Experimental Investigation and Modeling of Inertance Tubes

Abstract

Three models describing and predicting the inertance effect in inertance tubes for pulse tube refrigerators are presented in this thesis. All models take the inertance, the compliance, and the resistance associated with oscillating gas flow in the inertance tube into account. The first model is based on electrical transmission line theory, the second is based on lumped components in which the resistance, compliance, and inertance are each To verify these models, the mass flow rate and pressure characteristics are measured for a number of different inertance tube geometries at different experimental conditions. The instantaneous pressure at various locations and the mass flow rate at the terminating end of the inertance tube can be measured relatively easily. The mass flow rate at the pulse tube end of the inertance tube presented a challenge and several techniques were used to measure this quantity. A commercially available, hot film anemometer was modified in order to measure the rather high mass flow into the inertance tube and withstand the high operating pressure.Unfortunately this modified anemometer failed to measure the true mass flow under oscillating flow conditions. Two methods of indirectly measuring the mass flow exiting the compressor were introduced. Although these two methods were shown to be capable of measuring the mass flow correctly under certain limiting conditions, they failed to accurately ii measure the mass flow rate consistently over a range of operating conditions. Therefore, the models are ultimately verified primarily through careful comparison with those quantities that can be easily and reliably measured; specifically the pressure variation along the length of the inertance tube and the mass flow rate into the reservoir. These experimental quantities are shown to be in good agreement with the model's predictions over a range of operating conditions.Design charts are generated with the experimentally verified, distributed component model and are presented for various operating conditions in order to ease the design of inertance tubes for pulse tube refrigerators with kW-level refrigeration power. These design charts enable the designer to select inertance tube geometry that achieves a desired phase shift for a given level of acoustic power.iii ACKNOWLEDGEMENTS

Approximate Design Method for Single Stage Pulse Tube Refrigerators

“…Although this condition is approximate, it is a key requirement for minimizing the regenerator losses and therefore maximizing the COP of the pulse tube refrigerator. Whatever phase shifting mechanisms are available in the pulse tube refrigerator should therefore be used to produce the condition given by equation (7). From FIGURE 1 it may be noticed that with the condition defined by equation (7) , 1 sin 2…”

“…Various inertance tube models are available [7,8] and show that for smaller values of acoustic power (less than 1 watt), the inertance tube does not provide any appreciable phase shift.. For the example calculation carried out here, the model by Schunk etal. [7] finds that for P o = 2.5 MPa, P r = 1.3, W ac = 57.1 watts and θ = 57°, the required length and diameter for the inertance tube are 2.92 m and 4.67 mm respectively.…”

“…The material presented in this report summarizes the lectures presented to the first year graduate students in the Cryogenics course and provides a step-by-step procedure for an approximate design of the pulse tube refrigerator. This approach relies on design charts for the regenerator and inertance tube that have been developed independently through the use of REGEN3.2, and inertance tube models [4][5][6][7][8]. It utilizes optimized phase relations between the sinusoidal mass flow and pressure waves in a manner similar to that presented by Hoffman & Pan [9] and Radebaugh [10], and makes use of an empirical 'rule-of-thumb' to determine the pulse tube volume.…”

Analytical model for a pulse tube cryocooler bellows phase shifter and experimental results

“…The optimal mass-flow-to-pressure phase is also restricted by the requirements of the PTC. Schunk et al [1] summarized that for a PTC, losses are minimized and acoustic work is maximized when there is a zero phase shift at the center of the regenerator. This requirement leads to a phase shift at the cold end of approximately -30°, meaning the mass flow should lag the pressure at this location.…”

“…Though simplistic in construction, inertance tubes are typically rather large and heavy in relation to the rest of the PTC. In addition, for certain PTCs only limited phase-angles are achievable using an inertance tube [1]. Furthermore, unlike electric RLC circuits, which can be actively tuned with variable resistors, capacitors and inductors, once an inertance tube is designed and installed, active phase control is not generally possible by changing the resistance, inertance, or capacitance.…”