Thermal behaviour of materials in interrupted phase change
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Abstract
The most critical part and barrier of phase change material (PCM) applications are the accuracy of simulations and the control of the process. The state of the PCM and the momentarily stored energy cannot be estimated easily unless numerous temperature sensors are used. There are a lot of models used by researchers, but most of them focus solely on the full charging or discharging of the PCM thermal energy storage. In a real working environment, the phase change is often interrupted so this phenomenon should also be modelled with high accuracy. The aim of this paper is to present the newly developed diagonal model validated by differential scanning calorimetry measurements, which can model what occurs inside the hysteresis of the solid–liquid two-phase state. The model was created and validated by using paraffin wax (P53) and was further tested with coconut oil (C.oil20), which has a very wide hysteresis. The modelling accuracy of the different models was compared with each other, and the evaluations were carried out.
Keywords
Phase change materials (PCM) Differential scanning calorimetry (DSC) Enthalpy–temperature curves Phase transition Interrupted melting/solidificationList of symbols
- c
Specific heat capacity (kJ kg^{−1} K^{−1})
- c_{app}
Apparent heat capacity of the PCM during melting (kJ kg^{−1} K^{−1})
- f
Liquid fraction of PCM (1)
- H
Enthalpy (kJ kg^{−1})
- L_{f}
Latent heat of fusion (kJ kg^{−1})
- T
Temperature (°C)
Superscripts
- m
Melting
- s
Solidification
- c
Corner/break point
Subscripts
- l
Liquid
- m
Melting
- melting, range
Melting temperature range
- s
Solid
- x
Interruption point
Introduction
Phase change materials (PCM) are widely used as thermal energy storage (TES) materials. As heat is stored through the phase change of materials and not through their temperature, it is possible to expand the commonly used TES solutions. PCM TES systems could be used for heating/cooling of residential or office buildings, industrial processes, renewable energy storages, cold chain management applications or consumer products to increase their energy efficiency or thermal comfort [1, 2, 3, 4, 5, 6, 7, 8, 9].
Researchers use different methods to model the thermal behaviour of PCMs [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. Ideally, phase change occurs completely and that is why most researchers only investigate full cycles of phase change [8, 9, 10, 11, 12, 13], but in practice a partial phase change frequently occurs. The solid–liquid two-phase state is investigated with the use of models inconsistent with each other [24, 25]. Their accuracy has to be analysed, and a new, more accurate model should be established. Thermal calorimetry measurements are needed for the examination of the two-phase state.
Differential scanning calorimetry (DSC) was used to carry out temperature–heat capacity measurements. DSC is a thermoanalytical technique, where the difference in the amount of heat required to increase a unit of temperature of a sample and a reference is measured as a function of temperature [26, 27, 28]. The sample and the reference are kept on the same temperature during the measurement. The heat capacity in the function of reference temperature is well-known. The heat capacity of the sample is calculated from the difference of the values between the sample and the reference.
Based on the recommendations in the articles [26, 27, 28, 29, 30, 31, 32, 33], every DSC measurement was carried out with 0.5 K min^{−1} heating/cooling rate, so thermal equilibrium was ensured and the temperature of the material sample was homogeneous and uniform. As a control, the result is investigated with three measurements carried out with 2 K min^{−1}.
First, paraffin wax (P53) was measured and a model was developed to ensure an accurate simulation of phase change materials. To validate and to extend the applicability of the model, further tests were made with coconut oil (C.oil20), which has a very wide hysteresis.
Every measurement was carried out with three different samples, and three different cycles were measured in each sample. The presented results show the average of the most accurate series of data measured.
Physical properties of the used PCMs
Property | P53 | C.oil20 |
---|---|---|
T_{m}—melting temperature/°C | 53.5 | 20 |
L_{f}—heat of fusion/kJ kg^{−1} | 196.2 | 107.6 |
c_{s}—solid specific heat capacity/kJ kg^{−1} K^{−1} | 4.1 | 1.8 |
c_{l}—liquid specific heat capacity/kJ kg^{−1} K^{−1} | 3.1 | 2.1 |
Modelling the full phase change
In this chapter, the existing thermal behaviour models are reviewed. The most commonly used techniques to model the thermal behaviour of PCMs are the effective heat capacity method [10, 11, 12, 13, 14, 15, 16, 17, 18] and the enthalpy method [18, 19, 20, 21, 22, 23]. The effective heat capacity method considers the heat of fusion as heat capacity in the range of the phase change, and the effective heat capacity is defined as the slope of the enthalpy–temperature curve. The enthalpy method defines a continuous and reversible function for a given volume and material, which will return the temperature depending on the calculated enthalpy.
I/1 curve estimates the melting point as an exact temperature; in this case, it is 53.5 °C.
I/2 curve models the heat of fusion with a 1 °C difference, 53–54 °C.
I/3 has a 3 °C difference, 52–55 °C.
I/4 shows the heat of fusion when it is modelled in the full range of 6 °C, 50.5–56.5 °C [12].
Model IV: Enthalpy method with fitted lines.
This model has high accuracy with a low computation demand.
Summary of the models and their describing factors
Model | Describing factors |
---|---|
I | c_{s}, c_{l}, L_{f}, T_{m}, ΔT_{m} |
II | c_{s}, c_{l}, L_{f}, \(T_{{\mathrm{s}}}^{{\mathrm{m}}}\), \(T_{{\mathrm{l}}}^{{\mathrm{m}}}\), T_{m} |
c_{s}, c_{l}, \(T_{{\mathrm{s}}}^{{\mathrm{m}}}\), \(T_{{\mathrm{l}}}^{{\mathrm{m}}}\), T_{m}, line1, line2 | |
III | Temperature–enthalpy data pairs |
IV | c_{s}, c_{l}, L_{f}, \(T_{{\mathrm{s}}}^{{\mathrm{m}}}\), \(T_{{\mathrm{l}}}^{{\mathrm{m}}}\) |
c_{s}, c_{l}, \(T_{{\mathrm{s}}}^{{\mathrm{m}}}\), \(T_{{\mathrm{l}}}^{{\mathrm{m}}}\), line1 | |
line1, line2, line3 |
Modelling the interrupted phase change
The ‘Transition scenario’ (TS), suggested by Bony and Citherlet [24], is a transition to the cooling curve using a slope equivalent to the solid or liquid specific heat; in this case, the end point is at point T_{1},
The ‘Stay scenario’ (SS), in which Chandrasekharan et al. [25] have suggested another option, consists of staying on the heating curve to reach point T_{2}.
As the scenarios based on the literature show a different approach in modelling the two-phase state of a material, it should be decided which model is to be used in which case for the highest accuracy or whether there is a need for a new model. The aim of this paper is to evaluate the accuracy of the presented two-phase scenarios.
The research presented in [34] is addressing the same problem, and their results show quite different behaviour of the investigated process. The authors demonstrate an optimization model, which can calculate the interrupted phase change process. According to their model, the interrupted phase change follows an enthalpy curve which is located between the heating and cooling curves. The article does not describe their optimization algorithm in detail, so it was not possible to compare our results with their models.
Methods
Three different samples were measured three times with 0.5 K min^{−1} heating/cooling rate, and the presented results show the average of the measurements. The aim was to determine the behaviour of materials in the solid–liquid two-phase state. Researchers use different models, and the measurements have to determine which model is the most accurate to use for which case.
Based on earlier measurements carried out on the presented P53 [35], the full melting range is between 50.5 and 56.5 °C. Interrupted melting was measured in cycles with a minimum temperature of 40 °C and a maximum temperature of 50.9, 51.8, 52.7, 53.9, 54.9, 55.8 °C (signed with Melting 1–6). The maximum temperature points equally divide the heating curve in the mean of enthalpy.
The interrupted solidification of the P53 was investigated in the following temperature points: 54.3, 53.4, 52.4, 51.35, 50.3 °C (signed with Solidification 1–5), and these points also equally divide the cooling curve.
The modelling was carried out in MATLAB R2016A software.
Results and discussion
Physical properties of the used PCMs
P53 | C.oil20 | |
---|---|---|
\(T_{{\mathrm{s}}}^{{\mathrm{m}}}\)—Beginning temperature of melting/°C | 50.5 | 13 |
\(T_{\text{l}}^{\text{m}}\)—End temperature of melting/°C | 56.5 | 24.5 |
\(T_{{\mathrm{s}}}^{{\mathrm{s}}}\)—end temperature of solidification/°C | 49.5 | 8 |
\(T_{\text{l}}^{\text{s}}\)—Beginning temperature of solidification/°C | 55.7 | 14 |
Lines were fitted to describe the two-phase process more accurately.
R^{2} values comparing the measurement with different models for P53
T_{x}/°C | H_{x}/kJ kg^{−1} | SS/% | TS/% | Diagonal model/% | |
---|---|---|---|---|---|
Heating full | – | – | 0.998 | 0.983 | – |
Cooling full | – | – | 0.982 | 0.998 | – |
Melting 1 | 50.9 | 16.7 | 0.979 | 0.943 | 0.989 |
Melting 2 | 51.8 | 33.3 | 0.976 | 0.961 | 0.992 |
Melting 3 | 52.7 | 69.2 | 0.973 | 0.963 | 0.995 |
Melting 4 | 53.9 | 121.2 | 0.967 | 0.967 | 0.996 |
Melting 5 | 54.9 | 160.3 | 0.965 | 0.971 | 0.997 |
Melting 6 | 55.8 | 183.4 | 0.956 | 0.973 | 0.998 |
Solidification 1 | 54.3 | 166.7 | 0.957 | 0.990 | 0.998 |
Solidification 2 | 53.4 | 126.5 | 0.962 | 0.987 | 0.998 |
Solidification 3 | 52.4 | 84.1 | 0.965 | 0.982 | 0.996 |
Solidification 4 | 51.35 | 45.2 | 0.971 | 0.981 | 0.995 |
Solidification 5 | 50.3 | 16.2 | 0.972 | 0.978 | 0.993 |
The SS column shows the value of R^{2} when the modelling was based on the two-phase model SS. This also applies for the TS column. The diagonal model shows the model that was developed for the accurate simulation.
In Table 4R^{2} gets lower with the SS modelling when the temperature is getting higher and the use of the TS model gets more accurate. This can also be seen concerning the interrupted solidification.
R^{2} values comparing the measurement with different models for C.oil20
T_{x}/°C | H_{x}/kJ kg^{−1} | SS/% | TS/% | Diagonal model/% | |
---|---|---|---|---|---|
Heating full | – | – | 0.997 | 0.714 | – |
Cooling full | – | – | 0.738 | 0.998 | – |
Melting 1 | 15.3 | 37.55 | 0.951 | 0.718 | 0.991 |
Melting 2 | 19.25 | 75 | 0.894 | 0.798 | 0.984 |
Melting 3 | 23.09 | 127.2 | 0.791 | 0.885 | 0.986 |
Solidification 1 | 13.35 | 101.3 | 0.80 | 0.942 | 0.989 |
Solidification 2 | 12 | 76 | 0.911 | 0.881 | 0.985 |
Solidification 3 | 9.5 | 41.6 | 0.926 | 0.789 | 0.982 |
Tables 4 and 5 present the accuracy of the different two-phase scenarios. At higher temperatures, the TS (interrupted phase change using the cooling curve) is more accurate, while on the other hand, at lower temperatures the certainty of the SS (material uses the heating curve after an interruption in phase change) is higher. The diagonal model has the highest accuracy for all measured points.
R^{2} values comparing the measurement with different models for P53 with 2 K min^{−1} heating/cooling rate
T_{x}/°C | H_{x}/kJ kg^{−1} | SS/% | TS/% | Diagonal model/% | |
---|---|---|---|---|---|
Heating full | – | – | 0.993 | 0.987 | |
Cooling full | – | – | 0.985 | 0.991 | |
Melting 1 | 51 | 15.4 | 0.982 | 0.951 | 0.989 |
Melting 2 | 53 | 74.6 | 0.975 | 0.966 | 0.995 |
Melting 3 | 55 | 154.7 | 0.968 | 0.973 | 0.997 |
Solidification 1 | 54.5 | 166.7 | 0.961 | 0.992 | 0.998 |
Solidification 2 | 52.5 | 89.8 | 0.968 | 0.986 | 0.996 |
Solidification 3 | 50 | 16.2 | 0.977 | 0.970 | 0.993 |
Conclusions and further research aims
This paper presents a new method to model the thermal behaviour of materials in the solid–liquid two-phase state. Measurements were taken with a DSC TA Q2000 device on P53 paraffin and C.oil20 coconut oil for full phase change and for interruption in melting and solidification. Four existing models were presented, and their accuracy was investigated.
Two different scenarios were taken from the literature and investigated to model the thermal behaviour of materials in the solid–liquid two-phase state when an interrupted phase change occurs. The new diagonal model was established evaluating the measurements of the investigated P53 paraffin wax material and was tested for the C.oil20 coconut oil. According to the measurements, the interrupted phase change could be modelled accurately with fitting two lines, where the corner points of the lines are located near the \(T_{{\mathrm{s}}}^{{\mathrm{m}}} - T_{{\mathrm{l}}}^{s}\) diagonal of the hysteresis. Modelling accuracy was highly increased, which means a more accurate simulation of phase change material applications. The results of the model show high accuracy for the two investigated materials and for the different heating/cooling rates.
After further research, results can be extended to all paraffin waxes and also for fatty acids due to their similar chemical structure. Further research will be aimed at investigating the interrupted phase change with a higher resolution, with more interruption points in the melting and solidification process. Furthermore, the accuracy of the model should be investigated if the fitted lines in the interrupted lines were calculated from the liquid fraction of the PCM.
Notes
Acknowledgements
Open access funding provided by Budapest University of Technology and Economics (BME). The DSC measurements were taken with the help of the Department of Polymer Engineering, Budapest University of Technology and Economics.
Funding
The measurements were supported by Balázs Pinke and the Department of Polymer Engineering at Budapest University of Technology and Economics. This study was supported by the ÚNKP-17-3 New National Excellence Program of the Ministry of Human Capacities. The research reported in this paper was supported by the Higher Education Excellence Program of the Ministry of Human Capacities in the frame of Water science & Disaster Prevention research area of Budapest University of Technology and Economics (BME FIKP-MI).
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