In the previous section, evaporation behaviour of non-polar homologous series binary hydrocarbon droplet was discussed. In this section, evaporation behaviour of polar binary mixtures comprising varying compositions of water and glycerol is analysed. Operating con- ditions of the three cases studied are listed in Table 4.3.
Table 4.3 Operating conditions of the water-glycerol systems
Case Reference Component (wt %) d0
(mm) TG
(K) pG
(kPa) Td0
(K) Red0 u∞
(m/s) 4 Davies et al.
(2012)
Water 82.5 %-
glycerol 17.5 % 0.042 298 1.01 298 1.07 0.4 5 This study Water 100 %-
glycerol 0 % 2.61 353 1.26 310 714 4.3 6 This study Water 70 %-
glycerol 30 % 2.61 353 1.26 317 708 4.3
a) Reduction in droplet size
For polar binary system, the proposed evaporation model was first validated with the published experimental data of Davies et al. (2012) μm sized water-glycerol mixture droplet with the operating conditions of Case 5. Figure 4.9 illustrates the temporal variation in drop- let diameter for this system. Clearly, a distinct transition point could be observed at t ~ 1.2 second due to the very different boiling points of the two species (373 K for water and 563 K for glycerol). Both model prediction and experimental data were in close agreement before transition point however some discrepancy is apparent afterwards (NRMSD ~ 0.041). Alt- hough the model used by Davies et al. (2012) provides good agreement with own experi-
Chapter 4 100 mental data (NRMSD ~ 0.017), it lacks the energy balance equation hence cannot be used for predicting droplet temperature.
Figure 4.9 Model prediction of temporal change of droplet size and temperature of water- glycerol mixture droplet (82.5 % water and 17.5 % glycerol), validated against experimental data of Davies et al. (2012). Operating conditions of Case 4: d0 = 0.042 mm; TG = 298 K; Td0
= 298 K; Red0 = 1.07.
Experiments in the present study were conducted at relatively high Reynolds number, Red0 = 708 for Case 5 and Red0 = 714 for Case 6. Reynolds number was based on the initial droplet diameter with an average relative velocity of 4.3 m/s which was determined from six time-averaged readings from the manometer with a standard deviation of ~ 0.3 m/s.
Figure 4.10 shows both measured and model predictions of the droplet diameter varia- tion with time under the operating conditions of Case 5. Droplet diameter decreases from ~ 2.61 mm to 1.65 mm within the first 160 s. Corresponding high-speed visualizations of drop- let size reduction are shown in the inset. This time series indicates slight deviation from
Chapter 4 101 spherical shape due to droplet size being close to the capillary length of ~ 2.7 mm. Further, due to significant contribution shear flow early in the evaporation lifetime, inertial force dom- inates over the surface tension force which also contributes to deviation from spherical shape.
However at later stage (t = 134 s onwards) when droplet size substantially shrinks due to evaporation of most of the high volatile component (water), spherical shape is regained due to dominance of surface tension force. The trend of the measured d2(t) (not shown) in this duration was found linear with a correlation coefficient R2 of 0.9979 in spite of highly con- vective condition. Overall, the model prediction had a good agreement with experimental da- ta with NRMSD of 0.07.
Chapter 4 102 Figure 4.10 Comparison of model predicted droplet diameter reduction for pure water system with present experimental data. (Error: 0.073 – 0.104 mm in 95% confidence interval). Inset plot shows high-speed visualizations of droplet size reduction. Operating conditions of Case
5: d0 = 2.61 mm; TG = 353 K; Td0 = 310 K; Red0 = 714.
Evaporation of a water-glycerol mixture with operating conditions corresponding to Case 6 (70 % water and 30 % glycerol) is described in Figure 4.11 which shows the predicted change in droplet diameter (Figure 4.11a) and liquid mole fraction of the two species (Figure 4.11b) with time. Three different evaporation stages could be noted here - (i) evaporation of water (high volatile component) that occurs from t = 0 - 130 seconds with rapid change in droplet diameter (ii) the transition stage of approximately 30 s that occurs within 130-160 seconds wherein mole fraction of water reduces to zero while glycerol (low volatile compo- nent) mole fraction reaches unity and (iii) evaporation of glycerol at the last stage when water completely disappears from the mixture and the droplet diameter reduces insignificantly with
4.10
Chapter 4 103 time due to very low vapour pressure of glycerol.
Figure 4.11 Model prediction for temporal droplet (70 % water and 30 % glycerol) diameter reduction validated against present experimental data (a); and predicted change in species mass fraction with time (b). (Error: 0.013 to 0.05 mm in 95% confidence interval); inset plot
shows a complete evaporation. Operating conditions of Case 6: d0 = 2.61 mm; TG = 353 K;
Td0 = 317 K; Red0 = 708.
Comparison of model prediction with experimentally measured droplet diameter in Figure 4.11a indicates some deviations towards the end of transition period at ~ 125 s. Re- duction in droplet diameter is explained in terms of mole fraction of the two species in Figure 4.11b which shows a declining trend of the water mole fraction with time reaching almost zero concentration at ~ 160 s. It could be noted that actual evaporation rate is slower than the model prediction which is attributed to the increasing diffusional resistance in the liquid mix-
4.10
Chapter 4 104 ture not accounted in the present modelling framework.
This can be explained by a simple time scale analysis. The mass diffusion characteristic time, for water-glycerol mixture was found to be ~ 391 s which was significantly higher than the thermal diffusion time 5.3 s (see Table 4.5). It is quite clear that mass diffusion time is the slowest step that governs the evaporation of the residual more volatile content (water) due to the fact that liquid phase diffusion coefficient varies inversely with the system viscosity which is significantly high (àglycerol ~ 0.27 Pa.s at Td0 = 317 K) in the remaining mixture. A similar observation was also reported for methanol-water droplet system in Sefiane et al.
(2008) wherein a small amount of methanol (more volatile component) remained in the mix- ture even after the first stage of evaporation and evaporated at a slower rate. Nevertheless, the mass diffusional time of ~ 391 s is insignificant compared with the total evaporation time of
~ 54 hours (the inset of Figure 4.11a) which could not be validated due to the limited record- ing capability of the camera. Broadly, except the transition regime which constitutes a small fraction of the complete evaporation life time, the proposed model produced reasonable agreement with the experimental data. Although not experimentally verified, the model pre- diction is expected to match glycerol evaporation data as well which can be deduced from the reduction trend of droplet size and increasing droplet temperature trend. A detailed time scale analysis is presented later in section 4.4.
Chapter 4 105 Figure 4.12 Temporal variation of evaporation rate for a) water species and b) glycerol spe- cies in the binary mixture droplet (70 % and 30 % glycerol). Operating conditions of Case 6:
d0 = 2.61 mm; TG = 353 K; Td0 = 317 K; Red0 = 708.
Figure 4.12 presents model predicted evaporation rate of individual species in the bina- ry system of water-glycerol given in Case 6 (TG = 353 K, Red0 = 708). Water evaporation rate drastically decreases within the first 10 seconds and then decreases gradually to zero at ~ 160 s. Glycerol evaporation rate remains close to zero up to ~ 140 s and then gradually increases in the transition zone and finally reaches a steady state. At these operating conditions, an av- erage evaporation rate for water species is predicted as ~ 0.5 × 10-7 kg/s which is three order of magnitudes higher than that for the glycerol component.
The modelling results for water-glycerol mixtures in this study (Figure 4.9, Figure 4.11 and Figure 4.12) were obtained by including non-unity activity coefficient. Figure 4.13 (Case 6) shows slight deviation in the predicted diameter obtained from the ideal and non-ideal as-
Chapter 4 106 sumption. Estimated activity coefficients for the water and glycerol mixture using Eq. (4.8) to (4.9) indicate negative deviation from the Raoult’s law (Figure 4.13b) which was consistent with the values reported in Soujanya et al. (2010) for water-glycerol system. Deviation from the ideal solution behaviour occurs in the middle of the evaporation process from t ~ 120 to 160 s wherein concentration of the species water and glycerol are both significant in the mix- ture (mole fraction of both species ~ 0.5, see Figure 4.11b). Apparently, inclusion of non- unity activity coefficients for the water-glycerol system in Case 6 produces larger deviation in prediction (SD ~ 0.0023 for diameter and 0.7 for temperature) leads to a larger change in the predicted evaporation rate compared with the heptane-decane system in Case 2 (SD ~ 0.0008 for diameter and 0.024 for temperature). This is because both water and glycerol are polar in nature and have strong hydrogen bond which does not exist in the non-polar hydro- carbons.
Figure 4.13 Transient change in droplet diameter predicted using ideal and non-ideal assump- tions. A max. standard deviation of 0.0023 mm (more visible in the inset) is the difference
Chapter 4 107 between these two assumptions (a), variation in activity coefficient of each species in the liq-
uid mixture (b). Operating conditions of Case 6: d0 = 2.61 mm; TG = 353 K; Td0 = 317 K;
Red0 = 708.
Although, inclusion of non-ideal behaviour for the water-glycerol system shows higher relative deviation compared to the heptane-decane system, it does not affect the predicted droplet size and temperature profile significantly when compared with the prediction based on ideal solution assumption. Insignificant role of non-unity activity coefficients on evapora- tion rate can be theoretically explained as follows: (4.1) relates the evaporation rate mito Spalding mass transfer number BM i, which in turn is connected to vapour mole fraction at the droplet interface χs i, [Eq. (4.3)] hence activity coefficient [Eq. (4.6)]. Although in theory, any change in activity coefficient, should directly affect the evaporation rate, a careful look at Eq. (4.1) reveals this contribution is relatively insignificant compared to the Sh number which increases with Reynolds number. It can therefore be concluded that the effect of activity coef- ficient is less significant in heat transfer dominated evaporation. It would be relevant to men- tion that negligible influence of the non-unity activity coefficient on the evolution of droplet temperature was also reported in the study of Sazhin et al. (2010) for a slightly polar system comprising acetone-ethanol mixture.
Chapter 4 108 Figure 4.14 Model prediction in temporal droplet size and temperature of pure glycerol drop- let evaporation. Inset plot shows the initial short heating period. Operating conditions of Case
6: d0 = 2.61 mm; TG = 353 K; Td0 = 317 K; Red0 = 708.
Figure 4.14 presents the evaporation performance of a pure glycerol droplet under the same operating conditions as Case 6 (TG = 353 K; PG = 126325 Pa; d0 = 2.61 mm; Td0 = 316.8 K; u∞ = 4.32 m/s). In this case, no cooling phenomenon was found due to the low latent heat of liquid glycerol as described. The droplet temperature reached thermal equilibrium with ambient gas within the first 10 seconds (see inset) although the droplet lifetime was pre- dicted to be very long (~ 94 hours). Note that the droplet size reduction in ~ 54 hours in Case 6 was computed using two different time steps. A lower time step of 0.2 seconds was used for the first 400-second period within which the mole fraction high volatile species (water) in the droplet reached ~ 0.001 and afterwards a time step of 2 seconds was used for the rest of the
Chapter 4 109 remaining droplet life time. Evidently, the droplet lifetime in Case 6 falls between the evapo- ration time of pure water droplet (289 s) and pure glycerol droplet (94 hours) at the same op- erating conditions.
b) Variation in droplet temperature
Figure 4.15 presents experimentally measured droplet temperature (local temperature) profile for case 5 (100 % water and 0 % glycerol) at flowing gas temperature of 353 K. The measured droplet temperature profile shows an initial cooling process of approximately 20 seconds that leads to temperature reduction of ~ 5.6 K. Corresponding model prediction is 13 seconds for the initial cooling stage and temperature decrease is 4.1 K. The predicted temper- ature profile has the same trend with measurement, involving a cooling and then heating stages and has reasonable agreement (NRMSD ~ 0.18) with experiment. A re-heating stage follows next leading temperature increase up to 3.5 K which is greater than the model predic- tion of 0.5 K. Deviation could be attributed to possible effect of heat conduction from the nozzle and the thermocouple not accounted in the model.
Chapter 4 110 Figure 4.15 Comparison of model predicted transient droplet (pure water) temperature profile with present experimental data (a); and heat ratio interpreting the change in droplet tempera-
ture (b). Operating conditions of Case 5: d0 = 2.61 mm; TG = 353 K; Td0 = 310 K; Red0 = 714. (Error: 0.07 – 1.57 K in 95% confidence interval).
The heat ratio expressed as LV∑md i, πdGNukG(TG −Td) - ratio of heat loss to heat gain - decreases from 1.45 down to 1.00 (unity) that results in a noticeable decrease in droplet temperature from 310 K to 304.4 K in the first cooling stage. Afterwards, this ratio insignifi- cantly changes to 0.996 resulting in a slight increase in droplet temperature from 304.4 K to 306.3 K. Due to large latent heat of vaporization in this case, the heat loss term therefore dominates and explains the observed cooling period.
Chapter 4 111 Figure 4.16 Model prediction of transient droplet (70 % water and 30 % glycerol) tempera- ture validated against present experimental data (a); and heat ratio interpreting the change in
droplet temperature (b). Operating conditions of Case 6: d0 = 2.61 mm; TG = 353 K; Td0 = 317 K; Red0 = 708. (Error: 0.09 – 4.95 K in 95% confidence interval).
Figure 4.16 presents droplet temperature profile and corresponding heat ratio for Case 6 (70 % water and 30 % glycerol). The experimental data showed similar temperature profile to that of Case 5 (pure water) comprising three regimes i.e. a short temperature reduction period due to cooling effect, a slight increase thereafter up to the transition period, and a rapid rise afterward finally reaching equilibrium at 386 s. Although following a similar profile to the experiment, the predicted temperature changed faster than the actual one, as was the case with evaporation rate (Figure 4.11a); this mismatch could result from disregard of diffusional resistance caused by the high viscosity glycerol in the mixture, which certainly decelerates
Chapter 4 112 the heat and mass transfer rate of the mixture. Within the first 10 seconds, the cooling effect dominated the process (heat ratio is greater than unity) representing a temperature reduction which was found shorter than that of pure water that occurred within 13 seconds (Figure 4.15a) because of the smaller latent heat of vaporization of the water-glycerol mixture.
The heat ratio profile before transition showed almost near unity value up to t ~ 125 s preceded by the initial cooling effect. A distinct temperature jump is apparent at the transition stage (t~160 s) when the heat gain term through convective heat transport dominates the heat loss term due to latent heat of vaporization and leads to less than unity heat ratio. The heat ratio then increases slightly above unity indicating rise in droplet temperatue finally reaching thermal equilibrium with the ambient gas.
The low latent heat of the mixture due to presence of glycerol also attributed for the significant rise in droplet temperature, at the end of the process, compared with pure water evaporation. Similar to Case 1 and Case 3, thermal equilibrium was obtained when the lighter component (heptane for heptane-decane mixtures and water for water-glycerol mixture) fin- ished evaporating and only the heavier component (decane or glycerol) was left in the drop- let. However, this is the only case among the six cases in which the liquid phase reached thermal equilibrium with the surrounding gas stream this is because of the very high boiling point of the glycerol species (563 K), which leads to a very small evaporation rate m̄d. A small m̄d and a low latent heat of vaporization (LV = 960 kJ/kg at room temperature) would attribute to a very small value of the second term in the right hand side of Eq. (4.11), result- ing in a significant increase in temperature (left hand side term).