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Statistics and experiments reveal that the mechanical failure of a spherical catalyst under single-particle crushing strength and bulk crushing strength tests are both brittle fracture and the fact that the single particle strength data follow a Weibull distribution. A model is proposed for the packing of spherical particles, which assumes tetrahedral contact and force transmission between the spheres. A deduction from this packing is that the broken percentage of spheres during bulk strength measurement follows the relationship of the Weibull equation. Experimental results confirm this conclusion, which relates the single particle property and the bulk behavior of the packed bed of catalytic materials.

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A density functional theory (DFT) study has been conducted to investigate the structural and electronic properties of PdO/HZSM-5 and thereby the relationship between PdO and the acid sites of HZSM-5. Different cluster models of PdO/HZSM-5 based on experimental and theoretical works are presented. The study shows that the local structure of PdO supported on HZSM-5 is very similar to the four-coordinated square planar structure of bulk PdO. This is consistent with the reported EXAFS analysis in the literature. The average distances between the Pd and the four coordinated oxygen atoms obtained by DFT calculations with different cluster models are close to the reported experimental results. The analysis of the Mulliken population and the electron density difference of the cluster models shows that there exists a charge transfer from the four coordinated oxygen atoms to the Pd. The components of the frontier molecular orbital of these cluster models come mainly from the Pd and the four coordinated oxygen atoms. The DFT study confirms that the acid sites in HZSM-5 keep the PdO well dispersed due to the bond interaction between the acidic proton atom and the oxygen atom of PdO. In addition, after the formation of PdOH + , the acidity of HZSM-5 complex remains high or in some instances increases.