Stirling and Pulse-tube Cryo-coolers

Appendix A

Conditions for equivalence of volume variations

A.1 Kinematics

It is assumed that a condition for optimum performance is that minimum compression-space dead volume (minimum unswept volume) is zero. This implies in turn that the target cycle minimum for variable distance, s, between the lowermost face of the displacer and the uppermost face of the piston (Fig. 2.3) is zero. (It is conceivable that the absolute minimum of compression-space dead volume over a cycle is achieved for some minimum of s other than zero, but the algebra becomes unmanageable.)

In the figure, is the angle, measured from outer dead centre, of the crank pin driving the displacer. This pin sits at radius (semi-stroke) r_d. The pin driving the piston lags the displacer pin by kinematic phase angle β, and its radius is r_p. In the notation of the diagram

The term (L_e – L_c) is eliminated by setting s_min = 0. Differentiating and setting equal to zero determines ₀ - the value of for which s=0

Expanding by the ...