Research Papers

Analytical and Computational Methods for Solidification Problems of Liquid That Expands During Freezing

[+] Author and Article Information
Deqi Liu

University of Lyon, INSA-Lyon,
F-69621 France
e-mail: deqi.liu@insa-lyon.fr

Hubert Maigre

University of Lyon, INSA-Lyon,
F-69621 France
e-mail: hubert.maigre@insa-lyon.fr

Fabrice Morestin

University of Lyon, INSA-Lyon,
F-69621 France
e-mail: fabrice.morestin@insa-lyon.fr

Philippe Géoris

Plastic Omnium, Clean Energy Systems,
60280 Venette, France
e-mail: philippe.georis@plasticomnium.com

1Corresponding author.

Contributed by the Heat Transfer Division of ASME for publication in the Journal of Thermal Science and Engineering Applications. Manuscript received December 18, 2018; final manuscript received February 15, 2019; published online May 20, 2019. Assoc. Editor: Gerard F. Jones.

J. Thermal Sci. Eng. Appl 11(6), 061012 (May 20, 2019) (13 pages) Paper No: TSEA-18-1679; doi: 10.1115/1.4043261 History: Received December 18, 2018; Accepted February 16, 2019

Both analytical and computational methods for solidification problems are introduced. First, the inward solidification process in a spherical vessel is studied. Expressions of the stress, displacement in the solid phase, and the liquid pressure are deduced based on the solidification interface position. A phase-change expansion orientation factor is introduced to characterize the nonisotropic expansion behavior at the freezing interface. Then, an efficient coupled thermomechanical finite-element method is proposed to evaluate the thermal stress, strain, displacement, and pressure in solidification problems with highly nonlinear constitutive relations. Two particular methods for treating the liquid phase with fixed-grid approaches are introduced. The thermal stress is computed at each integration point by integrating the elastoviscoplastic constitutive equations. Then, the boundary value problem described by the global finite-element equations is solved using the full Newton–Raphson method. This procedure is implemented into the finite-element package abaqus via a FORTRAN subroutine UMAT. Detailed implementation steps and the solution procedures are presented. The numerical model is validated first by the analytical solutions and then by a series of benchmark tests. Finally, an example of solidification in an open reservoir with a free liquid surface is introduced. Potential industrial applications of the numerical model are presented.

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Wiesche, S., 2007, “Numerical Heat Transfer and Thermal Engineering of Adblue (scr) Tanks for Combustion Engine Emission Reduction,” Appl. Therm. Eng., 27(11–12), pp. 1790–1798. [CrossRef]
Morgan, K., 1981, “A Numerical Analysis of Freezing and Melting With Convection,” Comput. Methods Appl. Mech. Eng., 28(3), pp. 275–284. [CrossRef]
Nithiarasu, P., 2000, “An Adaptive Finite Element Procedure for Solidification Problems,” Heat Mass Transfer, 36(3), pp. 223–229. [CrossRef]
Voller, V. R., Cross, M., and Markatos, N. C., 1987, “An Enthalpy Method for Convection/Diffusion Phase Change,” Int. J. Numer. Methods Eng., 24(1), pp. 271–284. [CrossRef]
Voller, V. R., 1990, “Fast Implicit Finite-Difference Method for the Analysis of Phase Change Problems,” Heat Transfer, 17(2), pp. 155–169.
Kurtze, D. A., 1991, “A Fixed-Grid Finite Element Method for Solidification,” Numer. Methods Free Bound. Problems, 99(1), pp. 235–241. [CrossRef]
Usmani, A. S., Lewis, R. W., and Seetharamu, K. N., 1992, “Finite Element Modelling of Natural Convection Controlled Change of Phase,” Int. J. Numer. Methods Fluids, 14(9), pp. 1019–1036. [CrossRef]
Wintruff, I., Gunther, C., and Class, G., 2001, “An Interface-Tracking Control-Volume Finite Element Method for Melting and Solidification Problems,” Numer. Heat Transfer B, 39(2), pp. 101–125.
Michalek, T., and Kowalewski, T. A., 2003, “Simulations of the Water Freezing Process - Numerical Benchmarks,” TASK Q., 7(3), pp. 389–408.
Bellet, M., Decultieux, F., Menai, M., Bay, F., Levaillant, C., and Svensson, I. L., 1996, “Thermomechanics of the Cooling Stage in Casting Processes: Three-Dimensional Finite Element Analysis and Experimental validation,” Metallurgical Mater. Trans. B, 27(1), pp. 81–99. [CrossRef]
Bellet, M., Jaouen, O., and Poitrault, I., 2005, “An ALE-FEM Approach to the Thermomechanics of Solidification Processes With Application to Prediction of Pipe Shrinkage,” J. Numer. Methods Heat Fluid Flow, 15(2), pp. 120–142. [CrossRef]
de Saracibar, C. A., Cervera, M., and Chiumenti, M., 2001, “On the Constitutive Modeling of Coupled Thermomechanical Phase-Change Problems,” Int. J. Plast., 17(12), pp. 1565–1622. [CrossRef]
Cervera, M., de Saracibar, C. A., and Chiumenti, M., 1999, “Thermo–Mechanical Analysis of Industrial Solidification Processes,” Int. J. Numer. Methods Eng., 46, pp. 1575–1591. [CrossRef]
Chiumenti, M., Cervera, M., and de Saracibar, C. A., 2006, “Coupled Thermomechanical Simulation of Solidification and Cooling Phases in Casting Processes,” MCWASP 2006, Opio, France, 4-6 June 2006 M. Schlesinger, ed., TMS 2006.
Koric, S., and Thomas, B. G., 2006, “Efficient Thermo-Mechanical Model for Soldification Processes,” Int. J. Numer. Methods Eng. 66(12), pp. 1955–1989. [CrossRef]
Koric, S., and Thomas, B. G., 2007, “Thermo-Mechanical Models of Steel Solidification Based on Two Elastic Visco-Plastic Constitutive Laws,” J. Mater. Process. Technol., 197(1–3), pp. 408–418.
Koric, S., Hibbeler, L. C., and Thomas, B. G., 2008, “Explicit Coupled Thermo-Mechanical Finite Element Model of Steel Solidification,” Int. J. Numer. Methods Eng., 78(1), pp. 1–31. [CrossRef]
Zhu, H., 1993, “Coupled Thermal-Mechanical Finite-Element Model With Application to Initial Solidification,” Ph.D Thesis. University of Illinois, Champaign, IL.
Li, C., and Thomas, B. G., 2004, “Thermo-Mechanical Finite-Element Model of Shell Behaviour in Continuous Casting of Steel,” Metallur. Mater. Trans. B, 35B(6), pp. 1151–1172. [CrossRef]
Weeks, W. F., and Wettlaufer, J. S., 1996, “Crystal Orientations in Floating Ice Sheets,” The Johannes Weertman Symposium, Anaheim, CA, Feb. 4–8, TMS 1996, pp. 337–350.
Schulson, E. M., and Duval, P., 2009, Creep and Fracture of Ice, Cambridge University Press, Cambridge.
Weiner, J. H., and Boley, B. A., 1963, “Elastic-Plastic Thermal Stresses in a Solidifying Body,” J. Mech. Phys. Solids, 11(3), pp. 145–154. [CrossRef]
Zhekamukhov, M. K., and Shokarov, K. B., 2003, “Calculation of Stresses and Strains Developing in Freezing of Water in Closed Vessels,” J. Eng. Phys. Thermophys., 76(1), pp. 210–221. [CrossRef]
Tao, L. C., 1967, “Generalized Numerical Solutions of Freezing a Saturated Liquid in Cylinders and Spheres,” AIChE J., 13(1), pp. 165–169. [CrossRef]
McCue, S. W., Wu, B., and Hill, J. M., 2008, “Classical Two-Phase Stefan Problem for Spheres,” Proc. R. Soc. London A, 464(2096), pp. 2055–2076. [CrossRef]
Bellet, M., and Thomas, B. G., 2007, “Solidifcation Macroprocesses–Thermal-Mechanical Modeling of Stress, Distortion and Hot Tearing,” Materials Processing Handbook, CRC Press, Taylor and Francis, Boca Raton.
Koric, S., Hibbeler, L. C., Liu, R., and Thomas, B. G., 2010, “Multiphysics Model of Metal Solidification on the Continuum Level,” Numer. Heat Transfer, Part B: Fundamentals, 58(6), pp. 371–392. [CrossRef]
Li, J., Saharan, A., Koric, S., and Ostoja-Starzewski, M., 2012, “Elastic-Plastic Transition in Three-Dimensional Random Materials: Massively Parallel Simulations, Fractal morphogenesis and Scaling Functions,” Philos. Mag., 92(22), pp. 2733–2758. [CrossRef]
abaqus 6.14 documentation,2014, Analysis User's Guide.
Chalmers, B., 1970, “Principles of Solidification,” Applied Solid State Physics, Springer, Boston, MA.
Budd, W. F., and Jacka, T. H., 1989, “A review of Ice Rheology for Ice Sheet Modelling,” Cold Regions Sci. Technol., 16(2), pp. 107–144. [CrossRef]
Weertman, J., 1983, “Creep Deformation of Ice,” Annu. Rev. Earth. Planet. Sci., 11(1), pp. 215–240. [CrossRef]
Dunne, F., and Petrinic, N., 2005, Introduction to Computational Plasticity, Oxford University Press, New York.
de Souza Neto, E., Peric, D., and Owen, D., 2008, Computational Methods for Plasticity: Theory and Applications, John Wiley & Sons Ltd., Chichester.
Schulson, E. M., 2001, “Brittle Failure of Ice,” Eng. Fract. Mech., 68(17–18), pp. 1839–1887. [CrossRef]
Lush, A. M., Weber, G., and Anand, L., 1989, “An Implicit Time-Integration Procedure for a Set of Integral Variable Constitutive Equations for Isotropic Elasto-Viscoplasticity,” Int. J. Plast., 5(5), pp. 521–549. [CrossRef]
Simo, J. C., and Taylor, R. L., 1985, “Consistent Tangent Operator for Rate-Independent Elastoplasticity,” Comput. Methods Appl. Mech. Eng., 48(1), pp. 101–118. [CrossRef]
Gupta, S. C., 1987, “Analytical and Numerical Solutions of Radially Symmetric Inward Solidification Problems in Spherical Geometry,” Int. J. Heat. Mass. Transfer., 30(12), pp. 2611–2616. [CrossRef]
Borja, R. I., 2013, Plasticity –Modeling & Computation, Springer, New York.
Glen, J. W., 1955, “The Creep of Polycrystalline Ice,” Proc. R. Soc. London A, 228(1175), pp. 519–538. [CrossRef]


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Fig. 1

Freezing urea solution in a tank: (a) thermal simulation (conduction) and (b) freezing test of urea solution in a tank

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Fig. 2

Isotropic and non-isotropic phase-change expansion (PCE)

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Fig. 3

Spherical inward solidification model

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Fig. 4

Liquid pressure versus solidification interface position

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Fig. 5

Stress profiles with different interface positions, κL/κs = 1

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Fig. 6

Ideal coupled system for solidification problems

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Fig. 7

Definition of thermal volume expansion (TVE)

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Fig. 8

Thermal strain components in the mushy zone

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Fig. 9

Flowchart of the solution procedure

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Fig. 10

Rigid spherical model

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Fig. 11

Validation: radial temperature distribution

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Fig. 12

Validation: solidification interface position versus time

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Fig. 13

Fluid cavity (FC) method for spherical solidification model

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Fig. 14

Validation: liquid pressure with elastic solid phase

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Fig. 15

Liquid pressure error to analytical solution for η = 1 case

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Fig. 16

Validation: stress profiles at t = 1500 s: (a) elastic solid and (b) isotropic hardening solid

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Fig. 17

Validation: liquid pressure with plastic solid phase

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Fig. 18

Stress profiles with different liquid treatment methods

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Fig. 19

Experiment setup and FE model for the steel spherical vessel test

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Fig. 20

Temperature and pressure evolution

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Fig. 21

Free liquid surface model

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Fig. 22

Contour plots of the free liquid surface reservoir model

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Fig. 23

Liquid pressure and maximal vertical displacement of the free surface



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