A miniature Joule-Thomson (JT) cooler with a recuperative heat exchanger is studied, both theoretically and experimentally, for its steady-state behaviors down to 96 K. Two unique aspects of analysis are presented in this paper. Firstly, the simulation employs a distributed approach (as opposed to a black-box) and it is performed with dynamic updating of the thermo-physical properties of the working fluid when the governing differential equations are solved simultaneously. Secondly, the conduction heat leak effects in the longitudinal direction of the exchanger are included along with those from the ambient. The properties of Argon, expressed as a function of the local values of P and T, have been found to compare well with data obtained from tables of IUPAC. Our experiments on the JT cooler indicate good agreement with the predictions to within a relative error of +/-0.3% across a large span of Argon pressures.