The Growth of Ice Crystals by Molecular Diffusion

Hyun Youk Department of Physics, University of Toronto, Toronto, Ontario, Canada

Search for other papers by Hyun Youk in
Current site
Google Scholar
PubMed
Close
,
Roland List Department of Physics, University of Toronto, Toronto, Ontario, Canada

Search for other papers by Roland List in
Current site
Google Scholar
PubMed
Close
, and
Theophilus Ola Department of Physics, University of Toronto, Toronto, Ontario, Canada

Search for other papers by Theophilus Ola in
Current site
Google Scholar
PubMed
Close
Restricted access

Abstract

The mass transfer of water molecules by diffusion onto ice particles is best described by their Sherwood number (Sh), a dimensionless quantity, which combines molecular and convective effects and depends on the airflow as represented by the Reynolds number (Re). While Sh (Re > 0) has been previously measured in experiments for typical crystal shapes, the limiting case of pure molecular diffusion (Sh0) for zero flow with Re = 0 is not known well and needs independent determination.

The direct numerical solution of the controlling Laplace equation links diffusion with electric fields through the electrostatic analogy. It will be solved for the electrostatic potential V around a crystal-shaped conductor of capacitance C. The results will then be converted by similarity theory. This led to the first numerical determination of Sh0 for hexagonal plates, hexagonal columns, stellar crystals, capped columns, and broad-branched crystals. The new data represent another necessary step in the formulation of an experiment-based theory of the growth of freely falling ice crystals in the atmosphere.

A discrete version of Gauss's flux law is developed to compute the flux generated by a crystal-shaped conductor in a finite Cartesian grid box, using a Gauss–Seidel iterative scheme. This method is general and can be applied to compute Sh0 for any rectilinear shapes to any degree of accuracy. The dimensionless mass transfer by molecular diffusion, Sh0, is identical to the diffusion of heat characterized by the Nusselt number Nu0.

Corresponding author address: Prof. Roland List, Dept. of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada. Email: list@atmosp.physics.utoronto.ca

Abstract

The mass transfer of water molecules by diffusion onto ice particles is best described by their Sherwood number (Sh), a dimensionless quantity, which combines molecular and convective effects and depends on the airflow as represented by the Reynolds number (Re). While Sh (Re > 0) has been previously measured in experiments for typical crystal shapes, the limiting case of pure molecular diffusion (Sh0) for zero flow with Re = 0 is not known well and needs independent determination.

The direct numerical solution of the controlling Laplace equation links diffusion with electric fields through the electrostatic analogy. It will be solved for the electrostatic potential V around a crystal-shaped conductor of capacitance C. The results will then be converted by similarity theory. This led to the first numerical determination of Sh0 for hexagonal plates, hexagonal columns, stellar crystals, capped columns, and broad-branched crystals. The new data represent another necessary step in the formulation of an experiment-based theory of the growth of freely falling ice crystals in the atmosphere.

A discrete version of Gauss's flux law is developed to compute the flux generated by a crystal-shaped conductor in a finite Cartesian grid box, using a Gauss–Seidel iterative scheme. This method is general and can be applied to compute Sh0 for any rectilinear shapes to any degree of accuracy. The dimensionless mass transfer by molecular diffusion, Sh0, is identical to the diffusion of heat characterized by the Nusselt number Nu0.

Corresponding author address: Prof. Roland List, Dept. of Physics, University of Toronto, 60 St. George Street, Toronto, ON M5S 1A7, Canada. Email: list@atmosp.physics.utoronto.ca

Save
  • Beard, K. V., and H. R. Pruppacher, 1971: A wind tunnel investigation of the rate of evaporation of small water drops falling at terminal velocity in air. J. Atmos. Sci, 28 , 14551464.

    • Search Google Scholar
    • Export Citation
  • Chiruta, M., and P. K. Wang, 2003: The capacitance of rosette ice crystals. J. Atmos. Sci, 60 , 836846.

  • Houghton, H. G., 1950: A preliminary quantitative analysis of precipitation mechanisms. J. Meteor, 7 , 363369.

  • Jayaweera, K. O. L. F., 1971: Calculations of ice crystal growth. J. Atmos. Sci, 28 , 728736.

  • List, R., and R. S. Schemenauer, 1971: Free-fall behavior of planar snow crystals, conical graupel and small hail. J. Atmos. Sci, 28 , 110115.

    • Search Google Scholar
    • Export Citation
  • McDonald, J. E., 1963: Use of electrostatic analogy in studies of ice crystal growth. Z. Agnew. Math. Phys, 14 , 610619.

  • Pasternak, I. S., and W. H. Gauvin, 1960: Turbulent heat and mass transfer from stationary particles. Can. J. Chem. Eng, 38 , 3542.

  • Schemenauer, R. S., 1972: The convective mass transfer of snow crystal, conical graupel, and conical small hail models. Ph.D. thesis, University of Toronto, 146 pp.

  • Schemenauer, R. S., and R. List, 1978: Measurements of the convective mass transfer of planar and columnar ice crystals. Borovikov Memorial Issue, Academy of Sciences of the USSR, 217–232.

    • Search Google Scholar
    • Export Citation
  • Schuepp, P. H., and R. List, 1969: Mass transfer of rough hailstone models in flows of various turbulence levels. J. Appl. Meteor, 8 , 254263.

    • Search Google Scholar
    • Export Citation
  • Strauss, W. A., 1992: Partial Differential Equations: An Introduction. John Wiley and Sons, 425 pp.

  • Todd, C., 1964: A system for computing ice phase hydrometeor development. Rep. ARG 64, Pa-121, Meteorology Research Inc., 30 pp.

  • Zheng, G., and R. List, 1996: Convective heat transfer of rotating spheres and spheroids with non-homogenous surface temperatures. Int. J. Heat Mass Transfer, 39 , 18151826.

    • Search Google Scholar
    • Export Citation
All Time Past Year Past 30 Days
Abstract Views 0 0 0
Full Text Views 177 45 8
PDF Downloads 130 35 6