A closed-loop cryogenic cooling system for high field magnets is presented. This design is motivated by our recent development of cooling system for 21 tesla Fourier Transform ion Cyclotron Resonance (FT-ICR) superconducting magnets without any replenishment of cryogen. The low temperature superconducting magnets are immersed in a subcooled 1.8 K bath, which is connected hydraulically to the 4.2 K reservoir through a narrow channel. Saturated liquid helium is cooled by Joule-Thomson heat exchanger and flows through the JT valve, isenthalpically dropping its pressure to approximately 1 6 kPa, corresponding saturation temperature of 1.8 K. Helium gas exhausted from pump is now recondensed by two-stage cryocooler located after vapor purify system. The amount of cryogenic Heat loads and required mass flow rate through closed-loop are estimated by a relevant heat transfer analysis, from which dimensions of JT heat exchanger and He II heat exchanger are determined. The detailed design of cryocooler heat exchanger for helium recondensing is performed. The effect of cryogenic loads, especially superfluid heat leak through the gap of weight load relief valve, on the dimensions of cryogenic system is also investigated.