3.1. Invariance of Solutions in Adiabatic Systems
The first part of this section is devoted to the discussion of the phase change process in adiabatic systems. In this section, the HTPCM considered is the salt. A wide temperature range is used to highlight the breaking of the invariance when or is chosen as the moving boundary, according to the solutions obtained through a local energy balance at the interface. Two examples are shown where the initial energy of the liquid–solid sample produces melting on the one hand and solidification on the other hand. The initial temperature at is () for the melting of with an initial interface position and initial size of . The initial temperature at is () for the solidification example, where . These conditions are used to determine the initial temperature distribution in the liquid and solid domains as quadratic functions of the spatial variable x. The system is subjected to the adiabatic boundary conditions given by Equation (1) and the isothermal boundary condition at the liquid–solid interface, given by Equation (2). The total energy is a constant of the motion since the system is thermally isolated. The bar is expected to reach thermodynamic equilibrium where the system growth (shrinkage) and the fraction of melted (solidified) solid (liquid) is given by Equations (10) and (11), respectively.
Figure 1 shows the time evolution of
and
(
) for
according to the solutions obtained by assuming the classical local thermal balance at the interface.
Figure 1a,b shows the system growth and fraction of melted solid
upon melting of
. The numerical solutions to the classical model when
or
is chosen as the moving boundary do not have the same time-dependent behavior for
and
. These solutions are not invariant when the volume of the system is allowed to change from the right or left boundary. Additionally, it is realized that solutions for
and
(
) reach entirely different thermodynamic equilibrium values when the sample is fixed at the left boundary and when the sample is fixed at the right boundary. The lack of invariance of the numerical solutions to the classical model is in contradiction with the predicted thermodynamic equilibrium values shown through Equations (10) and (11).
Figure 1c,d also illustrates this anomalous behavior when the
sample shrinks upon solidification of liquid in adiabatic systems.
Equation (5a) or Equation (5b) are solved through the FEM by imposing energy conservation through the apparent latent heat
given by Equation (8a) or Equation (8b), whether
or
is the time-dependent boundary. Additionally, total mass conservation is imposed through Equation (6a) or Equation (6b). The numerical solutions to
and the fraction of melted (solidified) solid (liquid) are independent of which boundary is chosen as the dynamical variable, as illustrated in
Figure 2. According to the proposed model, the FEM solutions are observed to reach the predicted thermodynamic equilibrium values given by Equations (10) and (11). The solutions during the melting and solidification processes in
are independent of which boundary is chosen as the dynamical variable, as illustrated in
Figure 2.
Figure 2a,b show the FEM solutions when the initial conditions produce melting of
. The time evolution of
and the fraction of melted solid
are independent of which boundary is chosen as the dynamical variable. Additionally the thermodynamic equilibrium values for
and
are well reproduced by the FEM solution at
as:
and
.
Figure 2c,d corresponds to the solidification example of
illustrated in
Figure 1. The numerical solutions are also invariant and well behaved near the thermodynamic equilibrium state of the system. Finite element method solutions reach the predicted values at thermodynamic equilibrium
and
at
as:
and
.
Total energy in adiabatic systems is a constant of the motion. The initial energy of the system
must be conserved throughout the melting or solidification process. Therefore, energy conservation can be used to determine the performance of the numerical solutions. An average energy error (AEE) for the total energy has been defined as follows for this purpose:
where
is the initial energy of the system,
is the total energy of the system at some time level
, and
n is the number of time partitions.
Three versions of the FEM were implemented: the first one labeled as
uses linear Lagrange functions,
is an implementation of the FEM with quadratic Lagrange functions, and
uses the cubic Lagrange shape functions described in
Section 2.4.
Table 2 shows the AEE for each implementation of the FEM during melting and solidification of
. Results are shown for ten elements that were used to discretize the entire spatial domain. The number of elements within the liquid and solid domains was varied during the phase change process. The number of elements at each phase and in a given time level was determined from the volume of the liquid and solid phases in order to have a total number of ten elements in the whole system with the same length. A total of
time partitions were used on each of the examples illustrated in
Figure 1 and
Figure 2 and
Table 2. Increasing the size of
n produces negligible changes to the AEE given by Equation (34) and shown in
Table 2. Local matrices obtained for the
,
, and
implementations have dimensions of 2, 3, and 4, respectively. Increasing the degree of the shape functions produces lower values for the AEE, as expected.
Table 2 illustrates how total energy conservation is best behaved through a
implementation of the FEM.
3.2. Energy Absorbed/Released
The liquid–solid system is subjected to mixed boundary conditions to produce melting on the one hand and solidification of the liquid phase on the other hand. Melting examples are designed through the boundary conditions given by Equation (12), and solidification of liquid phase is achieved by imposing the boundary conditions through Equation (14). The system will absorb (release) thermal energy until the solid (liquid) is almost completely melted (solidified). The total amount of energy absorbed or released by the PCM can be obtained through Equation (16a) or Equation (16b).
The sensible heat absorbed by a PCM can be conceived in four stages, as discussed in Refs. [
14,
25]. These stages can be used to show that latent heat is absorbed through the bulk latent heat
of the PCM and not the apparent latent heat previously defined [
29]. The first stage considers the thermal energy absorbed between
and some time
t by the initial mass of liquid as
where it is assumed that
is constant and equal to
, and
is the dynamical variable used to impose total mass conservation. During a second stage, the amount of solid mass
that will be eventually melted will increase its temperature from its initial value
to the melting temperature
, absorbing thermal energy as follows:
where
is the volume of this amount of solid that will be melted between
and any later time
t and is related with
as
. The value of
can be obtained through mass conservation as follows:
then, solving for
the following expression is obtained
, where
. The third stage considers the energy absorbed by
once it has transformed into liquid phase. Through this stage, this mass of liquid absorbs thermal energy from the melting temperature
, to the temperature of the liquid phase
at time
t, as follows:
where
is the volume of
, but now in its liquid state. Finally, the mass of solid that was not melted between the initial state and the state of the system at some later time
t, will absorb thermal energy as sensible heat by raising its temperature from its initial value
to the temperature
at some later time
t as follows:
Adding the contributions from each stage to the total sensible heat absorbed, the following expression is obtained:
The first two terms correspond to the total enthalpy absorbed by a melting PCM as shown by Equation (16a) for constant and equal to zero, and as the moving boundary. The last term is exactly the latent heat absorbed by the PCM during the phase change process, which is proportional to the bulk latent heat and not the apparent latent heat. The above discussion shows that even though solidification or melting rates can be pictured through an apparent latent heat, the latent heat storage capacity of the PCM depends only on the bulk latent heat and not . A similar analysis can be performed for solidification scenarios or when is chosen as the dynamical variable.
The present work estimates the contributions from the sensible and latent heat stored or released through Equations (16a) and (16b), according to the numerical and semi-analytical solutions to the old and new models.
Figure 3 shows the total energy absorbed and released by the
salt. Melting (solidification) examples are illustrated until the fraction of melted (solidified) solid (liquid) is
(
). Numerical and semi-analytical solutions according to a local thermal balance at the interface and the proposed total thermal balance are shown. Melting of
is produced through the boundary conditions given by Equation (12). The initial position of the liquid–solid interface is
in
Figure 3a,b. The temperature at
is kept at a constant value of
, and the initial temperature at the right boundary is
.
Figure 3a,b shows the numerical and semi-analytical estimations of the total energy absorbed by the salt according to both models and for each case, when
or
is the moving boundary. The relative difference between the estimations of both models is practically negligible upon melting of the solid phase. Alternatively, significant differences are observed when the salt is releasing thermal energy.
Figure 3c,d illustrates the numerical and semi-analytical solutions when the liquid experiences solidification and the system is subjected to the boundary conditions given by Equation (14). Initially, a small volume of solid is considered with
. The right boundary is kept at a constant temperature value of
and the initial temperature at
is
.
The relative percent difference (RPD) between the old and new models in the total energy absorbed by the
salt is shown in
Table 3. The small difference can be understood in terms of the apparent latent heat, which depends on the boundary conditions. On the one hand, according to Equation (13a) the corrections to the bulk latent heat are proportional to the difference between the melting temperature and the temperature at the right boundary. The temperature at
is expected to approach
as the solid melts; therefore, this term becomes smaller as the system evolves in time. On the other hand, the contributions from the apparent latent heat when
is the moving boundary can be negligible according to Equation (13b) if the temperature at this boundary is close to
. The RPD between the predictions obtained from the old and new models was determined as follows:
where
with
corresponds to which boundary is considered as the dynamical variable.
Higher RPD values are expected for the solidification of
as illustrated in
Table 4 since the temperature at the right boundary is much smaller than the melting point of the salt. According to Equation (15a), when the system releases energy and
is the moving boundary, the corrections introduced through the apparent latent heat are proportional to
. The example shown in
Figure 3c illustrates the solidification of liquid at a very low temperature value of
, well below the melting point of the salt. However, when
is the dynamical variable, the contributions from the apparent latent heat are much smaller since
approaches the bulk latent heat of fusion
as the system evolves in time [
29].
Finally, the energy absorbed and released by the
salt is estimated through the numerical and semi-analytical solutions to both models discussed in this work. The RPD between both models is highly related to the thermodynamic properties of the material and the boundary conditions. The salt can be exposed to higher temperature values, and due to its lower melting temperature as shown in
Table 1, the expected difference between both models should be higher when the salt is absorbing thermal energy.
Figure 4 shows the numerical and semi-analytical solutions for the thermal energy absorbed and released by the
salt. The charging process is shown in
Figure 4a,b. The initial interface position during the melting process of the salt is
. The temperature at
is kept constant at
, well above the melting temperature
of the salt. The left boundary is thermally isolated, and its initial temperature is
. The discharging process starts with a small volume of solid, where the initial interface position is
. Heat is removed from the right boundary which is kept at a constant temperature value of
, and the left boundary is initially set at
. The solutions for the thermal energy released during the discharging process are shown in
Figure 4c,d.
According to Equation (13b), due to the high difference between the melting temperature of the salt and a maximum charging temperature
, the contributions from the apparent latent heat are significant in the example shown in
Figure 4b. The RPD for the energy absorbed by the salt and according to both models is shown in
Table 5. The apparent latent heat, according to the new model, predicts lower energy absorption rates than those estimated by assuming local thermal balance at the interface. This behavior is observed in
Figure 4b, which according to Equation (13b) the PCM should take longer time values to absorb thermal energy until the sample is completely melted. The asymptotic time behavior of the apparent latent heat when
is the moving boundary predicts lower RPD values, as illustrated in
Figure 4a and
Table 5.
Lower RPD values are expected upon solidification of the
salt since thermal energy is drained from the right boundary at temperatures close to the melting point of the salt. According to Equation (15a) the contributions from total thermal balance are significantly lower due to the operating temperature value at the right boundary. The effects of total thermal balance on the released energy are even smaller when
is the moving boundary since the apparent latent heat approaches asymptotically to the bulk latent heat
, as illustrated in
Table 6.
Charging and discharging times are also an important parameter when HTPCMs are used as backup systems in thermoelectric generation applications. The proposed model estimates different energy densities of the salts used in this work and different charging or discharging times. According to the FEM and RHBIM solutions for the thermal energy absorbed or released by the
salt, the highest RPD between both models is found during a discharging process. The result is consistent with Equation (15a) that predicts significantly smaller energy releasing rates. Then according to the new model, the PCM should release thermal energy at a faster rate. The behavior can be observed in
Figure 3c. The maximum RPD between discharging times according to the numerical and semi-analytical solutions is
and
, respectively. Thermodynamic properties and operating temperatures used for the example shown in
Figure 4 increase the RPD between both models in a charging process. According to the proposed model, the apparent latent heat is increased by total thermal balance, as shown through Equation (13b). Maximum RPD between charging times is expected, as shown in
Figure 4b. Equation (13b) predicts lower energy absorption rates, as illustrated in
Figure 4b. The estimated RPD between charging times according to the numerical and semi-analytical solutions in this example is
and
, respectively.