The Molten Salt Reactor (MSR) is one of the most revolutionary Gen-IV reactors and it can be operated, especially with chloride salts, in the so-called breed and burn fuel cycle. In this type of fuel cycle the fissile isotopes from spent fuel do not need to be reprocessed, because the excess bred fuel covers the losses. The liquid phase of the MSR fuel assures its instant homogenization, and the reactor can be operated with batch-wise refueling thus reaching an equilibrium state. At the same time, the active core of the chloride fast MSR needs to be bulky to limit neutron leakage. In this study, the code Serpent 2 was coupled to the Python script BBP to simulate batch-wise operation of the breed and burn MSR fuel cycle. The script, previously developed for solid assemblies shuffling, was modified to simulate fuel homogenization after fertile material addition. Several fuel salts and fission products removal strategies were simulated and their impact was analyzed. Similarly, the influence of blanket volume was assessed in a two-fluid core layout. The results showed that the reactivity initially grows during the irradiation period and later decreases. The blanket has a large impact on the performance and it can be used to further increase the fuel burnup or to shrink the active core size. The breed and burn fuel cycle in MSR can reach high fuel utilization without fuel reprocessing and a multi-fluid layout can help to decrease the core size.