Determination of one unknown thermal coefficient through a mushy zone model with a convective overspecified boundary condition
aa r X i v : . [ m a t h . A P ] M a r Determination of one unknown thermal coefficientthrough a mushy zone model with a convectiveoverspecified boundary condition
Ceretani, Andrea N. and Tarzia, Domingo A.
CONICET - Depto. Matem´atica, Facultad de Ciencias Empresariales, Universidad Austral,Paraguay 1950, S2000FZF Rosario, Argentina.E-mail: [email protected]; [email protected]
Abstract
A semi-infinite material under a solidification process with the Solomon-Wilson- Alex-iades’s mushy zone model with a heat flux condition at the fixed boundary is considered.The associated free boundary problem is overspecified through a convective boundarycondition with the aim of the simultaneous determination of the temperature, the twofree boundaries of the mushy zone and one thermal coefficient among the latent heatby unit mass, the thermal conductivity, the mass density, the specific heat and the twocoefficients that characterize the mushy zone. Bulk temperature and coefficients whichcharacterize the heat flux and the heat transfer at the boundary are assumed to be de-termined experimentally. Explicit formulae for the unknowns are given for the resultingsix phase-change problems, beside necessary and sufficient conditions on data in order toobtain them. In addition, relationship between the phase-change process solved in thispaper with an analogous process overspecified by a temperature boundary condition ispresented, and this second problem is solved by considering a large heat transfer coeffi-cient at the boundary in the problem with the convective boundary condition. Formulaefor the unknown thermal coefficients corresponding to both problems are summarized intwo tables.
Keywords : Phase Change, Convective Condition, Lam´e-Clapeyron-Stefan Problem, MushyZone, Solomon-Wilson-Alexiades Model, Unknown Thermal Coefficients. : 35R35 - 35C06 - 80A22
Heat transfer problems with a phase-change such as melting and freezing have been studied inthe last century due to their wide scientific and technological applications. Some books in thesubject are [1, 4–6, 10, 12, 16, 26, 35].In this paper we consider a phase-change process for a semi-infinite material, which is char-acterized by x >
0, that is initially assumed to be liquid at its melting temperature (whichwithout loss of generality we assume equal to 0 ◦ C). We consider this material under a solidi-fication process with the presence of a zone where solid and liquid coexist, known as ”mushyzone”, with a heat flux boundary condition imposed at the fixed face x = 0. We follow [30, 33]in considering three different regions in this type of solidification process:1. liquid region at temperature T ( x, t ) = 0: D l = { ( x, t ) ∈ R / x > r ( t ) , t > } ,2. solid region at temperature T ( x, t ) < D s = { ( x, t ) ∈ R / < x < s ( t ) , t > } ,3. mushy region at temperature T ( x, t ) = 0: D p = { ( x, t ) ∈ R / s ( t ) < x < r ( t ) , t > } ,being x = s ( t ) and x = r ( t ) the functions that characterize the free boundaries of the mushyzone. We also follow [30] in making the following assumptions on the structure of the mushyzone, which is considered as isothermal:1. the material contains a fixed portion of the total latent heat per unit mass (see condition(3) below),2. its width is inversely proportional to the gradient of temperature (see condition (4) below).Parameters involved in this problem are: l >
0: latent heat by unit mass, k >
0: thermal conductivity, ρ >
0: mass density, c >
0: specific heat,0 < ǫ <
1: one of the two coefficients which characterize the mushy zone, γ >
0: one of the two coefficients which characterize the mushy zone, q >
0: coefficient that characterizes the heat flux at x = 0, h >
0: coefficient that characterizes the heat transfer at x = 0, − D ∞ <
0: bulk temperature at x = 0.We suppose that five of the six thermal coefficients l , k , ρ , c , ǫ and γ of the solid phase areknown and that, by means of a change of phase experiment (solidification of the material at itsmelting temperature) we are able to measure the quantities q , h and − D ∞ .Encouraged by the recent works [31,34] and with the aim of the simultaneous determinationof the temperature T = T ( x, t ), the two free boundaries x = r ( t ) and x = s ( t ), and one unknownthermal coefficient among l , k , ρ , c , ǫ and γ , we impose an overspecified boundary condition [4]which consists of the specification of a convective condition at the fixed face x = 0 (see condition(7) below) of the material undergoing the phase-change process. This lead us to the followingfree boundary problem: ρcT t ( x, t ) − kT xx ( x, t ) = 0 0 < x < s ( t ) , t > T ( s ( t ) , t ) = 0 t > kT x ( s ( t ) , t ) = ρl [ ǫ ˙ s ( t ) + (1 − ǫ ) ˙ r ( t )] t > T x ( s ( t ) , t )( r ( t ) − s ( t )) = γ t > r (0) = s (0) = 0 (5) kT x (0 , t ) = q √ t t > kT x (0 , t ) = h √ t ( T (0 , t ) + D ∞ ) t > x = 0instead of the convective condition (7) considered in this paper. Moreover, the determinationof one unknown thermal coefficient for the one-phase Lam´e-Clapeyron-Stefan problem with anoverspecified heat flux condition at the fixed face x = 0 without a mushy zone was done in [32].2ther papers related to determination of thermal coefficients are [2, 3, 7–9, 11, 13–15, 17–25, 27–29, 36–40].The goal of this paper is to obtain the explicit solution to phase-change process (1)-(7)with one unknown thermal coefficient, and the necessary and sufficient conditions on datain order to obtain an explicit formula for the unknown thermal coefficient. In addition, weare interested in analysing the relationship between problem (1)-(7) and the phase-changeprocess given by (1)-(6) beside the Dirichlet boundary condition overspecified at x = 0 givenby (35) (see below). In particular, we are interested in solving the problem with Dirichletboundary condition through problem with convective boundary condition when large values ofthe coefficient h that characterizes the heat transfer at x = 0 are considered.The organization of the paper is as follows. In Section 2 we prove a preliminary result wherenecessary and sufficient conditions on data for the phase-change process (1)-(7) are given inorder to obtain the temperature T = T ( x, t ) and the two free boundaries x = r ( t ) and x = s ( t ).Based on this preliminary result, in Section 3 we present and solve six different cases for thephase-change process (1)-(7) according the choice of the unknown thermal coefficient among l , k , ρ , c , ǫ and γ . In Section 4 we discuss the relationship between the phase-change process(1)-(6) with the Dirichlet boundary condition (35) and the same process with the convectiveboundary condition (7). We show that the temperature T D = T D ( x, t ), the free boundaries x = r D ( t ) and x = s D ( t ), and the explicit formula for the unknown thermal coefficient l , k , ρ , c , ǫ or γ for the phase-change process (1)-(6) with the Dirichlet condition (35) can be obtainedthrough the phase-change process with convective condition given by (1)-(7) when h tends to+ ∞ . Explicit formulae for the unknown thermal coefficient for problems (1)-(7) and (1)-(6)and (35), beside restrictions on data that guarantees their validity, are summarizes in Table 1and 2 respectively. The following lemma represents the base on which the work in this Section will be structured.
Lemma 2.1.
The solution to problem (1)-(7) is given by: T ( x, t ) = q √ παk (cid:20) erf (cid:18) x √ αt (cid:19) − erf( ξ ) (cid:21) < x < s ( t ) , t > s ( t ) = 2 ξ √ αt t > r ( t ) = 2 (cid:20) ξ + γk q √ α exp ( ξ ) (cid:21) √ αt t > if and only if the parameters involved in problem (1)-(7) satisfy the following two equations: (cid:20) ξ + γk (1 − ǫ )2 q √ α exp ( ξ ) (cid:21) exp ( ξ ) = q ρl √ α (11)erf( ξ ) = kD ∞ q √ πα (cid:18) − q h D ∞ (cid:19) (12) where α = kρc represents the thermal diffusivity.Proof. The kind of phase-change processes considered in this article have the following general3olution [30, 31, 33]: T ( x, t ) = A + B erf (cid:18) x √ αt (cid:19) < x < s ( t ) , t > s ( t ) = 2 ξ √ αt t > r ( t ) = 2 µ √ αt t > A , B , ξ and µ depend on the particular specifications of the phase-changeprocess.In orden to have the solution to problem (1)-(7), we impose conditions (2)-(4), (6) and (7)on (13)-(15) and obtain that coefficients A , B and µ must be given by: A = − q √ παk erf( ξ ) , B = q √ παk and µ = ξ + γk exp ( ξ )2 q √ α , (16)which corresponds to solution (8)-(10), and the parameters involved in the problem must satisfyequations (11) and (12). (cid:4) As a consequence of Lemma 2.1, we know that we can solve the phase-change process(1)-(7) with one unknown thermal coefficient through the determination of the parameter ξ that characterizes one of the two free boundaries of the mushy zone and the unknown thermalcoefficient among l , k , ρ , c , ǫ and γ . In addition, we also know from Lemma 2.1 that we cando that by solving the system of equations (11)-(12). In this Section we present and solve six different cases for the phase-change process (1)-(7)according the choice of the unknown thermal coefficient among l , k , ρ , c , ǫ and γ .With the aim of organizing our work, we classify each case by making reference to the co-efficients which is necessary to know in order to solve it (see Lemma 2.1):Case 1: Determination of l and ξ , Case 2: Determination of γ and ξ ,Case 3: Determination of ǫ and ξ , Case 4: Determination of k and ξ ,Case 5: Determination of ρ and ξ , Case 6: Determination of c and ξ .In addition, with the goal of make our presentation more readable, in the following statementsand proofs we introduce several functions. We name this functions with a subscript accordingthe case where they arise. Theorem 3.1 (Case 1: determination of l and ξ ) . If in problem (1)-(7) we consider the thermalparameter l as an unknown, then its solution is given by (8)-(10) with l and ξ given by: l = r cρk q exp ( − ξ ) h ξ + γ (1 − ǫ ) √ kρc q exp ( ξ ) i (17) ξ = erf − D ∞ q r kρcπ (cid:18) − q h D ∞ (cid:19)! (18)4 f and only if the parameters q , h , D ∞ , k , ρ and c satisfy the following two inequalities: − q h D ∞ > D ∞ q r kρcπ (cid:18) − q h D ∞ (cid:19) < . (R2) Proof.
Due to properties of the error function, it follows that a necessary and sufficient conditionfor the existence and uniqueness of a positive solution to equation (12) is:0 < D ∞ q r kρcπ (cid:18) − q h D ∞ (cid:19) < l is the positive thermalcoefficient given by (17). (cid:4) Theorem 3.2 (Case 2: determination of γ and ξ ) . If in problem (1)-(7) we consider the thermalparameter γ as an unknown, then its solution is given by (8)-(10) with γ given by: γ = 2 q (1 − ǫ ) √ kρc (cid:18) q l r cρk − f ( ξ ) (cid:19) exp ( − ξ ) (19) and ξ given by (18), if and only if the parameters q , h , D ∞ , k , ρ , c and l satisfy inequalities(R1), (R2) and: f erf − D ∞ q r kρcπ (cid:18) − q h D ∞ (cid:19)!! < q l r cρk , (R3) where the real function f is defined by: f ( x ) = x exp ( x ) , x > . (20) Proof.
As we see in the proof of Theorem 3.1, a necessary and sufficient condition that guar-antees the existence and uniqueness of solution to equation (12) is that inequalities (R1) and(R2) hold, and in that case, the coefficient ξ is given by (18).On the other hand, it follows from equation (11) that γ is given by (19). This coefficient ispositive if and only if: f ( ξ ) < q l r cρk , (21)where f is the real function defined in (20). Taking into account the expression of ξ given in(18), we have that inequality (21) is equivalent to inequality (R3). (cid:4) Theorem 3.3 (Case 3: determination of ǫ and ξ ) . If in problem (1)-(7) we consider the thermalparameter ǫ as an unknown, then its solution is given by (8)-(10) with ǫ given by: ǫ = 1 − q γ √ kρc (cid:18) q l r cρk − f ( ξ ) (cid:19) exp ( − ξ ) (22) and ξ given by (18), if and only if the parameters q , h , D ∞ , k , ρ , c and γ satisfy inequalities(R1), (R2), (R3) and: γ √ kρc q exp " erf − D ∞ q r kρcπ (cid:18) − q h D ∞ (cid:19)! + f erf − D ∞ q r kρcπ (cid:18) − q h D ∞ (cid:19)!! > q l r cρk , (R4)5 here f is the real function defined in (20).Proof. Conditions (R1) and (R2), and the expression of ξ given in (18) arise in the same waythat in the precedent proofs. On the other hand, it follows from equation (11) that ǫ is givenby (22), being f the real function defined in (20). This coefficient is positive if and only if:2 q γ √ kρc (cid:18) q l r cρk − f ( ξ ) (cid:19) exp ( − ξ ) <
1. (23)Taking into account the expression of ξ given in (18), we have that inequality (23) is equivalentto inequality (R4). Finally, we have that ǫ given in (22) is less than 1 if and only if f ( ξ ) < q ρl √ α ,which, as we see in the proof of Theorem 3.2, is equivalent to condition (R3). (cid:4) Theorem 3.4 (Case 4: determination of k and ξ ) . If in problem (1)-(7) we consider the thermalparameter k as an unknown, then its solution is given by (8)-(10) with k given by: k = πρc q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (24) and ξ the unique solution of the equation: f ( x ) = cD ∞ l √ π (cid:18) − q h D ∞ (cid:19) , x > , (25) where f is the real function defined by: f ( x ) = x + γ √ π (1 − ǫ )2 D ∞ (cid:16) − q h D ∞ (cid:17) erf( x ) exp ( x ) erf( x ) exp ( x ) , x > , (26) if and only if the parameters q , h and D ∞ satisfy inequality (R1).Proof. The system of equations (11)-(12) is equivalent to: √ k = r πρc q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (27) f ( ξ ) = cD ∞ l √ π (cid:18) − q h D ∞ (cid:19) , (28)where f is the real function defined in (26). A necessary condition for existence of solution tothis system is that inequality (R1) holds. Then, if we assume that (R1) holds, we inmediatelyobtain that k is given by (24). To complete the proof only remains to demonstrate that equation(25) admits a unique positive solution. This follows from the fact that f is an increasingfunction such that f (0 + ) = 0 and f (+ ∞ ) = + ∞ . (cid:4) Theorem 3.5 (Case 5: determination of ρ and ξ ) . If in problem (1)-(7) we consider the thermalparameter ρ as an unknown, then its solution is given by (8)-(10) with ρ given by: ρ = πkc q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (29) and ξ the unique solution of the equation (25), if and only if the parameters q , h and D ∞ satisfy inequality (R1). roof. It is similar to the proof of Theorem 3.4. (cid:4)
Theorem 3.6 (Case 6: determination of c and ξ ) . If in problem (1)-(7) we consider the thermalparameter c as an unknown, then its solution is given by (8)-(10) with c given by: c = πρk q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (30) and ξ the unique solution of the equation: f ( x ) = q √ πρlkD ∞ (cid:16) − q h D ∞ (cid:17) , x > , (31) where f is the real function defined by: f ( x ) = x erf( x ) + γ √ π (1 − ǫ )2 D ∞ (cid:16) − q h D ∞ (cid:17) exp ( x ) exp ( x ) , x > , (32) if and only if the parameters q , h and D ∞ satisfy inequalities (R1) and: − q h D ∞ < D ∞ (cid:20) q ρlk − γ (1 − ǫ ) (cid:21) . (R5) Proof.
The system of equations (11)-(12) is equivalent to: √ c = r πρk q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (33) f ( x ) = q √ πρlkD ∞ (cid:16) − q h D ∞ (cid:17) (34)where f is the real function defined in (32). A necessary condition for existence of solution tothis system is that inequality (R1) holds. Then, if we assume that (R1) holds, we immediatelyobtain that c is given by (30). To complete the proof only remains to demonstrate that equation(31) admits a unique positive solution. Since f is an increasing function such that f (0 + ) = π + γ (1 − ǫ ) √ π D ∞ (cid:16) − q h D ∞ (cid:17) and f (+ ∞ ) = + ∞ , it follows that a necessary and sufficient condition forexistence (and uniqueness) of solution to equation (31) is that: π γ (1 − ǫ ) √ π D ∞ (cid:16) − q h D ∞ (cid:17) < q √ πρlkD ∞ (cid:16) − q h D ∞ (cid:17) ,which is equivalent to inequality (R5). (cid:4) Table 1 summarizes the results of this section, corresponding to 6 cases.7ase Thermal coefficient Coefficient ξ that characterizes Restrictions on datathe free boundary x = s ( t )1 l = q cρk q exp ( − ξ ) h ξ + γ (1 − ǫ ) √ kρc q exp ( ξ ) i ξ = erf − (cid:18) D ∞ q q kρcπ (cid:16) − q h D ∞ (cid:17)(cid:19) (R1), (R2)2 γ = q (1 − ǫ ) √ kρc (cid:18) q l q cρk − ξ exp ( ξ ) (cid:19) exp ( − ξ ) ξ = erf − (cid:18) D ∞ q q kρcπ (cid:16) − q h D ∞ (cid:17)(cid:19) (R1), (R2), (R3)3 ǫ = 1 − q γ √ kρc (cid:18) q l q cρk − ξ exp ( ξ ) (cid:19) exp ( − ξ ) ξ = erf − (cid:18) D ∞ q q kρcπ (cid:16) − q h D ∞ (cid:17)(cid:19) (R1), (R2), (R3), (R4)Unique positive solution of:4 k = πρc (cid:20) q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (cid:21) f ( x ) = cD ∞ l √ π (cid:16) − q h D ∞ (cid:17) , with f defined by (26) (R1)Unique positive solution of:5 ρ = πkc (cid:20) q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (cid:21) f ( x ) = cD ∞ l √ π (cid:16) − q h D ∞ (cid:17) , with f defined by (26) (R1)Unique positive solution of:6 c = πρk (cid:20) q erf( ξ ) D ∞ (cid:16) − q h D ∞ (cid:17) (cid:21) f ( x ) = q √ πρlkD ∞ (cid:16) − q h D ∞ (cid:17) , with f defined by (32) (R1), (R5)Table 1: Formulae for problem (1)-(7) . Explicit formulae for the unknown thermal coefficient l , γ , ǫ , k , ρ or c and coefficient ξ (or the equation that it must satisfy) and the corresponding restrictions on data that guarantees their validity. The phase-change process with large heat transfer co-efficient
A similar phase-change process to (1)-(7) with one unknown thermal coefficient have beenstudied in [33]. In that paper, the author consider a fusion process with a mushy zone given by(1)-(6) overspecified with a temperature boundary condition and obtain the explicit solutionto some cases. Encouraged by [33], let us consider the solidification process (1)-(6) with oneunknown thermal coefficient overspecified with the Dirichlet boundary condition: T (0 , t ) = − D ∞ , t >
0. (35)We can see this condition as the limit case of the convective boundary condition (7) whenthe heat transfer coefficient h tends to + ∞ . From a physical point of view, overspecify thephase-change process (1)-(6) by imposing the convective boundary condition (7) seems to bemore appropriate than imposing the Dirichlet condition (35). This Section is devoted to showthat the temperature T D = T D ( x, t ), the free boundaries x = r D ( t ) and x = s D ( t ), and theexplicit formula for the unknown thermal coefficient l , k , ρ , c , ǫ or γ for the phase-changeprocess with Dirichlet boundary condition given by (1)-(6) and (35) can be obtained throughthe phase-change process with convective boundary condition given by (1)-(7) when h tendsto + ∞ .We begin with a result related to the solution to the phase-change process (1)-(6) and (35).This result may be shown in much the same manner as Lemma 2.1, thus we do not give itsproof here. Lemma 4.1.
The solution to problem (1)-(6) and (35) is given by (8)-(10), that is: T D ( x, t ) = q √ παk (cid:20) erf (cid:18) x √ αt (cid:19) − erf( ξ ) (cid:21) < x < s D ( t ) , t > s D ( t ) = 2 ξ √ αt t > r D ( t ) = 2 (cid:20) ξ + γk q √ α exp ( ξ ) (cid:21) √ αt t > ,where α = kρc represents the thermal diffusivity, if and only if the parameters involved in problem(1)-(6) and (35) satisfy the following two equations: (cid:20) ξ + γ (1 − ǫ ) √ π D ∞ erf ( ξ ) exp ( ξ ) (cid:21) erf ( ξ ) exp ( ξ ) = cD ∞ l √ π (36)erf( ξ ) = D ∞ q r kρcπ . (37)From Lemma 4.1 we know that, in order to have the temperature T D = T D ( x, t ), the freeboundaries x = r D ( t ) and x = s D ( t ), and the unknown thermal coefficient l , k , ρ , c , ǫ or γ for problem (1)-(6) and (35), it is enough to find the unknown thermal coefficient and theparameter that characterizes the free boundary s D ( t ). Proceeding analogously to the work donein [33] or in Section 3, we can obtain the thermal coefficient l , k , ρ , c , ǫ or γ and the parameter ξ for problem (1)-(6) and (35). Formulae for those quantities, besides restrictions on data thatguarantee their validity, are summarized in Table 2 (restrictions on data and definitions onfunctions mentioned in Table 2 are listed below the table).9ase Thermal coefficient Coefficient ξ that characterizes Restrictions on datathe free boundary x = s D ( t )1 l = q cρk q exp ( − ξ ) h ξ + γ (1 − ǫ ) √ kρc q exp ( ξ ) i ξ = erf − (cid:18) D ∞ q q kρcπ (cid:19) (R6)2 γ = q (1 − ǫ ) √ kρc (cid:18) q l q cρk − ξ exp ( ξ ) (cid:19) exp ( − ξ ) ξ = erf − (cid:18) D ∞ q q kρcπ (cid:19) (R7)3 ǫ = 1 − q γ √ kρc (cid:18) q l q cρk − ξ exp ( ξ ) (cid:19) exp ( − ξ ) ξ = erf − (cid:18) D ∞ q q kρcπ (cid:19) (R7), (R8)Unique positive solution of:4 k = πρc h q erf( ξ ) D ∞ i F ( x ) = cD ∞ l √ π , with F defined by (40)Unique positive solution of:5 ρ = πkc h q erf( ξ ) D ∞ i F ( x ) = cD ∞ l √ π , with F defined by (40)Unique positive solution of:6 c = πρk h q erf( ξ ) D ∞ i F ( x ) = q √ πρlkD ∞ , with F defined by (41) (R9)Table 2: Formulae for problem (1)-(6) and (35) . Explicit formulae for the unknown thermal coefficient l , γ , ǫ , k , ρ or c and coefficient ξ (or the equation that it must satisfy) and the corresponding restrictions on data that guarantees their validity. ist of restrictions on data for problem (1)-(6) and (35) mentioned in Table 2: D ∞ q r kρcπ < D ∞ q r kρcπ < erf ( η ) (R7)where η is the unique positive solution to the equation: f ( x ) = q l r cρk , x >
0, (38)being f the real function defined in (20). D ∞ q r kρcπ > erf ( η ) , x >
0, (R8)where η is the unique positive solution to the equation: f ( x ) + γ √ kρc q exp (2 x ) = q l r ckρ , x >
0, (39)being f the real function defined in (20). lkρD ∞ q (cid:18) γ (1 − ǫ ) D (cid:19) < F ( x ) = erf ( x ) f ( x ) + (1 − ǫ ) γ √ π D ∞ [erf ( x )] exp (2 x ) , x >
0, (40)being f the real function defined in (20). F ( x ) = (cid:18) x erf ( x ) + γ (1 − ǫ ) √ π D ∞ exp ( x ) (cid:19) exp ( x ) , x > h tending to + ∞ .This fact, besides Lemmas 2.1 and 4.1, allow us to conclude that we can solve the phase-changeprocess (1)-(6) with one unknown thermal coefficient overspecified by the Dirichlet condition(35) through the phase-change process (1)-(7), which is overspecified by the more physicallyappropriate convective boundary condition (7), when the heat transfer coefficient h tends to+ ∞ . In this paper, we consider a semi-infinite material under a solidification process with a mushyzone caused by an initial heat flux boundary condition. We solve the associated free boundaryproblem overspecified with a convective boundary condition and obtain the temperature, thetwo free boundaries of the mushy zone and one thermal coefficient among the latent heat by11nit mass, the thermal conductivity, the mass density, the specific heat and the two coefficientsthat characterize the mushy zone, when the bulk temperature and the coefficients that char-acterize the heat flux and the heat transfer at the boundary are assumed to be known. As aconsequence, we give formulae for the temperature, the two free boundaries and the unknownthermal coefficient, beside necessary and sufficient conditions on data in order to obtain them.In addition, we present the relationship between the phase-change process studied in this pa-per with another similar phase-change process which is overspecified by a Dirichelt boundarycondition. From this relationship, we solve the problem with the Dirichlet condition by con-sidering a large heat transfer coefficient in the problem with the convective condition. In thisway, we solve the phase-change process overspecified with a temperature boundary conditionthrough the more physically appropriate phase-change problem overspecified with a convectiveboundary condition. We summarize explicit formulae for the unknown thermal coefficient forboth problems in Tables 1 and 2.
Competing interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgements
This paper has been partially sponsored by the Project PIP No. 0534 from CONICET-UA(Rosario, Argentina) and AFOSR-SOARD Grant FA 9550-14-1-0122.
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