Vaporisation of droplets on a substrate is important to many applications including spray drying, granulation in spouted bed (Mann, 1983), or particle coating (Lech et al., 1997), etc. In pharmaceutical industries, a rotating drum drier is often used for spray coating of drug substrates wherein the particles are pre-heated to ~ 50°C before interacting with atomized liquid droplets containing the coating material, e.g. polymer in a solvent. In these real situa- tions, the liquid droplet(s) are in contact with solid particles and heat transfer takes place un- der a low temperature condition. This section briefly reviews the previous studies on the va- porisation of a droplet deposited on a substrate heated below saturation conditions.
A number of studies have been conducted to investigate the evaporation rate of sessile droplets deposited on flat substrate under room temperature conditions, (Dash & Garimella, 2013; Hu & Larson, 2002, 2005; Nguyen & Nguyen, 2012; Nguyen et al., 2012; Picknett &
Bexon, 1977; Rowan et al., 1995), which is reflected in the review work of Erbil (2012). Va- porisation of an assumed spherical-cap droplet resting on a substrate can be described using the mass transfer equation (vaporisation rate) at the liquid-vapour interface the energy equa-
Chapter 2 30 tion at the solid-liquid interface, e.g. sensible heat equals to the difference between heat gained from conduction and heat loss due to evaporation. Droplet evaporation rate ( )md can be obtained using Fick’s law and assuming thermodynamic vapour-liquid equilibrium exists at the droplet interface and radially outward vapour flux. The resultant expression formdis given as (full derivations can be seen in the Appendix B):
2 ln 1
d t 1
S
m h D C
π −C∞
= − −
(2.4)
where D is binary diffusion coefficient of the vapour; C∞is the vapour concentration far from the liquid-vapour interface and CSis the vapour concentration which is a function of tempera- ture at the liquid-vapour interface obtained from the energy equation; htis the height of the assumable spherical cap.
Hu and Larson (2002) carried out both experimental and numerical computations us- ing a finite element method to investigate the evaporation behaviour of a sessile droplet on a hydrophilic surface (contact angle less than 90°) with a pinned contact area. Evaporation flux was shown to increase from the top of the droplet to the contact line where the contact angle decreases with time due to evaporation. A simple approximation for droplet evaporation was obtained as a function of contact angle which is given as:
(1 ) (0.27 2 1.30)
d S S t
m = −πR DC −RH θ + (2.5)
where RH is relative humidity of the surrounding air and Cs is the vapour concentration at the liquid vapour interface and θtis the apparent contact angle. It is noted that Eq. (2.5) applies only for room temperature of the substrate (and θ less than 90°).
Chapter 2 31 Picknett and Bexon (1977) and Erbil (2012) pointed out that the suspended spherical droplet would evaporate faster than a droplet deposited on a non-wetting substrate (contact angle ~ 180°) due to presence of the substrate wall preventing the vapour diffusing down- ward. However, in the study of Hu and Larson (2002), the evaporation flux was found to be much larger at the three-phase contact line compared with the top of the droplet. For a heated substrate, Marangoni flows driven by surface tension difference due to inhomogeneous tem- perature distribution over the droplet surface may also accelerate the droplet evaporation rate (P. Chen et al., 2017; Diddens et al., 2017).
When heat is supplied underneath the liquid drop at the wetted contact area, the ther- mo-capillary or Marangoni effect take place (see section 2.4) that has impact on droplet evaporation (P. Chen et al., 2017; Diddens et al., 2017; Erbil, 2012; Hu & Larson, 2006). P.
Chen et al. (2017) utilised the result proposed by Hu and Larson (2002) as a base case [Eq. (2.5)] to investigate the effects of Marangoni convections on the droplet evaporation un- der heated conditions. The evaporation rate accounting for Marangoni flows was reported as
(1 ) (0.27 2 1.30 1.4) ( 04 1.0)
d S S t
m = −πR DC −RH θ + e− Ma+ (2.6)
where Ma is thermal Marangoni number which was calculated based on the temperature dif- ference measured at the droplet interface (infrared camera used in the study of Chen et al., 2017);
The apparent contact angle, θt, appearing in Eq. (2.5) and (2.6) varies with time dur- ing the evaporation. The contact angle and wetted area represents the spreading which con- trols the subsequent evaporation of the deposited droplet. In previously reported studies, the spreading behaviour of droplet on solid surface and subsequent (non-boiling) evaporation have been investigated mostly for two different solid surface temperature cases, namely at:
Chapter 2 32 (1) room temperature (Nguyen et al., 2012; Nguyen and Nguyen, 2012; Dash and Garimela, 2013); and (2) below droplet saturation temperature. Below the saturation temperature, solid surface temperature is an additional consideration for the wettability or contact angle of a droplet on solid substrate. The experimental work of Newmann et al. (1971) investigated temperature dependence of the contact angle for long-chain alkane (C10 – C16) droplets on a PTFE surface. It was noted that the contact angle remained unchanged over the surface tem- perature range from 20 to 80 oC for alkanes involving dodecane (43o) to hexadecane (46o).
Consequently, it was reasoned that no significant adsorption of long-chain alkanes occurred on the PTFE surface. In a later study, Neumann et al. (1974) noted a slight increase in the contact angle from 60 to 62o for a glycerol droplet on a vinyl chloride-vinyl acetate copoly- mer surface over a temperature range of ~20 to 45 oC; whilst a reduction in contact angle from 82o to 79o was observed for a water droplet under the same conditions. Di Marzo and Evans (1986) found experimentally that the contact angle of a water droplet from ~ 90o to 60o with increasing surface (aluminium) temperature.
A summary of the reported apparent contact angles of the different liquids on various flat substrates measured over a range of temperature is presented in Table 2.5 which shows an average contact angle of ~ 82o for water droplet deposited on a metal substrate heated under saturation temperature.
Table 2.5 Experimental contact angle of water droplet on metal surface at different tempera- ture.
Ref Surface Surface prepa-
ration
Solid temp oC
Apparent contact angle Bourges-Monnier and
Shanahan (1995) stainless steel polish epoxy
surface 25 61
Bernardin et al. (1997) aluminium polished 40 88
Chapter 2 33 Di Marzo and Evans
(1986) aluminium non polish sur-
face 75 92
Chandra and Avedisian
(1991a) stainless steel - 80 87.5
Di Marzo and Evans
(1986) aluminium non polish sur-
face 100 60
Boyes and Ponter (1973) copper sanded and pol-
ished 100.8 72
Erb (1965) gold commercial
rolled 101 85
Erb (1965) silver heated to 600 oC 101 85
Erb (1965) stainless steel plated on stain-
less steel 101 89
Erb (1965) rhodium plated on silver 101 65
Erb (1965) gold electro polished 101 63
Bernardin et al. (1997) aluminium polished < 100 ~ 90
Crafton and Black (2004) aluminium - 60 103
Crafton and Black (2004) copper - 60 112
Misyura (2017) copper structured wavy
shape 67-74 75-95
Present study brass - 50 76
Present study brass - 70 85
Present study Brass - 85 85.3
The two different modes of evaporation for a droplet deposited on a substrate (Picknett & Bexon, 1977) has been well investigated in the literature: (1) at a constant wet- ting radius and varying contact angle (pinning or CCR mode) and (2) at constant contact an- gle and varying wetting radius (depinning or CCA mode). The pinning mode of evaporation has been observed in many experimental studies in which the contact radius remains un- changed during most of the evaporation whist contact angle reduces with time according to the reduction in droplet volume due to evaporation (Chandra et al., 1996; Di Marzo & Evans, 1986; Hu & Larson, 2002; Liu et al., 2008; Nguyen et al., 2012). For example, Chandra et al.
(1996) and Di Marzo and Evans (1986) reported invariance in the wetted contact area (CCR
Chapter 2 34 mode) up to 95 % of the droplet lifetime. However, Sefiane et al. (2003) found three different stages of the contact behaviour of a water-ethanol droplet during evaporation: the CCR mode occurred only in a very short time followed by a complex mode and ended in a CCA mode that was longest in the entire droplet lifetime. The experimental study of Nguyen et al. (2012) found two different modes of evaporation: CCR mode (water droplet on Oct–silicon, OTS–
silicon, and Teflon flat surface) occurred only less than half of the total evaporation time fol- lowed by a CCA mode lasting for a longer time until the end. Therefore, the duration of pin- ning mode appears to be dependent on a variety of parameters such as physical properties and composition of the liquid phase, surface properties and temperature of the solid phase.
The above review of previous work on the vaporisation of sessile droplets at low tem- perature identifies the necessity of further studies on the vaporization of a binary mixture droplet deposited on a spherical surface. Specifically, the important aspects of droplet evapo- ration such as the effect of solid surface temperature on contact angle, wetted radius and the transient increase in the droplet temperature therefore also remain largely unexplored. Details of the work which address these specific aims are discussed in Chapter 5.
It is worth to notice that the evaporation of sessile droplet mentioned above has been examined in a macro level. However, heat and mass transfer flux at the three-phase contact line was found significant compared with entire macro droplet (Hu & Larson, 2002). There- fore, attempt was made to investigating the evaporation at the triple line (micro level) which can be seen Appendix F.