According to embodiments of the invention, a normally recursive ray tracing algorithm may be partitioned to form an iterative ray tracing algorithm. An open loop injection quantity is first determined based on operation conditions. The metallic body defines a shank portion and an airfoil portion having opposed pressure and suction sides, and is integral with the hub. The toner can be sufficiently mixed and agitated on a cartridge main body side without depending on a developing apparatus side, and the toner cartridge is prevented from being large-sized.