# Determination of One Unknown Thermal Coefficient through a Mushy Zone Model with a Convective Overspecified Boundary Condition

Authors: Andrea N. Ceretani; Domingo A. Tarzia
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

A semi-infinite material under a solidification process with the Solomon-Wilson-Alexiades mushy zone model with a heat flux condition at the fixed boundary is considered. The associated free boundary problem is overspecified through a convective boundary condition with the aim of the simultaneous determination of the temperature, the two free boundaries of the mushy zone and one thermal coefficient among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat, and the two coefficients that characterize the mushy zone, when the unknown thermal coefficient is supposed to be constant. Bulk temperature and coefficients which characterize the heat flux and the heat transfer at the boundary are assumed to be determined experimentally. Explicit formulae for the unknowns are given for the resulting six phase-change problems, besides necessary and sufficient conditions on data in order to obtain them. In addition, relationship between the phase-change process solved in this paper and an analogous process overspecified by a temperature boundary condition is presented, and this second problem is solved by considering a large heat transfer coefficient at the boundary in the problem with the convective boundary condition. Formulae for the unknown thermal coefficients corresponding to both problems are summarized in two tables.


---

## Body

## 1. Introduction

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. Some books in the subject are [1–9].In this paper we consider a phase-change process for a semi-infinite material, which is characterized byx>0, that is initially assumed to be liquid at its melting temperature (which without loss of generality we assume to be equal to 0∘C). We consider this material under a solidification process with the presence of a zone where solid and liquid coexist, known as “mushy zone,” with a heat flux boundary condition imposed at the fixed face x=0. We follow [10, 11] in considering three different regions in this type of solidification process:(1)
Liquid region at temperatureT(x,t)=0: Dl=(x,t)∈R2/x>r(t),t>0.(2)
Solid region at temperatureT(x,t)<0: Ds=(x,t)∈R2/0<x<s(t),t>0.(3)
Mushy region at temperatureT(x,t)=0: Dp=(x,t)∈R2/s(t)<x<r(t),t>0.x=s(t) and x=r(t) are the functions that characterize the free boundaries of the mushy zone. We also follow [10] in making the following assumptions on the structure of the mushy zone, which is considered as isothermal:(1)
The material contains a fixed portion of the total latent heat per unit mass (see condition (3)).(2)
Its width is inversely proportional to the gradient of temperature (see condition (4)).Thermal coefficients involved in the solidification process are assumed to be constant. They arel>0: latent heat by unit mass,k>0: thermal conductivity,ρ>0: mass density,c>0: specific heat,0<ϵ<1: one of the two coefficients which characterize the mushy zone,γ>0: one of the two coefficients which characterize the mushy zone,q0>0: coefficient that characterizes the heat flux at x=0,h0>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 coefficientsl, k, ρ, c, ϵ, and γ of the solid phase are known and that, by means of a change of phase experiment (solidification of the material at its melting temperature), we are able to measure the quantities q0, h0, and -D∞.Encouraged by the recent works [12, 13] and with the aim of the simultaneous determination of temperature T=T(x,t), the two free boundaries x=r(t) and x=s(t), and one unknown thermal coefficient among l, k, ρ, c, ϵ, and γ, we impose an overspecified boundary condition [2] which consists of the specification of a convective condition at the fixed face x=0 (see condition (7)) of the material undergoing the phase-change process. This leads us to the following free boundary problem:(1)ρcTtx,t-kTxxx,t=00<x<st,t>0,(2)Tst,t=0t>0,(3)kTxst,t=ρlϵs˙t+1-ϵr˙tt>0,(4)Txst,trt-st=γt>0,(5)r0=s0=0,(6)kTx0,t=q0tt>0,(7)kTx0,t=h0tT0,t+D∞t>0.This problem was first studied in [11] with a temperature boundary condition at x=0 instead of the convective condition (7) considered in this paper. Moreover, the determination of one unknown thermal coefficient for the one-phase Lamé-Clapeyron-Stefan problem with an overspecified heat flux condition at the fixed face x=0 without a mushy zone was done in [14]. Other papers related to determination of thermal coefficients are [15–40].The goal of this paper is to obtain the explicit solution to the phase-change process (1)–(7) with one unknown thermal coefficient independent of position and time and the necessary and sufficient conditions on data in order to obtain an explicit formula for the unknown thermal coefficient. In addition, we are interested in analysing the relationship between problem (1)–(7) and the phase-change process given by (1)–(6) besides the Dirichlet boundary condition overspecified at x=0 given by (31) (see below). In particular, we are interested in solving the problem with Dirichlet boundary condition through problem with convective boundary condition when large values of the coefficient h0 that characterizes the heat transfer at x=0 are considered.The organization of the paper is as follows. In Section2 we prove a preliminary result where necessary and sufficient conditions on data for the phase-change process (1)–(7) are given in order 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 the phase-change process (1)–(7) according to the choice of the unknown thermal coefficient among l, k, ρ, c, ϵ, and γ. In Section 4 we discuss the relationship between the phase-change process (1)–(6) with the Dirichlet boundary condition (31) and the same process with the convective boundary condition (7). We show that temperature TD=TD(x,t), free boundaries x=rD(t) and x=sD(t), and the explicit formula for the unknown thermal coefficient l, k, ρ, c, ϵ, or γ for the phase-change process (1)–(6) with the Dirichlet condition (31) can be obtained through the phase-change process with convective condition given by (1)–(7) when h0 tends to +∞. Explicit formulae for the unknown thermal coefficient for problems (1)–(7) and (1)–(6) and (31), besides restrictions on data that guarantees their validity, are summarized in Tables 1 and 2, respectively.Table 1
Formulae for problem (1)–(7). Explicit formulae for the unknown thermal coefficient l, γ, ϵ, k, ρ, or c and coefficient ξ (or the equation that it must satisfy) and the corresponding restrictions on data that guarantee their validity.
Case
Thermal coefficient
Coefficientξ that characterizes the free boundary x=s(t)
Restrictions on data
1
l
=
c
ρ
k
q
0
e
x
p
-
ξ
2
ξ
+
γ
1
-
ϵ
k
ρ
c
/
2
q
0
e
x
p
ξ
2
ξ
=
er
f
-
1
D
∞
q
0
k
ρ
c
π
1
-
q
0
h
0
D
∞
R
1, R2
2
γ
=
2
q
0
1
-
ϵ
k
ρ
c
q
0
l
c
ρ
k
-
ξ
e
x
p
ξ
2
e
x
p
-
2
ξ
2
ξ
=
er
f
-
1
D
∞
q
0
k
ρ
c
π
1
-
q
0
h
0
D
∞
R
1, R2, R3
3
ϵ
=
1
-
2
q
0
γ
k
ρ
c
q
0
l
c
ρ
k
-
ξ
e
x
p
ξ
2
e
x
p
-
2
ξ
2
ξ
=
er
f
-
1
D
∞
q
0
k
ρ
c
π
1
-
q
0
h
0
D
∞
R
1, R2, R3, R4
4
k
=
π
ρ
c
q
0
erf
⁡
(
ξ
)
D
∞
1
-
q
0
/
h
0
D
∞
2
Unique positive solution off4x=cD∞lπ1-q0h0D∞, with f4 defined by (23)
R
1
5
ρ
=
π
k
c
q
0
erf
⁡
(
ξ
)
D
∞
1
-
q
0
/
h
0
D
∞
2
Unique positive solution off4x=cD∞lπ1-q0h0D∞, with f4 defined by (23)
R
1
6
c
=
π
ρ
k
q
0
erf
⁡
(
ξ
)
D
∞
1
-
q
0
/
h
0
D
∞
2
Unique positive solution off6x=q02πρlkD∞1-q0/h0D∞, with f6 defined by (28)
R
1, R5Table 2
Formulae for problem (1)–(6) and (31). Explicit formulae for the unknown thermal coefficient l, γ, ϵ, k, ρ, or c and coefficient ξ (or the equation that it must satisfy) and the corresponding restrictions on data that guarantee their validity.
Case
Thermal coefficient
Coefficientξ that characterizes the free boundary x=sD(t)
Restrictions on data
1
l
=
c
ρ
k
q
0
e
x
p
-
ξ
2
ξ
+
γ
1
-
ϵ
k
ρ
c
/
2
q
0
e
x
p
ξ
2
ξ
=
er
f
-
1
D
∞
q
0
k
ρ
c
π
R
6
2
γ
=
2
q
0
1
-
ϵ
k
ρ
c
q
0
l
c
ρ
k
-
ξ
e
x
p
ξ
2
e
x
p
-
2
ξ
2
ξ
=
er
f
-
1
D
∞
q
0
k
ρ
c
π
R
7
3
ϵ
=
1
-
2
q
0
γ
k
ρ
c
q
0
l
c
ρ
k
-
ξ
e
x
p
ξ
2
e
x
p
-
2
ξ
2
ξ
=
er
f
-
1
D
∞
q
0
k
ρ
c
π
R
7, R8
4
k
=
π
ρ
c
q
0
erf
⁡
(
ξ
)
D
∞
2
Unique positive solution ofF4(x)=cD∞lπ, with F4 defined by (36)
5
ρ
=
π
k
c
q
0
erf
⁡
(
ξ
)
D
∞
2
Unique positive solution ofF4(x)=cD∞lπ, with F4 defined by (36)
6
c
=
π
ρ
k
q
0
erf
⁡
(
ξ
)
D
∞
2
Unique positive solution ofF6(x)=q02πρlkD∞, with F6 defined by (37)
R
9


## 2. Explicit Solution to the Phase-Change Process

The following lemma represents the base on which the work in this section will be structured.Lemma 1.
The solution to problem (1)–(7) is given by(8)Tx,t=q0παkerf⁡x2αt-erf⁡ξ0<x<st,t>0,st=2ξαtt>0,rt=2ξ+γk2q0αexp⁡ξ2αtt>0if and only if the parameters involved in problem (1)–(7) satisfy the following two equations:(9)ξ+γk1-ϵ2q0αexp⁡ξ2exp⁡ξ2=q0ρlα,(10)erf⁡ξ=kD∞q0πα1-q0h0D∞,where α=k/ρc represents the thermal diffusivity.Proof.
The kind of phase-change processes considered in this paper has the following general solution [10–12]:(11)Tx,t=A+Berf⁡x2αt0<x<st,t>0,st=2ξαtt>0,rt=2μαtt>0,where coefficients A, B, ξ, and μ depend on the particular specifications of the phase-change process.
In order to have the solution to problem (1)–(7), we impose conditions (2)–(4), (6) and (7) on (11) and obtain that coefficients A, B, and μ must be given by(12)A=-q0παkerf⁡ξ,B=q0παk,μ=ξ+γkexp⁡ξ22q0α,which corresponds to solution (8), and the parameters involved in the problem must satisfy (9) and (10).As a consequence of Lemma1, we know that we can solve the phase-change process (1)–(7) with one unknown thermal coefficient through the determination of parameter ξ that characterizes one of the two free boundaries of the mushy zone and the unknown thermal coefficient among l, k, ρ, c, ϵ, and γ. In addition, we also know from Lemma 1 that we can do that by solving the system of (9)-(10).


## 3. Explicit Formula for the Unknown Thermal Coefficient

In this section we present and solve six different cases for the phase-change process (1)–(7) according to the choice of the unknown thermal coefficient among l, k, ρ, c, ϵ, and γ.With the aim of organizing our work, we classify each case by making reference to the coefficients which is necessary to know in order to solve it (see Lemma1):Case1: determination of l and ξ.Case2: determination of γ and ξ.Case3: determination of ϵ and ξ.Case4: determination of k and ξ.Case5: determination of ρ and ξ.Case6: determination of c and ξ.In addition, with the goal of making our presentation more readable, in the following statements and proofs we introduce several functions. We name these functions with a subscript according the case where they arise.Theorem 2 (Case1: determination of l and ξ).
If in problem (1)–(7) one considers the thermal parameter l as an unknown, then its solution is given by (8) with l and ξ given by(13)l=cρkq0exp⁡-ξ2ξ+γ1-ϵkρc/2q0exp⁡ξ2,(14)ξ=erf-1⁡D∞q0kρcπ1-q0h0D∞if and only if the parameters q0, h0, D∞, k, ρ, and c satisfy the following two inequalities:R11-q0h0D∞>0R2D∞q0kρcπ1-q0h0D∞<1.Proof.
Due to properties of the error function, it follows that a necessary and sufficient condition for the existence and uniqueness of a positive solution to (10) is(15)0<D∞q0kρcπ1-q0h0D∞<1,which is equivalent to inequalities R1 and R2. In that case, the positive solution to (10) is given by (14). Finally, it follows from (9) that l is the positive thermal coefficient given by (13).Theorem 3 (Case2: determination of γ and ξ).
If in problem (1)–(7) one considers the thermal parameter γ as an unknown, then its solution is given by (8) with γ given by(16)γ=2q01-ϵkρcq0lcρk-f2ξexp⁡-2ξ2and ξ given by (14), if and only if the parameters q0, h0, D∞, k, ρ, c, and l satisfy inequalities R1 and R2 andR3f2erf-1⁡D∞q0kρcπ1-q0h0D∞<q0lcρk,where the real function f2 is defined by(17)f2x=xexp⁡x2,x>0.Proof.
As we see in the proof of Theorem2, a necessary and sufficient condition that guarantees the existence and uniqueness of solution to (10) is that inequalities R1 and R2 hold, and, in that case, the coefficient ξ is given by (14).
On the other hand, it follows from (9) that γ is given by (16). This coefficient is positive if and only if(18)f2ξ<q0lcρk,where f2 is the real function defined in (17). Taking into account the expression of ξ given in (14), we have that inequality (18) is equivalent to inequality R3.Theorem 4 (Case3: determination of ϵ and ξ).
If in problem (1)–(7) one considers the thermal parameter ϵ as an unknown, then its solution is given by (8) with ϵ given by(19)ϵ=1-2q0γkρcq0lcρk-f2ξexp⁡-2ξ2and ξ given by (14), if and only if the parameters q0, h0, D∞, k, ρ, c, and γ satisfy inequalities R1, R2, and R3 andR4γkρc2q0·exp2erf-1⁡D∞q0kρcπ1-q0h0D∞2+f2erf-1⁡D∞q0kρcπ1-q0h0D∞>q0l·cρk,where f2 is the real function defined in (17).Proof.
ConditionsR1 and R2 and the expression of ξ given in (14) arise in the same way as that in the precedent proofs. On the other hand, it follows from (9) that ϵ is given by (19), f2 being the real function defined in (17). This coefficient is positive if and only if(20)2q0γkρcq0lcρk-f2ξexp⁡-2ξ2<1.Taking into account the expression of ξ given in (14), we have that inequality (20) is equivalent to inequality R4. Finally, we have that ϵ given in (19) is less than 1 if and only if f2(ξ)<q0/ρlα, which, as we see in the proof of Theorem 3, is equivalent to condition R3.Theorem 5 (Case4: determination of k and ξ).
If in problem (1)–(7) one considers the thermal parameter k as an unknown, then its solution is given by (8) with k given by(21)k=πρcq0erf⁡ξD∞1-q0/h0D∞2and ξ is the unique solution to the equation(22)f4x=cD∞lπ1-q0h0D∞,x>0,where f4 is the real function defined by(23)f4x=x+γπ1-ϵ2D∞1-q0/h0D∞erf⁡xexp⁡x2·erf⁡xexp⁡x2,x>0,if and only if the parameters q0, h0, and D∞ satisfy inequality R1.Proof.
The system of (9)-(10) is equivalent to(24)k=πρcq0erf⁡ξD∞1-q0/h0D∞,f4ξ=cD∞lπ1-q0h0D∞,where f4 is the real function defined in (23). A necessary condition for existence of solution to this system is that inequality R1 holds. Then, if we assume that R1 holds, we immediately obtain that k is given by (21). To complete the proof it only remains to demonstrate that (22) admits a unique positive solution. This follows from the fact that f4 is an increasing function such that f(0+)=0 and f(+∞)=+∞.Theorem 6 (Case5: determination of ρ and ξ).
If in problem (1)–(7) one considers the thermal parameter ρ as an unknown, then its solution is given by (8) with ρ given by(25)ρ=πkcq0erf⁡ξD∞1-q0/h0D∞2and ξ is the unique solution to (22), if and only if the parameters q0, h0, and D∞ satisfy inequality R1.Proof.
It is similar to the proof of Theorem5.Theorem 7 (Case6: determination of c and ξ).
If in problem (1)–(7) one considers the thermal parameter c as an unknown, then its solution is given by (8) with c given by(26)c=πρkq0erf⁡ξD∞1-q0/h0D∞2and ξ is the unique solution to the equation(27)f6x=q02πρlkD∞1-q0/h0D∞,x>0,where f6 is the real function defined by(28)f6x=xerf⁡x+γπ1-ϵ2D∞1-q0/h0D∞exp⁡x2·exp⁡x2,x>0,if and only if the parameters q0, h0, and D∞ satisfy inequality R1 andR51-q0h0D∞<1D∞2q02ρlk-γ1-ϵ.Proof.
The system of (9)-(10) is equivalent to(29)c=πρkq0erf⁡ξD∞1-q0/h0D∞,f6ξ=q02πρlkD∞1-q0/h0D∞,where f6 is the real function defined in (28). A necessary condition for existence of solution to this system is that inequality R1 holds. Then, if we assume that R1 holds, we immediately obtain that c is given by (26). To complete the proof it only remains to demonstrate that (27) admits a unique positive solution. Since f6 is an increasing function such that f(0+)=π/2+γ(1-ϵ)π/2D∞1-q0/h0D∞ and f(+∞)=+∞, it follows that a necessary and sufficient condition for existence (and uniqueness) of solution to (27) is that (30)π2+γ1-ϵπ2D∞1-q0/h0D∞<q02πρlkD∞1-q0/h0D∞,which is equivalent to inequality R5.Table1 summarizes the results of this section, corresponding to 6 cases.


## 4. The Phase-Change Process with Large Heat Transfer Coefficient

A similar phase-change process to (1)–(7) with one unknown thermal coefficient has been studied in [11]. In that paper, the author considers a fusion process with a mushy zone given by (1)–(6) overspecified with a temperature boundary condition and obtains the explicit solution to some cases. Encouraged by [11], we consider the solidification process (1)–(6) with one unknown thermal coefficient overspecified with the Dirichlet boundary condition:(31)T0,t=-D∞,t>0.We can see this condition as the limit case of the convective boundary condition (7) when the heat transfer coefficient h0 tends to +∞. From a physical point of view, overspecifying the phase-change process (1)–(6) by imposing the convective boundary condition (7) seems to be more appropriate than imposing the Dirichlet condition (31). This section is devoted to showing that the temperature TD=TD(x,t), the free boundaries x=rD(t) and x=sD(t), and the explicit formula for the unknown thermal coefficient l, k, ρ, c, ϵ, or γ for the phase-change process with Dirichlet boundary condition given by (1)–(6) and (31) can be obtained through the phase-change process with convective boundary condition given by (1)–(7) when h0 tends to +∞.We begin with a result related to the solution to the phase-change process (1)–(6) and (31). This result may be shown in much the same manner as Lemma 1; thus we do not give its proof here.Lemma 8.
The solution to problem (1)–(6) and (31) is given by (8); that is,(32)TDx,t=q0παkerf⁡x2αt-erf⁡ξ0<x<sDt,t>0,sDt=2ξαtt>0,rDt=2ξ+γk2q0αexp⁡ξ2αtt>0,where α=k/ρc represents the thermal diffusivity, if and only if the parameters involved in problem (1)–(6) and (31) satisfy the following two equations:(33)ξ+γ1-ϵπ2D∞erf⁡ξexp⁡ξ2erf⁡ξexp⁡ξ2=cD∞lπ,erf⁡ξ=D∞q0kρcπ.From Lemma8 we know that, in order to have temperature TD=TD(x,t), the free boundaries x=rD(t) and x=sD(t), and the unknown thermal coefficient l, k, ρ, c, ϵ, or γ for problem (1)–(6) and (31), it is enough to find the unknown thermal coefficient and the parameter that characterizes the free boundary sD(t). Proceeding analogously to the work done in [11] or in Section 3, we can obtain the thermal coefficient l, k, ρ, c, ϵ, or γ and the parameter ξ for problem (1)–(6) and (31). Formulae for those quantities, besides restrictions on data that guarantee their validity, are summarized in Table 2 (restrictions on data and definitions on functions mentioned in Table 2 are listed as follows).List of restrictions on data for problem (1)–(6) and (31) mentioned in Table 2 isR6D∞q0kρcπ<1,R7D∞q0kρcπ<erf⁡η,where η is the unique positive solution to the equation(34)f2x=q0lcρk,x>0,f2 being the real function defined in (17);R8D∞q0kρcπ>erf⁡η,x>0,where η is the unique positive solution to the equation(35)f2x+γkρc2q0exp⁡2x2=q0lckρ,x>0,f2 being the real function defined in (17);R9lkρD∞2q01+γ1-ϵD0<1.List of definitions of functions related to problem (1)–(6) and (31) mentioned in Table 2 is(36)F4x=erf⁡xf2x+1-ϵγπ2D∞erf⁡x2exp⁡2x2,x>0,f2 being the real function defined in (17);(37)F6x=xerf⁡x+γ1-ϵπ2D∞exp⁡x2exp⁡x2,x>0.On the other hand, it is not difficult to verify that formulae and restrictions on data given in Table2 correspond to formulae and restrictions on data given in Table 1 for h0 tending to +∞. This fact, besides Lemmas 1 and 8, allows us to conclude that we can solve the phase-change process (1)–(6) with one unknown thermal coefficient overspecified by the Dirichlet condition (31) through the phase-change process (1)–(7), which is overspecified by the more physically appropriate convective boundary condition (7), when the heat transfer coefficient h0 tends to +∞.


## 5. Conclusions

In this paper, we consider a semi-infinite material under a solidification process with a mushy zone caused by an initial heat flux boundary condition, when the thermophysical parameters involved in the phase-change process are assumed to be constant. We solve the associated free boundary problem overspecified with a convective boundary condition and obtain the temperature, the two free boundaries of the mushy zone and one thermal coefficient among the latent heat by unit mass, the thermal conductivity, the mass density, the specific heat, and the two coefficients that characterize the mushy zone, when the bulk temperature and the coefficients that characterize the heat flux and the heat transfer at the boundary are assumed to be known. As a consequence, we give formulae for the temperature, the two free boundaries, and the unknown thermal coefficient, besides 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 paper and another similar phase-change process which is overspecified by a Dirichelt boundary condition. From this relationship, we solve the problem with the Dirichlet condition by considering a large heat transfer coefficient in the problem with the convective condition. In this way, we solve the phase-change process overspecified with a temperature boundary condition through the more physically appropriate phase-change problem overspecified with a convective boundary condition. We summarize explicit formulae for the unknown thermal coefficient for both problems in Tables1 and 2.



---

*Source: 637852-2015-10-18.xml*