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Transient analysis of sub-critical evaporation of fuel droplet in non-isothermal stagnant gaseous mixtures: effects of radiation and thermal expansion

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Abstract

Transient sub-critical droplet evaporation in non-isothermal stagnant gaseous mixtures taking into account the effects of radiation, liquid volumetric expansion and droplet heating is investigated numerically. We obtained equations for Stefan velocity and the rate of change of the droplet radius taking into account liquid volumetric expansion, and derived the boundary conditions taking into account the effect of liquid thermal expansion. It is shown that in the case of sub-critical evaporation neglecting the liquid volumetric expansion causes underestimation of the evaporation rate at the initial stage and overestimation of the evaporation rate at the final stage of droplet evaporation.

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Notes

  1. In general the Equations (4) are introduced for allj=1,...,K. However, only K − 1 of the quantities D j can be specified independently. Thus the K-th species can be treated differently from the others and may be found consistently using the following identity \(\sum\nolimits_{j = 1}^K {Y_j }=1\)

Abbreviations

c p :

Specific heat at a constant pressure

c :

Sound velocity

D d :

Droplet diameter

D j :

Diffusion coefficient of j-th species

D jk :

Binary diffusion coefficient

k :

Thermal conductivity

K V :

Evaporation constant

\( \hbox{Kn}= \frac{\lambda}{R}\) :

Knudsen number

L :

latent heat of evaporation

Le:

Lewis number

M j :

Molecular weight of j-th species

m ℓ :

Mass of the droplet

\(\ifmmode\expandafter\dot\else\expandafter\.\fi{m}\) :

Droplet vaporization rate

p :

Pressure

q R :

Radiation flux vector

r :

Radial coordinate

R :

Radius of the droplet

R g :

Universal gas constant

T :

Temperature

t :

Time

t D :

Diffusion relaxation time

t T :

Thermal relaxation time

u, \( v \) :

Velocity

x :

\(= \frac{r}{{R{\left(t \right)}}}\)

X j :

Mole fraction of the j-th species

Y j :

Mass fraction of the j-th species

αℓ :

Liquid thermal diffusivity

ζT :

\(= \frac{1}{{\alpha _{\ell} \rho c_{p}}}\)

ζD :

=ζ T c p

η:

Coefficient of thermal expansion

θ:

\(= \frac{T}{{T_{\infty}}}\)

λ:

Mean free path of the molecules

ξ:

\(= \frac{{R{\left(t \right)}}}{{R_{0}}}\)

τ:

\(= \frac{{\alpha _{{\ell}} t}}{{R^{2}_{0}}}\)

1:

Volatile species

e :

Value outside a droplet

i :

Value inside a droplet

j :

Number of a species (1-st is the volatile species)

ℓ:

Liquid

s:

Value at the droplet surface

∞:

Value at infinity

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Authors and Affiliations

  1. Department of Mechanical Engineering, The Pearlstone Center for Aeronautical Engineering Studies, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel

    T. Elperin & B. Krasovitov

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  1. T. Elperin
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  2. B. Krasovitov
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Correspondence to T. Elperin.

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Elperin, T., Krasovitov, B. Transient analysis of sub-critical evaporation of fuel droplet in non-isothermal stagnant gaseous mixtures: effects of radiation and thermal expansion. Heat Mass Transfer 42, 427–436 (2006). https://doi.org/10.1007/s00231-005-0028-z

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