The theory for thermoacoustic mixture separation is extended to include the effects of a nonzero concentration gradient. New data are presented, which are in excellent agreement with this theory. The maximum concentration gradient which may be achieved in a binary mixture of gases through this separation process is intrinsically limited by the fractional pressure amplitude, by the tidal displacement, and by the size of the thermal diffusion ratio. Ordinary diffusion further detracts from the attainable final concentration gradient and can become the dominant remixing process as the cross section of the duct is increased. Rayleigh streaming also works against thermoacoustic separation, and an estimate of the molar flux from streaming is given.

1.
G. W.
Swift
, and
P. S.
Spoor
, “
Thermal diffusion and mixture separation in the acoustic boundary layer
,”
J. Acoust. Soc. Am.
106
,
1794
(
1999
);
G. W.
Swift
, and
P. S.
Spoor
,
J. Acoust. Soc. Am.
107
,
2299
(E) (
2000
);
G. W.
Swift
, and
P. S.
Spoor
,
J. Acoust. Soc. Am.
109
,
1261
(E) (
2001
).
2.
P. S.
Spoor
and
G. W.
Swift
, “
Thermoacoustic separation of a He–Ar mixture
,”
Phys. Rev. Lett.
85
,
1646
(
2000
).
3.
Thermal diffusion was first discovered theoretically from the kinetic theory of gases by D. Enskog and S. Chapman independently. An account including the original references can be found in K. E. Grew and T. L. Ibbs, Thermal Diffusion in Gases (Cambridge University Press, Cambridge, 1952).
4.
B. E.
Atkins
,
R. E.
Bastick
, and
T. L.
Ibbs
, “
Thermal diffusion in mixtures of inert gases
,”
Proc. R. Soc. London, Ser. A
172
,
142
158
(
1939
).
5.
L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon, New York, 1982).
6.
The hydraulic radius rh of a tube or duct is defined as the ratio of its cross section to its perimeter. For a right circular cylinder, the hydraulic radius is equal to one half of the cylinder radius.
7.
Readers who are familiar with standing-wave thermoacoustic engines and refrigerators will recognize that the above discussion is similar to the description of the critical temperature gradient, which differs from (dcm/dx)sat by the factor kT/Tm.
8.
Note that Eqs. (15) and (16) may equally well be expressed in terms of the mole fraction. Use of Eqs. (2) and (5) shows that these are equivalent to
9.
For consistency with the prior literature, we denote the longitudinal direction along the tube by x. Where we refer to cylindrical coordinates, then, we consider the set to be (x,r,φ).
10.
E. J.
Watson
, “
Diffusion in oscillating pipe flow
,”
J. Fluid Mech.
133
,
233
244
(
1983
).
11.
W. L. M. Nyborg, “Acoustic Streaming,” in Physical Acoustics, edited by W. P. Mason (Academic, New York, 1965), Vol. II B, pp. 265–331.
12.
F. W.
Giacobbe
, “
Estimation of Prandtl numbers in binary mixtures of helium and other noble gases
,”
J. Acoust. Soc. Am.
96
,
3568
3580
(
1994
).
This content is only available via PDF.
You do not currently have access to this content.