the buoyancy effect in fluid phases must be taken into account in models resolving sessile drop evaporation. The competition of the flow induced by the privileged evaporation near the contact line, the thermo-capillarity and the buoyancy effects, is strong during drop evaporation [43–45], and the prevalence of one or two effects rules the flow direction inside the drop [8,33,46–48]. Several works were conducted to study the influencing parameters, which define flow patterns in the evaporating sessile drops [40,49–53]. The intensity of thermo-capillary convection depends on temperature gradient along the liquid-gas inter-
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face, and its direction is imposed by temperature profile at the drop surface. Hu and Larson [28] found that the surface temperature gradient changes sign at a critical contact angle of 14°. Xu et al. [54] and Ristenpart et al. [55] associated the critical value of the contact angle to solid/liquid thermal conductivity ratio and substrate thickness. Zhang et al. [56] showed that, in addition to the inversion of temperature profile sign at drop surface, this later could be non-monotonic at specific contact angles. Barash [57] showed that the non-monotonic temperature profile at liquid-gas interface induces a multi-cellular flow pattern inside the droplet according to the thermal conductivity, thickness and wall temperature of the substrate. This complex cellular flow structure was previously shown by Brutin et al. [58] in their experiments and associated the number of cells to wall temperature. Numerical results of Bouchenna et al. [34] predicted also the existence of 3 cell-flow pattern in the drop when it is deposited on a substrate under heating conditions. The above review is far from being exhaustive because of the huge number of publications on sessile drop evaporation. For the sake of brevity, we tried to present the most useful works, which help understanding this complex phenomenon. From this literature review, we may conclude that with regard to experimental works, there are no methods to measure accurately some local variables inside the drop such as local temperature and velocity. In addition, the multitude of parameters to consider (different fluids, different substrates and different evaporation conditions) and the large time required to experiment each case study makes the numerical tool a good alternative solution to attempt to generalize and extend the conclusions. Several studies involved with the flow inside an evaporating drop considered that the drop is deposited on a solid surface and the evaporation occurs in an isothermal state [18–23]. In that case, only the non-uniform evaporation at the drop surface is considered as the driving effect. Other studies included thermo-capillary effect associated or not to thermal buoyancy [32,35,43,56–58]. The authors analyzed heated or non-heated drop on a substrate with high thermal conductivity in most cases. To our knowledge, few studies focused on the effect of the thermal resistance of the substrate on fluid flow in the heated case, or the effect of the surrounding air relative humidity [34,43,52,59]. As a result, from the literature review, it was difficult to understand the combined effects on the evaporation kinetics. In the present work, a numerical model is implemented to investigate the flow patterns inside an evaporating water drop on either isothermal or non-isothermal solid surfaces. Despite the numerous works on the subject, the coupling of the different effects controlling the internal flow still requires further exploration for a good understanding. Therefore, a generalized formulation is used to account for fluid flow, heat and mass transfer as well as the interactions between the different phases (solid, liquid and gas). The analysis carried out in this paper aims to generalize the flow inside a drop induced by the coupling heat and mass transfer phenomena in the three media solid (substrate), liquid (drop) and gas (air). Three coordinate systems, toroidal, spherical and cylindrical, are combined in sub-domains to handle accurately the interactions at interfaces, particularly the liquid-gas interface where heat and mass exchanges are strong. The flow within the drop is mainly governed by the privileged evaporation near the contact line and the thermo-capillarity as well as the thermal buoyancy induced by the combined effects of evaporative cooling at the drop surface and heating of the drop from below. The objective is to evaluate the relative importance of each driving effect and to indicate the consequences on the flow patterns appearing during evaporation under several operating conditions i.e. heating or non-heating, conditions of relative humidity in the surrounding air and the thermal properties of the substrate. Based on the governing parameters,
maps are proposed to predict the flow pattern at any stage of evaporation. Results of evaporation rate obtained with the developed model are compared with those obtained with diffusion model. Until now, the evaporation rate correlations available in the literature are specific to isothermal solid surface cases and do not take into account the effect of evaporative cooling at the drop surface. Exception is the work of Sefiane and Bennacer [17] who proposed an expression based on a dimensionless number dependent on solid and liquid thermal conductivities. However, this expression is not simple because of the complexity of the dimensionless number. Hence, in the present work, the semi-empirical correlation of Hu and Larson [14] for computing the evaporation rate is extended to non-isothermal surfaces, and for any condition of temperature, thermal conductivity of the substrate, and relative humidity of the surrounding air. 2. Mathematical model We consider a water drop deposited on a solid surface or a finite substrate of given thickness. Fig. 1 illustrates (a) the global physical domain and (b) a zoom for the zone containing the water drop, part of surrounding air and the wetted part of the solid. The complexity of the domain geometry encompassing three different phases of different properties and the need for accurate assessment of heat and mass transfer requires a careful handling of the liquid-gas interface. Therefore, we use three different coordinate systems namely cylindrical coordinates in the solid phase (zone I), toroidal coordinates in the liquid phase and part of the gas phase around the drop surface in zone II, which is a hemisphere of radius R and spherical coordinates in the surrounding gas (zone III). We point out that toroidal coordinates are used in zone II because it allows locating easily the liquid-gas interface with a single coordinate b0 during evaporation. To maintain the same initial wetting angle of 78° for the different solid walls used, a very thin sheet of aluminum covers the top surface of the substrate. Based on experimental data in the literature, the drop evaporation is assumed to occur with pinned contact line because of the wetting angle lower than 90° [2,11,36]. The studied water drop is of spherical cap shape with a contact radius of about 1.86 mm, which is smaller than the capillary length pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi ‘c ¼ r=ðgq‘ Þ for water (2.69 mm) allowing then to neglect the effect of gravitational forces on the drop shape. The flow inside the evaporating sessile drop is mainly governed by three effects with different importance. The strong mass loss at the contact line yields the liquid flow from the drop center to the edge. The evaporation cooling effect imposes a temperature gradient and consequently a surface tension gradient, which induces thermo-capillary convection in the liquid phase. The temperature field in the drop induces liquid density variation that involves a thermal buoyancy flow. The thermo-capillarity is modeled by introducing surface tension dependence on temperature at liquid-gas interface as r ¼ r1 ðdr=dTÞðT T1 Þ, where r1 is the surface tension at T1 and (dr/dT) a physical property of water. Thermal buoyancy effect is included in the model with the Boussinesq approximation. The surrounding air is assumed quiescent and has an ambient temperature of 25 °C. The heat transfer in both gas and solid phases occurs by conduction mode alone. The saturated vapor resulting from phase change at the liquid-gas interface is transferred by diffusion to ambient moist air. In the mathematical model written in dimensionless form, the drop contact radius (R) is used as a reference length and the characteristic thermo-capillary velocity jdr=dTjDT=l‘ at liquid-gas interface as a reference velocity. The reference temperature difference DT is equal to (Tw T1) in the case of a heated substrate and represents the drop cooling due to the evaporation in the case of a non-heated substrate. In the present study, the quasi steady state is invoked and this is justified by the fact that the ratio of the drop
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485
(a)
h0
(b) Fig. 1. (a) Physical domain enclosing the liquid drop, the solid substrate and the surrounding air with associated boundary conditions, (b) a zoom for the zone containing the water drop, part of surrounding air and the wetted part of the solid. For non-heated substrate Tw = T1, for heated substrate Tw > T1. A zonal grid is applied in the computational domain.
height based characteristic velocity to the characteristic thermocapillary velocity and the ratio of the drop height based characteristic velocity to the characteristic velocity based on temperature
evolution are very small compared to unity. Convective flow inside the drop is governed by the equations of conservation of mass, momentum and energy. They are written in a general formulation as:
@ @ ðh h/ u Þ þ ðh h/ v Þ ¼ 0 @a @b @ @a
ð1Þ
2 Pr @u @ Pr h h/ u u Ma h h/ h @ a þ @b h h/ v u Ma hh/ h@u þ sin b u v þ sinh a v 2 h h/ ¼ h h/ @P @a @b h i h i acos b a cos b Pr @ Pr @ Ma h h/ 2 sinh a þ coshtgh h h/ 3 sinh a 1cosh v u þ 2h h/ sin b v þ Ma a @a @b sinh a h i 2 2 2 2 2 2 a cos b b Pr Pr 33 cosh a cos bþ2sinh a Ma u h h/ þ Ma 3sin b þ 2sinh a þ 2 1cosh cos sin b v h h/ sinh a sinh a h h i 2 RaPr h sinh a sin b ðT ‘ Þ h h/ Ma2
2 sinh au v þ sin bu2 h h/ ¼ h h/ @P @b i 2 Pr @ Pr @ Pr þ Ma 2h h/ sinh au þ 2h h/ sin bv þ Ma 4h h/ sin bu þ Ma ½ sinh a sin bu h h/ @b @a h i h i 2 2 2 2 Pr þ Ma sinh a 4 sin b þ 2ð1 cosh a cos bÞ v h h/ þ RaPr h ðcosh a cos b 1ÞðT ‘ Þ h h/ Ma2 @ ðh h/ u @a
h
Pr @ v @ Pr @ v v Ma h h/ h @ aÞ þ @b ðh h/ v v Ma h h/ h @bÞ þ
i
h
ð2Þ
ð3Þ
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@ 1 @T ‘ @ 1 @T ‘ ¼0 þ h h/ u T h h/ h h/ v T h h/ @a Ma @b Ma h @a h @b ð4Þ
where h = ha = hb = h/ are dimensionless metric coefficients. The equations of thermal and mass diffusion are solved in the surrounding air in toroidal coordinates in zone II and in spherical coordinates in zone III. These equations read: In zone II,
@ @ @T g @T g ¼0 þ h h/ h h/ @a @b h @a h @b
ð5Þ
@ @ @C @C ¼0 þ h h/ h h/ @a @b h @a h @b
ð6Þ
In zone III,
@T g @T g @ @ 02 0 þ ¼0 r sin u r sin u @r 0 @u @r 0 r 0 @ u
ð7Þ
@ @C @ @C 02 0 þ ¼0 r sin u r sin u @r 0 @u @r0 r 0 @ u
ð8Þ
The dimensionless concentration C is defined by ðC C1 Þ=DC, with DC equal to (Cv(Tw) Ha Cv(T1)) in the case of a heated substrate and equal to (1 Ha) Cv (T1) in the case of a non-heated substrate. Cv is the saturated vapor concentration and Ha the relative humidity of air. Thermal conduction equation in the solid phase is formulated in cylindrical coordinates as:
@ @ @T s @T s þ ¼0 r r @r @z @r @z
ð9Þ
Three dimensionless numbers appear in the governing equations and constitute the key parameters of the problem: the Prandtl number Pr ¼ ðm=aT Þl , the Rayleigh number Ra ¼ gbT l DTR3 =aT l , and the Marangoni number Ma ¼ jdr=dTjDTR=ðlaT Þl . Applied boundary conditions are given in Fig. 1 where the substrate can be replaced by an isothermal wall when the thermal conductivity of the substrate is very high. At the liquid-gas interface, the saturated vapor concentration is dependent on temperature according to a polynomial relationship of fourth order:
Cv ðTÞ ¼
4 X ai Ti
ð10Þ
i¼0
The coefficients ai in Eq. (10) are calculated using the least square method fitted with experimental data of Raznjevic [60]. The local evaporation flux is calculated through concentration gradient at the drop surface. Its dimensionless form is:
J ¼
J @C ¼ DDC=R h @b b0
ð11Þ
Mass conservation, momentum and energy balance at the liquidgas interface are expressed hereafter: i. Mass conservation,
~ W ~ IÞ ~ ðW n¼
RagT ðDC=q‘ Þ ~ n J ~ LeMa
ð12Þ
~I ~ ~ n is the dimensionless velocity of the n is a normal unit vector, W ~ ~ moving interface and W:n is the velocity of a liquid particle. Four dimensionless numbers appear in Eq. (12): the ratio DC=q‘ , the Lewis number, Le = D/aTg , the ratio of gas/liquid thermal diffusivity, RagT ¼ aTg =aT‘ and the Marangoni number Ma.
ii. Shear-stress balance,
~ T ~ Þ ~ ð~ ns t¼r t
ð13Þ
is the dimensionless stress tensor. ~ t is a tangential unit vector and s The dimensionless temperature gradient in Eq. (13) represents the term of thermo-capillary effect. iii. Energy balance,
@T g Ja DC @T J ‘ þ Rkg ¼ 0 Le q‘ hb @b hb @b
ð14aÞ
where Rgk ¼ kg =k‘ is the ratio of gas/liquid thermal conductivity, Ja ¼ h‘g =ðcP‘ DTÞ is the Jacob number and h‘g is the latent heat of vaporization. In addition, the temperature must be continuous at the liquid-gas interface, this reads:
T ‘ ¼ T g
ð14bÞ
The governing equations are solved for velocity, temperature and concentration in the corresponding phases for different values of the influencing parameters: Pr, Ra, Ma, Rk (gas/liquid and solid/ liquid), RagT , Le, Ja and DC=q‘ . The concentration gradients at the drop surface are then used to evaluate the evaporation rate and _ deduce the drop lifetime. The dimensionless evaporation rate M is determined by:
_¼ M
Z 1 _ M @C ¼ h h da 2pDRDC hb @b / a 0
ð15Þ
b0
3. Numerical procedure The finite volume method is used on a computation domain extended to a radius of 200xR to keep the boundary condition far enough from the drop [61]. After a grid sensitivity analysis, the computational mesh is composed of different element sizes and geometry depending on the coordinate system used in the different sub-domains (zone I, II and III). It is refined close to the different interfaces, near the vertical axis of the drop and near the domain boundaries to improve the accuracy of the numerical computations, as shown in Fig. 2. A fine mesh with a step size of R/105 is used around the contact line in order to handle this singular point. This yields an accurate evaluation of the evaporation flux near the contact line [26,38]. Coupling of velocity and pressure fields in the liquid phase is addressed by using the SIMPLE algorithm associated to a staggered grid. A power law differencing scheme (PLDS) is used to consider the contribution of convection and diffusion in the transport phenomena. The algebraic equations resulting from the finite volume discretization are solved using a combination of the tridiagonal matrix algorithm (TDMA) and the Gauss-Seidel iterative method along with under-relaxation. Solutions of velocity, temperature and concentration fields reach satisfactory convergence during the iterative process once the maximum relative error on the dependent variable (u, v, T, C) is smaller than 0.1%. The maximum allowable absolute residue in the mass conservation equation is less than 1010 and less than 105 in other conservation equations. At a given value of h(t) corresponding to an unknown time t, the volume of the drop VD(t) is computed using the following equation:
V D ¼
VD 2pR3
¼
ð1 cos hÞ2 ð2 þ cos hÞ 3
6 sin h
ð16Þ
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(a) 1
0.8
0.6
z/R 0.4
0.2
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
(b)
1.1
ZOOM
r/R
1
0.0003 0.00025 0.0002 0.00015 0.0001 5E-005 0 -5E-005 -0.0001 0.9997
0.9998
0.9999
1
1.0001
Fig. 2. Typical mesh elements in the vicinity of contact line and liquid-gas interface. (a) Different element sizes and geometry depending on the coordinate system used in the sub-domains I, II and III. (b) zoom of the part containing solid, liquid and gas phases near the contact line.
Once the concentration profile is obtained, the evaporation rate at instant t is determined from Eq. (15). Then, the time step Dt necessary to reach that position imposed by the value of h(t) is calculated from Eq. (17).
Dt ¼
Dt DV ¼ D _ ðR =DÞ=ðDC=q‘ Þ M 2
ð17Þ
A computational program is elaborated based on the developed numerical model. Validation of the results from the implemented computer program with experimental data is a vital and unavoidable step. Despite the existence of several studies on sessile drop evaporation, as previously reported in the introduction section, it
is not easy to access experimentally to local variables such as velocity and temperature variables inside the drop, chiefly for micrometer drops. Therefore, only comparisons with accessible variables such as drop volume, contact radius, wetting angle, evaporation rate and average velocity and temperature in liquid phase could be made. In the present study, comparison is made with the experimental data of Chen et al. [45] for a small droplet of 20 mL. Fig. 3a plots the average evaporation rate according to the temperature difference between solid surface and ambient air (DT), where ambient temperature is equal to 20 °C. Numerical results of the present study are in very good agreement with these experimental data with a maximum relative deviation less than 26%. Another validation with the analytical solutions of Masoud and Felske
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(a) Evaporation rate
(b) Dimensionless velocity profiles Fig. 3. (a) Comparison with experimental data of Chen et al. [45]. Effect of wall temperature on the evaporation rate (water drop of 20 lL deposited on an aluminum substrate with temperature difference varying from 0 to 60 °C). (b) Comparison with analytical solutions of Masoud and Felske[23]. Dimensionless radial and vertical velocities versus vertical position at different radial positions in an evaporating sessile drop. The contact angle is of 40°.
[23] is carried out. Fig. 3b shows dimensionless radial and vertical velocity profiles at different radial locations for a situation where the thermo-capillarity is neglected. The numerical results are in perfect agreement with those of Masoud and Felske. The same concordance is also obtained by comparison with the analytical solutions of Hu and Larson [22] (see Fig. 5 in [22]). 4. Results and discussion Results are presented for a water drop of 10 mm3 initial volume deposited on a substrate covered by a very thin sheet of aluminum that imposes an initial contact angle of 78°. The substrate is subject to heating with constant temperature boundary condition on its bottom surface. The effects of wall temperature, relative humidity of air and thermal conductivity of the substrate on the flow inside the drop are analyzed and documented. Two cases are invoked: the first one concerns the drop evaporation on an isothermal surface beneath the drop, which may correspond to a substrate with a very high thermal conductivity and the second one deals with a nonisothermal surface. This latter refers to a substrate of 1 mm thickness with a relatively small thermal conductivity, which makes the temperature on its upper surface not constant. It is worth noting that in the following paragraphs, when we refer to monocellular or multicellular flow patterns, we mean in half of the drop as it is presented in the figures.
4.1. Evaporation on an isothermal solid surface The drop is deposited on an isothermal surface at room temperature i.e. Tw = 25 °C. Streamlines, velocity profiles, temperature and concentration fields are plotted in Fig. 4 for three different models. In model (a), only the flow induced by the non-uniform evaporation at the liquid-gas interface is considered; in model (b), thermo-capillary effect due to temperature gradient along the drop surface is added and in model (c), all effects are considered that include thermal buoyancy, thermo-capillarity and non-uniform evaporation. The three models illustrate how each driving mechanism contributes to the liquid flow patterns inside the drop. The non-uniform evaporation along the drop surface with an intense mass loss near the pinned contact line induces a radial flow directed outward from the drop apex to the contact line (model a). Velocity profiles show that the flow accelerates with decreasing height of the drop. With regard to heat transfer, conduction is the prevailing mode as illustrated by the isotherms on the left part of the figures. The addition of the thermo-capillary effect in model (b) show a different flow pattern with cellular flow in a counterclockwise motion for a contact angle of 50°; that is indicated by the velocity profile at the radial position of the cell center (r⁄0). In this case, the heat is transferred by convection, which is indicated by isotherms distortion. When local evaporation, thermo-capillary and thermal buoyancy effects are included in model (c), the velocity
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Fig. 4. Streamlines in liquid phase, iso-concentrations in gas phase (right: Cmax = 0.023 lg/mm3, DC = 0.0005 lg/mm3) and isotherms in both liquid and gas phases (left, DTmax = 0.03 °C) at different stages of evaporation for Tw = T1 = 25 °C and Ha = 40%. (a) With non-uniform evaporation effect alone, (b) with non-uniform evaporation and thermo-capillary effects, (c) with all effects. At each cell center position (r0 ), a velocity profile is also represented according to drop height.
field is found identical to that obtained by the model (b), which means that the buoyancy effect is negligible. For the three models, when the contact angle declines to 30° or 20°, the observations about
temperature and velocity fields are the same as the case of 50°, but the flow intensity and temperature values are different. A decrease of the maximum value of the stream-function is observed during
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evaporation. As a result, velocity profiles according to drop height at center of cells have smaller values in time. At a contact angle of 30°, the thermo-capillarity induces convective effects, but this thermocapillary effect becomes negligible at a contact angle of 20°. Towards the evaporation end (h = 3°), the flow becomes radial and oriented outwards, which means that the effect of strong mass loss near the contact line dominates both thermo-capillary and buoyancy effects. The important result shown in Fig. 4 is that the thermal buoyancy effect is very weak during the whole lifetime of the drop on a non-heated solid surface. To analyze the contribution of these different effects, we introduce dimensionless numbers characterizing the thermo-capillary induced flow, the buoyancy induced flow and the liquid motion induced by the privileged evaporation near the contact line, namely, Marangoni number (Ma), Rayleigh number (Ra) and modified Richman number (Ri), respectively. The first two numbers appear explicitly in the Navier-Stokes and energy equations in a dimensionless form in the liquid phase and they are defined in Section 2. These numbers are based on a reference temperature difference (Tw T1) applied on the domain boundaries. However, a more rigorous analysis suggests that the flow inside the drop is influenced by the temperature difference applied on the drop boundaries i.e. between the drop base and the drop surface. Therefore, we reintroduce modified Marangoni and Rayleigh numbers that are variable during evaporation at each contact angle. They are written as:
Mar ¼ Ma
DT r DT
(a) Modified Rayleigh number
ð18Þ
where DTr is the temperature difference between the contact line and the apex i.e. along the drop surface.
Rar ¼ Ra
DT r DT
ð19Þ
where DTr is the temperature difference between the base center of the drop and the apex i.e. across the liquid thickness. The flow induced by non-uniform evaporation plays a crucial role towards the end of the evaporation, as shown in Fig. 4. For a general analysis of the importance of this flow, the modified Richman number Ri is defined as follows:
(b) Modified Marangoni and Richman numbers
ð20Þ
Fig. 5. Evaluation of the importance of buoyancy convection, thermo-capillary convection and flow due to the strong mass loss at drop edge during evaporation by comparing modified Rayleigh, Marangoni and Richman numbers, respectively, according to drop height and for different wall temperatures (Ha = 40%).
where Jr!R is the local evaporation flux near the contact line and h0 the height of the drop apex. This dimensionless number is identical to Reynolds number representing the intensity of the inertial forces due to the evaporation comparatively to the viscous forces in the liquid phase. The evolutions of the modified Rayleigh, Marangoni and Richman numbers (Rar, Mar and Ri) as function of the height of the drop apex are displayed in Fig. 5 for different values of wall temperature. It is depicted in Fig. 5 a that values of Rar, reported to Grashoff number Rar/Pr, are smaller than the critical value of 2400 corresponding to the onset of natural convection in a cavity [62]. Hence, the buoyancy effect is negligible during evaporation of the drop on heated and non-heated solid surfaces. This explains why the streamline, temperature and concentration distributions are the same for the models (b) and (c) in Fig. 4. In Fig. 5b, the values of the Marangoni number are high at the beginning of evaporation, but undergo progressively a decrease with time towards the end of evaporation. This confirms that the flow is mainly governed by thermo-capillary effect in most of the drop lifetime. In parallel, during a large period of the drop evaporation, the values of the modified Richman number remain very small compared to those of the modified Marangoni number. Exceptionally, this general situation is inverted towards the end of the evaporation where a significant increase of the values of Ri is noted. In
this case, the drop declined to be very thin and the temperature gradient at the surface becomes very small. As a result, the nonuniform evaporation at the drop surface prevails over the thermo-capillary effect inducing an outward radial flow. Furthermore, the effects of thermo-capillarity and non-uniform evaporation increase with the heating temperature Tw. The contribution of thermo-capillary effect is predominating for a large period during evaporation, except towards the end when this effect becomes weak and lower than the effect of the intense evaporation at the contact line. Fig. 6a shows the flow patterns in the evaporating drop on an isothermal solid surface for three levels of temperature Tw (30, 40 and 50 °C). When the drop is sufficiently heated, as in Fig. 6a (ii) and (iii), the flow pattern shown previously evolves to a 3-cell flow structure when the drop height reaches a certain value during evaporation corresponding to h = 20°. Flow direction in each cell is given by the velocity profile according to (z/R) at cell center. The effect of more heating is shown at this value of the contact angle; it is clear that the increase of Tw intensifies the liquid motion as it is depicted by the values of Wmax that increase with Tw. Nevertheless, the flow pattern due to thermo-capillary effect remains mono-cellular for contact angles higher than 20° and becomes outward radial for h < 10°, due to the high evaporation
2
Ri ¼
J r!R DR ðaT lÞ‘ h0
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Fig. 6. (a) Flow pattern in the evaporating drop on an isothermal solid surface. For h = 20°, velocity profile at each flow cell center is presented: Tw = 30 °C (a0: r⁄0 = 0.55), Tw = 40 °C (a0: r⁄0 = 0.14, b0: r⁄0 = 0.35, c0: r⁄0 = 0.50), Tw = 50 °C (a0: r⁄0 = 0.22, b0: r⁄0 = 0.50, c0: r⁄0 = 0.63). (b) Effect of convective flow on local evaporation flux distribution at contact angle of 20°.
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Fig. 6 (continued)
Fig. 7. Influence of relative humidity of air on the flow pattern in the evaporating drop on an isothermal solid surface of different wall temperatures. The contact angle is of 20°.
rate at the contact line. Furthermore, at a given value of Tw, for example Tw = 40 °C in Fig. 6a (ii), when the drop evaporates and its height diminishes, the velocity magnitude of the liquid reduces and it is illustrated by values of the stream function that decrease. At h = 20°, the flow direction of the different cells in the 3-cellstructure is imposed by the temperature gradient along the drop surface that regulates the surface tension as it will be discussed later in this section. The convective effects are more important in the inner cells and this is shown by the velocity profiles, particularly, of same appearance and same magnitude order.
The relation between multi-cellular convection and the local evaporation flux at the liquid-gas interface is analyzed in Fig. 6b, which highlights the effect of wall temperature at the evaporation stage corresponding to the contact angle of 20°. At low temperature Tw = 30 °C where the flow structure is represented by a single counter-clockwise cell, the local evaporation flux is relatively constant at the drop surface and increases significantly towards the contact line. It is weakly influenced by the flow in the drop as compared with the result of the diffusion model in which we neglected convection inside the drop. When the wall is heated
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(a) Tw =30°C
(b) Tw =40°C Fig. 8. Influence of air relative humidity and convective flow on the local evaporation flux at the drop surface. The wall temperature is of (a) 30 °C, (b) 40 °C and the contact angle of 20°.
more, at 40, 50 and 70 °C where a tri-cellular flow structure sets up, the profile of the mass evaporation flux becomes nonmonotonic influenced locally by the flow direction and here again it increases near the contact line. The clockwise cell settles in between the two counter-clockwise cells; it creates a small disturbance and a slight increase of the local evaporation flux due to the heat received from the hot solid surface. This intermediate cell moves towards the drop edge and decreases in size when the heating temperature Tw increases, and the cell near the drop axis becomes dominant. The drop heating produces an increase of the evaporation rate and consequently leads to increasing thermo-capillary flow and the possibility of appearance of multi-cellular convection. This state occurs at a stage of evaporation where the drop is somewhat flattened. Similarly to the drop heating effect, it is expected to have the same effect with decreasing the moisture of the surrounding air (Ha) far from the drop. The evaporation rate increases with both
drop heating and dehumidification of the surrounding air. Fig. 7 allows analyzing the effect of relative humidity of air far away from the drop on the flow inside the drop. In addition to the case of Ha = 40%, given in Fig. 6a for h = 20°, we show in Fig. 7a the results of velocity and temperature fields for two other values of relative humidity of air: Ha = 0% and 100% (h = 20°). The decrease in the relative humidity of air (from 40%) implies increasing the flow magnitude and the flow structure may undergo some changes. In the case of wall temperature equal to 30 °C, the drop evaporation when Ha = 100% is very slow and as a result, the flow in the drop is very weak and has the same pattern as observed when Ha = 40%, corresponding to a counter-clockwise 1-cell flow. However, when the relative humidity is decreased to 0%, a 2-cell flow pattern is observed due to the great importance of the evaporation rate, which induces high convective effects inside the drop. For a drop deposited on a solid surface at 40 °C, the flow pattern evolves from one cell when Ha = 100% to three cells when Ha = 0%. Whereas, at a
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Fig. 9. Drop evaporation rate versus the drop contact angle. Effect of air relative humidity, and wall temperature on the evaporation rate of the drop deposited on an isothermal solid surface. Results are obtained using the model with or without internal flow.
(a) Evaporation rate for Ha = 40%
higher wall temperature (Tw = 50 °C), the flow keeps the same structure of three cells. Globally, we may say that both effects of higher temperature difference (Tw T1) and higher concentration difference (Cinterface C1) may emphasize the transition to multicellular flow pattern. In fact, the increase of the concentration gradient, i.e. Ha = 0%, increases the evaporation, resulting then in a cooling of the drop surface, whereas, the increase of Tw yields more heating of the drop. When the heating effect at the drop base is higher than the cooling effect at the drop surface, this results in the appearance of the 3-cell flow structure. Thus, the significant heating is the determining factor in settling a multi-cellular flow pattern even at low relative humidity of air. However, at a lower level of heating, the air dehumidification is responsible for the appearance of the 2-cell flow structure. Fig. 8 shows the effect of air relative humidity on evaporation flux at the drop surface. The wall temperatures are of 30 and 40 °C and the contact angle is of 20°. Local evaporation is induced by drop heating and increases with decreasing air relative humidity. As a result, multi-cellular convection can appear at low values of Ha, for example 0% and can on the contrary not appear at high values of Ha, for example 100%. The 2-cell flow driven by the effect of air relative humidity occurs because of the decrease of the drop surface temperature while the 3-cell flow driven by the effect of the wall temperature is due to increasing drop surface temperature. The evaporation flux usually increases monotonically from the drop apex to the contact line; however, in the case of multicellular flow, it follows a non-monotonic variation characterized by local minimum with a decreasing trend at the counterclockwise cells and an increase at the clockwise cell. For a more general analysis, Fig. 9 shows the effect of both air relative humidity and wall temperature and the link between the internal flow and the drop evaporation rate on an isothermal solid surface. The evaporation rate undergoes a weak decrease with reduction of the contact angle. An increase of the wall temperature and/or a decrease of the relative humidity of air lead to a higher speed in the evaporation of the drop. At the beginning of the evaporation when the contact angle is higher than 30°, thermocapillary effect induces a 1-cell flow pattern. In this case, the entire surface of the drop receives heat from the solid surface by the convective motion of the liquid and thus affecting the evaporation rate. For lower contact angles, the moderate convective effects imply a negligible variation of the evaporation rate compared to that obtained with the model without considering the flow. The reason is that the drop surface is only heated at edge, while the remaining surface is rather cooled. Hence, the presence of multi-cellular flow modifies locally the evaporation flux but not in a global manner for all drop surface. For low contact angles, the outward radial flow is very weak and has no influence on the evaporation. As a result, the quantification of the loss in the drop mass can be carried out just by a diffusion model. Hu and Larson [14] proposed a semi-empirical correlation to calculate the evaporation rate in the case of vapor diffusion without including the fluid flow and the drop cooling. This correlation is based on a contact angle function representing the wetting effect on the solid surface, and the concentration difference between the liquid-gas interface and the surrounding air, it is expressed by:
_ HL ¼ pRDð1 HaÞC v ðT 1 Þð0:27h2 þ 1:30Þ M
(b) Dimensionless evaporation rate Fig. 10. Drop evaporation rate versus the contact angle. Comparison between evaporation rates of the developed numerical model and those calculated with Hu and Larson relationship [14]. Dimensionless evaporation rate in (b) is obtained by division on the reference value: 2pRDðCV ðTCL Þ HaCV ðT1 ÞÞ where TCL is the contact line temperature. The wall temperature is of 25, 40 and 50 °C and air relative humidity of 0%, 40% and 100%.
ð21Þ
Fig. 10(a) gives a comparison between numerical results of the present study and those obtained with the correlation of Hu and Larson [14] based on the concentration difference DC ¼ C v ðT w Þ HaC v ðT 1 Þ. Different wall temperatures are considered and the relative humidity of air is equal to 40%. Overall, a good agreement is found between the compared results even when taking into account the effects of the drop cooling and the internal
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Fig. 11. Flow pattern when evaporating drop declined to a contact angle of 15° in both heated (Tw = 50 °C) and non-heated (Tw = 25 °C) cases and according to substrate thermal conductivity: (a) ks = 237 W/m K, (b) ks = 0.25 W/m K, (c) ks = 0.025 W/m K (Ha = 40%).
convective flow, particularly, at high contact angles (h 50°) and high wall temperatures, as for Tw = 50 °C. Indeed, this is explained by the opposite effect of the drop cooling inducing a decrease of the evaporation rate against the effect of liquid convective flow implying on contrary an increase of the evaporation rate. Furthermore, the evaporation rate is scaled with a reference mass transfer rate given by 2p RD DC and it is recomputed in a dimensionless form. This interesting result is illustrated in Fig. 10b and shows the superposition of all the curves corresponding to different values of wall temperature (25, 40 and 50 °C) and different values of the relative humidity of air varying between 0 and 100%. One may conclude that this dimensionless evaporation rate is mainly dependent on the contact angle. The reason is that each of the thermo-capillary internal flow and the drop cooling has separately an effect on the evaporation rate, but when combined, these effects counteract and are somehow balanced. 4.2. Evaporation on a non-isothermal solid surface The effect of a susbtrate of finite thickness (1 mm) and a given thermal conductivity yields non-uniform temperature on the upper wall of the substrate. This is what we call non-isothermal solid surface. The lower side of the substrate is maintaned at constant wall temperature Tw. Fig. 11 exhibits flow structures inside an evaporating drop for different substrate thermal conductivities. The contact angle is of 15° corresponding to the evaporation stage where the internal flow can undergo noticeable changes as a function of Tw and Ha. Decreasing the substrate thermal conductivity implies a structure change in the internal flow and a decrease of its magnitude. For a drop deposited on a substrate with a high thermal conductivity, for example Aluminum with ks = 237 W/m K, the upper side of the substrate remains isothermal. Hence, for a non-heated substrate (Tw = 25 °C), the positive temperature gradient along the drop surface yields a negative surface tension gradient along the liquid-gas interface and therefore, thermo-capillary effect induces a counter-clockwise 1-cell flow pattern. When the substrate is heated at Tw = 50 °C, a 3-cell flow
pattern settles in the drop. When the thermal conductivity of the substrate is smaller than that of Aluminum, like for PTFE substrate with ks = 0.25 W/m K, the thermo-capillary effect diminishes because the drop is poorly supplied with heat from the substrate. The resulting cooling of the drop leads to a change in the direction of the temperature gradient along the liquid-gas interface [63,64]. This sets up a local minimum of temperature on the drop surface close to the contact line and accordingly, thermo-capillary effect creates a bicellular flow pattern with a dominant cell in the core of the drop presenting a negative surface temperature gradient and a micro-cell at drop edge presenting a positive surface temperature gradient. When the substrate is heated at Tw = 50 °C, normally the heat supplied to the drop promotes the 3-cell flow pattern. However, the high thermal resistance of the substrate due to its small thermal conductivity (ks = 0.025 W/m K) weakens the drop heating and yields the settlement of a 2-cell flow pattern of reduced magnitude. Thus, when the substrate has a small thermal conductivity, it is opposed to the drop heating and the relative importance of each effect with respect to the other imposes the flow structure inside the drop at an evaporation stage where the drop is somewhat flattened with a contact angle between 10 and 20°. Later, towards the evaporation end (when h < 10°), the thermo-capillary effect is lessened and the flow becomes unicellular with a single counter-clockwise cell. Fig. 12 shows clearly that the evolution of the evaporation rate as a function of the contact angle is not influenced by the bicellular or tricellular internal flow, contrary to the unicellular thermocapillary flow at the earlier stage of evaporation where the involved convective effects induce an increase of the drop surface temperature and hence the evaporation rate. Cases of evaporation on substrates with a certain thermal resistance are illustrated in Fig. 12. The higher the thermal resistance is, the smaller the evaporation rate due to the increase of the drop cooling and the reduction of the internal flow magnitude. On an perfectly adiabatic substrate, presenting a theoretical case (ks = 0), the heat supplied to liquid phase is very weak so that the drop remains almost isothermal and cold. Consequently, the velocity inside the drop is
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(a) ks = 0.25 W/m K
null. The evaporation rate computed for this case fits well with the values given by the correlation of Hu and Larson [14] using the saturated vapor concentration at the contact line temperature. For a real case, a thermally insulating substrate, as for ks = 0.25 or 0.025 W/m K, plays an important role with regard to heat transferand therefore its upper wall is no longer isothermal. The temperature of the wetted area is less than that of the dry zone, which is itself lower than that of the bottom side of the substrate. Compared to the case of an isothermal solid surface, the drop is better cooled with a more uniform temperature distribution at the drop surface involving smaller thermo-capillary effect. In this case, it is expected to obtain evaporation rates close to those calculated by the correlation of Hu and Larson [14] based on saturated vapor concentration evaluated at contact line temperature TCL. Whereas, the results are different if the saturated vapor concentration in the correlation is evaluated at the temperature of the bottom wall of the substrate Tw, which is higher than that of the contact line. With regard to the evaporation rate, we may say that it is strongly dependent on the contact angle (h), the saturated vapor concentration at the contact line (Cv(TCL)) and the ambiant vapor concentration (Ha Cv(T1)). An increase of the thermal resistance of the substrate results in a better cooling of the drop, a more uniform temperature distribution as well as a decrease in the flow magnitude. Whereas, heating the drop yields an increase of the internal flow magnitude that results in heating the drop surface, which compensate the cooling effect due to increasing evaporation.
4.3. Flow pattern map
(b) ks = 0.025W/m K
(c) Adiabatic substrate Fig. 12. Drop evaporation rate versus the contact angle. Comparison between results obtained with the developed numerical model and those obtained with the correlation of Hu and Larson [14] using the temperature imposed at lower face of the substrate (Tw) or the temperature of the contact line (TCL). The water drop evaporates on substrates of 1 mm thickness and different thermal conductivities: (a) ks = 0.25 W/m K, (b) ks = 0.025 W/m K and (c) adiabatic substrate. The correla_ ¼ 2pRDðCV ðTÞ HaCV ðT1 ÞÞð0:27h2 þ 1:3Þ, tion of Hu and Larson is expressed by: M where Cv is the concentration of the saturated vapor.
Different flow patterns are found during the evaporation of the drop on a solid surface, which is either isothermal or nonisothermal. They are illustrated in Fig. 13 in addition to the representation of the dimensionless temperature profile at the drop surface. The internal flow pattern could be due to a prevailing single effect or combined effects with relatively the same order among which the thermo-capillarity due to the temperature gradient along the liquid-gas interface, and the non-uniform evaporation at the drop surface due to concentration gradient. The one counterclockwise cell (CC) starts with evaporation and induces convective effects that decrease in time as the cooling effect at the drop surface is opposed to the heating from below. The flow pattern may evolve to a bi-cellular (C-CC) or tri-cellular convection (CC-C-CC) because of significant heating of the drop at a stage of the evaporation when the drop height reduces. For the case of a weak heating, dehumidifying the surrounding air or increasing the substrate thermal resistance can yield a bi-cellular flow pattern (C-CC). Towards the evaporation end, thermo-capillary effect becomes negligible and the liquid flow at very weak velocity is driven radially outward by the important evaporation near the contact line. Flow pattern maps are plotted in Fig. 14 based on implicated study parameters: drop geometry (drop apex height h0), wall temperature (Tw), and substrate thermal conductivity (ks). Air relative humidity is of a moderate value of 40%. The influence of the thermal properties of the substrate is taken into account in the dimensionless term of concentration difference between drop surface and ambient air, given by: [Cv(TCL)/Cv(T1)Ha]. Cv is the saturated vapor concentration and TCL is the temperature of the contact line. This flow map is calculated for the case of an isothermal solid surface (substrate with high thermal conductivity, ks?1) and the case of PTFE substrate (ks = 0.25 W/m K). The drop life time throughout the apex height variation is divided on two, three, four or five phases; each phase corresponds to a given flow pattern.
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i) 1-cell flow (CC)
ii) Tri-cellular flow (CC-C-CC)
iii) 2-cell flow (C-CC) due to Ha effect
iv) 2-cell flow (C-CC) due to ks effect
497
v) Outwards radial flow (Rad) Fig. 13. Dimensionless temperature profile at drop surface and corresponding flow pattern appearing during evaporation for different conditions of wall temperature, substrate thermal conductivity and air relative humidity. The importance of the convective effects in the drop is marked by the difference in surface temperature profile of the model with internal flow compared to the model without internal flow.
For the case of an isothermal solid surface (Fig. 14a) and when the dimensionless concentration difference is small (lower than 1.15, corresponding to the case of solid surface not heated sufficiently or the case of high values of air relative humidity), only
two flow structures exist during drop evaporation. In the first one, only one counterclockwise cell is found and this is due the positive temperature gradient where the coldest point is located at drop apex. This flow pattern starts from drop evaporation
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(a) Isothermal solid surface
directed radially outwards at evaporation end. For high values of dimensionless concentration difference (>1.65), when the solid is more heated or air relative humidity is very weak, five phases exist during evaporation. From drop evaporation beginning, the flow is unicellular and counterclockwise. For smaller values of the drop height, three alternated cells (counterclockwise-clockwise-counter clockwise) are present within the drop as result of a nonmonotonic temperature profile installed at drop surface where three different gradient exist from drop apex to the edge (positive-negative-positive). Then a two cells flow pattern (Clockwise-Counterclockwise) appears, followed by unicellular flow and the radial flow. In the case of non-isothermal solid surface, the effect of the thermal resistance of substrate is introduced, as for the example PTFE substrate given in Fig. 14b. In this case, for any value of the dimensionless concentration difference, the flow structure at evaporation beginning is one counterclockwise cell. But after this phase, it can appear either directly a bi-cellular structure (C-CC) for values of dimensionless concentration difference lower than 1.1, or a tricellular structure followed by a bi-cellular structure for values of dimensionless concentration difference higher than 1.1. In these two cases, when the bi-cellular structure appeared, the coldest point is neither at drop apex nor at the edge. Contrarily to the isothermal solid surface case where the coldest point moves to the drop top, here the coldest point moves to the drop edge. This is due to the strong cooling resulting from phase change near the contact line which overcomes heating from the thermal insulating substrate. The displacement of the coldest point continues until reaching drop edge where after a large clockwise cell dominates within the drop.
5. Conclusion
(b) PTFE substrate Fig. 14. Flow pattern maps based on drop geometry, wall temperature and substrate thermal conductivity. The case (a) corresponds to the evaporation on an isothermal solid surface and the case (b) to the evaporation on a non-isothermal solid surface which is the upper face of the PTFE substrate. Air relative humidity is of 40%. The flow patterns are noted: 1-cell flow (CC), Tri-cellular flow (CC-C-CC), 2-cell flow (C-CC), -outward radial flow (Rad). CC means Counter-Clockwise cell and C Clockwise cell.
beginning and endures till end of evaporation where the privileged mass loss at contact line induces a radial flow within the drop. For dimensionless concentration differences between 1.15 and 1.65, the flow structure is one counterclockwise cell at evaporation beginning, and then a phase of bi-cellular flow appears. This is due to temperature profile at drop surface, where the coldest point is no more at drop apex. Thus two temperature gradients are present at drop surface. The first is negative creating a clockwise flow, and the second is positive giving the counterclockwise cell. After this phase, the flow becomes again one counterclockwise cell, then
A generalized formulation is used to study the flow patterns inside an evaporating sessile drop deposited on either an isothermal or a non-isothermal solid surface under heating or nonheating conditions. The governing equations, Navier-Stokes, energy and mass diffusion equations, are solved by means of a numerical approach and developing a CFD tool. The implemented computer program takes into account the interactions between the different phases (solid, liquid and gas) and includes all the driving effects on the liquid flow inside the drop, namely: thermocapillarity, thermal buoyancy and non-uniform evaporation at the drop surface. Results are obtained for different values of the heating temperature, the relative humidity of the surrounding air and the thermal conductivity of the substrate. The numerical predictions showed that the thermal buoyancy effect is negligible and the thermo-capillarity governs mainly the flow pattern within the drop during evaporation, except towards the end where the effect of the strong evaporation near the contact line becomes prevailing. Different flow patterns are found, particularly, the unicellular flow, which becomes radial outwards at the end of evaporation when the drop is much flattened. Multicellular flow pattern of BénardMarangoni is induced when the drop height is moderately reduced and when the drop is subject to a significant heating. The same flow pattern is predicted under weak heating but strictly at low values of air relative humidity. Substrate thermal resistance tends to reduce the drop temperature while making it of more uniform distribution. Consequently, the unicellular flow can evolve to a bi-cellular flow when the drop is sufficiently flattened. Flow pattern maps are elaborated based on the dimensionless height of the drop apex (h0/R) and the dimensionless concentration difference [Cv(TCL)/Cv(T1) Ha], regrouping all study parameters: Tw, Ha and ks. The results of evaporation rate are almost identical to those obtained with the mass diffusion model because of the
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