Vaporisation of droplet in a convective environment is of interest in spray combustion or in FCC risers wherein droplets are injected at a high velocity into the hot gaseous or gas- solid environment respectively. Vaporisation occurs owing to the convective heat received from the hot gas stream. Studies of the vaporisation of a moving droplet in a stagnant gas en- vironment have been analogised to a suspended droplet vaporising in a flowing gas stream using identical relative velocity between droplet and the gas flow. This section first presents a review of typical numerical modelling studies of both single and multicomponent droplet va- porising in heated gas flows, followed by a review of selected experimental work in the same field.
Figure 2.1 Schematic of a suspended evaporating droplet
Chapter 2 19 Figure 2.1 shows a suspended droplet vaporising in a flowing gas wherein heat and mass transfer take place at the droplet interface. The vaporisation rate can be obtained using Fick’s law assuming thermodynamic vapour-liquid equilibrium exists at the liquid-gas inter- face whilst heat transfer represented by transient temperature of the droplet can be obtained from the energy balance between convective heat gained and heat loss due to evaporation (see Appendix A for the full derivation). Sherwood and Nusselt number via Ranz-Marshall (or modified Ranz-Marshall) correlations are utilised to represent mass and heat transfer of the droplet the respectively. Selected reports of Sherwood and Nusselt numbers in the litera- ture which based on the originally Frossling’s (Frossling, 1938) mass transfer correlation was reviewed in Table 2.2 with corresponding operating conditions.
Table 2.2 Development of heat and mass transfer correlations for droplet evaporation*
Authors Correlations Conditions
Frossling (1938)
0.5 0.33
2 0.552 Re
Sh= + ∞ Sc∞ 2<Re∞ <1300
Ranz and Marshall
(1952b)
0.5 0.33
2 0.6 Ref f Sh= + Sc
0.5 0.33
2 0.6 Ref Prf Nu= +
0<Ref <200
Air temperature up to 220°C Eisenklam et
al. (1967) Sh=(2+0.6 Re0.5f Sc0.33f ) (1+BM)
(2 0.6 Re0.5f Pr0.33f ) (1 T)
Nu= + +B
0.01<Ref <400
Air temperatures up to 1000 oC Yuen and Chen
(1978) Nu=(2+0.6 Re0.5m Pr0.33f ) (1+BT) 200<Ref <2000
Air temperature up to 960 °C (properties are evaluated at film conditions except for the densi- ty in the Reynolds number which is the free stream densi- ty)
Renksizbulut
and Yuen Nu=(2+0.6 Re0.5m Pr0.33f ) (1+BT)0.7 10<Ref <100Air tempera-
tures up to 1000K
Chapter 2 20
(*): the used subscript “m” means “modified” Reynolds number using free-stream density, “f” means film con- ditions (vapour film at the droplet interface) and “∞” means free-stream density
Numerical approaches for the gas phase characterised by Sherwood and Nusselt number is strongly coupled to the liquid phase calculations through the linked parameters, which is temperature and mass fraction at the liquid-vapour interface. According to Sirignano (2010), heat transfer inside the liquid droplet can be classified into six different modelling approaches of increasing complexity. (i) constant droplet temperature models which yield the well- known d2 law relating the linear relationship between droplet diameter or radius with time);
(ii) infinite liquid thermal conductivity models wherein droplet temperature is considered homogeneous but time-varying; (iii) spherically symmetric transient droplet heating model (diffusion limit) wherein droplet temperature is inhomogeneous and time-varying; (iv) effec- tive thermal conductivity model which considered droplet temperature to be inhomogeneous and time-varying and internal motions accounted by introducing an effective thermal diffu- sivity; (v) vortex model; (vi) complete solution of the Navier-Stokes equation.
Aggarwal et al. (1984) compared available droplet evaporation models and suggested the use of the diffusion limit model which assumes a temperature gradient in the liquid phase for negligible droplet Reynolds number. Also, suggested in this work was the simplified vor- (1983)
Abramzon and Sirignano (1989)
( )
0.5 0.33
2 0.6 Rem Scf / BM
Sh= + F
( )
0.5 0.33
2 0.6 Rem Prf / T
Nu= + F B
( ) ( )0.7ln 1( )
1 B
F B B
B
= + +
wide range of droplet sizes and Reynolds numbers
R. Miller et al.
(1998) Sh=(2 0.552 Re+ 0.5m Sc0.33f )ln 1( +BM) (2 0.552 Re0.5m Prf0.33)
Nu= + G
) 1 /( −
=β eβ
G and
( d Pr / 8G G) 2
β dd
ρ à dt
= − ;
wide range of droplet sizes and Reynolds numbers
Chapter 2 21 tex model which accounts for the internal motion within the droplet for high Reynolds num- ber conditions. Abramzon and Sirignano (1989) introduced an effective thermal conductivity model (type 4) in which the recirculation due to external shear stress was considered by simp- ly replacing the molecular diffusivity in the diffusion equation by effective diffusivity, which is given as αL eff, . = χ αα L where the factorχα is a function of the interior Reynolds and Prandtl numbers and it is bounded between 1 and 2.72 given as:
1.86 0.86 tanh[2.245 log (Re Pr / 30)]10 L L
χα = + . (2.1)
The two models suggested by Aggarwal et al. (1984) have been extensively applied in spray calculations (Aggarwal & Mongia, 2002; Kneer et al., 1993; Maqua et al., 2008; Sazhin et al., 2014; Sazhin et al., 2010; Sazhin et al., 2011; Stengele et al., 1996). Also, the Abramzon and Sirignano model has become the most popular choice for predicting the vaporization rates of droplets (Aggarwal & Peng, 1995; Kristyadi et al., 2010; R. Miller et al., 1998; Sazhin et al., 2006; Sazhin et al., 2004) because not only its simplicities (compare with exact numerical solutions) but it covers a wide range of droplet sizes and Reynolds numbers. However, the computational cost of these types of model becomes prohibitive for any practical system which involves multitude of droplets. This limitation is however often eliminated by using the less rigorous yet effective rapid mixing model which assumes temperature and concentra- tion remain spatially uniform. Rapid mixing model introduced by Law (1976) has been com- monly used in both batch distillation and spray vaporization for both single and binary mix- ture droplet evaporation with reasonable agreement with the experimental data (X.-Q. Chen
& Pereira, 1996; Tamim & Hallett, 1995; L. Zhang & Kong, 2010).
When the droplet is a multicomponent mixture, liquid-phase modelling becomes dif- ficult as each of the species needs to be separately accounted. Different species have different vaporisation rates which create a gradient of concentration in the liquid phase additional to
Chapter 2 22 the temperature gradient for single component liquid. There are two limits commonly exist- ing in the theory of multi-component droplet vaporisation: (1) rapid mixing model or infinite diffusion model, which assumes spatially uniform temperature and concentration within the droplet and (2) diffusion limit which assumes inhomogeneous temperature and concentration.
Landis and Mills (1974) proposed a diffusion limit model for multicomponent droplet wherein the concentration profile is induced from the preferential vaporisation of the different components in the droplets. The more volatile component is vaporized first leaving behind the higher boiling point less volatile components in the droplet core. Slow mass diffusional process might slow down the remaining more volatile component from moving outward to droplet surface and eventually the droplet attains a steady concentration profile. Results from the study of Landis and Mills (1974) indicate that the infinite diffusion model in the predic- tions of species vaporisation rates (compared with exact solutions) produce small deviations at only low ambient temperature (600 K in the study). However inaccuracy are found for pre- dictions under high ambient gas (2300 K).
In contrast to the diffusion limit concept, Law (1976) proposed a rapid mixing model for vaporisation of multi-component droplets. It was reasoned that the rapid mixing occurs when internal motion, which is either produced from violent atomization or forced convective gas flow, are so strong that temperature and concentration become spatially homogeneous within the droplet. Due to the shearing effect of surrounding gas flow, for most hydrocarbon droplets in combustion applications, the maximum liquid velocity at the droplet interface is found to be ~ 4 to 17% of the free stream velocity (Law & Law, 1982). A higher gas tem- perature and lower viscosity liquid lead to a larger velocity at the droplet surface, resulting in a stronger internal circulation inside the droplet. The rapid mixing model was believed to be valid for only less viscous mixtures which enable the fast motion in the liquid bulk, however
Chapter 2 23 no analysis was made for the high viscous liquids in the study of Law (1976). Delplanque et al. (1991) extended the Abramzon and Sirignano (1989) model for multicomponent droplet vaporisation considering the internal circulation and suggested an analogous effective mass diffusivity for the liquid phase as
,
L eff d L
D =χ D (2.2)
where
1.86 0.86 tanh[2.245log (Re10 / 30)
d LScL
χ = + (2.3)
and DL and DL,ef are liquid molecular and effective mass diffusion coefficients, respectively.
Equation (2.2) has been widely used for modelling multicomponent droplet vaporisation such as in Sirignano (2010), Sazhin et al. (2010) and Sazhin et al. (2014).
Aggarwal and Mongia (2002) investigated the vaporisation behaviour of droplets un- der high pressure conditions and suggested the use of finite diffusion model for combustion or gas turbine engines wherein high pressure operating conditions apply. The rapid mixing model was deemed not appropriate for such operating conditions. Although the use of finite and effective diffusion model is often suggested, the computational cost of these types of models becomes prohibitive for any practical system comprising multitude of droplets. The less rigorous yet effective rapid mixing model proposed by Law (1976) has been still widely used in spray calculations (X.-Q. Chen & Pereira, 1996; Tamim & Hallett, 1995; L. Zhang &
Kong, 2010, 2012). A summary of the numerical studies on multicomponent droplet vapori- zation is provided in Table 2.3
Chapter 2 24
Table 2.3 A comparative summary of modelling studies on multicomponent droplet evaporation.
Authors Liquid compounds Model approaches Analyses
Landis and Mills (1974)
Pentane-hexane Heptane-octane Ideal solution
Diffusion limit Concentration gradient exists within the liquid phase due to the preferential vaporization of the more volatile component.
However, the very slow mass diffusion transport causes a resistance for the component to replenish from the droplet surface.
Law (1976) Hexane-octane Ideal solution
Rapid internal mixing Internal recirculation inside droplet was assumed to be infi- nitely fast which led to a uniform temperature and concentra- tion within the droplet.
Law et al. (1977) Hexane-octane Ideal solution
Potential flow for gas phase.
Hill's Vortex for liquid phase.
Boundary layers at the droplet interface
Internal recirculation within the liquid phase was found to play an important role in the transport process. The more volatile species preferentially diffused outward while the less volatile compound diffused inward the liquid droplet alt- hough the net fluxes were directed outward.
Aggarwal et al.
(1984)
Hexane Decane Hexadecane (single component)
Rapid mixing Simplified vortex Diffusion limit (comparison)
The simplified vortex model predictions lied between the diffusion limit and rapid mixing model. Diffusion limit mod- el was recommended for a low Reynolds number and simpli- fied vortex model was recommended for high Reynolds number conditions.
Kneer et al. (1993) Heptane-dodecane Ideal solution
Diffusion limit Single component model could not describe the whole va- porization process of multicomponent models. Diffusional resistance within the liquid phase was marked in the first half of droplet lifetime for multicomponent evaporation.
Tamim and Hallett gasoline, diesel, do- Infinite diffusion Evaporation of a complex mixture droplet whose composi-
Chapter 2 25
(1995) decane
Ideal solution
tion, properties and vapour liquid equilibrium were described by the methods of continuous thermodynamics
Stengele et al.
(1996)
Heptane-hexadecane Ideality
Diffusion limit. Stagnant droplet.
Stagnant air, pressure of 20- 40bar. Temperature 2000K.
Ambient pressure was found to have insignificant effect on the composition of the liquid droplet during evaporation.
Droplet lifetime however was reported to depend strongly on temperature and pressure of the ambient gas.
Aggarwal and Mongia (2002)
Decane- tetradecane mixture and the equivalent: dodecane Ideality
Infinite diffusion. Diffusion limit Non-ideal gas. Pressure of 1-15 atm, temperature up to 1500 K
Diffusion limit was suggested for high pressure. Equivalent single component was suggested for high pressure. Quasi- steady vaporization model was reported to be invalid if pres- sure above critical value.
Brenn et al. (2007) Four/five components Non-ideality (UNI- FAC)
Diffusion limit
Extension of AS model
Effective velocity of the acoustic stream replaced convective gas flow velocity. Nu and Sh correlations were reported to be valid for any convective transport as long as it causes rela- tive motion between gas and droplet. The non-ideal behav- iour was considered in the model however its effect in its absence was not discussed.
(L. Zhang & Kong, 2010)
Heptane-decane, Biodiesel. Diesel- biodiesel.
Gasoline-ethanol Ideality
Infinite diffusion
(Atmospheric pressure, tempera- ture up to 1019K)
Model was successfully valid with single, binary mixture and biodiesel (five components mixture) before applied for petro- leum fuels.
Sazhin et al. (2010) Acetone-ethanol Non-Ideality
Effective diffusion
(Atmospheric pressure. Room temperature. Stagnant ambient gas)
Effective diffusion model was suggested to replace the com- plex vortex model (liquid phase). Non-unity activity coeffi- cient was reported to have negligible influence.
(Sazhin et al., 20 components C5- Effective diffusion Quasi-discrete approach was suggested for diesel fuel with assumption that all compositions are close in number of car-
Chapter 2 26
2011) C25 (diesel) bon atoms and properties vary relatively slowly. Effective
diffusion model was suggested.
L. Zhang and Kong (2012)
Bio oil (10 compo- nents)
Infinite diffusion The volatility of the fuel/fuel mixture depends on the compo- sition of the bio-oil and the presence of bio-oil in the fuel mixture spreads the vaporization time of drop lifetime (which was computed up to 1.5ms).
Banerjee (2013) Ethanol-iso-octane binary droplet Non-ideality
Simulation (Ansys Fluent 13) VOF method
Influence of the compositions of the liquid droplet on the evaporating performance via the overall heat capacity of the mixture was found; evaporation rate was reported to be strongly depends on droplet temperature. The non-ideal be- haviour included in the numerical simulation using UNIFAC method due to high polarity of ethanol molecule, however its effect in its absence was not discussed.
Sazhin et al. (2014) Biodiesel fuel Ideality
Quasi-discrete model Moving droplets evaporate much faster than stationary drop- lets. Multicomponent model was found to give longer evapo- ration duration and higher temperature prediction compared with equivalent single component model.
Chapter 2 27 Although the numerical studies on droplet vaporisation are well developed in the lit- erature as described, it is always challenging to experimentally obtain adequate quantifica- tions due to the fine droplet size and high temperature, pressure and relative velocity in the practical conditions
The earliest well-known evaporation experiment is reported by (Ranz & Marshall, 1952a, 1952b) for a stationary water droplet evaporating in convective environment in Reyn- olds number range from 0 to 200 which was later shown to be valid for extended Reynolds number range up to 1000. Later, Downingm (1966) and Wong and Lin (1992) investigated the evaporation performance of hexane and decane droplets respectively in forced convective environment. However, these experiments were carried out only at moderate droplet Reyn- olds numbers (up to 110) and for single component droplets only. On evaporation of the bina- ry hydrocarbon droplets in heated gas medium, Gửkalp et al. (1994) reported evaporation be- haviour of heptane-decane droplets in air at temperature of 372 K and gas velocity of 1.45 m/s while Daıf et al. (1998) examined both temporal evolution of droplet diameter and tem- perature for heptane-decane droplets in hot gas flow at velocity of up to 3.1 m/s (droplet Reynolds number ~ 215). Evaporation of droplets in higher gas temperature (up to ~ 1000 K) was performed by Ghassemi et al. (2006b), Hallett and Beauchamp-Kiss (2010), K. Han, Song, et al. (2016), Y. Zhang et al. (2017); however, these measurements were obtained at stagnant air condition. Table 2.4 summarizes the published experimental work on binary mix- ture droplet evaporation at zero to moderate droplet Reynolds number.
Chapter 2 28 Table 2.4 Summary of experimental studies on multicomponent droplet evaporation.
Authors Material Air tem-
perature
Initial diameter
Reynolds number/air velocities
Droplet tem- perature avail- able
Ranz and Marshall (1952b)
Pure water 298 1.1 0 - 200 No
Downingm (1966)
Hexane 437 1.78 110 No
Wong and Lin (1992)
decane 1000 2.0 17 Yes (thermo-
couple) Nomura et al.
(1996)
Heptane 741 0.8 0 No
Gửkalp et al.
(1994)
Heptane-decane 372 1.56 ~ 105 No
Daıf et al.
(1998)
Heptane-decane 348 1.38 ~ 215 Yes (infrared
camera J.-R. Yang and
Wong (2002)
Heptane, hexade- cane
490 750
0.70 1.0
5-17 Yes (thermo-
couple) Ghassemi et al.
(2006a)
Heptane- hexadecane
~ 873 1.28 0 No
Chauveau et al.
(2008)
Heptane 473, 973 0.5 0 No
Hallett and Beauchamp- Kiss (2010)
Ethanol
Ethanol & fuel oil
up to 1023 ~ 1.4- 1.8
0 No
Woo et al.
(2011)
Water 358 ~1.0-1.5 ~ 80 No
Davies et al.
(2012)
Water-glycerol 298 0.042 air velocity 0.4 m/s
No Javed et al.
(2013)
Kerosene 1073 ~1.0 0 No
K. Han et al.
(2015)
dodecane–
hexadecane
1046 1.23 0 Yes (thermo-
couple) K. Han, Yang, et
al. (2016)
ethanol–diesel Up to 723 0.7-1.3 0 Yes (thermo- couple)
Y. Zhang et al.
(2017)
biodiesel-butanol 1073 [-] 0 Yes (thermo-
couple) Volkov et al.
(2017)
Water Up to ~
773
2.0-4.0 air velocity 0-3.5m/s
Yes (PLIF)
Chapter 2 29 The available studies report measurements of change in droplet size with time but what lacks in the majority of these studies is the data of droplet temperature which is rather inadequately reported in the literature and indeed is critical to validate any numerical model- ling work in this area. Experiments for binary mixture droplets were done mostly for the non- polar hydrocarbon mixtures and studies on the polar component mixtures are relatively less.
Importantly, the dearth of data for polar binary systems evaporating in hot convective envi- ronment at higher gas stream velocity or droplet Reynolds are recognised from the above re- view. Details of the work which address these additional considerations are discussed in Chapter 4.