^{1,a)}and G. W. Swift

^{1}

### Abstract

The theory of thermoacoustic mixture separation is extended to include the effect of a nonzero axial temperature gradient. The analysis yields a new term in the second-order mole flux that is proportional to the temperature gradient and to the square of the volumetric velocity and is independent of the phasing of the wave. Because of this new term, thermoacoustic separation stops at a critical temperature gradient and changes direction above that gradient. For a traveling wave, this gradient is somewhat higher than that predicted by a simple four-step model. An experiment tests the theory for temperature gradients from 0 to 416 K/m in 50–50 He–Ar mixtures.

This work was supported by Locally Directed R&D funds at Los Alamos National Laboratory. We are grateful to Mike Torrez and Carmen Espinoza for fabrication and assembly of the apparatus.

I. INTRODUCTION

II. THE BUCKET-BRIGADE MODEL

III. DEVELOPMENT OF THE FIRST-ORDER EQUATIONS

IV. THE SEPARATION FLUX TO SECOND ORDER

V. EXPERIMENTAL APPARATUS

VI. RESULTS

### Key Topics

- Thermoacoustics
- 20.0
- Thermal diffusion
- 14.0
- Transducers
- 9.0
- Acoustic waves
- 7.0
- Copper
- 7.0

## Figures

Discrete time-step model of thermoacoustic separation in a binary mixture, assuming standing-wave phasing and including the effect of a temperature gradient along the duct. The three parcels of gas closest to the wall are locked in place by viscosity, but they each have different mean temperatures due to the thermal gradient along the duct. On the right-hand side of the figure, far from the wall of the duct, one parcel of gas is shown at the extrema of its motion. This parcel experiences no lateral temperature gradient during the motion, because it is outside the thermal boundary layer, as depicted in the bottom portion of the figure. In the middle of the figure, another parcel of gas is shown at the extrema of its motion near the edge of the thermoviscous boundary layer. At the extremes of its motion, there is a temperature gradient between this parcel and the parcels adjacent to the wall, driving thermal diffusion between this parcel and those at the wall. If for the phasing between motion and pressure considered here, then the lateral thermal gradient, and therefore also the thermal diffusion between the parcels, is lower than it would be for an isothermal boundary.

Discrete time-step model of thermoacoustic separation in a binary mixture, assuming standing-wave phasing and including the effect of a temperature gradient along the duct. The three parcels of gas closest to the wall are locked in place by viscosity, but they each have different mean temperatures due to the thermal gradient along the duct. On the right-hand side of the figure, far from the wall of the duct, one parcel of gas is shown at the extrema of its motion. This parcel experiences no lateral temperature gradient during the motion, because it is outside the thermal boundary layer, as depicted in the bottom portion of the figure. In the middle of the figure, another parcel of gas is shown at the extrema of its motion near the edge of the thermoviscous boundary layer. At the extremes of its motion, there is a temperature gradient between this parcel and the parcels adjacent to the wall, driving thermal diffusion between this parcel and those at the wall. If for the phasing between motion and pressure considered here, then the lateral thermal gradient, and therefore also the thermal diffusion between the parcels, is lower than it would be for an isothermal boundary.

(a) Comparison of to previously defined ’s, for He–Ar in the boundary-layer limit. (b) The calculated mole-fraction gradient at which thermoacoustic mixture separation saturates (i.e., ) versus oscillating pressure amplitude for various values of the temperature gradient along a duct similar to those used in the experiments. These curves are calculated for 80 kPa 50–50 He–Ar mixtures in a 3.3 mm diameter tube at a frequency of 200 Hz. The calculation assumes traveling-wave phasing and a mean temperature of 300 K, and it uses the functional forms for a finite-diameter circular tube.

(a) Comparison of to previously defined ’s, for He–Ar in the boundary-layer limit. (b) The calculated mole-fraction gradient at which thermoacoustic mixture separation saturates (i.e., ) versus oscillating pressure amplitude for various values of the temperature gradient along a duct similar to those used in the experiments. These curves are calculated for 80 kPa 50–50 He–Ar mixtures in a 3.3 mm diameter tube at a frequency of 200 Hz. The calculation assumes traveling-wave phasing and a mean temperature of 300 K, and it uses the functional forms for a finite-diameter circular tube.

The mixture-separation apparatus consists of a 3.3 mm diameter, 0.965 m long stainless steel tube, with hermetically sealed acoustic drivers at both ends. The connections to these drivers have ports for measuring the oscillating pressure and for withdrawing the mixture to a RGA, such that the length of the duct across which measurements are made is 0.975 m, slightly longer than the steel tube. The steel tube is sandwiched between two copper bars through which a circular groove was machined. This copper clamshell establishes the nearly linear thermal gradient along the duct, and it is held in place by the copper heatsinks at either end and by a copper clamp in the middle around which the heater is wound. The entire length of the copper clamshell is surrounded by fiberglass insulation (not shown) in order to minimize heat leaks to the surrounding room.

The mixture-separation apparatus consists of a 3.3 mm diameter, 0.965 m long stainless steel tube, with hermetically sealed acoustic drivers at both ends. The connections to these drivers have ports for measuring the oscillating pressure and for withdrawing the mixture to a RGA, such that the length of the duct across which measurements are made is 0.975 m, slightly longer than the steel tube. The steel tube is sandwiched between two copper bars through which a circular groove was machined. This copper clamshell establishes the nearly linear thermal gradient along the duct, and it is held in place by the copper heatsinks at either end and by a copper clamp in the middle around which the heater is wound. The entire length of the copper clamshell is surrounded by fiberglass insulation (not shown) in order to minimize heat leaks to the surrounding room.

Thermal diffusion for several applied gradients without acoustic excitation. Symbols are measurements and lines are calculations. At the ends of the duct, the temperature was held at 290 K by a recirculating chiller.

Thermal diffusion for several applied gradients without acoustic excitation. Symbols are measurements and lines are calculations. At the ends of the duct, the temperature was held at 290 K by a recirculating chiller.

Concentration versus position for a traveling wave propagating in the direction with pressure amplitude at the midpoint and a temperature gradient of ±416 K/m on either side of the midpoint. The solid triangles are measurements of helium mole fraction at the five microcapillaries along the duct. The solid curve is a calculation (Ref. 13) using Eq. (45) and the dotted curve is a corresponding calculation using the theory of Ref. 2, which omits the term in Eq. (45). The curves were calculated using as boundary conditions the values of acoustic pressure at each end of the duct and the requirement that the total helium concentration integrated over the apparatus was 0.5, because the fill valve was closed at the beginning of each experiment. Comparing the data and models in this way highlights the small differences in slope arising from the term with .

Concentration versus position for a traveling wave propagating in the direction with pressure amplitude at the midpoint and a temperature gradient of ±416 K/m on either side of the midpoint. The solid triangles are measurements of helium mole fraction at the five microcapillaries along the duct. The solid curve is a calculation (Ref. 13) using Eq. (45) and the dotted curve is a corresponding calculation using the theory of Ref. 2, which omits the term in Eq. (45). The curves were calculated using as boundary conditions the values of acoustic pressure at each end of the duct and the requirement that the total helium concentration integrated over the apparatus was 0.5, because the fill valve was closed at the beginning of each experiment. Comparing the data and models in this way highlights the small differences in slope arising from the term with .

A representation of all data and corresponding calculations, over the ranges and . The filled symbols are data for the finite-difference gradient in helium concentration versus the finite-difference gradient in temperature. The differences are from the ends of the duct to the middle, ignoring the measurements at the intermediate microcapillaries numbered 2 and 4. The curves are corresponding calculations, matched to the actual pressure amplitude of each measured point.

A representation of all data and corresponding calculations, over the ranges and . The filled symbols are data for the finite-difference gradient in helium concentration versus the finite-difference gradient in temperature. The differences are from the ends of the duct to the middle, ignoring the measurements at the intermediate microcapillaries numbered 2 and 4. The curves are corresponding calculations, matched to the actual pressure amplitude of each measured point.

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