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TECHNICAL PAPERS

Experimental Investigation and Modeling of Inertance Tubes

[+] Author and Article Information
L. O. Schunk, J. M. Pfotenhauer

Cryogenic Engineering Laboratory, Room 1339 Engineering Research Building, 1500 Engineering Drive, University of Wisconsin, Madison, WI 53706

G. F. Nellis1

Cryogenic Engineering Laboratory, Room 1339 Engineering Research Building, 1500 Engineering Drive, University of Wisconsin, Madison, WI 53706gfnellis@engr.wisc.edu

1

Corresponding author.

J. Fluids Eng 127(5), 1029-1037 (Apr 27, 2005) (9 pages) doi:10.1115/1.1989369 History: Received June 25, 2004; Revised April 27, 2005

Growing interest in larger scale pulse tubes has focused attention on optimizing their thermodynamic efficiency. For Stirling-type pulse tubes, the performance is governed by the phase difference between the pressure and mass flow, a characteristic that can be conveniently adjusted through the use of inertance tubes. In this paper we present a model in which the inertance tube is divided into a large number of increments; each increment is represented by a resistance, compliance, and inertance. This model can include local variations along the inertance tube and is capable of predicting pressure, mass flow rate, and the phase between these quantities at any location in the inertance tube as well as in the attached reservoir. The model is verified through careful comparison with those quantities that can be easily and reliably measured; these include the pressure variations 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 wide range of operating conditions. Design charts are subsequently generated using the model and are presented for various operating conditions in order to facilitate the design of inertance tubes for pulse tube refrigerators. These design charts enable the pulse tube designer to select an inertance tube geometry that achieves a desired phase shift for a given level of acoustic power.

Copyright © 2005 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic of the distributed component model of the inertance tube/reservoir system. Note that the inertance tube is broken into discrete segments that are represented using a capacitance, resistance, and inertance.

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Figure 2

Comparison of the phase predicted by the distributed component model with the transmission line model predictions presented by Luo (8)

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Figure 3

Comparison of the dimensionless acoustic power predicted by the distributed component model with the transmission line model predictions presented by Luo (8)

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Figure 4

Schematic of the experimental setup

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Figure 5

Phase between the pressure in the reservoir and the mass flow rate at the reservoir end of the inertance tube as a function of inertance tube length for a 7.9mm i.d. tube (15.3Hz) predicted by model and inferred from measurements

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Figure 6

Mass flow rate at the reservoir end of the inertance tube as a function of inertance tube length for a 7.9mm i.d. tube (15.3Hz) predicted by model and inferred from measurements

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Figure 7

Phase between the pressure in the reservoir and the mass flow rate at the reservoir end of the inertance tube as a function of inertance tube length for a 11.1mm i.d. tube (15.3Hz) predicted by model and inferred from measurements

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Figure 8

Mass flow rate at the reservoir end of the inertance tube as a function of inertance tube length for a 11.1mm i.d. tube (15.3Hz) predicted by model and inferred from measurements

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Figure 9

Measured and predicted pressure variation half-way along (i.e., at 3.05m) a 6.1m long inertance tube with an 11.1mm inner diameter (15.3Hz). Also shown is the measured pressure variation at the inertance tube inlet.

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Figure 10

Measured and predicted pressure variation at the reservoir inlet for a 6.1m long inertance tube with an 11.1mm inner diameter (15.3Hz). Also shown is the measured pressure variation at the inertance tube inlet.

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Figure 11

Phase between the mass flow rate and pressure at the inertance tube inlet as a function of length for various diameters (at 30Hz, a pressure ratio of 1.1, and a mean pressure of 25bars). Also shown are three combinations of diameter and length that yield a desired phase of −60°.

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Figure 12

Acoustic power as a function of length for various diameters (at 30Hz, a pressure ratio of 1.1, and a mean pressure of 25bars). Also shown are three combinations of diameter and length that yield a desired phase of −60°.

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Figure 13

Contours of constant length and constant diameter shown in the space of acoustic power and phase (assuming an operating frequency of 30Hz, a pressure ratio of 1.2, 25bar mean pressure, and a 7.6liter reservoir volume)

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Figure 14

Inertance tube design chart at 30Hz and 1.2 pressure ratio, 25bar mean pressure, and 7.6liter reservoir volume for short inertance tubes. Bold lines represent various lengths (m).

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Figure 15

Inertance tube design chart at 40Hz and 1.2 pressure ratio, 25bar mean pressure, and 7.6liter reservoir volume for short inertance tubes. Bold lines represent various lengths (m).

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Figure 16

Inertance tube design chart at 50Hz and 1.2 pressure ratio, 25bar mean pressure, and 7.6liter reservoir volume for short inertance tubes. Bold lines represent various lengths (m).

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Figure 17

Dimensionless inertance tube design chart at 1.1 pressure ratio and 7.6liter reservoir volume for short inertance tubes. Bold lines represent various dimensionless lengths.

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Figure 18

Dimensionless inertance tube design chart at 1.3 pressure ratio and 7.6liter reservoir volume for short inertance tubes. Bold lines represent various dimensionless lengths.

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