Heat transfer problems with a phase-change such as melting and freezing have been studied in the last century due to their wide scientific and technological applications. A review of a long bibliography on moving and free boundary problems for phase-change materials (PCM) for the heat equation is shown in [16].
We consider the following solidification problem for a semi-infinite material with an over specified condition on the fixed face x = 0 [1], [3], [4] and [7]:
equation(1)
View the MathML source{i)ρcTt(x,t)=(k(T)Tx(x,t))x,0<x<s(t),t>0ii)T(0,t)=To<Tf,t>0iii)k(To)Tx(0,t)=qot,t>0,qo>0iv)T(s(t),t)=Tf,t>0v)k(Tf)Tx(s(t),t)=ρhs˙(t),t>0
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where T(x,t) is the temperature of the solid phase, ρ > 0 is the density of mass, h > 0 is the latent heat of fusion by unity of mass, c > 0 is the specific heat, x = s(t) is the phase-change interface, Tf is the phase-change temperature, To is the temperature at the fixed face x = 0 and qo is the coefficient that characterizes the heat flux at x = 0 given by Eq. (1iii), which must be obtained experimentally through a phase-change process [2]. We suppose that the thermal conductivity has the following expression [5]:
equation(2)
View the MathML sourcek=k(T)=ko[1+β(T−To)/(Tf−To)],β∈R.
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Let αo = ko/ρc be the coefficient of the diffusivity at the temperature To. We observe that if β = 0, the problem (1) becomes the classical one-phase Lamé-Clapeyron-Stefan problem with an overspecified condition at the fixed face x = 0, and for this problem the corresponding simultaneous determination of thermal coefficients was studied in [13] and [14]. The phase-change process with temperature-dependent thermal coefficient of the type (2) was firstly studied in [5]. Other papers related to determination of thermal coefficients are [8], [10], [11], [17], [18], [19] and [20].
The solution to problem (1) is given by [5] and [15]:
equation(3)
View the MathML source{i)T(x,t)=To+(Tf−To)Φ(λ)Φ(η),η=x2αot,0<η<λii)s(t)=2λαot
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where Φ = Φ(x) = Φδ(x) is the modified error function, for a given δ > -1, the unique solution to the following boundary value problem in variable x, i.e:
equation(4)
View the MathML source{i)[(1+δΦ′(x))Φ′(x)]′+2xΦ′(x)=0,x>0,ii)Φ(0+)=0,Φ(+∞)=1
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and the unknown thermal coefficients must satisfy the following system of equations [15]:
equation(5)
View the MathML sourceβ−δΦ(λ)=0
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equation(6)
View the MathML source[1+δΦ(λ)]Φ′(λ)λΦ(λ)−2hc(Tf−To)=0
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equation(7)
View the MathML sourceΦ′(0)Φ(λ)−2qo(Tf−To)koρc=0.
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For the particular case δ = 0 we have that Φ(x) = erf(x) is the error function, which is defined by:
equation(8)
View the MathML sourceerf(x)=2π∫0xe−u2du.
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We remark that if problem (1) is a free boundary problem (this case can be considered as a Stefan problem) with an overspecified condition on the fixed face x = 0, then the coefficient λ > 0 is an unknown coefficient. On the other hand, if problem (1) is a moving boundary problem (this case can be considered as an inverse Stefan problem) with an overspecified condition on the fixed face x = 0, then the phase-change interface will be given by
equation(3iibis)
View the MathML sources(t)=2σt
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where σ must be obtained experimentally (View the MathML sourceσ=λαo) through a phase-change process [2] and [14].
When the coefficient δ = 0, the corresponding determination of formulas for one or two unknown thermal coefficients were obtained in [13] and [14] and the numerical-experimental determination was given in [2]. When the coefficient δ ≠ 0 is given, the corresponding problem was analyzed in [15]; in this case, the necessary and sufficient conditions on the data were obtained in order to ensure the existence of the solution.
The goal of the present paper is to make the sensitivity analysis of the free and moving boundary problems, analyzed in [15]. For a one-phase Stefan problem, the temperature T(x,t), the free boundary interface s(t) (i.e. the coefficient λ, defined in Eq. (3ii), is also an unknown coefficient) and the following parameters in four different cases:
FB: i) λ, β, ko ii) λ, β, ρ iii) λ, β, h iv) λ, β, c,
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were determined in [15]. For a one-phase inverse Stefan problem (i.e. the interface s(t) is given by Eq. (3iibis) for a given σ > 0), the temperature T(x,t) and the following parameters in six cases:
MB: i) β, ko, ρ ii) β, ko, c iii) β, ko, h iv) β, ρ, c v) β, ρ, h vi) β, c, h,
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were also determined in [15].
The explicit formulas corresponding to the ten cases for the unknown thermal coefficients were summarized in [15] (see Table 1). For cases FB (iii and iv) and MB (ii, iv, v and vi) the data must satisfy certain restrictions in order to obtain the solution of the corresponding thermal problem. These restrictions, called R1, R2, R3 and R4 in [15], are the following:
equation(R1)
View the MathML source(Tf−To)Φ′(0)2qokoρc<1
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equation(R2)
View the MathML source(Tf−To)koρh2qo2<1
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equation(R3)
View the MathML sourceρσhqo<1
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equation(R4)
View the MathML source(Tf−To)ko2σqo<1.
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Table 1.
Left and right normalized sensitivities in the four cases of free boundary problems.
Case number Unknown Coefficients δ ko ρ c h
1 λ 0.008 0.008 – – 0 0 0.48 0.48 − 0.48 − 0.48
β 1.01 1.01 – – 0 0 0.46 0.46 − 0.46 − 0.46
ko − 0.015 − 0.015 – – − 1.01 − 0.99 − 0.082 − 0.081 − 0.92 − 0.9
2 λ 0.008 0.008 0 0 – – 0.48 0.48 − 0.48 − 0.48
β 1.01 1.01 0 0 – – 0.46 0.45 − 0.46 − 0.45
ρ − 0.015 − 0.015 − 1.01 − 0.99 – – − 0.082 − 0.081 − 0.92 − 0.9
3 λ 0.016 0.016 0.52 0.52 0.52 0.52 – – 0.52 0.52
β 1.02 1.02 0.5 0.49 0.5 0.49 – – 0.5 0.49
h − 0.016 − 0.016 − 1.1 − 1.07 − 1.1 − 1.07 – – − 0.089 − 0.088
4 λ − 0.084 − 0.084 − 5.85 − 6.08 − 5.85 − 6.08 − 5.85 − 6.08 – –
β 0.92 0.92 − 5.52 − 5.78 − 5.52 − 5.78 − 5.52 − 5.78 – –
c − 0.19 − 0.19 − 12.4 − 12.1 − 12.4 − 12.1 − 11.3 − 11.2 – –
