We complete the Solomon-Wilson-Alexiades’s mushy zone model (Solomon, 1982) for the one-phase Lamé-Clapeyron-Stefan problem by obtaining explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get the explicit solutions for both cases which is new with respect to the original model. Moreover, when these conditions are satisfied, the two phase-change problems are equivalent to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces of the model is also obtained.
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 [1–9]. A review of a long bibliography on moving and free boundary problems for phase-change materials (PCM) for the heat equation is shown in [10]. The importance of obtaining explicit solutions to some free boundary problems was given in the works [11–26].
We consider a semi-infinite material, with constant thermal coefficients, that is initially assumed to be liquid at its melting temperature which is assumed to be equal to 0°C. At time t=0, a heat flux or a convective boundary condition is imposed at the fixed face x=0, and a solidification process begins where three regions can be distinguished [27, 28]:
liquid region at the temperature 0°C, in x>r(t), t>0;
solid region at the temperature T(x,t)<0, in 0<x<s(t), t>0(withs(t)<r(t));
mushy region at the temperature T(x,t)=0, in s(t)≤x≤r(t), t>0. The mushy region is considered isothermal and we make the following assumptions on its structure:
the material contains a fixed portion εl (with 0<ε<1) of the total latent heat l (see condition (3) in below);
the width of the mushy region is inversely proportional to the gradient of temperature (see condition (4) below).
Following the methodology given in [27–29] and the recent one in [30], we consider a convective boundary condition in Sections 2 to 4 and a heat flux condition in Sections 5 and 6 at the fixed face x=0, respectively. In both cases, we obtain explicit solutions for the temperature and the two free boundaries which define the mushy region. We also obtain, for both cases, the necessary and sufficient condition on data in order to get these explicit solutions given in Sections 2 and 5, respectively, which is new with respect to the original model when a temperature boundary condition at the face x=0 was imposed. Moreover, these two problems are equivalent to the same phase-change process with a temperature boundary condition on the fixed face x=0 studied in [27] and therefore an inequality for the coefficient which characterized one of the two free interfaces is also obtained in Sections 4 and 6. Moreover, in Section 3, we obtain the convergence of the solution of the phase-change process with a convective boundary condition to the solution given in [27] for a temperature boundary condition at the fixed face x=0 when the heat transfer coefficient goes to infinity, and we also give the order of the corresponding convergence when the coefficient that characterized the transient heat transfer at x=0 goes to infinity.
This paper completes the model given in [27] by considering two new boundary conditions (convective and heat flux) at the fixed face of the PCMs and obtaining explicit solutions for both cases when a restriction on data is satisfied.
2. Explicit Solution with a Convective Boundary Condition
The phase-change process consists in finding the free boundaries x=s(t) and x=r(t) and the temperature T=T(x,t) such that the following conditions must be verified (problem (P1)):(1)Tt-αTxx=0,0<x<st,t>0α=kρc,(2)Tst,t=0,t>0,(3)kTxst,t=ρlεs˙t+1-εr˙t,t>0,(4)Txst,trt-st=γ>0,t>0with γ>0,(5)s0=r0=0,(6)kTx0,t=h0tT0,t+D∞,t>0h0>0,D∞>0.
Condition (6) represents a convective boundary condition (Robin condition) at the fixed face x=0 [31–33] with a heat transfer coefficient which is inversely proportional to the square root of the time [29, 30, 34, 35]. Now, we will obtain the solution of problem (1)–(6) when data satisfy the restriction (7).
Theorem 1.
If the coefficient h0 satisfies the inequality(7)h0>1D∞γ1-ερlk2=h0∗,then the solution of problem (1)–(6) is given by(8)Tx,t=-h0D∞πα/kerfξ1+h0πα/kerfξ1-erfx/2αterfξ,0<x<st,t>0,(9)st=2ξαt,t>0,(10)rt=2μαt,t>0,with(11)μ=ξ+γk2D∞h0αeξ21+h0παkerfξ,and the coefficient ξ is given as the unique solution of the equation(12)D∞clπFx=Gx,x>0,where the real functions G and F are defined by(13)Fx=e-x2k/h0πα+erfx,Gx=x+γ1-επ2D∞1Fx,x>0.
Proof.
Taking into account that erfx/2αt is a solution of the heat equation (3) [3], we propose as a solution of problem (1)–(6) the following expression:(14)Tx,t=C1+C2erfx2αt,0<x<st,t>0,where the two coefficients C1 and C2 must be determined.
From condition (4), we deduce the expression (9) for the free boundary s(t), where the coefficient ξ must be determined. From conditions (6) and (2), we deduce the system of equations(15)C2=h0παkC1+D∞,C1+C2erfξ=0,whose solution is given by(16)C1=-h0πα/kD∞erfξ1+h0πα/kerfξ,C2=h0D∞παk11+h0πα/kerfξ,and then we get expression (8) for the temperature.
From condition (4), we deduce expression (10) for the interface r(t) and expression (11) for μ. From condition (3), we deduce (12) for the coefficient ξ. Functions F3 and G have the following properties:(17)F0+=h0παk>0,F+∞=0+,F′x<0,∀x>0,G0+=1-εγk2D∞h0α>0,G+∞=+∞,G′x>0,∀x>0.
Therefore, we deduce that (12) has a unique solution when the coefficient h0 satisfies the inequality(18)D∞clπF0+>G0+⟺h02>γ1-ερlk2D∞2;that is, inequality (7) holds.
Corollary 2.
If the coefficient h0 satisfies inequality (7), then the temperature, defined by (8), verifies the following inequalities:(19)-D∞<T0,t≤Tx,t<0,0<x<st,t>0.
Proof.
From (8), we obtain(20)T0,t=-h0D∞πα/kerfξ1+h0πα/kerfξ=-D∞1+k/h0παerfξ>-D∞,∀t>0.
Moreover, from (8) and (20), we also get(21)Tx,t+D∞=D∞1+h0πα/kerfξ1+h0παkerfx2αt≥D∞1+h0πα/kerfξ=T0,t+D∞>0,0<x<st,t>0;that is, (19) holds.
3. Asymptotic Behavior When the Coefficient h0→+∞
Now, we will obtain the asymptotic behaviour of solution (8)–(12) of problem (1)–(6) when the heat transfer coefficient is large, that is, when h0→+∞. From a physical point of view, it must be convergent to the solution of the same problem with a temperature boundary condition at the fixed face x=0 given by (23).
For any coefficient h0 satisfying inequality (7), we will denote the temperature T and the two free boundaries s and r (defined in (8), (9), and (10), resp.) by T=T(x,t,h0), x=s(t,h0), and x=r(t,h0), respectively, with coefficients ξ=ξ(h0) and μ=μ(h0). We will also denote by Fx,h0 and Gx,h0 the functions defined in (13). We have the following result.
Theorem 3.
One obtains the following limits:(22)limh0→∞Tx,t,h0=T∞x,t,limh0→∞st,h0=s∞t,limh0→∞rt,h0=r∞t,where T∞(x,t), s∞(t), and r∞(t) are the solutions of the following phase-change process with mushy region: (1)–(5) and (23)T0,t=-D∞,t>0,instead of the boundary condition (6).
Proof.
The solution of problem (1)–(5) and (23) is given by [27](24)T∞x,t=-D∞1-erfx/2αterfξ∞,0<x<s∞t,t>0,(25)s∞t=2ξ∞αt,t>0,(26)r∞t=2μ∞αt,t>0,with(27)μ∞=ξ∞+γπ2D∞eξ∞2erfξ∞,and the coefficient ξ∞ given as the unique solution of the equation(28)G1x=D∞clπ,x>0,where the real function G1 is defined by (29)G1x=G∞xF∞x,x>0,with(30)G∞x=x+γ1-επ2D∞1F∞x=limh0→∞Gx,h0,x>0,(31)F∞x=e-x2erfx=limh0→∞Fx,h0,x>0.Then,(32)limh0→∞ξh0=ξ∞,limh0→∞μh0=μ∞.And, therefore, the limits (22) hold.
Now, by studying the real functions Fx,h0 and Gx,h0 as functions of the variable h0, we can obtain the order of the convergence of solution (8)–(12) of problem (1)–(6) to solution (24)–(28) of problem (1)–(5) and (23) when h0→∞.
Theorem 4.
When the variable h0→∞, one obtains the following estimations:(33)0<ξ∞-ξh0=O1h0whenh0⟶∞,(34)0<s∞t-st,h0=O1h0,∀t≥0whenh0⟶∞,(35)μh0-μ∞=O1h0whenh0⟶∞,(36)rt,h0-r∞t=O1h0,∀t≥0whenh0⟶∞,(37)Tx,t,h0-T∞x,t=O1h0,∀x≥0,∀t>0whenh0⟶∞.
Proof.
As the variable h0→∞, we can consider that h0>h0∗ and then solution (8)–(12) of problem (1)–(6) is well defined.
Function F(x,h0) is an increasing function in variable h0; therefore, function G(x,h0) is a decreasing function in variable h0. Then, function G(x,h0)/F(x,h0) is a decreasing function in variable h0, ∀x>0, which is convergent to G∞(x)/F∞(x) as h0→∞ because (30) and (31) hold.
By using (13) and (31), we have(38)0<F∞x-Fx,h0=F∞xk/h0παerfx+k/h0πα<e-x2erf2xkh0πα=O1h0,∀x>0 when h0⟶∞.
By using (13) and (30), we have(39)0<Gx,h0-G∞x=γ1-επ2D∞F∞x-Fx,h0F∞xFx,h0<γ1-επ2D∞F∞x-Fx,h0F∞xFx,h0∗=O1h0,∀x>0 when h0⟶∞.
Therefore, we have(40)0<Gx,h0Fx,h0-G∞xF∞x<F∞x-Fx,h0x+γ1-επ/D∞1/Fx,h0∗F∞xFx,h0∗=O1h0,∀x>0 when h0⟶∞.
Then, the estimation (33) holds and(41)0<s∞t-st,h0=2αtξ∞-ξh0=O1h0,∀t≥0 when h0⟶∞.By using (11) and (27), we get(42)μ∞-μh0=ξ∞-ξh0-γπ2D∞αeξ2h0h0+γπ2D∞eξ∞2erfξ∞-eξ2h0erfξh0=O1h0when h0⟶∞,rt,h0-r∞t=2αtμh0-μ∞=O1h0,∀t≥0 when h0⟶∞.
Finally, by using (8) and (24), we get(43)Tx,t,h0-T∞x,t=D∞1+h0πα/kerfξh01+erfx/2αterfξ0h0παkerfξ∞-erfξh0=O1h0,∀x>0,∀t>0 when h0⟶∞,and the thesis holds. In the particular case, when x=0, we have(44)0<T0,t,h0-T∞0,t=D∞1+h0πα/kerfξh0<kD∞h0παerfξh0∗=O1h0,∀t>0 when h0⟶∞.
4. Equivalence between the Mushy Zone Models with Convective and Temperature Boundary Conditions
We consider problem (P2) defined by conditions (1)–(5) and temperature boundary condition(45)T0,t=-D0<0,t>0,at the fixed face x=0, whose solution was given in [27]. We have the following property.
Theorem 5.
If the coefficient h0 satisfies inequality (7), then problem (P1), defined by conditions (1)–(6), is equivalent to problem (P2), defined by conditions (1)–(5) and (45), when the parameter D0 in problem (P2) is related to parameters h0 and D0 in problem (P1) by the following expression:(46)D0=D∞erfξk/h0πα+erfξ>0,where the coefficient ξ is given as the unique solution of (12) for problem (P1) or as the unique solution of equation(47)G2x=D0clπ,x>0,for problem (P2), where the real function G2 is defined by (48)G2x=G0xF∞x,G0x=x+γ1-επ2D01F∞x,x>0.
Proof.
If the coefficient h0 satisfies inequality (7), then the solution of problem (P1) is given by (8)–(12). Taking into account that(49)T0,t=-h0πα/kD∞erfξ1+h0πα/kerfξ=-D∞erfξk/h0πα+erfξ<0,t>0,we can define problem (P2) by imposing the temperature boundary condition (45) with data D0 given in (46). By using this data D0 in problem (P2) and the method developed in [30], we can prove that the solutions of both problems (P1) and (P2) are the same and then the two problems are equivalent.
Corollary 6.
If the coefficient h0 satisfies inequality (7), then the coefficient ξ of the solid-mushy zone interface of problem (P2) verifies the following inequality:(50)erfξ<D∞D0D∞-D02cπγ1-εl,∀D∞>D0.Then,(51)erfξ<D02cπγ1-εl.
Remark 7.
The real functions G∞, defined in (30), and G0, defined in (48), are similar; the difference between them is the parameters D∞ or D0 used in each definition.
5. Explicit Solution with a Heat Flux Boundary Condition
Now, we will consider a phase-change process which consists in finding the free boundaries x=s(t) and x=r(t) and the temperature T=T(x,t) such that the following conditions must be verified (problem (P3)): conditions (1)–(5) and(52)kTx0,t=q0t,t>0q0>0.
Condition (52) represents the heat flux at the fixed face x=0 characterized by a coefficient which is inversely proportional to the square root of the time [34].
Theorem 8.
If the coefficient q0 satisfies the inequality(53)q0>γ1-ερlk2=q0∗,then the solution of problem (1)–(5) and (52) is given by(54)Tx,t=-q0παerfωk1-erfx/2αterfω<0,0<x<st,t>0,(55)st=2ωαt,t>0,(56)rt=2ναt,t>0,with(57)ν=ω+γk2q0αeω2,and the coefficient ω>0 given as the unique solution of the equation(58)G3x=q0ρlα,x>0,where the real function G3 is defined by(59)G3x=x+γ1-εk2q0αex2ex2,x>0.
Proof.
Following the proof of Theorem 1, we propose as a solution of problem (1)–(5) and (52) the following expression:(60)Tx,t=A1+A2erfx2αt,0<x<st,t>0,where the two coefficients A1 and A2 must be determined.
From condition (2), we deduce expression (55) for the free boundary s(t), with the coefficient ω to be determined. From conditions (2) and (52), we deduce(61)A1=-q0παkerfω,A2=q0παk,and then we get expression (54) for the temperature.
From condition (4), we deduce expression (56) for the interface r(t) and expression (57) for ν. From condition (3), we deduce (58) for the coefficient ω. Since function G3 has the following properties:(62)G30+=γ1-εk2q0α>0,G3+∞=+∞,G3′x>0,∀x>0,we can deduce that (58) has a unique solution when the coefficient q0 satisfies the inequality(63)q0ρlα>G30+⟺q02>γ1-ερlk2,which is inequality (53).
Remark 9.
We have a relationship between q0∗ (the lower limit for coefficient q0 in order to have a phase-change process with a mushy region with a heat flux boundary condition at x=0) and h0∗ (the lower limit for the coefficient h0 in order to have a phase-change process with a mushy region with a convective boundary condition at x=0) given by(64)q0∗=D∞h0∗.
6. Equivalence between the Mushy Zone Models with Heat Flux and Temperature Boundary Conditions
Following Section 4, we will now study the relationship between problems (P3) and (P2). We have the following property.
Theorem 10.
If the coefficient q0 satisfies inequality (53), then problem (P3), defined by conditions (1)–(5) and (52), is equivalent to problem (P2), defined by conditions (1)–(5) and (45), when the parameter D0 in problem (P2) is related to the parameter q0 in problem (P3) by the following expression:(65)D0=q0παkerfω>0,where the coefficient ω is given as the unique solution of (58) for problem (P3) or as the unique solution of (47) for problem (P2).
Proof.
If the coefficient q0 satisfies inequality (53), then the solution of problem (P3) is given by (54)–(58). Taking into account that(66)T0,t=-q0παkerfω<0,t>0,we can define problem (P2) by imposing the temperature boundary condition (45) with the data D0 given in (65). By using this data D0 in problem (P2) and the method developed in [30], we can prove that the solutions of both problems (P3) and (P2) are the same and then the two problems are equivalent.
Corollary 11.
If the coefficient q0 satisfies inequality (53), then the coefficient ξ of the solid-mushy zone interface of problem (P2) verifies inequality (51) which is the same as that we have obtained through the equivalence between problems (P1) and (P2).
Remark 12.
At last, for a suggestion of an anonymous referee, we will transform problem (P1), given by the equations and conditions (1)–(6), and inequality (7) in a dimensionless form. We define the following dimensionless change of variables:(67)η=xL,τ=αtL2,Sτ=stL,Rτ=rtL,θη,τ=Tx,tD∞,where L is a characteristic length. Therefore, the equations and conditions (1)–(6) are transformed as(68)θτ-θηη=0,0<η<Sτ,τ>0,θSτ,τ=0,τ>0,θηrτ,τ=1SteεS′τ+1-εR′τ,τ>0,θηrτ,τRτ-Sτ=γD∞,τ>0,S0=R0=0,θη0,τ=Bτθ0,τ+1,τ>0,where Ste is the Stefan number and B/τ is the Biot number defined by the following expressions:(69)Ste=cD∞l>0,(70)B=h0αk=h0ρkc>0.
Moreover, inequality (7) for the physical coefficient h0, which characterized the heat transfer coefficient in the boundary condition (6), is transformed in the following way: (71)B>1D∞γ1-εl2c=B∗.
Therefore, limit h0→∞ in problem (1)–(6) in physical variables is equivalent to limit B→∞ in problem (68) in dimensionless variables.
By using the results of this work, we can now obtain new explicit expression for the determination of one or two unknown thermal coefficients through a phase-change process with a mushy zone by imposing an overspecified convective boundary condition at the fixed face x=0. This will complete and improve the results obtained previously in [28].
7. Conclusions
The goal of this paper is to complete the solution of Solomon-Wilson-Alexiades’s model for a mushy zone model for phase-change materials when a convective or a heat flux boundary condition at the fixed face x=0 is imposed. In both cases, explicit solutions for the temperature and the two free boundaries which define the mushy region were obtained and, for both cases, the necessary and sufficient conditions on data in order to get these explicit solutions are also obtained which is new with respect to the original model when a temperature boundary condition at the face x=0 was imposed. Moreover, the equivalence of this two phase-change process with the one with a temperature boundary condition on the fixed face x=0 was obtained and an inequality for the dimensionless coefficient that characterizes the first free boundary is also given. On the other hand, the convergence of the phase-change process with mushy zone when the heat transfer coefficient goes to infinity was also obtained and the order of the convergence is also shown.
NomenclatureB(B∗):
Dimensionless coefficient that characterizes the transient heat transfer at x=0 (Biot number), defined in (70)
c:
Specific heat, J/(kg°C)
-D0(<0):
Temperature at the fixed face x=0, °C
-D∞(<0):
Bulk temperature at the fixed face x=0, °C
h0h0∗:
Coefficient that characterizes the transient heat transfer at x=0, kg/(C°s5/2)
k:
Thermal conductivity, W/(m°C)
l:
Latent heat of fusion by unit of mass, J/kg
L:
Characteristic length, m
P1:
Phase-change process defined by conditions (1)–(6)
P2:
Phase-change process defined by conditions (1)–(5) and (45)
P3:
Phase-change process defined by conditions (1)–(5) and (52)
q0(q0∗):
Coefficient that characterizes the transient heat flux at x=0, kg/s5/2
r=r(t)(>s(t)):
Position of the liquid-mushy zone interface at time t, m
R=R(τ)(>S(τ)):
Dimensionless position of the liquid-mushy zone interface at time τ
s=s(t):
Position of the solid-mushy zone interface at time t, m
S=S(τ):
Dimensionless position of the solid-mushy zone interface at time τ
Ste:
Stefan number, defined in (69)
t:
Time, s
T:
Temperature of the solid phase, °C
x:
Spatial coordinate, m.
Greek Symbolsα=k/ρc:
Diffusivity coefficient, m2/s
γ>0:
One of the two coefficients that characterizes the mushy zone, °C
ε∈(0,1):
One of the two coefficients that characterizes the mushy zone, being dimensionless
ν(>ω):
Coefficient that characterizes the free boundary r(t) in (45), being dimensionless
ρ:
Density of mass, kg/m3
μ(>ξ):
Coefficient that characterizes the free boundary r(t) in (10), being dimensionless
μ∞(>ξ∞):
Coefficient that characterizes the free boundary r(t) in (26), being dimensionless
ω>0:
Coefficient that characterizes the free boundary s(t) in (43), being dimensionless
ξ>0:
Coefficient that characterizes the free boundary s(t) in (9), being dimensionless
ξ∞>0:
Coefficient that characterizes the free boundary s(t) in (25), being dimensionless
τ:
Dimensionless time
θ:
Dimensionless temperature of the solid phase
η:
Dimensionless spatial coordinate.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
The authors would like to thank an anonymous referee for his constructive comments which improves the readability of the paper. The present work has been partially sponsored by the Projects PIP no. 0534 from CONICET, Universidad Austral, Rosario, Argentina and AFOSR-SOARD Grant FA9550-14-1-0122.
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