# Simulative Investigation of Thermal Capacity Analysis Methods for Metallic Latent Thermal Energy Storage Systems

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Simulated Specimen

_{s}= 5.688 kg. The chosen storage material was the eutectic alloy Al-12wt%Si, which was operated in a temperature range between 100 °C and 600 °C [4]. The storage material choice was made according to the storage concept proposed by Kraft and Klein Altstedde [4].

#### 2.2. Methods

_{fus}, specific heat capacity c

_{p}(ϑ) and melting temperature ϑ

_{fus}. The specific state variables are gained by normalising to the sample mass m

_{s}. The extrinsic values can be derived from the course of sample heat Q

_{S}over sample temperature ϑ

_{S}. Figure 1 shows a typical course of Q

_{S}for a sample with first-order phase transition and its context with the mentioned relevant thermophysical properties. Here, only systems with a sharp melting temperature—not a melting temperature range—are considered (e.g., pure metals or eutectics). In general, it is possible to define the sample, not only as the storage material, but as the full system, including further system components like the housing. Hence, the boundaries of the sample can be defined depending on the matter of interest.

_{S}is shown for a sample with a first-order phase transition. In Figure 2, it is qualitatively compared to a demonstrative “measured” Q

_{S}-curve. Deviations from the ideal course result from a temperature gradient in the sample and test setup, which in turn means that the entire sample volume is not passing through the phase transition at the same time. This effect becomes more dominant the bigger the sample is. However, this effect is not expected to be very severe in this case, as metallic samples with high thermal conductivity minimise the thermal gradient [27].

_{p}

^{(ϑ1: ϑ2)}in a temperature range from ϑ

_{1}to ϑ

_{2}can directly be deduced from the slope of Q

_{s}(ϑ):

_{p}can be derived from the relation to the sample mass m

_{s}via c

_{p}= C

_{p}/m

_{s}. As specific heat capacity is a temperature-dependent quantity, it is recommended to compare c

_{p}values of different samples for similar temperature ranges. Enthalpy of fusion H

_{fus}and melting temperature ϑ

_{fus}are derived geometrically by the routine shown in Figure 3. This is a standardised routine described in DIN 51007.

_{S}is extrapolated linearly from the temperature ranges for C

_{p}determination closest to the melting temperature (range for lower temperature: a; range for higher temperature: b) to ϑ

_{fus}. This linear course of Q

_{S}before the phase transition step is called the initial base line. ϑ

_{fus}is defined as the intercept between the extrapolated initial base line and the inflection tangent of the step. The equation to determine H

_{fus}is:

_{fus}is determined via h

_{fus}= H

_{fus}/m

_{s}.

- •
- Isothermal calorimeters can only be operated at a defined temperature, thus it is not possible to test the whole operating temperature range of the specimen system.
- •
- Isoperibol calorimeters with uncontrolled heat exchange cannot fulfill the criterion of controllable heat input and output.
- •
- Isoperibol flow calorimeters are only suitable for fluid specimens.
- •
- Calorimeters with linear or nonlinear temperature changes of the surroundings are called scanning calorimeters. An essential component in the operation of scanning calorimeters is the time needed for heat exchange between the furnace and specimen. Depending on the heat path from the sample to the surroundings, the calorimeter signal can show a thermal lag [28,29]. Therefore, scanning calorimeters are not recommended for use with large sample sizes [30] and inhomogeneous samples, which is how the THS specimen system can be classified [31].

_{S}at a certain sample temperature ϑ

_{S}, referring to the chosen reference temperature ϑ

_{Ref}. ϑ

_{S}is logged over time t, making it possible to describe either Q

_{S}(ϑ

_{S}) or Q

_{S}(t). Here, we chose to use the time-dependent values. The balancing equation, which can be used for every procedure, is:

_{S}(t) = Q

_{H}(t) + Q

_{C}(t) + Q

_{Loss}(t) + Q

_{WF}(t)

_{H}is the heat input of the electrical heater, Q

_{C}is the thermal energy in the components of the test bench, Q

_{Loss}is the heat loss to the surroundings and Q

_{WF}is the heat output by the working fluid. The difference between the procedures is in how the different contributions are determined. An overview is given in Table 1.

_{C}is always determined by measuring the difference of the components’ instantaneous temperature ϑ

_{C}and initial temperature ϑ

_{C,0}. With knowledge about the heat capacity of all test setup components C

_{C}, the heat put into the components is described by:

#### 2.2.1. Stepwise Adiabatic Procedure

_{H}can be determined by electrical heating power P

_{H}, which is supplied in the time interval between t

_{i}and t

_{i+1}:

_{Loss}cannot be directly measured but can be determined via calibration. To achieve this, the heater is power-controlled to hold the sample temperature constant for an equilibration time t

_{eq}until a steady state is reached. Under steady state conditions, the total heating rate equals the heat flow into the setup components and the heat loss rate to the surroundings ${\dot{Q}}_{\mathrm{Loss}}$. The heat loss rate is then correlated to the surface temperature via a second-degree polynomial. The heat loss to the surroundings Q

_{Loss}is finally received via integration over time. For this study, t

_{eq}was chosen to be 45 min for each temperature range. The temperature step between 500 °C and 600 °C, t

_{eq}, was chosen to be 60 min, as the phase change of the examined material Al-12wt%Si occurs in this temperature range.

#### 2.2.2. Isoperibolic Procedure with Loss Correlation

_{S}is determined by measuring the heat balance of the working fluid which is cooling the sample. This procedure belongs to the class of isoperibol calorimeters with controlled heat exchange. Therefore, the sample is again thermally isolated, but this time, the working fluid is conducted through the central tube of the sample system. At first, the sample is heated to a maximum temperature by an electrical heater. Then, the heater is shut off and the flow of the working fluid is started. The heat transferred from the sample to the working fluid Q

_{WF}is determined by measuring the temperature difference of the working fluid before passing sample ϑ

_{in}and after passing sample ϑ

_{out}. The mass flow rate $\dot{m}$ of the working fluid is set constant and the specific heat capacity of the working fluid c

_{p,WF}is known. Q

_{WF}can be determined via the following equation:

_{Loss}is determined via calibration and correlation during the heating phase, as described for the stepwise adiabatic procedure. Thus, calibration and measurement can be performed concurrently. For this procedure, t

_{eq}= 30 min was chosen for calibration. The mass flow rate $\dot{m}$ was set to 50 kg/h.

#### 2.2.3. Isoperibolic Procedure with Cooling Correlation

_{WF}and the heat loss to the surroundings Q

_{Loss}are not determined separately, but are combined in the expression Q

_{Cool}= Q

_{WF}+ Q

_{Loss}. Calibration is performed in order to correlate Q

_{Cool}to the temperature of the heat exchanger ϑ

_{ex}in the heat transfer zone of the sample to the working fluid. Therefore, the power of the electrical heater P

_{H}is logged for different steady state temperature steps during heating. According to the other calibration methods, the following balance is true: ${\dot{Q}}_{\mathrm{H}}$ − ${\dot{Q}}_{\mathrm{C}}$ = ${\dot{Q}}_{\mathrm{Cool}}$. Cooling rate ${\dot{Q}}_{\mathrm{Cool}}$ is then correlated to ϑ

_{ex}. It cannot be distinguished which amount of thermal energy is put out by the working fluid or by the loss to the surroundings. To keep the correlation as precise as possible, the working fluid’s mass flow $\dot{m}$ and the surrounding conditions need to be constant for both calibration and measurement. After the calibration and heating phase, the electrical heater is shut off, but the working fluid is kept flowing. Then, ϑ

_{ex}(t) is logged over the cooling phase. For this procedure, t

_{eq}= 30 min was chosen for calibration with active working fluid flow.

#### 2.3. Experimental Setup

_{s}= 5.688 kg Al-12wt%Si cast into it. The heating unit was a hollow cylindrical copper piece with bore holes, into which 8 electric cartridge heaters were inserted within the central tube of the containment. Each cartridge heater had a maximum electrical power of 500 W. Within the centre of the heating unit, air could flow through the specimen for discharging. In order to enable high heat transfer, a heat transfer structure with fins was present in the hollow shape of the copper cylinder. The components of the air duct through the centre of the storage were thermally decoupled from the rest of the setup by ceramic fittings. The thermal energy storage component was surrounded by two layers of microporous thermal insulation to reduce thermal losses. PID controllers controlled the air mass flow and the heating power.

#### 2.4. Simulation

## 3. Results

#### 3.1. Stepwise Adiabatic Procedure

_{S}over time t for the stepwise adiabatic procedure is shown in Figure 6a. The temperature plateaus show the six equilibration steps where the heat loss to the surroundings was determined by measuring the heating power. The melting process of the storage material can be observed at an additional temperature plateau at 576.5 °C. The course of Q

_{S}(ϑ) resulting from Equations (4) and (5) with temperature and power outputs of the simulation is shown in Figure 6b as well as the course of Q

_{S}(ϑ) calculated from the simulation input values for c

_{p}, ϑ

_{fus}and h

_{fus}. The slope of Q

_{S}(ϑ), which corresponds to c

_{p}, is observed to be lower in the simulation output than the input property. Thus, the overall amount of energy stored in the studied temperature range is underestimated.

#### 3.2. Isoperibolic Procedure with Loss Correlation

_{S}over time t for the isoperibolic procedure with loss correlation is shown in Figure 7a. The heating process for calibration and the cooling process for measurement can be seen. The course of Q

_{s}resulting from simulation and the input course of Q

_{s}are shown in Figure 7b. The simulated course of ϑ

_{s}again shows the equilibration steps for heat loss determination and reveals the solidification process with an additional plateau at 576.5 °C. Upon further cooling, the temperature decreased more slowly and approximated room temperature asymptotically. The simulated course of Q

_{s}revealed that c

_{p}(ϑ) was not constant, although this was assumed for the input value. This interesting observation is elaborated further in the discussion. Again, the overall energy amount stored in the studied temperature range is underestimated. In contrast to the stepwise adiabatic procedure, this time Q

_{s}resulted from a cooling half-cycle. This can be seen at the sharp edge at higher energy amounts, where solidification starts, and the diffuse transition to sensible cooling, as a solidification of the whole sample, takes some time.

#### 3.3. Isoperibolic Procedure with Cooling Correlation

_{S}over time t for the isoperibolic procedure with cooling correlation is shown in Figure 8a. The heating process for calibration and the cooling process for measurement can be seen. The course of Q

_{s}resulting from simulation and the input course of Q

_{s}are shown in Figure 8b. The simulation of ϑ

_{S}shows a similar course as in the procedure with loss correlation. The simulated course of Q

_{S}was nearly linear in the sensible range. However, the deviation of c

_{p}in the solid range was visibly larger compared to the other procedures, and the energy stored in the studied temperature range was more greatly underestimated.

#### 3.4. Deviation of Simulation Input Values

_{p}(ϑ), ϑ

_{fus}, h

_{fus}and the overall stored energy in sample Q

_{S}between 100 °C and 600 °C. The input values were determined via differential scanning calorimetry. An overview of the thermophysical properties generated by the different procedures in comparison to the input values is shown in Figure 9, with indication of the absolute deviation (primary ordinate) and relative deviation E (secondary ordinate).

_{p}, but the largest deviation for h

_{fus}. However, the isoperibolic procedure with cooling correlation shows contrasting behaviour. The assignment of stored energy to the sensible or the latent regime depends on the choice of extrapolation limits and the width of the now diffuse phase transition (compared with Figure 3). The stored heat in the whole temperature range Q

_{s}

^{100:600}is a better indication for the accuracy of the procedure, as it is not as affected by these evaluation artefacts. Regarding this quantity, the stepwise adiabatic procedure has the lowest deviation from the simulation input value. However, this procedure is only suitable for analysing heating processes, not half-cycles with active cooling.

## 4. Discussion

_{p}, which could result from a temperature gradient in the sample setup, meaning that steady state was not reached in the chosen time t

_{eq}. Thus, Q

_{Loss}was overestimated and c

_{p}underestimated. The deviation of c

_{p}for t

_{eq}= 45 min was smaller than for a tested alternative with t

_{eq}= 30 min, as the state between the temperature steps was closer to steady state conditions. This suggests that increasing t

_{eq}leads to increased accuracy in the determination of c

_{p}and especially heat losses. Increasing the time step allows large thermal inertias to equilibrate, but this increases experiment time. An optimum between correlation temperature and equilibration time should be determined when commissioning physical experiments. In the simulation, thermal inertia only results from bulk thermal resistance, whereas in the experiment, thermal contact resistance also needs to be considered. Thermal contacts are strongly geometry-dependent and are affected by thermal expansion and the properties of layers or coatings at interfaces. For storage systems with poor thermal contact, the accuracy of the procedures can be reduced if the equilibration time is not increased.

_{Loss}and stored thermal energy Q

_{S}increased. For the isoperibolic procedure with loss correlation, the biggest deviation between simulated and input c

_{p}occurred at low temperatures, where the cooling rate is slowest. The slower the cooling, the faster Q

_{WF}decreases compared to Q

_{Loss}and the larger the proportion of Q

_{Loss}in the total balance. The accuracy of this procedure might be increased by increasing the portion of Q

_{WF}in the total balance with an increased mass flow rate of coolant. Similar behaviour was observed for the isoperibolic procedure with cooling correlation. However, in this procedure, the working fluid mass flow cannot be increased without limit, as the thermal output is not allowed to be larger than the thermal input realisable by the installed electrical heaters.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Context of thermophysical properties enthalpy of fusion H

_{fus}, heat capacity C

_{p}(ϑ) and melting temperature ϑ

_{fus}with course of sample heat Q

_{S}over sample temperature ϑ

_{S}referred to reference temperature ϑ

_{Ref}.

**Figure 2.**Ideal (solid) and measured (broken) curve of Q

_{S}over ϑ

_{S}of sample with first-order phase transition.

**Figure 3.**Schematic routine to derive enthalpy of fusion H

_{fus}and melting temperature ϑ

_{fus}geometrically from Q

_{S}curve.

**Figure 4.**Schematic drawing of test bench. Component labelled “thermal energy storage” is the specimen for this study and consists of AISI 321 containment filled with mPCM Al-12wt%Si.

**Figure 6.**Results of stepwise adiabatic procedure. (

**a**) Course of simulated sample temperature over time with t

_{eq}= 45 min. (

**b**) Course of Q

_{S}curve resulting from simulation (solid) compared to curve resulting from simulation input values (broken).

**Figure 7.**Results of isoperibolic procedure with loss correlation. (

**a**) Course of simulated sample temperature over time with t

_{eq}= 30 min and $\dot{m}$ = 50 kg/h. (

**b**) Course of Q

_{S}curve resulting from simulation (solid) compared to curve resulting from simulation input values (broken).

**Figure 8.**Results of isoperibolic procedure with cooling correlation. (

**a**) Course of simulated sample temperature over time with t

_{eq}= 30 min and $\dot{m}$ = 20 kg/h. (

**b**) Course of Q

_{S}curve resulting from simulation (solid) compared to curve resulting from simulation input values (broken).

**Figure 9.**Absolute values of measuring procedures referring to simulation input values (maximum of primary ordinate) and relative deviation E (secondary ordinate). Procedure 1 = stepwise adiabatic procedure; 2 = isoperibolic procedure with loss correlation; 3 = isoperibolic procedure with cooling correlation. (

**a**) Specific heat capacity between 50 °C and 500 °C; (

**b**) specific heat capacity between 600 °C and 650 °C; (

**c**) temperature of fusion; (

**d**) specific heat of fusion; (

**e**) heat stored in specimen between 100 °C and 600 °C.

**Table 1.**Overview of determination of each contribution to heat flow balance equation for different calorimetric measurement procedures.

Procedure | Q_{H} | Q_{C} | Q_{Loss} | Q_{WF} |
---|---|---|---|---|

Stepwise adiabatic | Measure P_{H} | Measure (ϑ_{C} − ϑ_{C,0}) | Correlate to ϑ_{Surface} | Zero |

Isoperibolic with loss correlation | Zero | Measure (ϑ_{C} − ϑ_{C,0}) | Correlate to ϑ_{Surface} | Measure (ϑ_{out} − ϑ_{in}) and $\dot{m}$ |

Isoperibolic with cooling correlation | Zero | Measure (ϑ_{C} − ϑ_{C,0}) | Correlate Q_{Cool} to ϑ_{ex} |

_{H}, power of electrical heater (W); ϑ

_{C}, temperature of test bench components (°C); ϑ

_{C,0}, Initial temperature of test bench components (°C); ϑ

_{Surface}, temperature of test bench surface (°C); ϑ

_{Out}, temperature of working fluid after passing sample (°C); ϑ

_{In}, tempertuare of working fluid before passing sample (°C); Q

_{Cool}, Q

_{Loss}+ Q

_{WF}(J); ϑ

_{ex}, temperature of heat exchanger in heat transfer zone of sample to working fluid (°C).

**Table 2.**Temperature sensors used to measure relevant temperatures for procedure evaluation (for cylindrical components, logarithmic average was used). In italics are positions of sensors as marked in Figure 5.

Measure | Sensors |
---|---|

ϑ_{s} | Average of seven temperature sensors located in mPCM (s1–s7) |

ϑ_{surface} | Weighted average from three temperature sensors at the bottom, one side and top sheet (surface1–surface3) |

ϑ_{out} | One temperature sensor (out) |

ϑ_{in} | One temperature sensor (in) |

ϑ_{ex} | Average of two temperature sensors located at top and bottom of copper cylinder (ex1 and ex2) |

ϑ_{C} | Components of test bench were divided into: Copper cylinder: ϑ _{ex}Inner insulation: Average of ϑ _{S} and temperature sensor between insulationsOuter insulation: Average of temperature sensor between insulations and ϑ _{Surface}Outer sheets: ϑ _{Surface}Containment: ϑ _{S} |

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**MDPI and ACS Style**

Stahl, V.; Kraft, W.; Vetter, P.; Feder, F.
Simulative Investigation of Thermal Capacity Analysis Methods for Metallic Latent Thermal Energy Storage Systems. *Energies* **2021**, *14*, 2241.
https://doi.org/10.3390/en14082241

**AMA Style**

Stahl V, Kraft W, Vetter P, Feder F.
Simulative Investigation of Thermal Capacity Analysis Methods for Metallic Latent Thermal Energy Storage Systems. *Energies*. 2021; 14(8):2241.
https://doi.org/10.3390/en14082241

**Chicago/Turabian Style**

Stahl, Veronika, Werner Kraft, Peter Vetter, and Florian Feder.
2021. "Simulative Investigation of Thermal Capacity Analysis Methods for Metallic Latent Thermal Energy Storage Systems" *Energies* 14, no. 8: 2241.
https://doi.org/10.3390/en14082241