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Research Papers: Melting and Solidification

Solidification of Two-Dimensional Viscous, Incompressible Stagnation Flow

[+] Author and Article Information
Asghar B. Rahimi

Professor
e-mail: rahimiab@yahoo.com
Faculty of Engineering,
Ferdowsi University of Mashhad,
P. O. Box No. 91775- 1111,
Mashhad, Iran

Three diagonal matrix algorithm.

Alternating direction implicit.

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received July 2, 2012; final manuscript received March 1, 2013; published online June 17, 2013. Assoc. Editor: Ali Ebadian.

J. Heat Transfer 135(7), 072301 (Jun 17, 2013) (8 pages) Paper No: HT-12-1344; doi: 10.1115/1.4023936 History: Received July 02, 2012; Revised March 01, 2013

The history of the study of fluid solidification in stagnation flow is limited to a few cases. Among these few studies, only some articles have considered the fluid viscosity and yet pressure variations along the thickness of the viscous layer have not been taken into account and the energy equation has been assumed to be one-dimensional. In this study the solidification of stagnation flows is modeled as an accelerated flat plate moving toward an impinging fluid. The unsteady momentum equations, taking the pressure variations along viscous layer thickness into account, are reduced to ordinary differential equations by the use of proper similarity variables and are solved by using a fourth-order Runge-Kutta integrating method at each prescribed interval of time. In addition, the energy equation is numerically solved at any step for the known velocity and the problem is presented in a two-dimensional Cartesian coordinate. Comparisons of these solutions are made with existing special ranges of past solutions. The fluid temperature distribution, transient velocity component distribution, and, most important of all the rate of solidification or the solidification front are presented for different values of nondimensional Prandtl and Stefan numbers. The results show that an increase of the Prandtl numbers (up to ten times) or an increase of the heat diffusivity ratios (up to two times) causes a decrease of the ultimate frozen thickness by almost half, while the Stefan number has no effect on this thickness and its effect is only on the freezing time.

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References

Stefan, J., 1891, “Uber die theorie der eisbildung, insbesondere uber die eisbildung in polarmaere,” Ann. Phys. Chem., 42, pp. 269–286.
Goodrich, L. E., 1978, “Efficient Numerical Technique for One-Dimensional Thermal Problems With Phase Change,” Int. J. Heat Mass Transfer, 21, pp. 615–621. [CrossRef]
Sparrow, E. M., Ramsey, J. W., and Harris, S., 1981, “The Transition From Natural Convection Controlled Freezing to Conduction Controlled Freezing,” ASME J. Heat Transfer, 103, pp. 7–13. [CrossRef]
Lacroix, M., 1989, “Computation of Heat Transfer During Melting of a Pure Substance From an Isothermal Wall,” Numer. Heat Transfer, Part B15, pp. 191–210. [CrossRef]
Yeoh, G. H., Behnia, M., De Vahl Davis, G., and Leonardi, E., 1990, “A Numerical Study of Three-Dimensional Natural Convection During Freezing of Water,” Int. J. Numer.Methods Eng., 30, pp. 899–914. [CrossRef]
Hadji, L. and Schell, M., 1990, “Interfacial Pattern Formation in the Presence of Solidification and Thermal Convection,” Phys. Rev. A, 41, pp. 863–873. [CrossRef] [PubMed]
Hanumanth, G. S., 1990, “Solidification in the Presence of Natural Convection,” Int. Commun. Heat Mass Transfer, 17, pp. 283–292. [CrossRef]
Oldenburg, C. M., and Spera, F. J., 1992, “Hybrid Model for Solidification and Convection,” Numer, Heat Transfer, Part B, 21, pp. 217–229. [CrossRef]
Trapaga, G., Matthys, E. F., Valecia, J. J., and Szekely, J., 1992, “Fluid Flow, Heat Transfer and Solidification of Molten Metal Droplets Impinging on Substrates: Comparison of Numerical and Experimental Results,” Metall. Trans. B, 23B, pp. 701–718. [CrossRef]
Watanabe, T., Kuribayashi, I., Honda, T., and Kanzawa, A., 1992, “Deformation and Solidification of a Droplet on a Cold Substrate,” Cham. Eng. Sci., 47, pp. 3059–3065. [CrossRef]
San Marchi, C., Liu, H., Lavernia, E. J., and Rangel, R. H., 1993, “Numerical Analysis of the Deformation and Solidification of a Single Droplet Impinging on to a Flat Substrate,” J. Mater. Sci., 28, pp. 3313–3321. [CrossRef]
Brattkus, K., and Davis, S. H., 1988, “Flow Induced Morphological Instabilities: Stagnation-Point Flows,” J. Cryst. Growth, 89, pp. 423–427. [CrossRef]
Rangel, R. H., and Bian, X., 1994, “The Inviscid Stagnation-Flow Solidification Problem,” Int. J. Heat Mass Transfer, 39(8), pp. 1591–1602. [CrossRef]
Lambert, R. A., and Rangel, R. H., 2003, “Solidification of a Super-Cooled Liquid in Stagnation-Point Flow,” Int. J. Heat Mass Transfer, 46, pp. 4013–4021. [CrossRef]
Rangel, R. H., and Bian, X., 1996, “The Viscous Stagnation-Flow Solidification Problem,” Int. J. Heat Mass Transfer, 39(17), pp. 3581–3594. [CrossRef]
Yoo, J. S., 2000, “Effect of Viscous Plane Stagnation Flow on the Freezing of Fluid,” Int. J. Heat Fluid Flow, 21, pp. 735–739. [CrossRef]
Shokrgozar Abbasi, A., and Rahimi, A. B., 2012, “Investigation of Two-Dimensional Unsteady Stagnation Flow and Heat Transfer Impinging on an Accelerated Flat Plate,” ASME J. Heat Transfer, 134(6), p. 064501. [CrossRef]
Hong, C., Yamamoto, T., Asako, Y., and Suzuki, K., 2012, “Heat Transfer Characteristics of Compressible Laminar Flow Through Microtubes,” ASME J. Heat Transfer, 134, p. 011602. [CrossRef]
Alassar, R., and Abushoshah, M., 2012, “Hagen-Poiseuille Flow in Semi-Elliptic Microchannels,” ASME J. Fluids Eng., 134, p. 124502. [CrossRef]
Norouzi, M., Rezaei Niya, S. M., Kayhani, M. H., Shariati, M., Karimi Demneh, M., and Naghavi, M. S., 2012, “Exact Solution of Unsteady Convective Heat Transfer in Cylindrical Composite Laminates,” ASME J. Heat Transfer, 134, p. 101301. [CrossRef]
Pitts, D. R., and Sissom, L. E., 1977, Theory and Problems of Heat Transfer, Schaum's Outlines Series, McGraw-Hill, New York.

Figures

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Fig. 4

Comparison of the present study and the Ref. [15] results for (Pr = 1, St = 1, kr = 1, αr = 1 and θi = 1)

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Fig. 5

Solidification front for Pr = 10 and (St = 1, kr = 1, αr = 1, and θi = 1)

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Fig. 7

Solidification front for θi = 0.5 and (Pr = 1, St = 1, kr = 1, and αr = 1)

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Fig. 8

Effect of the St number upon the solidification front for (Pr = 1, St = 1, kr = 1, and αr = 1)

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Fig. 9

Effect of the Pr number upon the solidification front for (St = 1, kr = 1, αr = 1, and θi = 1)

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Fig. 6

Solidification front for kr = αr = 0.5 and (Pr = 1, St = 1, and θi = 1)

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Fig. 3

Ultimate solidification thickness after stability of the thermal boundary layer for various Pr numbers and αr = 1

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Fig. 2

Explanation of strain and stream lines

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Fig. 12

Effect of the inlet temperature variations to the freeze point versus freeze time for (Pr = 10, St = 1, kr = 1, αr = 1, and θi = 1)

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Fig. 13

Comparison of the conduction and convection terms contribution for (Pr = 10, St = 1, kr = 1, αr = 1, and θi = 1)

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Fig. 10

Effect of the kr and αr variations upon the solidification front for (Pr = 1, St = 1, and θi = 1)

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Fig. 11

Effect of the θi variations upon the solidification front for (Pr = 1, St = 1, kr = 1, and αr = 1)

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Fig. 14

Evolution of the solidification front for water with α° = 1

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Fig. 15

Thermal profile for (Pr = 1, St = 1, kr = 1, αr = 1, and θi = 1)

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Fig. 16

Velocity profile in the x direction for (Pr = 1, St = 1, kr = 1, αr = 1, and θi = 1)

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Fig. 17

Velocity profile in the z direction for (Pr = 1, St = 1, kr = 1, αr = 1, and θi = 1)

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