*5.1. Experimental Results*

The experimental tests were carried out for the outer diameters of 15, 22, and 28 mm, and the thermal-flow conditions obtained in the wind tunnel, as defined in Table 3. This combination of the outer diameters and air tunnel conditions necessitated 12 measurements. Figure 10 shows a representative example of the heating process for a module with an outer diameter of 28 mm and an air flow rate of 0.92 m/s (Case I).

Generally, from the performed experiments, it can be concluded that the heating process of a module filled with the PCM can be divided into five phases. This solution is unique in comparison to homogeneous PCM systems, where only three phases can be observed [49,52,53]. In the first phase, starting with a uniform initial temperature of −18 ◦C throughout, the mixture of paraffin wax and water coexisted in solid form is heated. This phase that is presented in Figure 10 as Phase I is characterised by a linear temperature change with a high value of the inclination angle of the characteristic. There are two reasons why temperature increases rapidly. The first reason is related to the fact that in this temperature range there is the highest temperature gradient between the tested module and the air flowing outside; therefore, the highest temperature driving force occurred. The second reason may also be that ice has half of the specific heat lower than water.

**Figure 10.** Measured temperatures evolution for *d* = 28 mm PCM module and 0.92 m/s value of the velocity.

Additionally in this temperature range, mention should be made of the condensation of water vapour that is contained in the humid air. This phenomenon also affects the thermal processes that take place during the heating of the module. Subsequently, starting from approximately −3 ◦C to 0 ◦C, flattened temperature curves can be observed (the first area marked in yellow called Phase II), and the temperature stratification process in the module begins. This phase is directly related to the process of changing the water phase from ice to water. The stratification of temperature curves also indicates the convective flow term in the module itself. Figure 10 shows the third phase as Phase III between the phase transitions of water and paraffin wax. It is characterised by an increase to a temperature of about 4 ◦C. From this temperature the fourth phase starts (Phase IV indicated in Figure 10 as the second area marked as the yellow). The curves reach a plateau and can be treated as a process associated with the phase transformation of the paraffin wax contained in the emulsion. This phase ends when the temperature reaches 6 ◦C. Subsequently, during the rest of the process, described as the fifth phase (Phase V), the temperature of the mixture steadily increases to an ambient condition.

The phase analysis performed above is mainly conducted for the temperature evolution recorded by the *T*3 thermocouple. However, during the measurements, some differences between the individual thermocouples were observed. Despite the fact that the design of the wind tunnel ensures the achievement of homogeneous thermal and flow conditions on the outer surface of the cylinder, in the upper part of the measuring section, as shown in Figure 7, the velocity values are decreased and, thus, the heat flux to PCM is also decreasing. A second major reason for the temperature evolution discrepancy may be that the thermocouples are not positioned in the module axis. This applies to the two bottom thermocouples, which were the most difficult to install in the module and during the cooling process of the module, could have moved closer to the inside of the module wall. However, regardless of the differences, all of the curves have similar trends and phases.

The identification of the aforementioned phase transitions for ice-water and paraffin wax can also be observed in Figure 11. If a derivative of the temperature function *dT*/*dτ* versus time is determined, then the first two "valleys" of the function with values smaller than 0.005 ◦C/s can be interpreted as the phase transitions (areas marked in yellow in Figure 11). The rest of the derivative close to zero starting around 6000 s describes the so-called steady-state process, i.e., the temperature in the module reached about 90% of the ambient value.

**Figure 11.** Derivation evolution of *T*3 thermocouple for *d* = 28 mm PCM module and 0.92 m/s value of the velocity.

Figure 12 shows the comparison of the changes in the recorded values of *T*3 thermocouple for different velocity and external diameters of the module. As the airflow rate flowing around the module increases, the heat transfer to the PCM also increases. This is evident in the dynamics of temperature increase and the time of phase transitions, and it can be seen that, for higher velocity values, the temperature increase is faster. However, as the outside diameter of the module increases, the amount of PCM used increases and, hence, the amount of cold accumulated. For this reason, the increase in temperature is less dynamic.

From a practical point of view, it is important to assess how fast the process is going through the characteristic phase. In the case of the tested module, two characteristic temperature ranges were selected: the first temperature range is the range from −4 ◦C to 0 ◦C (phase marked in Figures 10 and 11 as Phase II-ice–water phase transition region), the second is the phase change range of the tested PCM contained in temperatures from 4 ◦C to 6 ◦C, showed in Figures 10 and 11 as Phase IV-paraffin wax phase transition region. Table 6 presents the measured time interval for the various velocity and module diameter values that are defined in Table 3 as Case 0, Case I, Case II, and Case III for the water in the mixture, while Table 7 specifies this for paraffin wax.

**Table 6.** Water melting time.


**Table 7.** Paraffin melting time.


**Figure 12.** *T*3 thermocouple evolution for different diameter of PCM module and velocity: (**a**) Case 0–0.0 m/s, (**b**) Case I– 0.92 m/s, (**c**) Case II–2.27 m/s, and (**d**) Case III–3.18 m/s.

It is obvious that the longest time of phase transitions will occur for processes in which the velocity value is zero (Case 0) and it will increase with increasing diameter, i.e., mass of PCM. In order to show this process in a graphical form, it would be advisable to refer to the longest process (Case 0) and when a system with forced convection is used (Case I-yellow, Case II-cyan, and Case III-grey). In the article, a dimensionless parameter called Relative Time Increase (RTI) was introduced and defined by the following formula:

$$RTI = \frac{\pi\_0 - \pi}{\pi\_0} 100\% \,\tag{9}$$

where *<sup>τ</sup>*0-time of phase change for Case 0 and *τ*-time of phase change.

In the case of the phase transitions of water that are shown in Figure 13a, the smallest time increments occur for the lowest velocities and amount from 42% to 50%. With an increase in the value of the velocity of the flowing air, the time needed to achieve the phase changes is reduced from 66% to 71%, respectively.

**Figure 13.** Relative time increase (*RT I*) for analysed Cases: (**a**) water and (**b**) paraffin wax.

For paraffin wax, as presented in Figure 13b, the situation is very similar, only the values of the RTI are different. For the smallest air velocities, the RTI varies from 55% to 61%, and for the highest velocities from 85% to 87%.

Figure 13 also shows how effectively the time of the melting process changes. The analysis presents that for paraffin it is more "sensitive" than for water to alterations in external conditions, i.e., heat transfer coefficients. Interestingly, the effect of changing the diameter is less effective and it only causes a change in the accumulated heat value, not in the phase transition time.

## *5.2. Numerical Results*

In Figures 14–16, the comparison of the experimental and numerical results for the analysed values of air velocity 0.92 (Figure 14), 2.27 (Figure 15), and 3.18 m/s (Figure 16) and module diameters (a) 15, (b) 22, and (c) 28 mm are presented. All of the comparisons were carried out up to 1440 s, which allowed for the analysis of mainly phase changes taking place in the paraffin wax, due to the long computing time. Additionally, the initial temperature for all cases was 2 ◦C, because of the fact that the considered model did not take the ice-water phase change into account.

It is visible that, especially for low diameter (*d* = 15 mm) of the cylinder (see Figures 14a, 15a, and 16a), the numerical model agrees quite well with the experimental data. At the beginning of the simulation, an inaccurate prediction can be caused by the initial conditions for the temperature and velocity contours. In the experiment, the flow is fully developed while in the model, uniform distributions of velocity and temperature are assumed. For larger diameters, the time of heating is higher and the initial influence of the conditions is slightly larger. For *d* = 28 mm (Figures 14c, 15c, and 16c) numerical errors are marginally higher than in the case of *d* = 22 mm (Figures 14b, 15b, and 16b).

For the higher velocity of *v* = 2.27 m/s and diameter of *d* = 15 mm (Figure 15a), the accuracy of the model is similar to the case of *v* = 0.92 m/s (Figure 14a). The numerical errors slightly increase with the increase of the cylinder diameter.

 **Figure 14.** Comparison of the experimental and numerical results for the inlet velocity of 0.92 m/s and different cylinder diameters.

**Figure 15.** Comparison of the experimental and numerical results for the inlet velocity of 2.27 m/s and different cylinder diameters.

With the increase of the inlet velocity to 3.18 m/s (see Figure 16), higher discrepancies between the experimental and model results were recorded. In the case of a small diameter (*d* = 15 mm, Figure 16a), the model agrees fairly well with the experiment. For larger diameters, i.e., *d* = 22 (Figure 16b) and *d* = 28 mm (Figure 16c), the accuracy is still satisfactory and the model predicts the temperature evolution well; however, the errors are more visible. For higher velocities, the turbulence model may also have a greater impact on the results. On the other hand, probably, the influence of the turbulence model was not the goal of this work and further studies should be conducted.

**Figure 16.** Comparison of the experimental and numerical results for the inlet velocity of 3.18 m/s and different cylinder diameters.

Figures 17 and 18 show the exemplary numerical results of temperature distributions in the vertical and horizontal middle plane, respectively, for different times. It is visible from Figure 17a–d that the heating of the PCM firstly takes place in the top and bottom space of the cylinder. This is caused by the conduction heat flux at the contact surfaces of the cylinder and direct contact with the tunnel walls. Moreover, more mass is melted in the upper part of the cylinder when compared to the bottom one. The reason of this is natural convection that intensifies thermal-flow processes in the upper part of the solid [54]. Similar results and observations are also presented in the literature [55–58].

**Figure 17.** Numerical results of temperature distribution in the middle vertical plane for the inlet velocity of 0.92 m/s and cylinder diameter of 15 mm and different times of the simulation.

Over time, the isotherms of high temperature propagate into the interior of the PCM. Behind the cylinder, a wake is developing, which gives rise to change temperature distribution in the front of, as well as in the back, of the cylinder (see Figure 18). The differences in temperature distribution around cylinder are caused by increasing local heat transfer coefficient, which was described by Cengel [59] and Incropera et al. [60]. At the beginning (Figure 18a,b), behind the cylinder, one large area of lower temperature exists. For greater times (Figure 18c,d), this area slightly shrinks and its length behind the cylinder is shortened. This process is caused by changes in temperature difference between the cylinder and air. With an increase of the PCM temperature, the temperature difference decrease; therefore, the air faster reach surrounding temperature.

As it results from the comparison of the results that were experimentally obtained with the results obtained from the proposed numerical model, presented in Figure 19, the relative error in a wide temperature range does not exceed 10%. However, it has been observed that at the beginning of the heating process and in the temperature region where the phase change occurs, this error can reach even 20% (see enlarged analysis region in Figure 19 in the upper left corner). A few factors could cause the observed differences between the numerical and experimental results, in particular, in the initial heating phase and in the phase change region. During the numerical calculations, only pure paraffin

wax was analysed. In the experiment, paraffin wax that was capsuled in the melamineformaldehyde membrane micro-capsules was used. Such an interior structure is difficult to model and it was not considered in the simulations. Moreover, during the preparing of the experimental set-up, unintentional human errors could occur, e.g., imperfect placement of the thermocouples in the module axis or its movement due to a melted PCM, which can result, in particular, in the region of phase change. In the numerical investigations, boundary conditions were averaged and stable during the heat transfer processes. On the other hand, in the experiment, chiller hysteresis caused unstable inlet air temperature and, therefore, unstable boundary conditions. Another thing could be the air humidity impact on the heat transfer when the air dew point temperature was higher than the module temperature. During this situation, vapour from the air condensed on the module walls, which probably caused an intensification of heat transfer.

(**a**) *τ* = 60 s.

(**b**) *τ* = 120 s.

**Figure 18.** Numerical results of temperature distribution in the middle horizontal plane for the inlet velocity of 0.92 m/s and cylinder diameter of 15 mm and different times of the simulation.

**Figure 19.** The comparison of numerical and experimental temperatures and obtained relative error.
