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Quasi-steady diffusion-controlled droplet evaporation and condensation

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Abstract

This theoretical investigation deals with the vaporization or condensation rate of a motionless liquid droplet using the quasi-steady diffusion-controlled model. A single-component liquid droplet is considered to be undergoing a phase change within a binary mixture of ideal gases (vapor plus noncondensable gas). Droplet vaporization rates corresponding to specified ambient conditions have been calculated by numerical solution of the variable-property governing equations. Results are presented for water and a series of pure hydrocarbon liquids for a range of ambient conditions of interest. A dimensionless correlation is given for the hydrocarbon vaporization and condensation rates. The pressure variation in the region surrounding a droplet undergoing vaporization or condensation has been investigated by numerical integration of the momentum equation. The resulting calculations indicate that the pressure decreases with increasing distance from the droplet for condensation as well as vaporization. Finally, criteria are given for estimating when the pressure gradient and viscous dissipation may be of significance in the energy equation.

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Abbreviations

A, B:

coefficients in the vapor pressure expression, eq. (C3)

a j , b j :

coefficients in dimensionless correlation, eqns. (D2) and (D3); see Table D-II

C ji :

coefficients in the expression for C Pi , eq. (C2) and Table C-II

C P :

specific heat at constant pressure, energy/(mass, temperature)

D :

binary diffusion coefficient, D=D ij =D ji

D 1 :

collection of factors in expression (C12) for the diffusion coefficient, see Table C-IV

E :

relative difference between variable property and constant property calculation, see Table III

E i :

modified Eucken correction to thermal conductivity

G ij :

coefficient in eq. (C5)

h :

enthalpy

\(\mathop h\limits^\_\) :

dimensionless enthalpy, \(h/(C_{{\text{P}}_{\text{0}} } T_0 )\)

k:

thermal conductivity (energy/time length temp.) or Boltzmann constant

\( \bar k \) :

dimensionless thermal conductivity, k/k0

l :

mean free path

L(T):

latent heat of vaporization, energy/mass, evaluated at the temperature T

L 1, L 2 :

coefficients in expression (C4) for the latent heat of vaporization of water

Le :

dimensionless Lewis number, D ρ C P/k

m :

vaporization rate, mass/time

n :

number of components present in the gas phase

P :

absolute pressure

\(\hat \bar P\) :

dimensionless pressure, (P−P 0)/ρ 0(ν 0/r 0)2

\(\bar P\) :

dimensionless pressure, P/P 0

P 1(T):

vapor pressure of component 1 at temperature T

Q :

heat transfer rate for quasi-steady vaporization, mL(T 0)

Q R, Q S :

heat transfer rate (energy/time) due to radiative absorption or conduction via droplet support

q :

heat transfer rate (energy/time area)

R:

universal gas constant

r :

radial coordinate

S :

dimensionless radial coordinate, r/r 0

T :

absolute temperature

\(\bar T\) :

dimensionless temperature, T/T 0

V i :

diffusion velocity of component i

v :

mass average velocity

\( \bar \upsilon \) :

dimensionless velocity, v/v 0

W i :

molecular weight of component i, mass/mole

X i :

mole fraction of component i

Y i :

mass fraction of component i

α :

empirical correction to eq. (C12) for the diffusion coefficient, also absorptivity

β :

ratio of the rate of energy transfer by conduction from the gas phase to the rate of energy transfer required for vaporization, defined in eq. (A11)

Δ :

change in vaporization rate due to change in pressure, Table I

ε ij , σ ij :

intermolecular potential parameters characterizing the i–j interaction

σ :

Stefan-Boltzmann constant

η :

viscosity

\( \bar \eta \) :

dimensionless viscosity, η/η 0

μ :

dimensionless vaporization rate, m/4πr 0(D ρ)0, or micron (10−6 meter)

ν :

kinematic viscosity, η/ρ

ρ :

density

\(\bar \rho\) :

dimensionless density, ρ/ρ 0

Φ :

viscous dissipation function, eq. (A4)

Ω (1, 1)⋆, Ω (2, 2)⋆:

collision integrals for Lennard-Jones potential

0:

droplet surface (gas phase)

∞:

infinitely far from the droplet surface

B:

boiling point at pressure P 0

c:

thermodynamic critical point

D:

dew point

i :

index running from 1 to n, denoting the i th component

L:

liquid phase

R:

reference quantity, defined in eq. (D1)

⋆:

dimensionless vaporization rate μ⋆ formed with reference value (D ρ)R, or dimensionless temperature T⋆=kT/ε

References

  1. Fuchs, N. A., Evaporation and Droplet Growth in Gaseous Media, trans. by J. M. Pratt, Pergamon Press, New York, 1959.

    Google Scholar 

  2. Williams, F. A., Int. J. Heat Mass Transfer 8 (1965) 575.

    Article  MATH  Google Scholar 

  3. Sutton, G. W., AIAA J. 8 (1970) 1907.

    ADS  Google Scholar 

  4. Muggia, A., Aerotechnica Roma 36 (1956) 127.

    MathSciNet  Google Scholar 

  5. Fendell, F. E., D. C. Coats, and E. B. Smith, AIAA J. 6 (1968) 1953.

    Article  MATH  ADS  Google Scholar 

  6. Kassoy, D. R., and F. A. Williams, AIAA J. 6 (1968) 1961.

    ADS  Google Scholar 

  7. Faeth, G. M., and D. R. Olson, The Ignition of Hydrocarbon Fuel Droplets in Air, SAE Preprint 680465, Mid-Year Meeting, Detroit, Michigan, 1968.

  8. Manrique, J. A., and G. L. Borman, Int. J. Heat Mass Transfer 12 (1969) 1081.

    Article  Google Scholar 

  9. Lazar, R. S., and G. M. Faeth, 13th Int. Symposium on Combustion (1971) 801.

  10. Downing, C. G., The Effect of Mass Transfer in the Evaporation of Drops of Pure Liquids, Ph.D. Thesis, University of Wisconsin, 1960.

  11. Borman, G. L., M. M. El Wakil, O. A. Uyehara, and P. S. Myers, Graphs of Reduced Variables for Computing Histories of Vaporizing Fuel Drops, and Drop Histories under Pressure, NACA TN-4338, Sept. 1958.

  12. Lee, K., and D. J. Ryley, J. Heat Transfer 90 (1968) 445.

    Google Scholar 

  13. Savery, W., and G. Borman, AIAA Preprint No. 70-6, New York, 1970.

  14. Luchak, G., and G. O. Langstroth, Can. J. Res. A28 (1950) 574.

    MathSciNet  Google Scholar 

  15. Williams, F. A., J. Chem. Phys. 33 (1960) 133.

    Article  ADS  Google Scholar 

  16. Brzustowski, T. A., Can. J. Chem. Eng. 43 (1965) 30.

    Article  Google Scholar 

  17. Strahle, W. C., A Theoretical Study of Unsteady Droplet Burning: Transients and Periodic Solutions, Ph.D. Thesis, Princeton University, 1964.

  18. Romanelli, M. J., Runge-Kutta Methods for the Solution of Ordinary Differential Equations, Part 9 of Mathematical Methods for Digital Computers, Ed. by Ralston, A., and H. S. Wilf, John Wiley and Sons, Inc., New York, 1960.

    Google Scholar 

  19. Ingebo, R. D., Study of Pressure Effects on Vaporization Rate of Drops in Gas Streams, NACA TN-2850, Washington, D. C., 1953.

  20. Hall, A. R., and J. Diederichsen, 4th Int. Symposium on Combustion (1953) 837.

  21. Goldsmith, M., Jet Propulsion 26 (1956) 172.

    Google Scholar 

  22. Kumagai, S., T. Sakai, and S. Okajima, 13th Int. Symposium on Combustion, (1971) 779.

  23. Hoffman, T. W., and W. H. Gauvin, Can. J. Chem. Eng. 38 (1960) 129.

    Google Scholar 

  24. Ro, P. S., T. S. Fahlen, and H. C. Bryant, Applied Optics 7 (1968) 883.

    ADS  Google Scholar 

  25. Houghton, H. G., Physics 4 (1933) 419.

    Article  Google Scholar 

  26. Langstroth, G. O., C. H. H. Diehl, and E. J. Winhold, Can. J. Res. 28A (1950) 580.

    Google Scholar 

  27. Hirschfelder, J. O., C. F. Curtiss, and R. B. Bird, Molecular Theory of Gases and Liquids, John Wiley and Sons, Inc., New York, 1954.

    MATH  Google Scholar 

  28. Williams, F. A., Combustion Theory, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965.

    Google Scholar 

  29. Merk, H. J., Appl. Sci. Res. A8 (1958) 73.

    MathSciNet  Google Scholar 

  30. Keenan, J. H., Thermodynamics, John Wiley and Sons, Inc., New York, 1941.

    Google Scholar 

  31. Rossini, F. D., et al, Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds, A.P.I. Project 44, Carnegie Press, Pittsburgh, Penn., 1953.

    Google Scholar 

  32. Reid, R. C., and T. K. Sherwood, The Properties of Gases and Liquids, McGraw-Hill Book Co., Inc., New York, 1958.

    Google Scholar 

  33. Keenan, J. H., and F. G. Keyes, Thermodynamic Properties of Steam, John Wiley and Sons, Inc., New York, 1936.

    Google Scholar 

  34. Klein, V. A., Chem. Eng. Prog. 45 (1949) 675.

    Google Scholar 

  35. Haggenmacher, J. E., J. Am. Chem. Soc. 68 (1946) 1123.

    Article  Google Scholar 

  36. Martin, J. J., and J. B. Edwards, AIChE J. 11 (1965) 331.

    Article  Google Scholar 

  37. Mason, E. A., and S. C. Saxena, Phys. Fluids 1 (1958) 361.

    Article  MathSciNet  ADS  Google Scholar 

  38. Tondon, P. K., and S. C. Saxena, Appl. Sci. Res. 19 (1968) 163.

    Article  Google Scholar 

  39. Hirschfelder, J. O., J. Chem. Phys. 26 (1957) 274.

    Article  ADS  Google Scholar 

  40. Thermophysical Properties Research Center, Data Book, Vol. 2, Nonmetallic Elements, Compounds and Mixtures, Purdue Research Foundation, Lafayette, Indiana, 1964.

  41. Lee, C. Y., and C. R. Wilke, I. and E. C. 46 (1954) 2381.

    Article  Google Scholar 

  42. Grob, A. K., and M. M. El-Wakil, J. Heat Transfer 91 (1969) 259.

    Google Scholar 

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  1. J. C. Kent

    Present address: Lockheed Palo Alto Reserach Laboratory, 94304, Palo Alto, California

Authors and Affiliations

  1. Research Laboratories, General Motors Corporation, 48090, Warren, Michigan, USA

    J. C. Kent

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Kent, J.C. Quasi-steady diffusion-controlled droplet evaporation and condensation. Appl. Sci. Res. 28, 315–360 (1973). https://doi.org/10.1007/BF00413076

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  • DOI: https://doi.org/10.1007/BF00413076

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