Vapor solidification of saturated air in two-dimensional stagnation flow

Document Type : Article

Authors

1 Department of mechanical engineering, Payame Noor University, Iran, Mashhad, Iran. Tel: 0985138683887

2 Department of mechanical engineering, Payame Noor University, Iran.

Abstract

In the present paper, stagnation flow solidification of vapor from saturated air is investigated. Saturated air with strain rate a impinges on a flat plate and, because the plate temperature is below the freezing temperature of water, condensation occurs and an ice layer forms on the plate. The ice surface is modeled as an accelerated flat plate moving toward the impinging fluid. The unsteady Navier-Stokes equations were subjected to a similarity transformation to obtain a single ordinary differential equation for the velocity distribution. Two methods of solution were used for the energy equation: a finite-difference numerical technique and a numerical solution of a similarity equation; these two results were compared to establish accuracy. Freezing time first increases as the far-field temperature decreases from above zero degrees Celcius and then rapidly approaches zero as the far-field temperature approaches zero Celcius. Despite the physical experiment, here the size of the nearest cell to the substrate controls the time at which condensation begins. As a result, maximum time before freezing begins occurs at about 5℃ air temperature, with the cell size 0.01 or 0.02 mm. The air temperature distribution and the ultimate frozen thickness for two saturated air temperatures are also presented.

Keywords

Main Subjects


References
1. Stefan, J. About the theory of ice formation, in
particular on the ice formation in polar seas", [Uber
die Theorie der Eisbildung, Insbesondere uber die
Eisbildung in Polarmaere], A. Phys. Chem., 42, pp.
269{286 (1891).
2. Goodrich, L.E. Ecient numerical technique for one
dimensional thermal problems with phase change",
Int. Journal of Heat Mass Transfer, 21, pp. 615{621
(1978).
3. Sparrow, E.M., Ramsey, J.W., and Harris, J.S. The
transition from natural convection controlled freezing
to conduction controlled freezing", Journal of Heat
Transfer, 103, pp. 7{13 (1983).
4. Lacroix, M. Computation of heat transfer during
melting of a pure substance from an isothermal wall",
Numer. Heat Transfer B, 15, pp. 191{210 (1989).
5. Yeoh, G.H., Behnia, M., De Vahl Davis, G. et al. A
numerical study of three-dimensional natural convection
during freezing of water", Int. J. Numer. Mech.
Engng., 30, pp. 899{914 (1990).
6. Hadji, L. and Schell, M. Interfacial pattern formation
in the presence of solidi cation and thermal convection",
Phy., Rev. A, 41, pp. 863{873 (1990).
7. Hanumanth, G.S. Solidi cation in the presence of
natural convection", Int. Comm. Heat Mass Transfer,
17, pp. 283{292 (1990).
8. Oldenburg, C.M. and Spera, F.J. Hybrid model for
solidi cation and convection", Numerical Heat Transfer,
Part B: Fundamentals, 21(2), pp. 217-229 (1992).
9. Trapaga, G., Matthys, E.F., Valecia, J.J. et al. Fluid

ow, heat transfer and solidi cation of molten metal
droplets impinging on substrates: comparison of numerical
and experimental results", Metall. Trans. B,
23B, pp. 701-718 (1992).
10. Watanabe, T.I., Kuribayashi, and Honda, et al. Deformation
and solidi cation of a droplet on a cold
substrate", Cham. Engng. 47, pp. 3059-3065 (1992).
11. Marchi, C. San, Liu, H., Lavernia, et al. Numerical
analysis of the deformation and solidi cation of a single
droplet impinging on to a
at substrate", Journal
Mater., 28, pp. 3313{3321 (1993).
12. Weidman, P.D. and Mahalingam, S. Axisymmetric
stagnation-point
ow impinging on a transversely oscillating
plate with suction", J. Engineering Mathematics,
31, pp. 305{318 (1997).
13. Shokrgozar Abbasi, A. and Rahimi, A.B. Nonaxisymmetric
three-dimensional stagnation-point
ow
and heat transfer on a
at plate", Trans. ASME
J. Fluids Engineering, 131(7), pp. 074501.1{074501.5
(2009).
14. Shokrgozar Abbasi, A. and Rahimi, A.B. Threedimensional
stagnation-point
ow and heat transfer on
a
at plate with transpiration", J. Thermophys. Heat
Transfer, 23(3), pp. 513{521 (2009).
15. Sha q, A., Hammouch, Z. and Turab Impact of
radiation in a stagnation point
ow of Walters' B

uid towards a Riga plate", Thermal Science and
Engineering Progress, 6, pp. 33{37 (2018).
16. Hammouch, Z., Mekkaoui, T., and Sadki, H. Similarity
solutions of a steady MHD
ow over a semiin
nite surface", Mathematics in Engineering, Science
& Aerospace (MESA), 8(1), pp. 109{117 (2017).
17. Eriksson, D., Gasch, T., and Malm, R. Freezing of
partially saturated air-entrained concrete: A multiphase
description of the hygro-thermomechanical behavior",
International Journal of Solids and Structures,
152{153, pp. 294{304 (2018).
18. Naganthran, K., Nazar, R., and Pop, I. Unsteady
stagnation-point
ow and heat transfer of a special
third grade
uid past a permeable stretching/shrinking
sheet", Scienti c Reports, 6(6), p. 24632 (2016).
19. Farooq, M., Javed, M., Ijaz Khan, M., et al. Melting
heat transfer and double strati cation in stagnation

ow of viscous nano
uid", Results in Physics, 7, pp.
2296-2301 (2017).
20. Hayat, T., Ijaz Khan, M., Waqas, M., et al. Numerical
simulation for melting heat transfer and radiation
e ects in stagnation point
ow of carbon-water
nano
uid", Computer Methods in Applied Mechanics
and Engineering, 315, pp. 1011-1024 (2017).
21. Shokrgozar Abbasi, A., Rahimi, A.B. and Niazmand,
H. Exact solution of three-dimensional unsteady stagnation

ow on a heated plate", J. Thermophys. Heat
Transfer, 25(1), pp. 55{58 (2011).
22. Shokrgozar Abbasi, A. and Rahimi, A.B. Investigation
of two-dimensional unsteady stagnation
ow and
heat transfer impinging on an accelerated
at plate",
Trans. ASME J. Heat Transfer, 134(6), pp. 064501.1{
064501.5 (2012).
A. Shokrgozar Abbasi and M. Ghayeni/Scientia Iranica, Transactions B: Mechanical Engineering 27 (2020) 693{703 703
23. Shokrgozar Abbasi, A., Rahimi, A.B., and Mozayyeni,
H. Investigation of three-dimensional axisymmetric
unsteady stagnation-point
ow and heat transfer impinging
on an accelerated
at plate", J. Applied Fluid
Mechanics (JAFM), 9(1), Part 2, pp. 451{461 (2016).
24. Brattkus, K. and Davis, S.H. Flow induced morphological
instabilities: stagnation-point
ows", Journal
of Crystal Growth, 89, pp. 423-427 (1988).
25. Rangel, R.H. and Bian, X. The inviscid stagnation-

ow solidi cation problem", Int. Journal of Heat Mass
Transfer, 39(8), pp. 1591{1602 (1994).
26. Lambert, R.H. and Rangel, R.H. Solidi cation of
a supercooled liquid in stagnation-point
ow", Int.
Journal of Heat Mass Transfer, 46, pp. 4013{4021
(2003).
27. Yoo, J.S. E ect of viscous plane stagnation
ow on
the freezing of
uid", Int. Journal of Heat and Fluid
Flow, l 21, pp. 735{739 (2000).
28. Mohammadiun, H., Amerian, V., Mohammadiun, M.,
et al. Axisymmetric stagnation-point
ow and heat
transfer of nano-
uid impinging on a cylinder with
constant wall heat
ux", Thermal Science, 23(5B),
pp. 3153{3164 (2019). DOI: 10.2298/TSCI171124090M
(2018)
29. Rahimi, A.B. and Mozayeni, H. Similarity solutions of
axisymmetric stagnation-point
ow and heat transfer
of a viscous, boussinesq-related density
uid on a
moving
at plate", Scientia Iranica, B, 21(4), pp.
1440{1450 (2014).
30. Alizadeh, R., Rahimi, A.B., and Naja , M. Nonaxisymmetric
stagnation-point
ow and heat transfer
of a viscous
uid on a stationary cylinder", Scientia
Iranica, 23(5), pp. 2238{2246 (2016).
31. Shokrgozar Abbasi, A. and Rahimi, A.B. Solidi cation
of two-dimensional viscous, incompressible stagnation

ow", Int. Journal of Heat Transfer, 135, pp.
072301{8 (2013).
32. Shokrgozar Abbasi, A. Three-dimensional axisymmetric
solidi cation of a viscous incompressible
flow
in the stagnation point region", J. Applied Fluid
Mechanics (JAFM), 10(1), Part 2, pp. 413{420 (2017).
33. Carslaw, H.S. and Jaeger, J.C., Conduction of Heat in
Solids, 2nd Edn., Chapter 2, Oxford Science Publications
(2000).