#
Thermochemical Heat Storage in a Lab-Scale Indirectly Operated CaO/Ca(OH)_{2} Reactor—Numerical Modeling and Model Validation through Inverse Parameter Estimation

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Numerical Model

#### 2.1. Experimental Setup

#### 2.2. Model Setup

- The air density in the HTF channel is a function of temperature only (the influence of pressure variation on the density is negligible).
- Gas flow in the HTF channel is laminar with a fully developed viscous boundary layer.
- Heat conduction in the metal casing is much larger than in the HTF channel and the porous bulk. The casing is thus not considered in the model.
- There is local thermal equilibrium in the reactive bulk.
- Gas flow in the reactive bulk is creeping.

#### 2.2.1. Sub-Model of the Reactive-Bed Domain

#### 2.2.2. Formulation of the Heat Loss

#### 2.2.3. Model of the Heat-Transfer Channel

#### 2.2.4. Coupling of the Two Domains

#### 2.2.5. Software Framework of the Numerical Model

#### 2.3. Simulation Setup

## 3. Preliminary Calibration Attempt and Selection of Parameters for Optimization

#### 3.1. Porosity Induced Permeability Change

#### 3.2. Heat Conductivity

#### 3.3. Reaction Rate

#### 3.4. Summary of the Selected Parameters for Optimization

## 4. Statistical Optimization and Model Validation

#### 4.1. Bayesian Inference

#### 4.2. Speeding Up the Forward Propagation of Uncertainty Via a Surrogate Model

#### 4.3. Solution Procedure

#### 4.4. Errors and Uncertainties

#### 4.4.1. Measurement Error

#### 4.4.2. Numerical Error

#### 4.4.3. Surrogate Error

## 5. Results and Discussion

#### 5.1. Calibration

#### 5.2. Validation

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

HTF | Heat Transfer |

MCMC | Markov Chain Monte Carlo |

aPCE | arbitrary Polynomial Chaos Expansion |

PCA | Principal Component Analysis |

## Appendix A. Polynomial Chaos Expansion

#### Treating Time Dependency

**Figure A1.**Principal component analysis of the simulated temperature time series at the measurement point T1, compressed into the first 16 principal components.

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**Figure 1.**Reactor setup;

**Left**: Experimental setup described in Reference [32]. The solid bulk is kept between filter plates. Water is provided at constant pressure in the surrounding casing. Air flows through the heat transfer channel (HTF) to provide/remove heat. Thermocouples are placed at different locations, here T1, T3 and T7. For better visibility, the figure is not true to scale. The dimensions are given in mm.

**Top right**: In the model, the setup consists of two separate domains, the reactive bulk and the (HTF). The two domains are coupled by a conductive heat flux. Due to symmetry, only half of the bulk domain is simulated.

**Bottom right**: Domain coupling; a heat flux is calculated between the cell centers of the two-dimensional reaction bulk and the 1D HTF channel.

**Figure 2.**Preliminary simulation results to identify the optimization parameters. In black: experimental values from Reference [31] as reference; Simulation cases: “Only heat loss” is simulated with the standard parameters at constant permeability. “Permeability” applies a permeability change due to porosity alteration, “Heat conductivity” uses different values for solid heat conductivity for CaO and Ca(OH)${}_{2}$, “Reaction Param 1” applies a reduced reaction rate constant and “Reaction Param 2” introduces the constant term ${k}_{R,2}$ to the reaction kinetics. The parameters for each simulation case are listed in Table 4.

**Figure 3.**Histogramms for the parameter distribution after inverse modeling based on the experimental data of Reference [31]. On the diagonal are the histograms for the single parameters including mean and 95% confidence interval (indicated by the dashed lines). The scatter plots show the covariance between the parameters of the respective row and column.

**Figure 4.**Temperature distribution of the outer pressure vessel according to the parameters k${}_{1}$–k${}_{3}$ and at the top boundary of the reactive bed over the reactor length at different times simulated with the DuMu${}^{\mathrm{x}}$-model with the parameters of Table 6.

**Figure 5.**Simulation results of the calibrated surrogate model with its 95% confidence interval (CI); the surrogate model is based on 200 simulation runs of the numerical model plotted in gray. The experimental reference data are plotted in black including error-bars representing the uncertainties.

**Figure 6.**Simulation results (blue) for model validation according to the experimental data of Case R20.

**Table 1.**Boundary and initial conditions with: ${q}_{air}$ source term for the component air, ${q}_{\mathrm{H}{}_{2}\mathrm{O}}$ water source term, ${q}_{\mathrm{Ca}\left(\mathrm{OH}\right){}_{2}}$ Ca(OH)${}_{2}$ source term, ${q}_{\mathrm{CaO}}$ CaO source term, ${q}_{e}$ energy source term. The initial temperature for all simulation cases is ${T}_{init}=773\phantom{\rule{3.33333pt}{0ex}}\left[\mathrm{K}\right]$.

Domain | Location | PriVar and Value |
---|---|---|

HTF channel | $t=0$ | $p={p}_{HTF,init}$, $T={T}_{init}$ |

$x=0$ | ${q}_{air}={q}_{air}$; $T={T}_{init}$ | |

$x={x}_{max}$ | $p={p}_{HTF,init}$; ${q}_{energy}=2{q}_{e}$ | |

Porous Bulk | $t=0$ | ${p}_{\mathrm{H}{}_{2}\mathrm{O}}={p}_{bulk,init}$; $T={T}_{init}$; ${\varphi}_{\mathrm{CaO}}=0.1113\phantom{\rule{0.277778em}{0ex}}[-]$, ${\varphi}_{\mathrm{Ca}\left(\mathrm{OH}\right){}_{2}}=0.0[-]$ |

$y=0$ | ${q}_{\mathrm{H}{}_{2}\mathrm{O}}=0$; ${q}_{\mathrm{CaO}}={q}_{\mathrm{CaH}}=0$ | |

${q}_{energy}={q}_{e}$ | ||

$y={y}_{max}$ | ${p}_{\mathrm{H}{}_{2}\mathrm{O}}={p}_{bulk,init}$; ${q}_{\mathrm{CaO}}={q}_{\mathrm{CaH}}=0$; | |

${q}_{energy}=\frac{{\rho}_{\mathrm{H}{}_{2}\mathrm{O}}{K}_{bulk}}{{\mu}_{\mathrm{H}{}_{2}\mathrm{O}}}\nabla {p}_{y={y}_{max}}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}h(\mathrm{H}{}_{2}\mathrm{O},{T}_{y={y}_{max}})+(T(y={y}_{max})-{T}_{BC}){\alpha}_{loss}$ | ||

$x=0$ & $x=z$ | ${q}_{\mathrm{H}{}_{2}\mathrm{O}}={q}_{\mathrm{CaO}}={q}_{\mathrm{CaH}}=0$; ${q}_{energy}=(T\left(xy\right)-{T}_{BC}){\alpha}_{loss}$ |

Property | Symbol | Value |
---|---|---|

Reference porosity (according to Reference [20]) | ${\varphi}_{\mathrm{g}}$ | 0.773 |

Density Ca(OH)${}_{2}$ [26] | ${\varrho}_{\mathrm{Ca}}{\left(\mathrm{OH}\right)}_{2}$ | 2200 kg m${}^{-3}$ |

Density CaO [42] | ${\varrho}_{\mathrm{CaO}}$ | 3340 kg m${}^{-3}$ |

Permeability (according to Reference [20]) | $\mathbf{K}$ | 8.8 × 10${}^{-12}$ m${}^{2}$ |

Specific heat capacity Ca(OH)${}_{2}$ [26] | ${c}_{\mathrm{p}\phantom{\rule{0.166667em}{0ex}}\mathrm{Ca}{\left(\mathrm{OH}\right)}_{2}}$ | 1530 J kg${}^{-1}$ K${}^{-1}$ |

Specific heat capacity CaO [26] | ${c}_{\mathrm{p}\phantom{\rule{0.166667em}{0ex}}\mathrm{CaO}}$ | 934 J kg${}^{-1}$ K${}^{-1}$ |

Solid heat conductivity [8] | ${\lambda}_{\mathrm{s}}$ | 0.4 W m${}^{-1}$K${}^{-1}$ |

Reaction enthalpy [8] | $\Delta {h}_{\mathrm{R}}$ | 104.4 kJ mol${}^{-1}$ |

Case | Vapor Pressure ${\mathit{p}}_{\mathit{bulk},\mathit{init}}$ [kPa] | Air Flux ${\mathit{q}}_{\mathit{air}}$ $\left[\frac{\mathbf{kg}}{\mathbf{h}}\right]$ |
---|---|---|

Case 17 (according to case A in Reference [31]) | 470 | 16 |

Case 20 (according to reference case in Reference [20]) | 270 | 25 |

Case | ${\mathit{\alpha}}_{\mathit{HTF}}$ | ${\mathit{\alpha}}_{\mathit{loss}}$ | K | ${\mathit{\lambda}}_{\mathbf{CaO}}$ | ${\mathit{\lambda}}_{\mathbf{Ca}\left(\mathbf{OH}\right){}_{2}}$ | ${\mathit{k}}_{\mathit{R},1}$ | ${\mathit{k}}_{\mathit{R},2}$ |
---|---|---|---|---|---|---|---|

Heat losses | 275 | 30 | $8.8\times {10}^{-12}$ | 0.4 | 0.4 | 10 | 0 |

Permeability | 275 | 30 | $8.8\times {10}^{-12}$ | 0.4 | 0.4 | 10 | 0 |

$8.8\times {10}^{-11}$ | |||||||

Heat conductivity | 275 | 30 | $8.8\times {10}^{-12}$ | 0.3 | 0.5 | 10 | 0 |

Reaction param 1 | 275 | 30 | $8.8\times {10}^{-12}$ | 0.4 | 0.4 | 0.2 | 0 |

Reaction param 2 | 275 | 30 | $8.8\times {10}^{-12}$ | 0.4 | 0.4 | 10 | 2 |

Parameter | Symbol | Unit | Range Minimum | Range Maximum |
---|---|---|---|---|

Heat-transfer coefficient | ${\alpha}_{HTF}$ | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ | 150 | 350 |

Heat-loss coefficient | ${\alpha}_{loss}$ | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ | 0 | 350 |

Temperature distribution Param1 | ${k}_{1}$ | ${\mathrm{Km}}^{-2}$ | −15 | 15 |

Temperature distribution Param2 | ${k}_{2}$ | ${\mathrm{Km}}^{-1}$ | −20 | 20 |

Temperature distribution Param3 | ${k}_{3}$ | K | 740 | 780 |

Reaction coeffiecient | ${k}_{R,1}$ | s${}^{-1}$ | 0.1 | 20 |

Reaction exponent | ${k}_{R,2}$ | - | 0 | 3 |

Parameter | Symbol | Unit | Mean | + | − |
---|---|---|---|---|---|

Heat-transfer coefficient | ${\alpha}_{HTF}$ | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ | 337 | 1.30 | 1.38 |

Heat-loss coefficient | ${\alpha}_{loss}$ | $\mathrm{W}/{\mathrm{m}}^{2}\mathrm{K}$ | 13.2 | 0.40 | 0.38 |

Temperature distribution Param1 | ${k}_{1}$ | ${\mathrm{Km}}^{-2}$ | 7.54 | 1.51 | 1.49 |

Temperature distribution Param2 | ${k}_{2}$ | ${\mathrm{Km}}^{-1}$ | −4.71 | 1.34 | 1.35 |

Temperature distribution Param3 | ${k}_{3}$ | K | 739 | 1.39 | 1.41 |

Reaction coeffiecient | ${k}_{R,1}$ | s${}^{-1}$ | 20.9 | 1.46 | 1.45 |

Reaction exponent | ${k}_{R,2}$ | - | 5.58 | 0.26 | 0.26 |

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**MDPI and ACS Style**

Seitz, G.; Mohammadi, F.; Class, H.
Thermochemical Heat Storage in a Lab-Scale Indirectly Operated CaO/Ca(OH)_{2} Reactor—Numerical Modeling and Model Validation through Inverse Parameter Estimation. *Appl. Sci.* **2021**, *11*, 682.
https://doi.org/10.3390/app11020682

**AMA Style**

Seitz G, Mohammadi F, Class H.
Thermochemical Heat Storage in a Lab-Scale Indirectly Operated CaO/Ca(OH)_{2} Reactor—Numerical Modeling and Model Validation through Inverse Parameter Estimation. *Applied Sciences*. 2021; 11(2):682.
https://doi.org/10.3390/app11020682

**Chicago/Turabian Style**

Seitz, Gabriele, Farid Mohammadi, and Holger Class.
2021. "Thermochemical Heat Storage in a Lab-Scale Indirectly Operated CaO/Ca(OH)_{2} Reactor—Numerical Modeling and Model Validation through Inverse Parameter Estimation" *Applied Sciences* 11, no. 2: 682.
https://doi.org/10.3390/app11020682