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\begin{abstract}  We have studied magnetic properties of water-quenched 5 wt.% (Fe, Ti)particle-doped MgB$_2$ comparing with that of air-cooled one. Generally, grainrefinement is achieved by increasing cooling rate, which implies an increase ofgrainboundaries in the superconductor. Here we show that increasedgrainboundaries influence what kinds of effects on the field dependence ofmagnetization and what is the mechanism. As a result, they are served as apinning center at a high field whereas they are served as a pathway tofacilitate the movement of fluxes pinned on volume defects at a low field. Asmodeling grainboundaries in a superconductor, we explained that they had a fluxpinning effect as well as the flux-penetrating promotion effect. As temperatureincreases, the pinning ability of a grainboundaries decreases, which was causedby increased coherence length. Stacking fault planes and twin boundaries havealso been considered by using the model. It explained the reason for thatstacking fault planes of MgB$_2$ do not have any pinning effect and the twinboundary of HTSC have the strong pinning or strong flux-penetration effectdepending on the direction of the applied field.

Water-Quenched Effects of 5 wt.% (Fe, Ti) particle-doped MgB$_2$ Superconductor and Low Limit of Pinning Effect