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Abstract
This work focuses on uncertainty quantification of eight random parameters required as input for 1D modelling of methane catalytic partial oxidation within a highly dense foam reactor. Parameters related to geometrical properties, reactor thermophysics and catalyst loading are taken as uncertain. A widely applied 1D heterogeneous mathematical model that accounts for proper transport and surface chemistry steps is considered for the evaluation of deterministic samples. The non-intrusive spectral projection approach based on polynomial chaos expansion is applied to determine the stochastic temperature and species profiles along the reactor axial direction as well as their ensemble mean and error bars with a confidence interval of 95%. Probability density functions of relevant variables in specific reactor sections are also analysed. A different contribution is noticed from each random input to the total uncertainty range. Porosity, specific surface area and catalyst loading appear as the major sources of uncertainty to bulk gas and surface temperature and species molar profiles. Porosity and the mean pore diameter have an important impact on the pressure drop along the whole reactor as expected. It is also concluded that any trace of uncertainty in the eight input random variables can be almost dissipated near the catalyst outlet section for a long-enough catalyst, mainly due to the approximation to thermodynamic equilibrium.
Nomenclature
| English | = | |
| aV | = | specific surface area (m−1) |
| cXj | = | expansion mode coefficient number j of X PC expansion |
| Cp | = | specific heat under constant pressure (J kg−1 K−1) |
| cv | = | coefficient of variation |
| Dp | = | mean pore diameter (m) |
| f | = | stochastic model solution |
| Fcat/geo | = | ratio of the catalytic surface to the total geometric surface area |
| fd | = | deterministic model solution |
| = | total exchange area between volume zones j and i (m2) | |
| h | = | interphase heat transport coefficient (W m−2 K−1) |
| Hk | = | molar enthalpy of species k (J mol−1) |
| k | = | thermal conductivity (W m−1 K−1) |
| KK | = | total number of species |
| Kmat, k | = | interphase mass transport coefficient (m s−1) |
| N | = | total number of uncertain parameters |
| Ng | = | total number of volume zones |
| NR | = | total number of reactions |
| Ns | = | total number of surface zones |
| P | = | pressure (Pa) |
| qR | = | rate-of-progress variable of surface reaction R (mol m−2 s−1) |
| q′′′Rad | = | net radiative heat flux (W m−3) |
| Re | = | Reynolds number |
| Sk | = | Selectivity of species k |
| Sc | = | Schmidt number |
| = | total exchange area between surface zone j and volume zone i (m2) | |
| Sh | = | Sherwood number |
| t | = | time (s) |
| T | = | temperature (K) |
| u | = | interstitial flow velocity (m s−1) |
| Vk | = | diffusion velocity of species k (m s−1); volume of zone k (m3) |
| W | = | molecular weight (kg mol−1) |
| x | = | axial reactor dimension (m) |
| Xk | = | molar fraction of species k; uncertain parameter |
| [Xk] | = | molar concentration of gas or surface species k (mol m−3 or mol m−2) |
| Yk | = | mass fraction of species k |
| Greek | = | |
| β | = | extinction coefficient (m−1) |
| γ | = | skewness |
| δw | = | washcoat thickness (m) |
| ε | = | porosity |
| κi | = | absorption coefficient of volume zone i (m−1) |
| μ | = | dynamic viscosity (kg m−1 s−1); mean value |
| νk, R | = | stoichiometric coefficient of species k in reaction R |
| ν′k, R | = | stoichiometric coefficient of reactant species k in reaction R |
| ξ | = | random variable |
| ρ | = | specific mass (kg m−3) |
| σ | = | Stefan–Boltzmann constant (W m−2 K−4); standard deviation |
| τ | = | solid tortuosity |
| = | methane conversion | |
| ψj | = | one-dimensional orthogonal polynomial |
| Ψj | = | multi-dimensional orthogonal polynomial |
| ω | = | single scattering albedo |
| = | molar production/consumption rate (mol m−2 s−1) | |
| Subscripts | = | |
| cat | = | catalyst |
| f | = | stochastic model solution |
| g | = | gas phase; gas phase species in the bulk gas flow |
| gi | = | volume zone i |
| k | = | species k; random parameter k |
| s | = | solid phase; surface (adsorbed) species |
| si | = | surface zone i |
| w | = | gas phase species at the gas/wall interface |
| Acronyms | = | |
| CI | = | confidence interval |
| CPOx | = | catalytic partial oxidation |
| DEA | = | direct exchange area |
| FHS | = | front heat shield |
| MC | = | Monte Carlo |
| NISP | = | non-intrusive spectral projection |
| PC | = | polynomial chaos |
| = | probability density function | |
| ppi | = | pore per linear inch |
| TEA | = | total exchange area |
| UQ | = | uncertainty quantification |
| UR | = | uncertainty range |