Vapor solidification of saturated air in two-dimensional stagnation flow

Document Type : Article

Authors

Department of Mechanical Engineering, Payame Noor University, Mashhad, 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.

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Main Subjects


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. E_cient 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 nanouid", 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  nanouid", 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 ow  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).