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Whereas it is generally accepted that calcium-aluminum-rich inclusions (CAIs) from chondritic meteorites formed in a hot environment in the solar protoplanetary disk, the conditions of their formation remain debated. Recent laboratory studies of CAIs have provided new kind of data: their size distributions. We show that size distributions of CAIs measured in laboratory from sections of carbonaceous chondrites have a power law size distribution with cumulative size exponent between -1.7 and -1.9, which translates into cumulative size exponent between -2.5 and -2.8 after correction for sectioning. To explain these observations, numerical simulations were run to explore the growth of CAIs from micrometer to centimeter sizes, in a hot and turbulent protoplanetary disk through the competition of coagulation and fragmentation. We show that the size distributions obtained in growth simulations are in agreement with CAIs size distributions in meteorites. We explain the CAI sharp cut-off of their size distribution at centimeter sizes as the direct result from the famous fragmentation barrier, provided that CAI fragment for impact velocities larger than 10 m/s. The growth/destruction timescales of millimeter- and centimeter-sized CAIs is inversely proportional to the local dust/gas ratio and is about 10 years at 1300 K and up to 104 years at 1670K. This implies that the most refractory CAIs are expected to be smaller in size owing to their long growth timescale compared to less refractory CAIs. Conversely, the least refractory CAIs could have been recycled many times during the CAI production era which may have profound consequences for their radiometric age., Comment: Accepted in Icarus