Regenerator Optimization for Stirling Cycle Refrigeration, II
A cryogenic regenerator for a Stirling cycle is discussed using the gain in entropy per entropy transferred of a unit mass as the criterion of performance. Since the entropy of a unit mass (entropy times volume of a unit mass) remains constant in the transfer through an ideal regenerator, the fractional gain in entropy becomes a measure of the fractional loss or inefficiency of the non-ideal processes. The extra entropy gained is calculated first as the square of the thermal gradients over the volume including both the working fluid, a gas, and the stationary regenerator material. In addition the entropy generated by fluid friction is considered separately. It is argued that the optimum design corresponds to uniform channel flow with minimum turbulence where the gas velocity and channel width are optimized as a function of gas temperature. The maximization of heat transfer from the gas to the wall and the minimization of entropy production by friction leads to a gas flow velocity equal to sound speed times (the above) fractional loss, σ. This velocity and the axial thermal conductivity in the gas leads to a minimum channel width and characteristic length, L=T(dz/dT). A particular scaling of width, W2 = Wo 2T1/2, and length, L = LoT1/2 leads to a design where longitudinal conduction loss decreases as T3/2, and the remaining two losses, transverse conduction and friction are equal and constant. The fractional loss, σ, must be made quite small, ~(1/25) in order that the cumulative losses for a large temperature ratio like 75 i.e., 300K to 4K, be small enough, like 10% to 20%. This is because ~1/3 of the entropy generated as a loss must be transported first to the cold end and then returned to the hot end before being rejected. The dead volume ratio then determines the minimum frequency. This frequency along with the pressure determines the required wall properties. The thermal properties of the channel wall must then accommodate this cyclic heat flow without substantially increasing the entropy or fractional loss. This generation of entropy in the walls is due to the finite transverse thermal conductivity, which generates a transverse temperature gradient and the finite heat capacity of the wall, which effectively increases the axial thermal gradient. The lower temperature limit for ordinarily thermally isotropic regenerator materials like plastic or glass is roughly 30K. Below this temperature either more exotic materials or thermally anisotropic construction must be used.
KeywordsEntropy Generation Dead Volume Cycle Refrigeration Fractional Loss Transverse Conduction
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