# Efficient Characterization of Macroscopic Composite Cement Mortars with Various Contents of Phase Change Material

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

- the thermal conductivity is searched for in [31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46] using either hot wire [31,45], hot plate [32,34,35,36,39,42,43,44,46], or laser flash [33,40] techniques.Let us mention here that the thermal contact resistances and their influence on the measured conductivity are rarely taken into account.
- the heat capacity is mainly investigated as an apparent capacity, which is supposed to integrate the latent heat and goes continuously from the solid value to the liquid one [31,33,34,36,37,38,40,44,45,47,48,49,50,51,52,53]. However, many of these studies determine this property from a differential scanning calorimetry (DSC) experiment [33,34,38,40,44,45,47,48,49,50,51,53,54,55], where the size of the sample is such that it is not very representative of the macroscopic material [5,31,36,52]. As far as real scales are concerned, one can also note the possible use of the dynamic heat flow meter method (DHFM), which resembles the isothermal mode of calorimeters [56,57]. Practically, this apparent capacity is very often obtained by directly integrating the heat-flow rate. Unfortunately, it can potentially lead to inconsistent or imprecise characterization [58,59,60,61,62] and be responsible for erroneous predictions when modeling the macroscopic material [60,63,64].
- in recent studies, a more generic approach based on inversion methods has been developed in order to determine separately the main thermo-physical properties, such as the heat capacities (solid and liquid) and/or the latent heat etc. [36,52,65]. The main idea is to combine some experimental measurements, usually the temporal evolution of the heat-flow rate, with a numerical model of the composite sample. The correct values for the intrinsic parameters of the material are retrieved by minimizing an objective function.

## 2. Method

#### 2.1. Experimental Protocol

^{®}DS 5001 X, where the PCM is contained in the core of micro-capsules ranging from $2\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ to $20\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{m}$ in diameter. Practically, BASF claims that the micro-capsules are completely sealed, safe to process, and free of formaldehyde. It is also specified that the encapsulation process protects the wax in its very pure form, resulting in a high heat storage capacity of $110\phantom{\rule{0.166667em}{0ex}}\mathrm{k}\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ being permanently guaranteed. The melting point of Micronal

^{®}DS 5001 X, given by the manufacturer, is about $26{\phantom{\rule{0.166667em}{0ex}}}^{\circ}\mathrm{C}$. We realized the characterization of the PCM included in our composite materials with a pyris Diamond Perkin-Elmer DSC and we concluded that the latent heat is around $99\pm 10\phantom{\rule{0.166667em}{0ex}}\mathrm{k}\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$. This result is detailed within [68].

^{2}. Practically, these are CAPTEC products, whose thicknesses are about 0.2 mm and sensitivity are $120\phantom{\rule{0.166667em}{0ex}}\mathsf{\mu}\mathrm{V}\phantom{\rule{0.166667em}{0ex}}{\mathrm{W}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$. The calibration procedure, described in [69,70], permits us to calibrate these sensors with a precision of $\pm 3\%$ as demonstrated for the example in [71]. Secondly, the entire sample is positioned between two exchange plates in order to impose the temperature on the two main faces. The plates are held compressed against the sample by means of clamps, as shown in Figure 2b.

#### 2.2. Numerical Method

- The PCM being micro-encapsulated, the thermal expansion during the phase transition can be neglected, which implies that solid and liquid densities are equal.n.b. in such a case, there is no difference between the mass and volume fractions, which are therefore both denoted by $\chi $.
- The thermodynamic behavior of the PCM, i.e., its enthalpy function, obeys a binary solution model [68]:$$\begin{array}{cc}\hfill \mathrm{d}\chi & ={\displaystyle \frac{{T}_{M}^{\star}-{T}_{M}}{{\left(\right)open="("\; close=")">{T}_{M}^{\star}-T}^{}}}\phantom{\rule{0.166667em}{0ex}}\mathrm{d}T\hfill \end{array}$$$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}h}{\mathrm{d}T}}& ={\displaystyle \frac{{T}_{M}^{\star}-{T}_{M}}{{\left(\right)open="("\; close=")">{T}_{M}^{\star}-T}^{}}}{L}_{M}(T)+{c}_{s}\left(\right)open="("\; close=")">1-{\displaystyle \frac{{T}_{M}^{\star}-{T}_{M}}{{T}_{M}^{\star}-T}}\hfill & +{c}_{l}{\displaystyle \frac{{T}_{M}^{\star}-{T}_{M}}{{T}_{M}^{\star}-T}}\end{array}$$Here, the involved latent heat for the composite mortar depends on the mass fraction Y of the PCM included in this latter one:$${L}_{M}=Y{L}_{M}^{\star}$$

#### 2.3. Inversion Process

`GenOpt`free tool from the Lawrence Berkeley National Laboratory [79,80]. It indeed implements several classic algorithms, including the ‘

`GPSCoordinateSearch`’ that has been used here: the basic principle of the generalized pattern search (GPS) is to link monodimensional minimizations according to the different directions of the search space (dimension N). The version implemented in

`GenOpt`automatically adapts the search accuracy to obtain an acceptable solution without requiring too much computation time. As any optimization method,

`GenOpt`requires to define all the N parameters to be identified ($\overrightarrow{p}\in {\mathbb{R}}^{N}$) so as to minimize the cost function $f(\overrightarrow{p}):{\mathbb{R}}^{N}\u27f6\mathbb{R}$ given by relation (6). Let us mention here that

`GenOpt`always delegates the evaluation of the cost function f to an external program, which corresponds here to the code implementing the aforementioned system of PDE. From a practical point of view, the raw text files that ensure the communication between these two threads are configured following the documentation [81]. As a rule of thumb, all identified parameters are described by their name, used only for display purpose, and by an initial value (first guess), as well as by lower and upper bounds. As an example, the latent heat and solid thermal conductivity are initialized, respectively, at $16.5\phantom{\rule{0.166667em}{0ex}}\mathrm{k}\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ and $0.4\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$, and their minimal and maximal authorized values are 14 and $18\phantom{\rule{0.166667em}{0ex}}\mathrm{k}\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$, and 0.3 and $0.5\phantom{\rule{0.166667em}{0ex}}\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$. Moreover, each run is repeated three times: in one case, the initial seed is set by the user, and in the two others, they are randomly chosen inside the search space. In these latter cases, it is obviously checked that the chosen first guess will lay inside the respective bounds defined hereinbefore.

`GenOpt`therefore proposes to re-launch the same optimization several times (three times in this work) by randomly selecting the initial estimate in the search space: this lengthens computation times but increases the chances of converging towards the global minimum.

## 3. Results

#### 3.1. First Sample

#### 3.2. All Samples

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Latin symbols | |

c | specific heat capacity, $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ |

D | duration, $\mathrm{h}$ |

e | thickness, $\mathrm{m}$ |

f | cost function, ${\mathrm{W}}^{2}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-4}$ |

h | specific enthalpy, $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ |

L | latent heat, $\mathrm{J}\phantom{\rule{0.166667em}{0ex}}{\mathrm{kg}}^{-1}$ |

$\overrightarrow{p}$ | set of parameters, − |

R | thermal resistance, ${\mathrm{W}}^{-1}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{2}$ |

T | temperature, ${}^{\circ}\mathrm{C}$ |

t | time, $\mathrm{s}$ or min |

$\overrightarrow{X}$ | vector of coordinates, $\mathrm{m}$ |

Y | mass fraction of PCM, − |

Greek symbols | |

$\epsilon $ | error, % |

$\phi $ | heat flux, $\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-2}$ |

$\lambda $ | thermal conductivity, $\mathrm{W}\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-1}$ |

$\rho $ | density, $\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3}$ |

$\chi $ | liquid fraction, − |

Subscripts and superscripts | |

$\star $ | pure component (solvent phase) |

$ex$ | execution |

$\mathrm{exp}$ | experimental |

L | left |

l | liquid |

M | melting |

$\mathrm{num}$ | numerical |

p | exchange plate |

R | right |

s | solid |

w | weight |

Acronyms | |

DSC | differential scanning calorimetry |

GPS | generalized pattern search |

PCM | phase change material |

## Appendix A. Additional Data

**Table A1.**19.2%

_{w}-PCM mortar: values of the identified parameters for each heating/cooling ramp and for the first, second, third, and all cycles together cases.

D | f | R_{L} | R_{R} | λ_{s} | λ_{l} | c_{s} | c_{l} | T_{M} | ${\mathit{T}}_{\mathit{M}}^{\star}$ | L_{M} | t_{ex} |
---|---|---|---|---|---|---|---|---|---|---|---|

22,577 | 1.098 · 10^{−2} | 4.638 · 10^{−3} | 0.606 | 0.6 | 1094 | 1094 | 25.57 | 27.18 | 12,322 | 0:22:39 | |

2 h | 22,596 | 1.085 · 10^{−2} | 4.39 · 10^{−3} | 0.594 | 0.589 | 1094 | 1094 | 25.57 | 27.17 | 12,317 | 1:05:41 |

23,028 | 1.084 · 10^{−2} | 4.39 · 10^{−3} | 0.594 | 0.589 | 1100 | 1094 | 25.58 | 27.17 | 12,267 | 1:12:19 | |

68,175 | 1.084 · 10^{−2} | 4.412 · 10^{−3} | 0.594 | 0.589 | 1094 | 1094 | 25.57 | 27.17 | 12,322 | 0:56:43 | |

20,403 | 1.125 · 10^{−2} | 4.737 · 10^{−3} | 0.594 | 0.589 | 1089 | 1089 | 25.57 | 27.17 | 12,478 | 2:05:37 | |

3 h | 19,899 | 1.154 · 10^{−2} | 5.028 · 10^{−3} | 0.611 | 0.6 | 1089 | 1089 | 25.58 | 27.19 | 12,461 | 2:06:14 |

19,774 | 1.139 · 10^{−2} | 4.918 · 10^{−3} | 0.606 | 0.594 | 1094 | 1094 | 25.54 | 27.09 | 12,294 | 2:05:01 | |

59,337 | 1.139 · 10^{−2} | 4.891 · 10^{−3} | 0.606 | 0.594 | 1089 | 1094 | 25.56 | 27.14 | 12,400 | 1:45:29 | |

18,095 | 1.111 · 10^{−2} | 5 · 10^{−3} | 0.6 | 0.589 | 1094 | 1094 | 25.53 | 27.06 | 12,328 | 2:27:47 | |

4 h | 18,642 | 1.111 · 10^{−2} | 4.918 · 10^{−3} | 0.594 | 0.583 | 1089 | 1094 | 25.54 | 27.11 | 12,422 | 0:44:17 |

18,230 | 1.125 · 10^{−2} | 5 · 10^{−3} | 0.6 | 0.589 | 1094 | 1094 | 25.53 | 27.07 | 12,328 | 2:50:45 | |

55,793 | 1.111 · 10^{−2} | 4.891 · 10^{−3} | 0.594 | 0.583 | 1089 | 1094 | 25.54 | 27.11 | 12,422 | 2:05:56 | |

17,821 | 1.2 · 10^{−2} | 5.455 · 10^{−3} | 0.606 | 0.594 | 1094 | 1094 | 25.53 | 27.03 | 12,300 | 4:12:57 | |

5 h | 17,924 | 1.2 · 10^{−2} | 5.455 · 10^{−3} | 0.606 | 0.594 | 1094 | 1094 | 25.52 | 27.01 | 12,283 | 4:31:10 |

17,518 | 1.216 · 10^{−2} | 5.625 · 10^{−3} | 0.617 | 0.6 | 1089 | 1094 | 25.54 | 27.08 | 12,361 | 1:20:27 | |

52,845 | 1.2 · 10^{−2} | 5.455 · 10^{−3} | 0.606 | 0.594 | 1094 | 1094 | 25.52 | 27 | 12,261 | 3:28:31 |

**Table A2.**19.2%

_{w}-PCM mortar: values of the identified parameters for each heating/cooling ramp and for the first, second, third, and all cycles together cases.

D | f | R_{L} | R_{R} | λ_{s} | λ_{l} | c_{s} | c_{l} | T_{M} | ${\mathit{T}}_{\mathit{M}}^{\star}$ | L_{M} | t_{ex} |
---|---|---|---|---|---|---|---|---|---|---|---|

30,313 | 1.159 · 10^{−2} | 3.127 · 10^{−2} | 0.435 | 0.442 | 1096 | 1035 | 25.59 | 27.31 | 18,135 | 01:43:24 | |

2 h | 23,390 | 1.224 · 10^{−2} | 3.14 · 10^{−2} | 0.431 | 0.436 | 1067 | 1037 | 25.57 | 27.28 | 18,233 | 00:50:28 |

21,859 | 1.24 · 10^{−2} | 3.127 · 10^{−2} | 0.431 | 0.436 | 1069 | 1036 | 25.58 | 27.28 | 18,168 | 00:59:37 | |

85,878 | 1.204 · 10^{−2} | 3.127 · 10^{−2} | 0.431 | 0.437 | 1074 | 1035 | 25.59 | 27.34 | 18,279 | 02:16:47 | |

20,785 | 1.304 · 10^{−2} | 3.176 · 10^{−2} | 0.427 | 0.437 | 1062 | 1035 | 25.7 | 27.51 | 18,352 | 06:43:38 | |

3 h | 20,551 | 1.321 · 10^{−2} | 3.189 · 10^{−2} | 0.427 | 0.436 | 1065 | 1031 | 25.68 | 27.43 | 18,241 | 08:27:05 |

21,697 | 1.33 · 10^{−2} | 3.189 · 10^{−2} | 0.428 | 0.438 | 1067 | 1030 | 25.69 | 27.46 | 18,265 | 10:36:33 | |

61,906 | 1.313 · 10^{−2} | 3.176 · 10^{−2} | 0.426 | 0.436 | 1066 | 1032 | 25.69 | 27.47 | 18,273 | 08:26:34 | |

19,051 | 1.399 · 10^{−2} | 3.24 · 10^{−2} | 0.427 | 0.443 | 1062 | 1031 | 25.76 | 27.59 | 18,343 | 13:48:26 | |

4h | 19,216 | 1.397 · 10^{−2} | 3.24 · 10^{−2} | 0.428 | 0.442 | 1067 | 1031 | 25.74 | 27.54 | 18,207 | 11:21:10 |

19,216 | 1.397 · 10^{−2} | 3.24 · 10^{−2} | 0.428 | 0.442 | 1067 | 1031 | 25.74 | 27.54 | 18,207 | 08:59:56 | |

57,062 | 1.399 · 10^{−2} | 3.24 · 10^{−2} | 0.427 | 0.442 | 1065 | 1031 | 25.75 | 27.57 | 18,270 | 08:19:15 | |

18,094 | 1.434 · 10^{−2} | 3.266 · 10^{−2} | 0.429 | 0.443 | 1073 | 1035 | 25.77 | 27.54 | 17,978 | 10:20:06 | |

5 h | 19,170 | 1.467 · 10^{−2} | 3.293 · 10^{−2} | 0.433 | 0.45 | 1066 | 1032 | 25.78 | 27.61 | 18,190 | 06:04:20 |

18,679 | 1.473 · 10^{−2} | 3.306 · 10^{−2} | 0.433 | 0.451 | 1063 | 1031 | 25.78 | 27.63 | 18,260 | 06:47:33 | |

56,006 | 1.465 · 10^{−2} | 3.293 · 10^{−2} | 0.433 | 0.449 | 1064 | 1032 | 25.78 | 27.63 | 18,222 | 07:19:31 |

**Table A3.**22.1%

_{w}-PCM mortar: values of the identified parameters for each heating/cooling ramp and for the first, second, third, and all cycles together cases.

D | f | R_{L} | R_{R} | λ_{s} | λ_{l} | c_{s} | c_{l} | T_{M} | ${\mathit{T}}_{\mathit{M}}^{\star}$ | L_{M} | t_{ex} |
---|---|---|---|---|---|---|---|---|---|---|---|

17,511 | 7.893 · 10^{−3} | 1.364 · 10^{−2} | 0.384 | 0.384 | 1154 | 1088 | 25.52 | 26.91 | 20,679 | 00:54:19 | |

2 h | 18,899 | 7.825 · 10^{−3} | 1.364 · 10^{−2} | 0.384 | 0.384 | 1151 | 1084 | 25.52 | 26.94 | 20,778 | 04:15:25 |

21,183 | 7.826 · 10^{−3} | 1.364 · 10^{−2} | 0.382 | 0.382 | 1137 | 1084 | 25.54 | 27.02 | 21,027 | 3:43:08 | |

57,935 | 7.895 · 10^{−3} | 1.364 · 10^{−2} | 0.384 | 0.384 | 1150 | 1087 | 25.52 | 26.93 | 20,757 | 8:31:31 | |

21,014 | 8.333 · 10^{−3} | 1.429 · 10^{−2} | 0.384 | 0.387 | 1132 | 1081 | 25.72 | 27.37 | 21,336 | 18:28:41 | |

3 h | 20,915 | 8.257 · 10^{−3} | 1.429 · 10^{−2} | 0.382 | 0.387 | 1142 | 1081 | 25.63 | 27.19 | 21,094 | 41:58:24 |

23,850 | 8.333 · 10^{−3} | 1.429 · 10^{−2} | 0.384 | 0.387 | 1153 | 1081 | 25.62 | 27.1 | 20,848 | 10:45:50 | |

66,027 | 8.333 · 10^{−3} | 1.429 · 10^{−2} | 0.384 | 0.389 | 1144 | 1083 | 25.66 | 27.21 | 21,052 | 40:53:30 | |

28,680 | 8.571 · 10^{−3} | 1.452 · 10^{−2} | 0.382 | 0.384 | 1144 | 1081 | 25.67 | 27.26 | 21,053 | 42:29:36 | |

4 h | 26,641 | 8.654 · 10^{−3} | 1.452 · 10^{−2} | 0.384 | 0.387 | 1151 | 1081 | 25.67 | 27.23 | 20,983 | 108:57:29 |

25,323 | 8.571 · 10^{−3} | 1.452 · 10^{−2} | 0.382 | 0.387 | 1144 | 1082 | 25.67 | 27.26 | 21,067 | 32:22:59 | |

80,789 | 8.571 · 10^{−3} | 1.452 · 10^{−2} | 0.382 | 0.387 | 1146 | 1081 | 25.67 | 27.26 | 21,079 | 59:49:49 | |

28,322 | 8.824 · 10^{−3} | 1.475 · 10^{−2} | 0.384 | 0.387 | 1160 | 1086 | 25.67 | 27.16 | 20,682 | 9:59:39 | |

5 h | 26,548 | 8.738 · 10^{−3} | 1.475 · 10^{−2} | 0.382 | 0.387 | 1159 | 1084 | 25.67 | 27.2 | 20,763 | 44:44:31 |

27,889 | 8.654 · 10^{−3} | 1.475 · 10^{−2} | 0.382 | 0.387 | 1163 | 1087 | 25.67 | 27.16 | 20,587 | 13:43:08 | |

80,419 | 8.738 · 10^{−3} | 1.475 · 10^{−2} | 0.382 | 0.387 | 1147 | 1086 | 25.69 | 27.28 | 21,000 | 74:28:11 |

**Figure A1.**12.4%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 3 h heating/cooling ramp experiment.

**Figure A2.**12.4%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 4 h heating/cooling ramp experiment.

**Figure A3.**12.4%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 5 h heating/cooling ramp experiment.

**Figure A4.**19.2%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 2 h heating/cooling ramp experiment.

**Figure A5.**19.2%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 3 h heating/cooling ramp experiment.

**Figure A6.**19.2%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 4 h heating/cooling ramp experiment.

**Figure A7.**19.2%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 5 h heating/cooling ramp experiment.

**Figure A8.**22.1%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 2 h heating/cooling ramp experiment.

**Figure A9.**22.1%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 3 h heating/cooling ramp experiment.

**Figure A10.**22.1%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 4 h heating/cooling ramp experiment.

**Figure A11.**22.1%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 5 h heating/cooling ramp experiment.

**Figure A12.**12.4%

_{w}-PCM mortar: comparisons of the identified enthalpy functions with respect to the heating/cooling ramp (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure A13.**19.2%

_{w}-PCM mortar: comparisons of the identified enthalpy functions with respect to the heating/cooling ramp (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure A14.**22.1%

_{w}-PCM mortar: comparisons of the identified enthalpy functions with respect to the heating/cooling ramp (n.b. unique formulation pertaining for both fusion and crystallization).

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**Figure 3.**Detailed description of the experimental set-up. (

**a**) Schematic view; (

**b**) Detailed view (with boundary conditions).

**Figure 5.**12.4${\%}_{w}$-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the first cycle of the $2\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ heating/cooling ramp experiment.

**Figure 6.**12.4${\%}_{w}$-PCM mortar: identified enthalpy function for the first cycle of the $2\phantom{\rule{0.166667em}{0ex}}\mathrm{h}$ heating/cooling ramp (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure 7.**12.4%

_{w}-PCM mortar: comparisons of the experimental fluxes with the numerical ones after identification for the 2 h heating/cooling ramp experiment

**Figure 8.**12.4%

_{w}-PCM mortar: comparisons of the identified enthalpy functions for 2 h heating/cooling ramp (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure 9.**Comparisons of the identified enthalpy functions for all the heating/cooling ramps (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure 10.**Comparisons of the identified enthalpy functions for all the mortars (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure 11.**Comparisons of the enthalpy functions obtained with minimal (blue) and maximal (red) values of the identified parameters, together with the absolute discrepancy, for all the mortars (n.b. unique formulation pertaining for both fusion and crystallization).

**Figure 12.**Comparisons of the present estimations for the thermal conductivities for the solid (blue) and liquid (red) state with similar studies (black data mean that only an unique value has been given, without specifying the physical state (solid or liquid)).

**Figure 13.**Comparisons of the present estimations for the specific heat capacities for the solid (blue) and liquid (red) state with similar studies (black data mean that only an unique value has been given, without specifying the physical state (solid or liquid)).

Material | Sand | Cement | Water | PCM |
---|---|---|---|---|

$\rho \phantom{\rule{0.166667em}{0ex}}(\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}{\mathrm{m}}^{-3})$ | 1600 | 3060 | 1000 | 300 |

Mortar | $\mathit{e}\phantom{\rule{0.166667em}{0ex}}(\mathbf{m})$ | $\mathit{\rho}$ (kg m${}^{-3}$) |
---|---|---|

12.4${\%}_{w}$ | 0.039 | 1412 |

19.2${\%}_{w}$ | 0.0388 | 1257 |

22.1${\%}_{w}$ | 0.0385 | 1231 |

**Table 3.**Minimal, maximal, average values, and discrepancy between each identified parameter for each mortar (considering all ramps and various cycles).

Mortar | Stat. | R_{L} (W^{−1}K m^{2}) | R_{R} (W^{−1}K m^{2}) | λ_{s} (W K^{−1} m^{−1}) | λ_{l} (W K^{−1} m^{−1}) | c_{s} (J K^{−1} kg^{−1}) | c_{l} (J K^{−1} kg^{−1}) | T_{M} (°C) | ${\mathit{T}}_{\mathit{M}}^{\star}$ (°C) | L_{M} (J kg^{−1}) |
---|---|---|---|---|---|---|---|---|---|---|

Min | 1.216 · 10^{−2} | 5.625 · 10^{−3} | 0.594 | 0.583 | 1089 | 1089 | 25.52 | 27 | 12,26 | |

12.4%_{w} | Max | 1.084 · 10^{−2} | 4.39 · 10^{−3} | 0.617 | 0.6 | 1100 | 1094 | 25.58 | 27.19 | 12,478 |

Avg. | 1.135 · 10^{−2} | 4.921 · 10^{−3} | 0.602 | 0.592 | 1093 | 1094 | 25.55 | 27.11 | 12,348 | |

ε | 12.16% | 28.12% | 3.74% | 2.86% | 1.02% | 0.51% | 0.22% | 0.7% | 1.77% | |

Min | 1.473 · 10^{−2} | 3.306 · 10^{−2} | 0.426 | 0.436 | 1062 | 1030 | 25.57 | 27.28 | 17,978 | |

19.2%_{w} | Max | 1.159 · 10^{−2} | 3.127 · 10^{−2} | 0.435 | 0.451 | 1096 | 1037 | 25.78 | 27.63 | 18,352 |

Avg. | 1.338 · 10^{−2} | 3.21 · 10^{−2} | 0.43 | 0.441 | 1068 | 1033 | 25.7 | 27.48 | 18,227 | |

ε | 27.09% | 5.71% | 2.03% | 3.54% | 3.26% | 0.72% | 0.85% | 1.3% | 2.08% | |

Min | 8.824 · 10^{−3} | 1.475 · 10^{−2} | 0.382 | 0.382 | 1132 | 1081 | 25.52 | 26.91 | 20,587 | |

22.1%_{w} | Max | 7.826 · 10^{−3} | 1.364 · 10^{−2} | 0.384 | 0.389 | 1163 | 1088 | 25.72 | 27.37 | 21,336 |

Avg. | 8.362 · 10^{−3} | 1.429 · 10^{−2} | 0.383 | 0.386 | 1149 | 1084 | 25.63 | 27.15 | 20,924 | |

ε | 12.75% | 8.2% | 0.58% | 1.74% | 2.75% | 0.62% | 0.78% | 1.69% | 3.64% |

**Table 4.**Overview of the phase change material (PCM) possibilities for use in combination with mortars.

Fatty Acids | Alcohols | Ethers | Paraffins | |||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Micronal | Mikrathermic | Devan | ||||||||||||||||||||||

Reference | Hexadecanoic Acid (Butyl Palmitate) | Octadecanoic Acid (Butyl Stearate) | Decanoic Acid (Capric Acid) | Dodecanoic Acid (Lauric Acid) | Tetradecanoic Acid (Myristic Acid) | Dodecanol | Tetradecanol | PEG400 | PEG600 | PEG1000 | DS5001 | DS5001 X | DS 5008 X | DS 5039 X | DS5040X | D18 | D24 | D28 | Microtek MPCM28 | MC18 | MC24 | MC28 | Octadecan | Unicere 55 |

[47] | ✓ | ✓ | ✓ | |||||||||||||||||||||

[48] | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||

[49] | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||

[83] | ✓ | ✓ | ||||||||||||||||||||||

[51] | ✓ | ✓ | ✓ | ✓ | ||||||||||||||||||||

[84] | ✓ | |||||||||||||||||||||||

[31] | ✓ | |||||||||||||||||||||||

[85] | ✓ | |||||||||||||||||||||||

[33] | ✓ | |||||||||||||||||||||||

[34] | ✓ | ✓ | ||||||||||||||||||||||

[35] | ✓ | |||||||||||||||||||||||

[37] | ✓ | |||||||||||||||||||||||

[82] | ✓ | |||||||||||||||||||||||

[39] | ✓ | |||||||||||||||||||||||

[40] | ✓ | |||||||||||||||||||||||

[41] | ✓(eutectic mixture) | |||||||||||||||||||||||

[54] | ✓ | ✓ | ✓ | |||||||||||||||||||||

[52] | ✓ | |||||||||||||||||||||||

[42] | ✓ | |||||||||||||||||||||||

[44] | ✓ | |||||||||||||||||||||||

[53] | ✓ | |||||||||||||||||||||||

[45] | ✓ | |||||||||||||||||||||||

[86] | ✓ | ✓ | ✓ | ✓ | ✓ | |||||||||||||||||||

[87] | ✓ | |||||||||||||||||||||||

[46] | ✓ |

**Table 5.**Estimated values of the micronal latent heat and comparison with the experimental value obtained by DSC [68].

12.4${\%}_{\mathit{w}}$ | 19.2${\%}_{\mathit{w}}$ | 22.1${\%}_{\mathit{w}}$ | |
---|---|---|---|

Avg. (J kg${}^{-1}$) | 99,696 | 94,930 | 94,792 |

$\epsilon $ | 0.7% | 4.11% | 4.25% |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Zalewski, L.; Franquet, E.; Gibout, S.; Tittelein, P.; Defer, D.
Efficient Characterization of Macroscopic Composite Cement Mortars with Various Contents of Phase Change Material. *Appl. Sci.* **2019**, *9*, 1104.
https://doi.org/10.3390/app9061104

**AMA Style**

Zalewski L, Franquet E, Gibout S, Tittelein P, Defer D.
Efficient Characterization of Macroscopic Composite Cement Mortars with Various Contents of Phase Change Material. *Applied Sciences*. 2019; 9(6):1104.
https://doi.org/10.3390/app9061104

**Chicago/Turabian Style**

Zalewski, Laurent, Erwin Franquet, Stéphane Gibout, Pierre Tittelein, and Didier Defer.
2019. "Efficient Characterization of Macroscopic Composite Cement Mortars with Various Contents of Phase Change Material" *Applied Sciences* 9, no. 6: 1104.
https://doi.org/10.3390/app9061104