Evaporation of Hydrocarbon Fuel Droplet under Elevated Temperatures

Extended model is a t heoretical and analytical model for evaporation of hydrocarbon fuel droplet. This model assumes that there is a moving hydrocarbon fuel droplet in quasis teady environmental air. Four types of fuel (n-heptane, n-hexane, n-decan


Introduction
For modern combustion engines using liquid fuels, several processes play important roles in reaching high efficiency in the combustion cycle and low emissions in the exhaust gas.One of these processes is the evaporation of the fuel in the combustion chamber.For direct injection systems used in aircraft or car engines, the fuel enters the combustion chamber in the liquid state.During injection, the liquid disintegrates into single droplets by atomization.In addition, the droplets evaporate before combustion occurs, figure (1) shows the state of droplet evaporation [1].The theory of fuel droplet vaporization has been intensively developed during the past several decades.The classical droplet vaporization mode1 explained in many literature [2,3], deals with an isolated pure component single droplet suddenly exposed to a hot environment at low pressure.Since 5404 https://doi.org/10.30684/etj.28.17.2 2412-0758/University of Technology-Iraq, Baghdad, Iraq This is an open access article under the CC BY 4.0 license http://creativecommons.org/licenses/by/4.0 the droplet temperature is lower than the surrounding atmosphere temperature, a net driving force due to temperature difference will transfer heat into the droplet, which will be used to supply energy for vaporization as well as for heating the liquid.As the droplet heats up, the vapour concentration will start to build up at the liquid-vapour interface where equilibrium between the liquid and vapour exists; this will be a function of interface temperature and the total atmospheric pressure.The vapour will start to diffuse into the surrounding air, resulting a net mass flux outward from the droplet.Sirignano and Law [4], studied the effect of non-uniform liquid-phase temperatures on a single component fuel droplet by considering that internal motion does not exist and diffusion is the only heat transport mechanism.They compared this case with that of a uniform liquid-phase temperature (complete mixing model) and they concluded that the droplet size variations and consequently the droplet vaporization time can be predicted with good accuracy regardless of the mode1 of internal heat transfer.Law [5], presented a review for the progresses on understanding the fundamental mechanisms governing droplet vaporization and combustion.Topics include the d2-law and its limitations; the major transient processes of droplet heating and fuel vapor accumulation; effects due to variable transport property assumptions; combustion of multi-component fuels including the miscible fuel blends, immiscible emulsions, and coal-oil mixtures, finite-rate kinetics leading to ignition and extinction; and droplet interactions, have been presented.

2
Extension to convective environment model: The purpose of this model is to study the effect of the convective transport caused by the relative velocity between the droplet and the free stream on the evaporation process of a single component droplet suddenly exposed to a hot environment.The essence of film theory is the replacement of the heat and mass transfer boundary conditions at infinity with the same conditions moved in ward to the so-called film For droplet burning with forced convection, at low Reynolds number limit [7]: (Nu or Sh)=2+1/2(Re.Pr or Re.Sc) +higher order terms (3) Other studies of evaporation in creeping motion show that the coefficient, 1/2, depends on the variable properties model and degree of overall properties variation in the flow field around the drop, and neglecting the higher order terms in Eqn.(3) yields a prediction that is within 2% of the more complete expansion.
PDF created with pdfFactory Pro trial version www.pdffactory.com(5) For (10<Re<1800) At the higher Reynolds numbers, Yuge suggests a higher power for the Reynolds number, in agreement with other measurement.Combining Eqn.
(3) and ( 5) to obtain a synthesized correlation which approaches the correct limiting values at low and high Reynolds numbers (Re<1800) For outer region: (12)

Energy conservation:-
For spherical droplet and axisymmetric geometries, the Shvab-Zeldovich energy equation is given in the form [9], (13) With the assumptions of constant properties, unity Lewis number, zero reaction rate and pure evaporation equation ( 13), after arrangement can be written as, For the outer region bounded by ( ) , the temperature distribution is obtained from the application of the boundary conditions:- (16) The equation of the temperature distribution for general solution at the gas-phase of the burning droplet is: Where, 1 C and 2 C , the constants of integration.For convenience, if Then, the temperature distribution equation becomes:- Applying the boundary condition, Eqn (15), in Eqn. ( 18) to obtain:-2 . .
Then, the temperature distribution becomes:-  ( ) PDF created with pdfFactory Pro trial version www.pdffactory.com

Energy balance at flame sheet:-
The surface energy balance at the flame sheet can be written : By differentiation the temperature distribution in Eqn. ( 22) at the flame radius in the outer region:- ( ) Then, the energy balance at the flame yields, The droplet mass evaporation rate in terms of the transfer number q o B , can be expressed by: ( ) As in pure evaporation analysis, assume the fuel is at the boiling point.The problem is greatly simplified with this assumption and by using Eqn.( 12 Then, the flame temperature becomes: The droplet life time is obtained using the mass balance for the classical theory which states that the rate at which the mass of the droplet decreases, is equal to the rate at which the liquid is vaporized i.e.;

Extension to convective environment model and the classical model
The calculations of both the classical and extension to convective environment models for the four types of fuel under study, are based on three variables (size ratio, vaporization time, and environmental temperature), where the other parameters have been evaluated according to these three variables.Then, the comparison between the two models is held.

(a) Droplet size variation
Figures ( 3) and ( 4) show that increasing the amount of fuel vaporized from the droplet surface which in turn increases the mass vaporization rate.Higher mass vaporization has been noted for extended model due to the droplet movement that lead to a force convection case , and this in turn causes an increase in the sensible heat quantity that entering the droplet, subsequently, increasing the mass evaporation rate.Regarding the classical model, the droplet is considered as quasi-steadiness.In extended model, n-decan gives higher mass vaporization than other fuel while in classical model, light diesel has higher mass vaporization and other fuel give approximately the same behavior due to the physical properties of fuel.

(b) Burning time variation
Figures ( 5) and ( 6) show the variation of droplet size with dimensionless time of evaporation.The two models give approximately the same behavior.The curves could be divided into two parts of droplet life time, At the early stages of its lifetime the curves decrease in the droplet size with the increases of the time and this is higher than that for classical model because the droplet lifetime decreases with increase of the heat gain transfer.During the early part, a large value should be used (higher ambient temperature and higher Reynolds number), while during the later part a smaller value should be used.Therefore, at the later stages of droplet life time, the curves indicate that the behavior of nheptane and n-hexane are the same for the classical model.

(c) Environmental temperature variation
Figures ( 7) and ( 8) for classical model, are directly related to the environmental temperature, so, any increase in the environmental temperature will lead to a noticeable increase in the flame conditions.While, in extended model, the relationship is constant due to independency on transfer number which is a function of temperature and the kind of reaction.

Comparison between the classical and extended models:-
The comparisons are carried out for characteristic parameters of nheptane for the two models.

Comparison the result of extended model using different Reynolds number
Figure (12) presents that the flame stand-off ratio decreasing with increasing Reynolds number and decreasing will droplet life time due to the flame radius depending for Nusselt number.

Comparison with experimental work
A comparison between present work and experimental work of Nomura [12] is presented in figure (15) for droplet size variation and life-time.A good agreement was obtained between the two works.

Conclusions
The following conclusions are obtained from the analysis of the obtained results: 1-Extended model has proven to be a successful model for single component calculation and has not yet been applied to droplet or other evaporation and combustion problems.The method has not been applied to transport process at all.2-The heat up process using extended model causes the mass evaporation rate at the early stages to be much than that in the classical theory of droplet evaporation.Prog. Energy Combust. Sci., Vol. 3, pp. 191-224, 1977. [8].Ranz, W. E. and Marshall, W .R., "Evaporation from Drops", Part II, Chemical Engineering Progress, Prog. 48, pp. 173-180, 1952. [9] radius, Mδ for species, and T δ for energy[6].The film radii are defined in terms of the Nusselt number Nu, for heat transfer, and the Sherwood number Sh, for mass transfer, given by,

C
,respectively, back into the general solution given by Eqn.(18), radius in the inner region gives [6], is a reasonable assumption when the droplet is burning vigorously after its initial heat-up transient and can be evaluated by the mass fraction Clausius-Clapeyron equation, by solving the Eqn.(12), (22), and ( ) to evaluate the flame temperature after arrogant:PDF created with pdfFactory Pro trial version www.pdffactory.com Then the flame stand-off ratio (ratio of the flame size to the droplet size), is given by: Type of Fuel: four types of fuel have been used in the calculations of the classical and extension to convective environment models which are; n-heptane, nhexane, n-decan, and light diesel Droplet Size: The droplet size (d) used in calculations =100µm Environmental Conditions: The environmental conditions assumed in the calculations are: a) Temperature: (300, 600, 1000, 1200, and 1500) K. PDF created with pdfFactory Pro trial version www.pdffactory.comb) Pressure: (1) bar c) Reynolds number at range (0.1-1800) (a) Droplet size variation Figure (9) shows the variation of the system Damköhler number with size for the two models.Damköhler number represents the ratio between the flow time and reaction time.So, as it is shown in the figure, Damköhler number is proportional to size, where it decreases with the decrease in size, while the variation of the system Damköhler number with size remain constant for the classical model.PDF created with pdfFactory Pro trial version www.pdffactory.com(b) Vaporization time variation Figure (10) shows that the flame temperature similarity in behavior for the two models is due to the fact that flame temperature is not affected by Nusselt number since it depends totally on the fuel properties and the environmental temperature.(c) Environmental temperature variation Figure (11) which show that the flame position obtained from the classical model is the most affected from temperature variation between them.And that is due to the flame position in the classical model is directly

Figure
deviation between the two models is due to forced convection used in the present model while Law's model based on conduction heat transfer only.Figure (14) reveals the comparison between present model and Law's work [11] for ( ) 3-The heat transfer to the drop increases with increase Re. 4-The flame conditions (position) are not constant but, function of Re. 5-The mass evaporation rate obtained by the classical model can be corrected with a useful relation for the mass evaporation rate by the extended model for different temperature and different kind of fuel and other for different Reynold's number and different kind of fuel .6-The flame temperature remains constant in the two models.7-Good agreements were obtained through comparison with other work.References [1].Jochen W., "Evaporation of Multicomponent Droplets", PH.D. PDF created with pdfFactory Pro trial version www.pdffactory.com

FigureFigure ( 8 )
Figure (1): Vapor mole fraction and temperature profiles in gas phase through droplet evaporation