^{1,2}

^{1}

^{2}

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.

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 [

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

liquid region at the temperature 0°C, in

solid region at the temperature

mushy region at the temperature

the material contains a fixed portion

the width of the mushy region is inversely proportional to the gradient of temperature (see condition (

Following the methodology given in [

This paper completes the model given in [

The phase-change process consists in finding the free boundaries

Condition (

If the coefficient

Taking into account that

From condition (

From condition (

Therefore, we deduce that (

If the coefficient

From (

Moreover, from (

Now, we will obtain the asymptotic behaviour of solution (

For any coefficient

One obtains the following limits:

The solution of problem (

Now, by studying the real functions

When the variable

As the variable

Function

By using (

By using (

Therefore, we have

Then, the estimation (

Finally, by using (

We consider problem (

If the coefficient

If the coefficient

If the coefficient

The real functions

Now, we will consider a phase-change process which consists in finding the free boundaries

Condition (

If the coefficient

Following the proof of Theorem

From condition (

From condition (

We have a relationship between

Following Section

If the coefficient

If the coefficient

If the coefficient

At last, for a suggestion of an anonymous referee, we will transform problem (

Moreover, inequality (

Therefore, limit

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

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

Dimensionless coefficient that characterizes the transient heat transfer at

Specific heat, J/(kg°C)

Temperature at the fixed face

Bulk temperature at the fixed face

Coefficient that characterizes the transient heat transfer at ^{5/2})

Thermal conductivity, W/(m°C)

Latent heat of fusion by unit of mass, J/kg

Characteristic length, m

Phase-change process defined by conditions (

Phase-change process defined by conditions (

Phase-change process defined by conditions (

Coefficient that characterizes the transient heat flux at ^{5/2}

Position of the liquid-mushy zone interface at time

Dimensionless position of the liquid-mushy zone interface at time

Position of the solid-mushy zone interface at time

Dimensionless position of the solid-mushy zone interface at time

Stefan number, defined in (

Time, s

Temperature of the solid phase, °C

Spatial coordinate, m.

Diffusivity coefficient, m^{2}/s

One of the two coefficients that characterizes the mushy zone, °C

One of the two coefficients that characterizes the mushy zone, being dimensionless

Coefficient that characterizes the free boundary

Density of mass, kg/m^{3}

Coefficient that characterizes the free boundary

Coefficient that characterizes the free boundary

Coefficient that characterizes the free boundary

Coefficient that characterizes the free boundary

Coefficient that characterizes the free boundary

Dimensionless time

Dimensionless temperature of the solid phase

Dimensionless spatial coordinate.

The author declares that there is no conflict of interests regarding the publication of this paper.

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.