Gas Dynamics of the Temperature-Determined Stirling Cycle

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Abstract
The Stirling cycle machine is modelled as a number of sections of duct in series, some tapered, some parallel. The working fluid assumes the temperature of the adjacent metal wall. Flow is defined by two conservation equations (mass and momentum) and the equation of state, p = ρRT. Friction is taken into account by using the steady-state correlation between friction factor, local instantaneous Reynolds number, and local hydraulic radius. The formulation permits frictional drag and frictional reheating to interact more or less as they do during operation of a Stirling cycle machine at high rotational speeds. The equations are converted to characteristic form and solved numerically with pressure, p, and velocity, u, as state variables rather than the more usual a (acoustic speed) and u. This formulation paves the way for a full characteristics solution incorporating the energy equation but avoiding the entropy gradient term ∂s/∂x which is inappropriate to conditions within the Stirling machine. The paper includes a Mach-line net plotted by computer for the first revolutions of the crankshaft after start-up. Indicator diagrams are presented corresponding to different angular speeds. It is found that the indicator diagram for the compression space is not greatly affected by angular speed, while that for the expansion space changes from positive, via figure-of-eight to negative over a relatively narrow speed range. An attempt is made to explain this unexpected finding in terms of the momentum equation for constant area flow with a severe temperature gradient. A comparison is included between the computed results and those predicted for the same operating conditions by the Schmidt isothermal analysis.

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