Explicit analytical equations for single port hybrid rocket combustion chamber sizing

Hybrid rockets have several advantages with respect to current propulsion systems like simplicity, safety, reliability, environmental friendliness, and lower cost. To size the combustion chamber, it is fundamental to understand how the length, the external diameter, the volume loading, and the length-to-diameter ratio vary with the design parameters like scale, burning time, average mixture ratio, initial oxidizer flux, and propellant combination. The equations available in the literature are not in explicit form with respect to the aforementioned design parameters, and sometimes they can be misinterpreted by hybrid rocket engineers. Moreover, it is not possible to determine the instantaneous and average characteristic velocities during the burn without a numerical time integration. To show explicitly the real trends, a set of analytical equations has been developed. The key step is the definition of the relation between initial and average mixture ratio and the asymptotic treatment with respect to the ratio between the external and internal diameters. Moreover, an approximate explicit semi-analytical expression of the instantaneous and average characteristic velocities is provided. The explicit analytical equations are validated with the exact implicit solutions showing good agreement and exact asymptotic behavior.