Steady state operation of the heat exchanger of a Joule-Thomson cryocooler is studied numerically through a one-dimensional model. Argon is used as the working fluid. The developed model is first verified using a cooler configuration that is studied extensively in the literature. Then the model is improved in several ways. A major mistake seen in many of the previous studies is related to the mismatch of the friction coefficient correlation and the conservation of momentum equation of the tube side flow. With the use of consistent correlations, the calculated tube side pressure drop turned out to be 81% smaller. Additionally, the effect of pressure dependency of enthalpy is considered in the energy conservation equation, resulting in 37% increase in the cooling power. Other improvements are; (i) use of Collins tubing friction correlation for the shell side, (ii) defining the exit pressure of the shell side as atmospheric pressure and calculating the inlet pressure from the pressure drop, (iii) determining the inlet temperature of shell side as saturation temperature at the calculated pressure, and (iv) defining the emissivity of the shield as a function of temperature. An optimization study is performed using the improved model with two objectives (maximization of the specific cooling power which allows decreasing flow rate and minimization of the shell side pressure drop which results in a decreased working temperature). Five design parameters (heat exchanger length, finned capillary inner diameter, fin pitch, fin length and fin thickness) are selected through a sensitivity analysis of all possible design parameters. The optimum geometry is obtained using grid search method to maximize the optimization function formed by a weighted sum of two contradicting objectives. The optimum geometry allows a decrease of the flow rate by 46% within the defined constraints. The final shell side pressure drop is decreased by 90%.