# Effective Heat and Mass Transport Properties of Anisotropic Porous Ceria for Solar Thermochemical Fuel Generation

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

**Figure 1.**Schematic of the solar reactor for the 2-step solar-driven thermochemical production of fuels. Concentrated solar radiation enters the cavity-receiver through a windowed aperture and is absorbed by the porous ceria (close-up). Reacting gases flow across the porous ceria while product gases exit the cavity.

## 2. Computed Tomography

**Figure 2.**SEM pictures (HORIBA TM1000) of the graphite powder, Alfa Aeasar 40769 (

**a**) and Alfa Aesar 10129 (

**b**), and volume-based particle-size distributions, f, of two random samples of each graphite powder type (

**c**).

**Table 1.**Volume percent ratio of ceria and graphite particles, numerically and experimentally determined porosity ε, numerically determined specific surface ${A}_{0}$, mean, mode and median diameter d, extinction coefficient β, fitting parameter a of Equation (9), Nu correlation, permeability K and Dupuit-Forchheimer coefficient ${F}_{\mathrm{DF}}$ of the three ceria foam samples in the x-, y-, z-directions.

Sample No. | 1 | 2 | 3 |
---|---|---|---|

vol% ratio ^{1} | 1:3:1 | 1:2:2 | 1:1:3 |

${\epsilon}_{\mathrm{numerical}}$ | 0.51 | 0.56 | 0.55 |

${\epsilon}_{\mathrm{experimental}}$ | 0.65 ± 0.1 | 0.65 ± 0.1 | 0.65 ± 0.1 |

${A}_{0}$ (mm${}^{-1}$) | 672 | 706 | 675 |

${d}_{\mathrm{m}}$ (μm) | 13.7 | 11.9 | 10.0 |

${d}_{\mathrm{mode}}$ (μm) | 13.3 | 11.8 | 8.9 |

${d}_{\mathrm{median}}$ (μm) | 13.1 | 11.5 | 9.7 |

β (m${}^{-1}$), x-direction | 30,003 ± 8,282 | 38,143 ± 6,277 | 45,173 ± 4,665 |

β (m${}^{-1}$), y-direction | 31,757 ± 7,067 | 35,042 ± 4,546 | 46,277 ± 6,485 |

β (m${}^{-1}$), z-direction | 69,018 ± 14,735 | 65,665 ± 9,809 | 74,835 ± 15,022 |

a, x-direction | 0.427 | 0.443 | 0.506 |

a, y-direction | 0.354 | 0.482 | 0.512 |

a, z-direction | 0.751 | 0.740 | 0.705 |

Nu, x-direction | 0.38 + 0.35Re${}^{0.75}$Pr${}^{0.64}$ | 1.09 + 0.50Re${}^{0.69}$Pr${}^{0.56}$ | 0.82 + 0.64Re${}^{0.66}$Pr${}^{0.51}$ |

Nu, y-direction | 0.37 + 0.41Re${}^{0.68}$Pr${}^{0.58}$ | 0.75 + 0.53Re${}^{0.68}$Pr${}^{0.59}$ | 1.13 + 0.56Re${}^{0.69}$Pr${}^{0.51}$ |

Nu, z-direction | 1.96 + 0.60Re${}^{0.80}$Pr${}^{0.52}$ | 1.28 + 0.65Re${}^{0.75}$Pr${}^{0.57}$ | 1.96 + 0.94Re${}^{0.68}$Pr${}^{0.61}$ |

K (m${}^{2}$), x-direction | 6.04 × 10${}^{-12}$ | 3.54 × 10${}^{-12}$ | 2.92 × 10${}^{-12}$ |

K (m${}^{2}$), y-direction | 7.97 × 10${}^{-12}$ | 3.80 × 10${}^{-12}$ | 3.03 × 10${}^{-12}$ |

K (m${}^{2}$), z-direction | 7.43 × 10${}^{-13}$ | 1.27 × 10${}^{-12}$ | 1.30 × 10${}^{-12}$ |

${F}_{\mathrm{DF}}$ (m${}^{-1}$) , x-dir. | 18.4 × 10${}^{4}$ | 16.8 × 10${}^{4}$ | 19.4 × 10${}^{4}$ |

${F}_{\mathrm{DF}}$ (m${}^{-1}$) , y-dir. | 12.4 × 10${}^{4}$ | 17.5 × 10${}^{4}$ | 19.3 × 10${}^{4}$ |

${F}_{\mathrm{DF}}$ (m${}^{-1}$) , z-dir. | 278.9 × 10${}^{4}$ | 93.5 × 10${}^{4}$ | 75.3 × 10${}^{4}$ |

^{1}vol% ratio: ceria - Alfa Aesar 40769 - Alfa Aesar 10129.

**Figure 3.**CT scan of the three ceria samples of Table 1: (

**a**) No. 1; (

**b**) No. 2; and (

**c**) No. 3. Edge length of the pictures is 376 μm. Dark is solid phase and bright is void phase.

## 3. Morphological Characterization

**Figure 5.**Pore-size distribution of the three ceria samples (

**a**), and porosity calculated for sample No. 3 in 20 subsequently growing volumes (

**b**). The dotted horizontal line indicates ${\epsilon}_{\mathrm{numerical}}$ to which the values converge.

**Figure 6.**Mean intercept length for ceria sample No. 1 as a function of θ at φ = 0 and 90${}^{\circ}$ (a), and three orthogonal planes of sample No. 1 with black bars indicating the direction of the elongated pores (b).

## 4. Heat Transfer Characterization

#### 4.1. Radiation Heat Transfer

#### 4.2. Conduction Heat Transfer

**Figure 7.**Effective normalized conductivity of the three samples (Table 1) as function of the ratio fluid-to-solid conductivities in the three directions.

#### 4.3. Convection Heat Transfer

^{3}, corresponding to × 1000 × 1000 × 500 voxels, is investigated. The smallest dimension lies in the main flow direction. Convergence of the numerical calculations is achieved for a terminal residual RMS of the iterative solution below 10

^{−5}and for a maximal mesh element length of 3 μm. The mesh is generated with an in-house mesh generator for unstructured body-fitted grids, which covers the domain by tetrahedral elements and subsequently refines the elements at the phase boundary by rounding, cutting, and smoothing [37]. The resulting grid has between 50 × 10

^{6}and 150 × 10

^{6}mesh elements. The calculated Nu as a function of Re for Pr = 0.1 and 1, is shown in Figure 8 in the three directions. Re = u

_{D}d

_{m}ρ/μ and Pr = c

_{p}μ/k

_{f}, where d

_{m}denotes the calculated mean pore diameter and u

_{D}the superficial velocity. The curves are fitted to a correlation of the form:

_{1}to a

_{4}given in Table 1. The Rms of the fitting is less than 0.6. The heat transfer coefficient increases in the z-direction because of the more tortuous path for fluid flow, increasing the accessible surface area for solid-fluid heat exchange. This trend is most pronounced for sample No. 1. Values for Nu lie above those experimentally measured for ceramic foams [38], but within the range of those for packed beds [39].

**Figure 8.**Nu number as a function of Re and Pr numbers (points) and fit (lines) for sample No. 1 in the three directions.

## 5. Mass Transfer Characterization

**Figure 9.**(

**a**) Normalized pressure drop as a function of Re number for the three directions in sample No. 1; (

**b**) Calculated permeability vs. Dupuit-Forchheimer coefficient for the three samples and in the three directions.

**Figure 10.**Tortuosity (

**a**) and residence time (

**b**) distributions for Re = 0.1 to 100 for sample No. 1, sample length of 0.17 mm, and for the three directions.

**Table 2.**Mean, mode, and median tortuosity and residence time for sample No. 1 in the three directions for a sample with length 0.17mm at Re = 1.

Direction | ${\tau}_{\mathrm{m}}$ | ${\tau}_{\mathrm{mode}}$ | ${\tau}_{\mathrm{median}}$ | ${t}_{\mathrm{m}}$ (s) | ${t}_{\mathrm{mode}}$ (s) | ${t}_{\mathrm{median}}$ (s) |
---|---|---|---|---|---|---|

x | 1.24 | 1.20 | 1.22 | 0.0026 | 0.0006 | 0.0018 |

y | 1.20 | 1.20 | 1.19 | 0.0028 | 0.0007 | 0.0020 |

z | 1.61 | 1.80 | 1.59 | 0.0018 | 0.0009 | 0.0011 |

## 6. Tailored Foam Design

**Figure 11.**2D slice through the artificial BDOTS samples with ξ = 1, and ${\epsilon}_{\mathrm{BDOTS}}$ = 0.6 (

**a**) and 0.8 (

**b**). Edge length is 540 μm.

**Figure 12.**(

**a**) Normalized pressure gradient as a function of Re; and (

**b**) Nu numbers as function of Re for Pr = 0.1, and 1, for the artificially generated porous sample (ξ = 1, and ${\epsilon}_{\mathrm{BDOTS}}$ = 0.6 and 0.8).

**Table 3.**Permeability, Dupuit-Forchheimer coefficient, and Nu correlation for the artificial porous sample with ξ = 1, and ${\epsilon}_{\mathrm{BDOTS}}$ = 0.6 and 0.8.

${\epsilon}_{\mathrm{BDOTS}}$ | K (m${}^{2}$) | ${F}_{\mathrm{DF}}$ (m${}^{-1}$) | Nu |
---|---|---|---|

0.6 | 6.62 × 10${}^{-11}$ | 219143 | 3.03 + 0.55Re${}^{0.71}$Pr${}^{0.54}$ |

0.8 | 2.13 × 10${}^{-10}$ | 23315 | 3.59 + 0.28Re${}^{0.76}$Pr${}^{0.55}$ |

## 7. Summary and Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Haussener, S.; Steinfeld, A.
Effective Heat and Mass Transport Properties of Anisotropic Porous Ceria for Solar Thermochemical Fuel Generation. *Materials* **2012**, *5*, 192-209.
https://doi.org/10.3390/ma5010192

**AMA Style**

Haussener S, Steinfeld A.
Effective Heat and Mass Transport Properties of Anisotropic Porous Ceria for Solar Thermochemical Fuel Generation. *Materials*. 2012; 5(1):192-209.
https://doi.org/10.3390/ma5010192

**Chicago/Turabian Style**

Haussener, Sophia, and Aldo Steinfeld.
2012. "Effective Heat and Mass Transport Properties of Anisotropic Porous Ceria for Solar Thermochemical Fuel Generation" *Materials* 5, no. 1: 192-209.
https://doi.org/10.3390/ma5010192