# Basic Analysis of Uncertainty Sources in the CFD Simulation of a Shell-and-Tube Latent Thermal Energy Storage Unit

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Set-Up

#### 2.2. Boundary and Initial Conditions

#### 2.3. Numerical Model

#### 2.4. Mesh and Solver

#### 2.5. Sensitivity Analysis

## 3. Results

#### 3.1. Validation and Independency Studies

#### 3.2. Sensitivity Analysis

#### 3.2.1. Temperature Measuring Position

#### 3.2.2. Material Properties

- Material properties that have no, or only little effect, if they are varied like indicated here: only ${C}_{\mathrm{I}}$ has an influence that is always below 1% (maximum of 0.12%). Here, one has to keep in mind that ${C}_{\mathrm{I}}$ is varied by several orders of magnitude in literature, and not only by ±10%.
- Material properties that have an influence on the power and the global liquid fraction, but none on the mean power (i.e., the total heat transferred): the material properties $\eta ,$ $\beta ,$ $\lambda $ and ${T}_{\mathrm{l}}-{T}_{\mathrm{s}}$ mainly influence the maximum deviations and not the mean power. The reason why $\lambda $ also has a slight influence on the average output is that the storage system can still absorb or release a certain amount of sensible heat at the end of charging and discharging.
- Material properties that also influence the mean power (i.e., the total heat transferred): all remaining material properties ($\rho ,$ $L,$ ${c}_{l},$ ${c}_{s},$ ${T}_{m}$) also affect the mean power when varied, but ${c}_{l},$ ${c}_{s}$ and ${T}_{m}$ only slightly.

#### 3.2.3. Boundary and Initial Conditions

## 4. Comparison with Previous Studies

## 5. Conclusions

- The variation of the material properties had a very large effect on the power and the global liquid fraction. In conclusion, uncertainties in the material properties may affect the results of a CFD simulation of a LTESS significantly.
- The material properties can be subdivided into three groups:
- ○
- Material properties that have no or only little effect: only ${C}_{\mathrm{I}}$ has an influence that is always below 1% (maximum of 0.12%), but one has to keep in mind that it is varied by several orders of magnitude in literature.
- ○
- Material properties that have an influence on the power and the global liquid fraction slope, but none on the mean power: the material properties $\eta ,$ $\beta ,$ $\lambda $ and ${T}_{\mathrm{l}}-{T}_{\mathrm{s}}$ mainly influence the maximum deviations and not the mean power.
- ○
- Material properties that also influence the mean power: all remaining material properties ($\rho ,$ $L,$ ${c}_{l},$ ${c}_{s},$ ${T}_{m}$) also affect the mean power when varied, but ${c}_{l},$ ${c}_{s}$ and ${T}_{m}$ only slightly.

- ${\dot{Q}}_{\mathrm{loss}}$ (up to 21.4%), $\rho $ (up to 9.2%) and $L$ (up to 6.4%) have the largest influence on the mean power.
- The ranking of the parameters and the magnitude of the influence is sometimes distinctively different in studies found in literature.
- Comparing the maximum deviation of the power can be misleading as the deviation may only occur as a short peak, and has no effect on the overall performance.
- For the investigated storage configuration and orientation, a radial variation of the temperature measurement position had a much higher impact than a vertical variation. A displacement of the measuring point in radial direction by $\pm 1\text{}\mathrm{mm}$ already led to a maximum deviation in the measured temperature of more than 5 K.

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

$A$ | lateral Surface of the storage unit | ${\mathrm{m}}^{2}$ |

$c$ | specific heat capacity | $\frac{\mathrm{J}}{\mathrm{kg}\xb7\mathrm{K}}$ |

${C}_{\mathrm{I}}$ | large constant in the Darcy term $D$ | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}\mathrm{s}}$ |

${C}_{\mathrm{I}\mathrm{I}}$ | small constant in the Darcy term $D$ | |

$D$ | Darcy term | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}\mathrm{s}}$ |

$d$ | air channel thickness | $\mathrm{m}$ |

$g$ | gravitational acceleration | $\frac{\mathrm{m}}{{\mathrm{s}}^{2}}$ |

$h$ | specific enthalpy | $\frac{\mathrm{J}}{\mathrm{kg}}$ |

$L$ | specific latent heat of fusion | $\frac{\mathrm{J}}{\mathrm{kg}}$ |

$p$ | pressure | $\mathrm{Pa}$ |

$\dot{Q}$ | heat flux | $\mathrm{W}$ |

$R$ | thermal resistance | $\frac{\mathrm{K}}{\mathrm{W}}$ |

$T$ | temperature | $\mathrm{K}$ |

${T}_{l}-{T}_{s}$ | mushy zone width | $\mathrm{K}$ |

$t$ | time | $\mathrm{s}$ |

$U$ | overall heat transfer coefficient | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

$u$ | velocity | $\frac{\mathrm{m}}{\mathrm{s}}$ |

$u$ | velocity vector | $\frac{\mathrm{m}}{\mathrm{s}}$ |

$\dot{V}$ | volume flow | $\frac{{\mathrm{m}}^{3}}{\mathrm{s}}$ |

$y$ | variable | |

$z$ | deviation | |

Greek symbols | ||

$\alpha $ | convective heat transfer coefficient | $\frac{\mathrm{W}}{{\mathrm{m}}^{2}\mathrm{K}}$ |

${\alpha}_{F}$ | global liquid fraction | |

$\beta $ | thermal expansion coefficient | $\frac{1}{\mathrm{K}}$ |

$\lambda $ | thermal conductivity | $\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ |

$\tau $ | stress tensor (without pressure) | $\mathrm{Pa}$ |

$\eta $ | dynamic viscosity | $\mathrm{Pa}\xb7\mathrm{s}$ |

$\rho $ | density | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ |

Subscripts | ||

${}_{0}$ | reference point | |

${}_{amb}$ | ambient | |

${}_{char}$ | charging | |

${}_{curve}$ | temporal curve | |

${}_{dis}$ | discharging | |

${}_{ext}$ | extern | |

${}_{HTF}$ | Heat Transfer Fluid | |

${}_{i}$ | inner | |

${}_{in}$ | inlet | |

${}_{init}$ | initial | |

${}_{l}$ | liquid, liquidus | |

${}_{loss}$ | heat loss | |

${}_{m}$ | melting | |

${}_{max}$ | maximum | |

${}_{mean}$ | mean | |

${}_{Plexi}$ | Plexiglas | |

${}_{profile}$ | spatial profile | |

${}_{ref}$ | reference case/simulation | |

${}_{s}$ | solid, solidus | |

${}_{var}$ | variation | |

${}_{W}$ | wall | |

${}_{y}$ | related to the variable y | |

Abbreviations | ||

abs | absolute value | |

CFD | Computational Fluid Dynamics | |

DSC | Differential Scanning Calorimetry | |

HTF | Heat Transfer Fluid | |

LTESS | Latent Thermal Energy Storage System | |

max | maximum value | |

PCM | Phase Change Material | |

PISO | Pressure-Implicit with Splitting of Operators | |

PRESTO! | PREssure STaggering Option | |

QUICK | Quadratic Upstream Interpolation for Convective Kinematics | |

SIMPLE | Semi-Implicit Method for Pressure Linked Equations | |

VVM | Variable Viscosity Method |

## Appendix A

Subfield | Parameter | Value |
---|---|---|

solution method | pressure-velocity coupling | PISO extrapolated |

discretization (pressure) | PRESTO! | |

discretization (momentum) | QUICK | |

discretization (energy) | QUICK | |

discretization (time) | first order implicit | |

residuals | continuity | ${10}^{-6}$ |

x-velocity | ${10}^{-6}$ | |

y-velocity | ${10}^{-6}$ | |

energy | ${10}^{-10}$ | |

simulation parameter | time step | 0.2 s |

${C}_{\mathrm{I}\mathrm{I}}$ | 0.001 |

**Table A2.**Temporal mean, maximum, and standard deviation of the temperature measurements with a varied measurement point position with respect to the reference position.

Point B-b | Point D-a | Point D-b | Point D-c | Point F-b | Mean | Unit | ||
---|---|---|---|---|---|---|---|---|

radial +1 mm | mean | −0.33 | −0.67 | −0.38 | −0.21 | −0.41 | −0.40 | K |

maximum | 4.30 | 2.24 | 2.84 | 5.21 | 2.16 | 3.35 | K | |

standard deviation | 0.74 | 0.84 | 0.62 | 0.70 | 0.49 | 0.68 | K | |

radial −1 mm | mean | 0.38 | 1.01 | 0.44 | 0.26 | 0.49 | 0.52 | K |

maximum | 3.61 | 3.10 | 2.09 | 4.32 | 1.47 | 2.92 | K | |

standard deviation | 0.74 | 1.10 | 0.60 | 0.68 | 0.48 | 0.72 | K | |

vertical +1 mm | mean | 0.03 | 0.02 | 0.02 | 0.02 | 0.03 | 0.03 | K |

maximum | 0.32 | 0.33 | 0.30 | 0.44 | 0.37 | 0.35 | K | |

standard deviation | 0.07 | 0.05 | 0.05 | 0.07 | 0.06 | 0.06 | K | |

vertical −1 mm | mean | −0.03 | −0.02 | −0.02 | −0.02 | −0.03 | −0.03 | K |

maximum | 0.32 | 0.33 | 0.30 | 0.44 | 0.37 | 0.35 | K | |

standard deviation | 0.07 | 0.05 | 0.05 | 0.07 | 0.06 | 0.06 | K |

**Table A3.**Overview of the influence on the magnitude of the mean power for all varied material properties as well as boundary and initial conditions. The charging and discharging power is always defined positive.

Charging | Discharging | |||
---|---|---|---|---|

+10% +1 K on | −10% −1 K off | +10% +1 K on | −10% −1 K off | |

$\rho $ | 8.64% | −9.20% | 7.09% | −8.02% |

${c}_{s}$ | 1.64% | −1.65% | 1.39% | −1.51% |

${c}_{l}$ | 1.08% | −1.09% | 0.70% | −0.70% |

$\lambda $ | 0.24% | −0.50% | 1.16% | −1.76% |

$L$ | 5.94% | −6.37% | 5.24% | −5.60% |

$\eta $ | −0.05% | 0.05% | 0.00% | 0.00% |

$\beta $ | 0.05% | −0.06% | 0.00% | 0.00% |

${C}_{\mathrm{I}}$ | −0.01% | 0.01% | 0.00% | 0.00% |

${T}_{\mathrm{m}}$ | 0.36% | −0.63% | 1.31% | −1.45% |

${T}_{l}-{T}_{s}$ | −0.06% | 0.04% | 0.12% | −0.20% |

inlet velocity | 0.09% | −0.12% | 0.21% | −0.25% |

initial temperature | −1.64% | 1.64% | 0.79% | −0.79% |

inlet temperature | 1.05% | −1.27% | −2.23% | 2.10% |

heat loss | 21.42% | − | −16.39% | − |

inlet temperature curve | 0.40% | − | 0.06% | − |

inlet velocity profile | 0.01% | − | −0.04% | − |

**Figure A1.**Power over time curve for charging of the standard case and the case with a decreased melting enthalpy as well as the regarding relative deviation.

**Figure A2.**Power over time curve for charging of the standard case and the case with a decreased heat conductivity as well as the regarding relative deviation.

**Figure A3.**Power over time curve for charging of the standard case and the case with heat losses as well as the regarding relative deviation.

**Figure A4.**Power over time curve for discharging of the standard case and the case with heat losses as well as the regarding relative deviation.

**Figure A5.**Power over time curve for charging of the standard case and the case with an increased inlet velocity as well as the regarding relative deviation.

## References

- Steinmann, W.-D.; Jockenhöfer, H.; Bauer, D. Thermodynamic Analysis of High-Temperature Carnot Battery Concepts. Energy Technol.
**2020**, 8, 1900895. [Google Scholar] [CrossRef][Green Version] - Mehling, H.; Cabeza, L.F. Heat and Cold Storage with PCM; Springer: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Faden, M.; Höhlein, S.; Wanner, J.; König-Haagen, A.; Brüggemann, D. Review of Thermophysical Property Data of Octadecane for Phase-Change Studies. Materials
**2019**, 12, 2974. [Google Scholar] [CrossRef] [PubMed][Green Version] - Dutil, Y.; Rousse, D.R.; Salah, N.B.; Lassue, S.; Zalewski, L. A review on phase-change materials: Mathematical modeling and simulations. Renew. Sustain. Energy Rev.
**2011**, 15, 112–130. [Google Scholar] [CrossRef] - Voller, V.R.; Markatos, N.C.; Cross, M. Techniques for account for the moving interface in convection/diffusion phase change. In Numerical Methods in Thermal Problems; Lewis, R.W., Morgan, K., Eds.; Pineridge Press: Swansea, UK, 1985; pp. 595–609. [Google Scholar]
- Morgan, K. A numerical analysis of freezing and melting with convection. Comput. Methods Appl. Mech. Eng.
**1981**, 28, 275–284. [Google Scholar] [CrossRef] - Gartling, D.K. Finite Element Analysis of Convection Heat Transfer Problems in Phase Change. In Computer Methods in Fluids; Morgan, K., Taylor, C., Brebbia, C.A., Eds.; Pentech: London, UK, 1980; pp. 257–284. [Google Scholar]
- Brent, A.D.; Voller, V.R.; Reid, K.J. Enthalpy-porosity technique for modeling convection-diffusion phase change: Application to the melting of a pure metal. Numer. Heat Transf. Part A Appl.
**1988**, 13, 297–318. [Google Scholar] - Kheirabadi, A.C.; Groulx, D. The effect of the mushy-zone constant on simulated phase change heat transfer. In Proceedings of the CHT-15: 6th International Symposium on Advances in Computational Heat Transfer, Begell House, Danbury, CT, USA, 25–29 May 2015; p. 22. [Google Scholar]
- Beust, C.; Franquet, E.; Bédécarrats, J.-P.; Garcia, P.; Pouvreau, J. Influence of the modeling parameters on the numerical CFD simulation of a shell-and-tube latent heat storage system with circular fins. In SOLARPACES 2018: International Conference on Concentrating Solar Power and Chemical Energy Systems; AIP Publishing: Melville, NY, USA, 2019; p. 200007. [Google Scholar]
- Fadl, M.; Eames, P.C. Numerical investigation of the influence of mushy zone parameter Amush on heat transfer characteristics in vertically and horizontally oriented thermal energy storage systems. Appl. Therm. Eng.
**2019**, 151, 90–99. [Google Scholar] [CrossRef] - Arena, S.; Casti, E.; Gasia, J.; Cabeza, L.F.; Cau, G. Numerical simulation of a finned-tube LHTES system: Influence of the mushy zone constant on the phase change behaviour. Energy Procedia
**2017**, 126, 517–524. [Google Scholar] [CrossRef] - Shmueli, H.; Ziskind, G.; Letan, R. Melting in a vertical cylindrical tube: Numerical investigation and comparison with experiments. Int. J. Heat Mass Transf.
**2010**, 53, 4082–4091. [Google Scholar] [CrossRef] - Ebrahimi, A.; Kleijn, C.R.; Richardson, I.M. Sensitivity of numerical predictions to the permeability coefficient in simulations of melting and solidification using the enthalpy-porosity method. Energies
**2019**, 12, 4360. [Google Scholar] [CrossRef][Green Version] - Voller, V.R.; Prakash, C. A fixed grid numerical modelling methodology for convection-diffusion mushy region phase-change problems. Int. J. Heat Mass Transf.
**1987**, 30, 1709–1719. [Google Scholar] [CrossRef] - Hu, H.; Argyropoulos, S.A. Mathematical modelling of solidification and melting: A review. Model. Simul. Mater. Sci. Eng.
**1996**, 4, 371–396. [Google Scholar] [CrossRef][Green Version] - Voller, V.R.; Swaminathan, C.R.; Thomas, B.G. Fixed grid techniques for phase change problems: A review. Int. J. Numer. Methods Eng.
**1990**, 30, 875–898. [Google Scholar] [CrossRef] - Voller, V.R. An overview of numerical methods for solving phase change problems. In Advances in Numerical Heat Transfer; Minkowycz, W.J., Sparrow, E.M., Eds.; Taylor & Francis: New York, NY, USA, 1997; pp. 341–380. [Google Scholar]
- König-Haagen, A.; Franquet, E.; Faden, M.; Brüggemann, D. Influence of the convective energy formulation for melting problems with enthalpy methods. Int. J. Therm. Sci.
**2020**, 158, 106477. [Google Scholar] [CrossRef] - Tittelein, P.; Gibout, S.; Franquet, E.; Johannes, K.; Zalewski, L.; Kuznik, F.; Dumas, J.-P.; Lassue, S.; Bédécarrats, J.-P.; David, D. Simulation of the thermal and energy behaviour of a composite material containing encapsulated-PCM: Influence of the thermodynamical modelling. Appl. Energy
**2015**, 140, 269–274. [Google Scholar] [CrossRef] - König-Haagen, A.; Franquet, E.; Pernot, E.; Brüggemann, D. A comprehensive benchmark of fixed-grid methods for the modeling of melting. Int. J. Therm. Sci.
**2017**, 118, 69–103. [Google Scholar] [CrossRef] - Galione, P.A.; Lehmkuhl, O.; Rigola, J.; Oliva, A. Fixed-grid numerical modeling of melting and solidification using variable thermo-physical properties—Application to the melting of n-Octadecane inside a spherical capsule. Int. J. Heat Mass Transf.
**2015**, 86, 721–743. [Google Scholar] [CrossRef][Green Version] - Ben-David, O.; Levy, A.; Mikhailovich, B.; Azulay, A. 3D numerical and experimental study of gallium melting in a rectangular container. Int. J. Heat Mass Transf.
**2013**, 67, 260–271. [Google Scholar] [CrossRef] - Günther, E.; Hiebler, S.; Mehling, H.; Redlich, R. Enthalpy of phase change materials as a function of temperature: Required accuracy and suitable measurement methods. Int. J. Thermophys.
**2009**, 30, 1257–1269. [Google Scholar] [CrossRef] - Arkar, C.; Medved, S. Influence of accuracy of thermal property data of a phase change material on the result of a numerical model of a packed bed latent heat storage with spheres. Thermochim. Acta
**2005**, 438, 192–201. [Google Scholar] [CrossRef] - Feng, G.; Huang, K.; Xie, H.; Li, H.; Liu, X.; Liu, S.; Cao, C. DSC test error of phase change material (PCM) and its influence on the simulation of the PCM floor. Renew. Energy
**2016**, 87, 1148–1153. [Google Scholar] [CrossRef] - Dolado, P.; Mazo, J.; Lázaro, A.; Marín, J.M.; Zalba, B. Experimental validation of a theoretical model: Uncertainty propagation analysis to a PCM-air thermal energy storage unit. Energy Build.
**2012**, 45, 124–131. [Google Scholar] [CrossRef] - Mazo, J.; el Badry, A.T.; Carreras, J.; Delgado, M.; Boer, D.; Zalba, B. Uncertainty propagation and sensitivity analysis of thermo-physical properties of phase change materials (PCM) in the energy demand calculations of a test cell with passive latent thermal storage. Appl. Therm. Eng.
**2015**, 90, 596–608. [Google Scholar] [CrossRef] - Soni, V.; Kumar, A.; Jain, V.K. Modeling of PCM melting: Analysis of discrepancy between numerical and experimental results and energy storage performance. Energy
**2018**, 150, 190–204. [Google Scholar] [CrossRef] - Zsembinszki, G.; Moreno, P.; Solé, C.; Castell, A.; Cabeza, L.F. Numerical model evaluation of a PCM cold storage tank and uncertainty analysis of the parameters. Appl. Therm. Eng.
**2014**, 67, 16–23. [Google Scholar] [CrossRef][Green Version] - Longeon, M.; Soupart, A.; Fourmigué, J.-F.; Bruch, A.; Marty, P. Experimental and numerical study of annular PCM storage in the presence of natural convection. Appl. Energy
**2013**, 112, 175–184. [Google Scholar] [CrossRef] - Rösler, F. Modellierung und Simulation der Phasenwechselvorgänge in Makroverkapselten Latenten Thermischen Speichern; Logos Verlag Berlin GmbH: Berlin, Germany, 2014. [Google Scholar]
- VDI. VDI-Wärmeatlas; Springer: Berlin/Heidelberg, Germany, 2013. [Google Scholar]
- Voller, V.R.; Swaminathan, C.R. General source-based method for solidification phase change. Numer. Heat Transf. Part B Fundam.
**1991**, 19, 175–189. [Google Scholar] [CrossRef] - Fluent, A. Ansys Fluent 15 User’s Guide; ANSYS: Canonsburg, PA, USA, 2013. [Google Scholar]

**Figure 1.**Basic scheme of the vertical thermocouple positioning in the phase change materials (PCM). HTF: heat transfer fluid.

**Figure 3.**Comparison of the height of the solid PCM for the experimental [31] and numerical results during charging (blue indicates solid, red indicates liquid, and yellow as well as green indicate the mushy region). In order to increase the visibility of the numerical results, they were stretched in the radial direction.

**Figure 4.**Comparison between experiment (exp) and standard simulation (num) for temperature curves during charging of measurement points with an identical height.

**Figure 5.**Comparison between experiment and standard simulation for temperature curves during charging of measurement points with an identical radius.

**Figure 6.**Absolute relative variation of the mean power for charging and discharging when varying the material properties by ±10% or ±1 K.

**Figure 7.**Absolute maxima of the relative variation of the power for charging and discharging when varying the material properties by ±10% or ±1 K.

**Figure 8.**Absolute maxima of the relative variation of the global liquid fraction for charging and discharging when varying the material properties by ±10% or ±1 K.

**Figure 9.**Power over time curve for charging of the standard case and the case with an increased melting enthalpy as well as the regarding relative deviation.

**Figure 10.**Power over time curve for charging of the standard case and the case with an increased heat conductivity as well as the regarding relative deviation.

**Figure 11.**Absolute relative variation of the mean power for charging and discharging when varying the boundary and initial conditions.

**Figure 12.**Absolute maxima of the relative variation of the power for charging and discharging when varying the boundary and initial conditions.

**Figure 13.**Absolute maxima of the relative variation of the global liquid fraction for charging and discharging when varying the boundary and initial conditions.

Measuring Point Identifiers | Radius in mm | Height in mm |
---|---|---|

B-b | 16 | 317 |

D-a | 13 | 217 |

D-b | 16 | 217 |

D-c | 19 | 217 |

F-b | 16 | 106 |

Physical Properties | Value | Unit | Source |
---|---|---|---|

$\rho $ | 760 | $\frac{\mathrm{kg}}{{\mathrm{m}}^{3}}$ | [31] |

${c}_{s}$ | 5.0 | $\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}$ | [32] |

${c}_{l}$ | 2.1 | $\frac{\mathrm{kJ}}{\mathrm{kg}\xb7\mathrm{K}}$ | [32] |

$\lambda $ | 0.2 | $\frac{\mathrm{W}}{\mathrm{m}\xb7\mathrm{K}}$ | [31] |

$L$ | 220 | $\frac{\mathrm{kJ}}{\mathrm{kg}}$ | [32] |

${T}_{s}$ | 34.5 | $\xb0\mathrm{C}$ | [32] |

${T}_{l}$ | 36.0 | $\xb0\mathrm{C}$ | [32] |

$\eta $ | see Equation (1) | $\mathrm{Pa}\times \mathrm{s}$ | own measurements |

$\beta $ | 0.001 | $\frac{1}{\mathrm{K}}$ | [31] |

Variable | Value | Unit |
---|---|---|

${T}_{init,char}$ | 23.79 | $\mathbb{C}$ |

${T}_{init,dis}$ | 47.15 | $\mathbb{C}$ |

${T}_{in,char}$ | 53.17 | $\mathbb{C}$ |

${T}_{in,dis}$ | 18.51 | $\mathbb{C}$ |

${t}_{char}$ | 8000 | $\mathrm{s}$ |

${t}_{dis}$ | 10,000 | $\mathrm{s}$ |

${u}_{in}$ | $10\times {10}^{-3}$ | $\frac{\mathrm{m}}{\mathrm{s}}$ |

${\dot{Q}}_{loss}$ | 0 | $\mathrm{W}$ |

Material Properties, Initial and Boundary Conditions | Variation |
---|---|

$\rho $ | $\pm 10\%$ |

${c}_{s}$ | $\pm 10\%$ |

${c}_{l}$ | $\pm 10\%$ |

$\lambda $ | $\pm 10\%$ |

$L$ | $\pm 10\%$ |

$\eta $ | $\pm 10\%$ |

$\beta $ | $\pm 10\%$ |

${C}_{\mathrm{I}}$ | $\pm 10\%$ |

${T}_{m}=\left({T}_{l}+{T}_{s}\right)/2$ | $\pm 1\text{}\mathrm{K}$ |

${T}_{l}-{T}_{s}$ | $\pm 1\text{}\mathrm{K}$ |

${u}_{in}$ | $\pm 10\%$ |

${T}_{init}$ | $\pm 1\text{}\mathrm{K}$ |

${T}_{in}$ | $\pm 1\text{}\mathrm{K}$ |

${\dot{Q}}_{loss}$ | on/off |

${T}_{in,curve}$ | on/off |

${u}_{in,profile}$ | on/off |

Study | Output | 1. | 2. | 3. | 4. | 5. |
---|---|---|---|---|---|---|

Dolado et al. [27] | mean power deviation | ${T}_{\mathrm{m}}$ $\pm 1\text{}\mathrm{K}$ | $\dot{V}$ $\pm 6.14\%$ | ${T}_{\mathrm{in}}$ $\pm 0.6\text{}\mathrm{K}$ | $d$ $\pm 10\%$ | $L$ $\pm 10\%$ |

Zsembinszki et al. [30] | mean absolute power deviation | ${T}_{\mathrm{in}}$ $\pm 1\text{}\mathrm{K}$ | ${T}_{\mathrm{m}}$ $\pm 1\text{}\mathrm{K}$ | $\rho $ * $\pm 5\%$ | $\frac{\partial h}{\partial T}$ * $\pm 5\%$ | $\alpha $ $\pm 30\%$ |

mean power deviation | ${T}_{\mathrm{in}}$ $\pm 1\text{}\mathrm{K}$ | $\rho $ ** $\pm 5\%$ | $\frac{\partial h}{\partial T}$ ** $\pm 5\%$ | ${T}_{\mathrm{m}}$ $\pm 1\text{}\mathrm{K}$ | ${\rho}_{\mathrm{HTF}}$ *** $\pm 5\%$ | |

Mazo et al. [28] | saving in yearly energy consumption | ${T}_{\mathrm{m}}$ $\pm 1\text{}\mathrm{K}$ | $\lambda $ $\pm 10\%$ | $L$ $\pm 10\%$ | $\rho $ $\pm 2\%$ | $c$ $\pm 5\%$ |

this study | mean power deviation | ${\dot{Q}}_{\mathrm{loss}}$ on/off | $\rho $ $\pm 10\%$ | $L$ $\pm 10\%$ | ${\mathrm{T}}_{\mathrm{in}}$ $\pm 1\text{}\mathrm{K}$ | ${c}_{s}$ $\pm 10\%$ |

maximum power deviation | ${\dot{Q}}_{\mathrm{loss}}$ on/off | ${u}_{\mathrm{in},\mathrm{profile}}$ On/off | ${u}_{\mathrm{in}}$ $\pm 10\%$ | ${T}_{\mathrm{in}}$ $\pm 1\mathrm{K}$ | $\rho $ $\pm 10\%$ | |

maximum global liquid fraction deviation | ${\dot{Q}}_{\mathrm{loss}}$ on/off | $\rho $ $\pm 10\%$ | $L$ $\pm 10\%$ | $\lambda $ $\pm 10\%$ | ${T}_{\mathrm{m}}$ $\pm 1\text{}\mathrm{K}$ |

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## Share and Cite

**MDPI and ACS Style**

König-Haagen, A.; Mühlbauer, A.; Marquardt, T.; Caron-Soupart, A.; Fourmigué, J.-F.; Brüggemann, D.
Basic Analysis of Uncertainty Sources in the CFD Simulation of a Shell-and-Tube Latent Thermal Energy Storage Unit. *Appl. Sci.* **2020**, *10*, 6723.
https://doi.org/10.3390/app10196723

**AMA Style**

König-Haagen A, Mühlbauer A, Marquardt T, Caron-Soupart A, Fourmigué J-F, Brüggemann D.
Basic Analysis of Uncertainty Sources in the CFD Simulation of a Shell-and-Tube Latent Thermal Energy Storage Unit. *Applied Sciences*. 2020; 10(19):6723.
https://doi.org/10.3390/app10196723

**Chicago/Turabian Style**

König-Haagen, Andreas, Adam Mühlbauer, Tom Marquardt, Adèle Caron-Soupart, Jean-François Fourmigué, and Dieter Brüggemann.
2020. "Basic Analysis of Uncertainty Sources in the CFD Simulation of a Shell-and-Tube Latent Thermal Energy Storage Unit" *Applied Sciences* 10, no. 19: 6723.
https://doi.org/10.3390/app10196723