An accurate numerical solution for the transient heating of an evaporating spherical droplet. (English) Zbl 1223.80009

Mitchell, S. L.; Vynnycky, M.; Gusev, I. G.; Sazhin, S. S.

Appl. Math. Comput. 217, No. 22, 9219-9233 (2011).

The authors consider the time-dependent Stefan problem which is extended in such a way to be convenient for the study of transient heating of an evaporating spherical droplet. The mathematical model is reduced by means of corresponding variable transformations to the classical one-dimensional Fourier heat conduction equation together with corresponding boundary conditions. Since the problem is numerically analyzed, the form of the equations obtained, is feasible for the numrical study and the second-order accuracy in time and space is achieved.
The authors demonstrate that a version of the Keller box finite-difference scheme, together with the boundary immobilization method can serve as a reliable vehicle for solving this kind of problems. The paper contains a very broad numerical study and demonstrates that the solutions obtained are highly accurate. The paper is written clearly and can be of use in many physical and engineering practical problems.

Reviewer: B. D. Vujanović (Novi Sad)

MSC:

80A22	Stefan problems, phase changes, etc.
80M20	Finite difference methods applied to problems in thermodynamics and heat transfer

Keywords:

droplets; diesel fuel; heating; evaporation; Stefan problem; keller box scheme

References:

[1]	Caldwell, J.; Kwan, Y. Y., Numerical methods for one-dimensional Stefan problems, Commun. Numer. Methods Eng., 20, 535-545 (2004) · Zbl 1048.65095
[2]	Caldwell, J.; Savović, S., Numerical solution of Stefan problem by variable space grid and boundary immobilization method, J. Math. Sci., 13, 67-79 (2002)
[3]	Caldwell, J.; Kwan, Y. Y., Starting solutions for the boundary immobilization method, Commun. Numer. Methods Eng., 21, 289-295 (2005) · Zbl 1112.80011
[4]	Savović, S.; Caldwell, J., Finite-difference solution of one-dimensional Stefan problem with periodic boundary conditions, Int. J. Heat Mass Transfer, 46, 2911-2916 (2003) · Zbl 1041.80004
[5]	Kutluay, S.; Bahadir, A. R.; Ozdes, A., The numerical solution of one-phase classical Stefan problem, J. Comput. Appl. Math., 81, 1, 135-144 (1997) · Zbl 0885.65102
[6]	Meek, P. C.; Norbury, J., Nonlinear moving boundary problems and a Keller box scheme, SIAM J. Numer. Anal., 21, 5, 883-893 (1984) · Zbl 0558.65087
[7]	Mitchell, S. L.; Vynnycky, M., Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems, Appl. Math. Comput., 215, 1609-1621 (2009) · Zbl 1177.80078
[9]	Esen, A.; Kutluay, S., A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method, Appl. Math. Comput., 148, 2, 321-329 (2004) · Zbl 1034.65070
[10]	Caldwell, J.; Chan, C.-C., Spherical solidification by the enthalpy method and the heat balance integral method, Appl. Math. Model., 24, 45-53 (2000) · Zbl 0944.80001
[11]	Liu, F.; McElwain, D. L.S., A computationally efficient solution technique for moving-boundary problems in finite media, IMA J. Appl. Math., 59, 1, 71-84 (1997) · Zbl 0958.80007
[12]	Rizwan-uddin, A nodal method for phase change moving boundary problems, Int. J. Comput. Fluid Dyn., 11, 211-221 (1999) · Zbl 0962.76074
[13]	Caldwell, J.; Chiu, C. K., Numerical solution of one-phase Stefan problems by the heat balance integral method. Part I - Cylindrical and spherical geometries, Commun. Numer. Methods Eng., 16, 569-583 (2000) · Zbl 0964.65109
[14]	Caldwell, J.; Chiu, C. K., Numerical solution of one-phase Stefan problems by the heat balance integral method. Part II - Special small time starting procedure, Commun. Numer. Methods Eng., 16, 585-593 (2000) · Zbl 0964.65110
[15]	Goodman, T. R., The heat-balance integral and its application to problems involving a change of phase, Trans. ASME, 80, 335-342 (1958)
[16]	Mitchell, S. L.; Myers, T. G., A heat balance integral method for one-dimensional finite ablation, AIAA J. Thermophys., 22, 2, 508-514 (2008)
[17]	Mitchell, S. L.; Myers, T. G., Approximate solution methods for one-dimensional solidification from an incoming fluid, Appl. Math. Comput., 202, 1, 311-326 (2008) · Zbl 1208.76142
[18]	Myers, T. G.; Mitchell, S. L.; Muchatibaya, G.; Myers, M. Y., A cubic heat balance integral method for one-dimensional melting of a finite thickness layer, Int. J. Heat Mass Transfer, 50, 5305-5317 (2007) · Zbl 1140.80389
[19]	Myers, T. G., Optimizing the exponent in the heat balance and refined integral methods, Int. Commun. Heat Mass Transfer, 36, 2, 143-147 (2009)
[20]	Mitchell, S. L.; Myers, T. G., Application of standard and refined heat balance integral methods to one-dimensional Stefan problems, SIAM Rev., 52, 1, 57-86 (2010) · Zbl 1188.80004
[21]	Sazhin, S. S.; Krutitskii, P. A.; Abdelghaffar, W. A.; Sazhina, E. M.; Mikhalovsky, S. V.; Meikle, S. T.; Heikal, M. R., Transient heating of diesel fuel droplets, Int. J. Heat Mass Transfer, 47, 3327-3340 (2004) · Zbl 1079.76659
[22]	Sazhin, S. S., Advanced models of fuel droplet heating and evaporation, Prog. Energy Combust. Sci., 32, 2, 162-214 (2006)
[23]	Sazhin, S. S.; Krutitskii, P. A.; Gusev, I. G.; Heikal, M. R., Transient heating of an evaporating droplet, Int. J. Heat Mass Transfer, 53, 2826-2836 (2010) · Zbl 1191.80028
[24]	Vynnycky, M., Concerning closed-streamline flows with discontinuous boundary conditions, J. Eng. Math., 33, 141-156 (1998) · Zbl 0902.76023
[25]	Sazhin, S. S.; Shishkova, I. N.; Kryukov, A. P.; Levashov, V. Y.; Heikal, M. R., Evaporation of droplets into a background gas: kinetic modelling, Int. J. Heat Mass Transfer, 50, 2675-2691 (2007) · Zbl 1119.80371
[26]	Sazhin, S. S.; Shishkova, I. N., A kinetic algorithm for modelling the droplet evaporation process in the presence of heat flux and background gas, Atom. Sprays, 19, 5, 473-489 (2009)
[27]	Cebeci, T.; Thiele, F.; Williams, P. G.; Stewartson, K., On the calculation of symmetric wakes. I - Two-dimensional flows, Numer. Heat Transfer, 2, 35-60 (1979)
[28]	Strikwerda, J. C., Finite Difference Schemes and Partial Differential Equations (2004), Society for Industrial Mathematics · Zbl 1071.65118
[29]	Cebeci, T.; Bradshaw, P., Momentum Transfer in Boundary Layers (1977), Hemisphere Publishing Corporation: Hemisphere Publishing Corporation Washington · Zbl 0424.76023
[30]	Cebeci, T.; Bradshaw, P., Physical and Computational Aspects of Convective Heat Transfer (1984), Springer: Springer Verlag · Zbl 0545.76090
[31]	Cebeci, T.; Cousteix, J., Modeling and Computation of Boundary-Layer Flows (2005), Springer
[33]	Abramzon, B.; Sazhin, S. S., Droplet vaporization model in the presence of thermal radiation, Int. J. Heat Mass Transfer, 48, 1868-1873 (2005) · Zbl 1189.76616
[34]	Abramzon, B.; Sazhin, S. S., Convective vaporization of fuel droplets with thermal radiation absorption, Fuel, 85, 1, 32-46 (2006)
[35]	Åberg, J.; Vynnycky, M.; Fredriksson, H., Heat-flux measurements of industrial on-site continuous copper casting and their use as boundary conditions for numerical simulations, Trans. Ind. Inst. Met., 62, 443-446 (2009)
[36]	Vynnycky, M., A mathematical model for air-gap formation in vertical continuous casting: the effect of superheat, Trans. Ind. Inst. Met., 62, 495-498 (2009)
[38]	Roday, A. P.; Kazmierczak, M. J., Analysis of phase-change in finite slabs subjected to convective boundary conditions. Part I - Melting, Int. Rev. Chem. Eng. (Rapid Commun.), 1, 87-99 (2009)
[39]	Roday, A. P.; Kazmierczak, M. J., Analysis of phase-change in finite slabs subjected to convective boundary conditions. Part II - Freezing, Int. Rev. Chem. Eng. (Rapid Commun.), 1, 100-108 (2009)
[40]	Roday, A. P.; Kazmierczak, M. J., Melting and freezing in a finite slab due to a linearly decreasing free-stream temperature of a convective boundary condition, Thermal Sci., 2, 141-153 (2009)