Chapter 5. Evaporation of sessile binary droplet on a heated spherical particle
5.2.6. Internal motion within droplet
As discussed in section 5.2.5, internal convections play a key role on the droplet evaporation, a brief analysis using various time scales is presented in this section to evaluate
Chapter 5 164 the significance of internal convections in the droplet on the evaporation phenomenon. Tem- perature difference between the solid surface and droplet induces a surface tension gradient along the liquid-vapour interface wherein surface tension is minimum at the three phase con- tact line and maximum at the apex of droplet. This gradient induces Marangoni flow along the interface from minimum to maximum surface tension region. The temperature gradient also induces a buoyancy driven Rayleigh flow from low density region near the solid-liquid interface to high density region in the liquid bulk. Both flows contribute to internal motion significantly.
The resulting convection time scales are thermal Marangoni convection
(τM =R0 /uM) where uMis the characteristic Marangoni velocity and Rayleigh convection
(τB = R0 /uB) where uB is the characteristic buoyancy velocity. These two characteristic time scales (Tam et al., 2009) are compared with the characteristic thermal diffusion time
(τdiff =ρLCp L, R02 /kL) to evaluate the effect of the internal motions on evaporation. Theoret-
ically, if the ratio (τdiff /τM)and (τdiff /τB)are both close to unity, then contribution from convection component is negligible, otherwise a greater than unity ratio would indicate dom- inant convection. It is worth mentioning that the ratios can also be expressed as
/ ; /
diff M MaT diff B Ra
τ τ ≡ τ τ ≡ and τ τB / M ≡ MaT /Ra where MaT is thermal Marangoni number and Ra is Rayleigh number and defined as:
, 0
L p L L
T
L L
C R T
Ma T k
σ ρ
à
∂ ∆
= − ∂ (5.6)
L M
L
d T
u dT
σ à
= ∆ (5.7)
where ∂σL /∂Tis surface tension variation with respect to temperature;ρL,Cp L, , àLand kL
Chapter 5 165 are density, heat capacity, dynamic viscosity and, thermal conductivity of the liquid phase, respectively; ∆Tis the temperature gradient in the liquid droplet which is the average initial temperature difference between pos. (2) and pos. (4) as shown in Figure 5.11 a.
Rayleigh number and the corresponding characteristic velocity can be described by the fol- lowing equations:
2 3
, 0
L p L
T
L L
gC R T
Ra k
β ρ
à
= ∆ (5.8)
2 0 L
B T
L
gR T
u β ρ
à
= ∆ (5.9)
where βT is thermal expansion coefficient of the liquid phase.
Figure 5.22 Reduction in normalised droplet volume with normalised time for three different binary mixtrue droplets compared with pure water droplet, at the same surface temperature at 323 K. Operating conditions: dp = 10 mm, Ta = 296 K, relative humidity RH =
~ 50 %;
Chapter 5 166 More reduction in evaporation rate of binary mixture droplets compared with single component droplet (at the same substrate temperature) shown in Figure 5.22 specifies the effects of solutal Marangoni flows which only take place for the mixture of different surface tension compositions. In this figure, droplet volume (V) is normalised by its initial volume (V0) while time is normalised by the time (t90) at which droplet volume has reduced 90 % (normalisation presented due to the difference in initial droplet volumes; the curves coincide if evaporation rate remains same.
Table 5.5 shows some quantifications of internal motions in which, dimensionless pa- rameters were calculated for case ‘gly10’ using the physical properties of the liquid phase evaluated at the average temperature given asT =(TS +Td0)/ 2.
Table 5.5 Parameters evaluating the convection inside droplet
Ts
(K)
∆T
(K) uM
(m/s) uB
(m/s) τM
(s) τB
(s) τdiff
(s)
diff M
τ τ
diff B
τ τ
B M
τ τ 323 2.51 0.46 0.016 0.0030 0.087 14.58 4869 168 29.15 343 3.78 0.69 0.023 0.0020 0.058 14.58 7354 253 29.07 358 3.95 0.78 0.027 0.0018 0.051 14.46 8236 283 29.10
It can be seen from Table 5.5 that the ratio (τdiff /τM)and (τdiff /τB)are indeed very large compared to unity which indicates the significance of these two motions driven by surface tension and density, respectively. Secondly, the ratio (τB /τM) is much greater than unity which confirms the contribution of Marangoni convection being dominant in the system. Us- ing the operating conditions, Marangoni numbers were obtained in the range of ~ 4869 to 8236 which were of the same order of magnitude recently reported in Chen et al. (2017) on the significant contribution of Marangoni flow on the droplet evaporation rate.
Chapter 5 167 In other cases, temperature difference ∆T for the pure water “gly00” droplet was found to be ~ 3.2 (±0.4) K (at surface temperature of 323 K) which resulted in thermal Ma- rangoni number of ~ 4679, and Rayleigh number of ~ 199. For the “gly35” case, ∆T was measured as ~ 4.1±0.5 K (at surface temperature of 323 K) and the corresponding thermal Ma number was ~ 5539 (much higher than the corresponding Rayleigh number ~ 218). The com- parisons confirm that the natural convection due to density difference (characterised by Ray- leigh number) is insignificant compared with the thermal Marangoni flow.
Another contribution to internal motions comes from the solutal Marangoni flows which is induced by the surface tension gradient due to concentration difference at the inter- face for the binary mixture systems. Solutal Marangoni number (MaS) is given as:
0 L S
L L
R c
Ma c D
σ à
∂ ∆
= − ∂ (10)
where ∂σL /∂cis the variation in surface tension due to concentration difference, DLis the diffusion coefficient of the liquid phase and ∆cis the concentration gradient at the droplet interface.
A comparison of the change in surface tension due to droplet temperature (∆σtemp) and glyc- erol concentration (∆σconc) is given in Table 5.6 which provides some insight to the approxi- mate effect of these two Marangoni numbers.
Table 5.6 Reduction in surface tensionsubject to temperature and glycerol concentration Glycerol
(Cwt%)
(2.5 )
temp K
σ
∆
mN/m MaTmin
(4.1 )
temp K
σ
∆
mN/m Tmax
Ma ∆σconc mN/m
,max
Mas
0 0.47 4,679 0.78 7,644 0 0
10 0.46 4,869 0.76 7,954 0.37 502,926
15 0.45 5,040 0.75 8,232 0.56 888,833
Chapter 5 168
20 0.44 4,677 0.74 7,639 0.75 1,091,654
25 0.44 4,272 0.73 6,978 0.98 1,246,507
35 0.43 3,391 0.72 5,539 1.43 1,412,473
From the measured temperature data, minimum and maximum temperature difference were found to be ~2.51 and 4.1 K (for the particle surface temperature of 323 K), respectively which were used to determine ∆σtemp and thermal Marangoni number. Initial concentrations of glycerol were used to determine ∆σconc and maximum solutal Marangoni number. Based on the surface tension difference given in Table 6, it is apparent that temperature effect is domi- nant when glycerol concentration is less than 10% (or “gly10” droplet), however at higher glycerol concentration cases, surface tension difference due to solute concentration is more significant. The corresponding maximum solutal Ma number (Mas,max) is at least two orders of magnitude larger than the thermal Ma number (MaT,max) at maximum temperature differ- ence case. The estimation suggests that solutal Ma flows should be considered in droplet evaporation at higher solute (glycerol) concentration cases. However, in the absence of any direct measurement of the solute concentration at the interface, actual contribution of the so- lutal Marangoni effect on the evaporation rate could not be quantified.