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Explicit Solutions to Phase-Change Problems and Applications

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Explicit Solutions to Phase-Change Problems and Applications 2016 Computational Mathematics Annual Program Review Arlington, VA (USA), August 8-11, 2016 EXPLICIT SOLUTIONS TO PHASE-CHANGE PROBLEMS AND APPLICATIONS Domingo Alberto Tarzia Grant AFOSR – SOARD FA 9550-14-1-0122 CONICET - Depto. Matemática, FCE, Univ. Austral, Paraguay 1950, S2000FZF Rosario, Argentina. E-mail: [email protected] Key Words: Free boundary problems, Lamé-Clapeyron-Stefan problem, Phase-change problems, Heat- Diffusion equation, Fractional diffusion, Caputo fractional derivative, Explicit solutions, Neumann solution, Unknown thermal coefficients, Over-specified boundary condition, Mushy region, Convective boundary condition. Mathematics Subject Classification 2010: 26A33, 35C05, 35R11, 35R35, 80A22. 2 ABSTRACT We obtain explicit solutions (in a closed form) for the following one-dimensional phase- change problems: • Two-phase solidification process with a convective boundary condition (T. Thermal Sci (2016), In Press); • One-phase solidification process with a mushy region and a convective boundary condition (T., J. Appl. Math. (2015)); • One-phase solidification process with a mushy region and a heat flux boundary condition (T., J. Appl. Math. (2015)); • Determination of one unknown thermal coefficient through a solidification problem with an over-specified boundary condition (Ceretani-T., Math. Problems in Eng. (2015)); AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 3 • Simultaneous determination of two unknown thermal coefficients through a solidification problem with an over-specified boundary condition (Ceretani-T., JP Journal Heat Mass Transfer (2016)); • Two-phase fractional Stefan problem with a temperature boundary condition (Roscani-T., Adv. Math. Sci. Appl. (2014)); • Two-phase fractional Stefan problem with a heat flux boundary condition (Roscani- T., In progress); • Determination of one unknown thermal coefficient through a one-phase fractional Stefan problem with an over-specified boundary condition (T. Appl. Math. (2015)); AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 4 INTRODUCTION Heat and mass transfer problems with a phase-change process (known in the literature as Stefan problem or Lamé-Clapeyron-Stefan problem) begins by modeling the solidification of the Earth planet (1831). Phase-change process such as melting, freezing, solidification, sublimation, desublimation, thawing and drying have been studied in the last century due to their wide scientific and technological applications. Some of them are: i) Continuous casting of steel (e.g., ingots, slabs); ii) Binary alloy solidification problem (liquidus and solidus curves); iii) Permafrost process (e.g., frozen ground in cold region of the Earth planet); iv) Manufacture of glass and crystal; v) Phase-change process with a mushy region; vi) Thawing in a saturated porous medium with the influence of the pressure on the melting temperature; vii) Coupled heat and mass transfer for a phase-change process (e.g., Luikov systems); viii) Coupled heat and mass transfer during the freezing of the high-water content materials with two free boundaries: the freezing and sublimation fronts (e.g., for food preservation); ix) Ablation process (e.g., entry of a spacecraft to the atmosphere of the Earth planet); x) Welding process; AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 5 xi) Solidification of a supercooled liquid (temperature of the liquid is below of the melting temperature); xii) Phase-change materials to store solar energy as the latent heat of melting (e.g., heating of a house); and some related free boundary problems as: xiii) Oxygen diffusion-consumption in a tissue (equivalent to the phase-change problem); xiv) Isothermal diffusion-reaction process of a gas with a solid; xv) Gas flow through a porous medium; xvi) Penetration of solvent in polymers; xvii) Filtration of water through oil in a porous medium; xviii) Nutrient uptake by a growing root system. AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 6 A review of a long bibliography on moving and free boundary problems for phase- change materials (PCM) for the heat equation is given in: T., "A bibliography on moving-free boundary problems for the heat-diffusion equation. The Stefan and related problems", MAT - Serie A, 2 (2000), 1-297 (with 5869 titles). Available from: http://web.austral.edu.ar/descargas/facultad-cienciasEmpresariales/mat/Tarzia-MAT- SerieA-2(2000).pdf Obtaining explicit solution we have: i) a tool to get super and sub-solution for general conditions; ii) a benchmark solution for checking on the correctness of a computer code A review on explicit solutions for heat and mass transfer problems is given in: T., “Explicit and Approximated Solutions for Heat and Mass Transfer Problems with a Moving Interface”, Chapter 20, In Advanced Topics in Mass Transfer, Mohamed El-Amin (Ed.), InTech Open Access Publisher, Rijeka (2011), pp. 439-484. Available from: http://www.intechopen.com/articles/show/title/explicit-and-approximated-solutions-for-heat- and-mass-transfer-problems-with-a-moving-interface AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 7 OUTLINE I. Classical two-phase Stefan problems Classical Neumann solution for a temperature boundary condition at the fixed face x= 0 Classical Neumann solution with a heat flux boundary condition at the fixed face x= 0 Classical Neumann solution with a convective boundary condition at the fixed face x= 0 II. Classical one-phase Stefan problem with a simple mushy region One-phase with a convective boundary condition One-phase with a heat flux boundary condition Applications: o Determination of one unknown thermal coefficient o Simultaneous determination of two unknown thermal coefficients AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 8 III. Phase-change problems for the fractional heat-diffusion equation Fractional derivatives o Caputo fractional derivative o Wright and Mainardi Functions o Basic properties Two-phases fractional Stefan problems: o Generalized Neumann solution with a temperature boundary condition o Generalized Neumann solution with a heat flux boundary condition One-phase fractional Stefan problem: o Determination of one unknown thermal coefficient IV. Papers which are not reviewed AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 9 I. Classical two-phase Stefan problems Classical Neumann solution with a temperature boundary condition at the fixed face x= 0 (Weber, Book (1901)). Classical Neumann solution with a heat flux boundary condition at the fixed face x= 0 (T., Quart. Appl. Math. (1981)). Classical Neumann solution with a convective boundary condition at the fixed face x= 0 (T. Thermal Science (2016), In Press) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 10 I.1) CLASSICAL NEUMANN SOLUTION (MELTING PROCESS) WITH A TEMPERATURE BOUNDARY CONDITION Problem (P1) The two-phase Lamé-Clapeyron-Stefan problem is given by: find the free boundary x= s() t , and the temperature T= T(,) x t defined by: T( x , t )> T if 0 < x < s ( t ), t > 0 f = = > T( x , t ) Tf if x s ( t ), t 0 (T1) < < > Ts( x , t ) T f if s ( t ) x , t 0 such that the following equations and conditions are satisfied: c T− k T =0, x > s ( t ), t > 0, (T2) s st s s xx c T− k T =0, 0 < x < s ( t ), t > 0, (T3) t xx s(0)= 0, (T4) = +∞ = < > > Ts(,0) x T s (,) t T i T f , x 0, t 0, (T5) = > Ts( s ( t ), t ) T f , t 0, (T6) = > Tl( s ( t ), t ) T f , t 0, (T7) kTstt( ( ),) − kTstt( ( ),) = st ( ), t > 0, (T8) s sx x = > > T (0, t ) T0 Tf , t 0. (T9) AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 11 AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 12 < < Theorem T1 [Weber, Book (1901)] Let TTTi f 0 be. The solution of the problem (P1) is the classical Neumann explicit solution given by: TT− = −0 f x ≤ ≤ > TxtT ( , )0 erf , 0 xstt ( ), 0, (T10) erf () 2 t TT− = +f i x ≤ > TxtTs( , ) i erfc , stxt ( ) , 0, (T11) erfc() 2 st = =k =2 =ks = 2 = s s( t ) 2 s t , , s s , , (T12) c cs where the dimensionless coefficient > 0 is the unique solution of the following equation: G( x )= x , x > 0 (T13) with AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 13 = − G()()() x b4 G 2 x b 3 G 1 x , (T14) 2 x erf( x )=∫ exp( − u2 ) du , erfc ( x ) = 1 − erf ( x ), (T15) 0 exp()()− x2 exp − x 2 G(),() x= G x = , (T16) 1erfc()() x 2 erf x c()() T− T k T − T =s f i > = 0 f > b30, b 4 0. (T17) s AFOSR - Comput. Math., Arlington (VA, USA), 8-11 August 2016 Tarzia, Explicit Solutions to Phase-Change Problems & Appl. 14 I.2) CLASSICAL NEUMANN SOLUTION (MELTING PROCESS) WITH A HEAT FLUX BOUNDARY CONDITION = = Problem (P2 ) Find the free boundary x s() t , and the temperature T T(,) x t defined by: T( x , t )> T if 0 < x < s ( t ), t > 0 f = = > T( x , t ) Tf if x s ( t ), t 0 (F1) < < > Ts( x , t ) T f if s ( t ) x , t 0 such that the following equations and conditions are satisfied: c T− k T =0, x > s ( t ), t > 0, (F2) s st s s xx c T− k T =0, 0 < x < s ( t ), t > 0, (F3) t xx s(0)= 0, (F4) = +∞ = < > > Ts(,0) x T s (,) t T i T f , x 0, t 0, (F5) = > Ts( s ( t ), t ) T f , t 0, (F6) = > Tl( s ( t ), t ) T f , t 0, (F7) kTstt( ( ),) − kTstt( ( ),) = st ( ), t > 0, (F8) s sx x q k T(0, t )= −0 , t > 0.
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