A methodology is developed for thermal-hydraulic analysis and design of a breed-and-burn molten salt reactor (BBMSR). By using separate fuel and coolant molten salts, the BBMSR is proposed to overcome key materials limitations of traditional breed-and-burn and molten salt reactor designs. The BBMSR fuel concept includes an inner wall that divides the ascending and descending flows of naturally convecting fuel salt. A finite-difference model (FDM) is developed to iteratively solve for the temperature and velocity distributions in both sections of the concentric fuel. The FDM is used to perform parametric studies of the effect of fuel geometry and heat generation rate on the heat transfer performance of the fuel. The FDM is then integrated into a design search algorithm that identifies the operational limits for a given BBMSR fuel geometry, within a set of defined constraints. A range of thermal-hydraulic fuel design options are evaluated, and trade-off studies are performed to identify the most promising fuel design space for competitive power production and neutronic efficiency in the BBMSR.