Isothermal Sinusoidal Analysis of Balanced Compound Vuilleumier Heat Pumps

Vuilleumier heat pumps with balanced compounding have been under investigation for about fifteen years but have not yet reached the stage of commercial utilization. Previously published analytical treatment based upon isothermal treatment of the variable-volume spaces resulted in closed form solution of considerable complexity of the coupled differential equations but applied only to free piston machines. In contrast, the procedure presented here is based on previously demonstrated sinusoidal excursions of the reciprocators. It is of the same order of accuracy, but much more simple and usable. It was found that there is only negligible difference between the results of the present and the previous approach. Additionally, the treatment presented here is applicable to kinematic machines, as well as to free piston machines. For the latter design, an equation for the natural frequency is also derived. Ideal proportions and practical expressions for the energy streams are derived. Gas forces are plotted versus displacement and it is shown that they are equivalent to a linear spring, which is of importance for the concept of a free-piston design.