In a fluid mixture in a channel with an axial time-averaged temperature gradient, high-amplitude oscillating flow can greatly increase the axial flux of thermal diffusion (Soret) separation of the components of the mixture. The enhancement occurs when the oscillating lateral temperature gradient greatly exceeds the axial gradient, causing a large oscillating concentration that can be favorably time-phased with the oscillating flow. This process can occur even with a negligible pressure oscillation or with a negligible temperature response to pressure, as is the case in most liquid solutions. The thermal boundary condition imposed by realistic solids on thermoacoustic liquids is imperfect, adding mathematical complications that are absent for typical gases, for which the solid surface is temporally isothermal. Compared with gas mixtures, the high Lewis number in typical liquid solutions reduces the separation flux associated with the time-averaged temperature gradient, but it also reduces the remixing associated with the time-averaged mole-fraction gradient. For large enough channels, the second-law separation efficiency is only slightly reduced from that of steady liquid Soret separation.

## I. INTRODUCTION

Separation of molecular mixtures is an enormous industrial enterprise,^{1,2} including separation of air into its elemental components, petroleum refining for fuel and petrochemicals, and the purification of contaminated air and water. The most energy-efficient techniques include distillation and membrane separation. Distillation, the industrial workhorse, uses about 3% of U.S. energy.^{3} Whether operating with continuous flows of feedstocks and products or in batches, understanding these widely used methods relies on analyses of fundamentally *steady* physical processes.

A number of less efficient, fundamentally steady mixture-separation methods, including crystallization, chromatography, gaseous diffusion, and thermal diffusion, have been used in the past, fill important niches today, and/or attract research attention. Steady thermal diffusion, closely related to the work described in this paper, has been applied to gas mixtures^{4} and liquid mixtures,^{5} including aqueous solutions of biological macromolecules and even impurities in molten semiconductors. In liquid mixtures and solutions, thermal diffusion is usually called the Soret effect.

A newer class of mixture-separation techniques is based on a *cyclic* or acoustic perspective, in which time oscillations of a fluid flow and a concentration-changing phenomenon, properly time-phased, result in a time-averaged separation effect. The most well-known of these is pressure-swing adsorption,^{2} in which the oscillating concentration change arises from an oscillating pressure change in a porous medium that preferentially adsorbs and desorbs one component of the mixture in response to that pressure. Traditional, low-frequency pressure-swing adsorption separation can be thought of as repetitive batch separation processing, but the adsorbing beds' time dependences are not spatially uniform, and the efficient use of adsorbent materials has driven increases in the cycle frequency. Both of these trends have made the cyclic perspective useful, although the fact that pressure oscillation amplitudes are usually comparable to mean pressure discourages the use of acoustic techniques of analysis despite frequencies sometimes on the order of 1 Hz. Recently, Weltsch *et al.*^{6} have pioneered a new realm of pressure-swing absorption separation, showing that it can occur at acoustic frequencies and with pressure oscillation amplitudes low enough to use acoustic methods of analysis.

Oscillating concentration changes that accomplish time-averaged mixture separation can also occur in a ternary mixture when two of the components have different mass-diffusion coefficients in the third component.^{7–10} Oscillating concentration changes yielding time-averaged separation can also arise in a binary mixture near a solid boundary from oscillating thermal diffusion into and out of the viscous boundary layer, caused by oscillating pressure.^{11–13} These acoustic separation methods build on earlier analyses of time-averaged transport of one component of a mixture from locations of higher concentration to locations of lower concentration by an acoustic wave near a boundary based on oscillating mass diffusion into and out of the viscous boundary layer^{14–16} or oscillating evaporation/condensation of one component at the boundary itself.^{17,18} These methods work in a channel or array of channels whose transverse dimensions are comparable to the viscous, thermal, and mass-diffusion boundary-layer thicknesses.

In some of the separation methods described above,^{11–13} the time phasing between concentration and motion oscillations is controlled by the deliberate time phasing between pressure and motion oscillations. Williams^{19} has suggested an alternative, illustrated in Fig. 1, using oscillating flow in a channel with a steady temperature gradient along the flow direction with viscous shear near the channel boundary interacting with that axial, steady temperature gradient to create a steep oscillating transverse temperature gradient, which, in turn, can create concentration oscillations via transverse thermal diffusion. Oscillating pressure and pressure-induced oscillating temperature are irrelevant, except insofar as a pressure gradient is needed to drive the oscillating flow. Thus, the Williams concept should work in a liquid solution as well as in a gas mixture.

This paper explores this new concept. The mathematics in our prior thermoacoustic separation publications^{11,13,20,21} is not directly applicable because those publications assumed a perfectly isothermal boundary condition imposed on the fluid by the channel's wall, but no real solid can impose such a perfect boundary condition on a liquid. We use the same notation as in our earlier thermoacoustic mixture-separation work, fully explaining the notation here so that this paper can be studied in isolation. We summarize some of the math from our previous publications to be clear about the assumptions here and to try to gain intuition about the processes. Sections II and III present derivations of the oscillating variables and time-averaged mole flux that result from them. The second-law energy efficiency of the method is estimated in Sec. IV. Appendix A consolidates mathematical identities that are used throughout this paper. Appendix B explains the relationships between our notation and that of other thermal diffusion research communities. Appendix C connects this work with a previous publication that had ambiguous notation in light of our present understanding.

## II. VELOCITY AND TEMPERATURE IN SMOOTH CHANNELS

Consider laminar oscillating flow of a binary liquid mixture at angular frequency, *ω*, in a channel with uniform cross-sectional geometry. Let the channel extend along the *x* direction. The transverse coordinates would be *y* and *z* for a rectangular channel or *r* for a circular channel. We use complex notation for all of the oscillating variables, e.g.,

where *p* is pressure, *u* is velocity in the *x* direction, *g* represents any of several other properties, such as temperature, *T*, density, *ρ*, and viscosity, *μ*, *t* is time, and *τ* stands for the transverse coordinates appropriate for the channel's geometry. The subscript *m* indicates the mean value, and the subscript “1” indicates the complex amplitude of the oscillating component. Re[-] signifies the real part of a complex quantity.

As derived, for example, in Ref. 22, the momentum equation at low velocities is

where the $\u2207\tau 2$ operator works on the transverse coordinates appropriate for the channel's geometry. Equation (4) results from substituting Eqs. (1)–(3) into the general momentum equation and neglecting any products of first-order variables because they are much smaller than the terms with only one first-order variable. This eliminates *ρ*_{1} and *μ*_{1} from further consideration while allowing *ρ _{m}* and

*μ*to be functions of

_{m}*x*. (We omit the subscript

*m*on the mean value of

*μ*for consistency with earlier thermoacoustics publications.) The velocity,

*u*

_{1}, must be zero at the channel's walls. With that boundary condition, the solution to Eq. (4) can be written

where $\u27e8\u2009\u27e9$ signifies the spatial average over the transverse coordinates, *h* is the homogeneous solution to the Helmholtz differential equation, normalized to unity at the transverse boundary, and $f=\u27e8h\u27e9.$ Expressing the solution as proportional to $\u27e8u1\u27e9$ instead of $dp1/dx$ keeps subsequent notation simple and reflects our focus on the average flow as the cause of other phenomena, with $dp1/dx$ as a less important requirement of that flow.

The function, $h\nu $, depends on the transverse coordinates. Both $h\nu $ and $f\nu $ depend on the channel geometry and viscous penetration depth, $\delta \nu =2\mu /\omega \rho m$, and are known for many channel geometries, some of which are summarized in Refs. 22 and 23. With *δ* representing any of the viscous, thermal, or mass-diffusion penetration depths, the functions are

in “boundary-layer approximation,” i.e., for *y* essentially unbounded away from a wall with negligible curvature in the cross section perpendicular to *x*, and with *y *=* *0 at the wall;

for a channel bounded by parallel planes separated by $2y0$ in the *y* direction and having essentially infinite extent in the *z* direction with *y *=* *0 in the center of the channel; and

for circular channels of radius *R*. The hydraulic radius, *r _{h}*, is the ratio of cross-sectional area to wetted perimeter in the channel. This definition implies that $rh=y0$ for a planar channel and $rh=R/2$ for a circular channel.

Later, we will also use the lowest-order Taylor-series expansions of *f* in Eqs. (8) and (9) for small arguments,

where $\beta f=2/3$ for planar channels, and $\beta f=1$ for circular channels.

To maintain the oscillating flow, acoustic power is dissipated at a time-averaged rate per unit length,

where *A* is the cross-sectional area of the channel, the tilde signifies complex conjugation, and Im[-] and $|$-$|$ denote the imaginary part and magnitude, respectively, of a complex quantity. This result and the background for understanding how time-averaged products of oscillating variables are expressed as the real part of a complex product are given in Ref. 22. In this paper, we neglect the $p1\u2009d\u27e8u1\u0303\u27e9/dx$ contribution to acoustic-power dissipation.

Below, as our differential equations and boundary conditions become more complicated than those in Eq. (4), deriving their solutions (or verifying them by substitution) is eased by frequent use of the general results listed in Appendix A.

If the channel has a nonzero $dTm/dx$, the transverse dependence of the velocity leads to a complicated transverse dependence in the oscillating temperature. For a channel wall with infinite heat capacity and thermal conductivity, the boundary condition on the liquid would be that *T*_{1} must be zero at the wall, and this is usually accurate for realistic wall materials when the thermoacoustic fluid is a gas. However, a liquid has enough heat capacity compared with realistic solids that the wetted surface of the channel has a significant oscillating temperature. The effect of the solid's properties on the fluid's temperature boundary condition was considered by Rott^{24,25} but the cases are inapplicable to our situation.

The heat-transfer problem in the solid wall is governed by

where *k* is thermal conductivity, *ρ* is density, *c* is heat capacity per unit mass, and the subscript *s* refers to the solid. The dry surface of the solid (if any) is assumed to be insulated; hence, $\u2207\tau ,sT1s|dry=0$.

For a single planar channel bounded by two identical planar solids, we take the coordinate $ys$ to be zero at the dry surface of the solid, increasing toward the wetted surface at $ys=s,$ where it is coincident with *y* = *y*_{0}. If a set of planar channels is created by a stack of equally spaced parallel plates, symmetry implies that $dT1s/dys=0$ in the *center* of each such plate; thus, we would simply take *s* to be half of the thickness of the plate. For a solid annulus surrounding a circular channel, we take the radial coordinate, *r _{s}*, in the annulus to be bounded by its wetted surface at

*R*and its dry surface at

*R*.

_{o}In the planar cases, the general solution to Eq. (12) can be written as a sum of sinh and cosh with coefficients to be determined by boundary conditions. The *y _{s}* = 0 boundary condition eliminates the sinh solution, leaving

where $\delta s=2ks/\omega \rho scs$ is the thermal penetration depth in the solid, and the wetted-surface temperature, $T1ws$, will be determined below.

In the annular case, the general solution to Eq. (12) can be written as a sum of zeroth-order Bessel and Neuman functions, *J*_{0} and *Y*_{0}, respectively. Imposing the *r _{s}* =

*R*boundary condition and normalizing

_{o}*h*to unity at the wetted surface yields

where

and the wetted-surface temperature, $T1ws$, is still undetermined. Later, we will also need the ratio of the annulus area to its wetted perimeter,

Henceforth, we restrict our analysis to a *liquid* mixture. With this assumption, the oscillating fluid temperature is independent of the oscillating mole fraction, and the heat-transfer equation in the liquid in the channel is simply

where the subscript *p* indicates the constant-pressure specific heat. For a gas mixture instead of a liquid, Eq. (17) would have two additional terms from $\u2202T/\u2202p$ at constant entropy and mole fraction and $\u2202T/\u2202n$ at constant entropy and pressure, as discussed in Refs. 11, 13, and 20 (and, here, in Appendix C), but we can neglect these effects for liquids.

To solve Eq. (17), we substitute Eq. (6) for *u*_{1} and follow the usual procedure for finding the particular and general solutions to a linear differential equation. The coefficient of the general solution and $T1ws$ in Eqs. (13) or (14) are then found algebraically and simultaneously from the two boundary conditions where the liquid and solid meet: The temperatures, *T*_{1} and $T1s$, and heat fluxes, $k\u2207\tau T1$ and $ks\u2207\tau ,sT1s$, must be equal there (and a minus sign must be used in the heat-flux equality for planar channels in which *y _{s}* points inward while

*y*points outward). The results are

where $\sigma =\mu cp/k$ is the Prandtl number of the liquid. (For water, *σ* = 7 at 20 °C and $\sigma =3$ at 60 °C.) These expressions introduce $h\kappa $ and $f\kappa $, which have the same functional forms as $h\nu $ and $f\nu $ but scale with the liquid's thermal penetration depth, $\delta \kappa =2k/\omega \rho mcp$. The properties of the solid appear in

for the planar case, and

for the cylindrical case.

In the cylindrical case, if either $\delta s\u226aR$ or $Ro\u2212R\u226aR$, then the radius of curvature is negligible in the solid, and Eq. (20) with $s=Ro\u2212R$ is a good approximation for Eqs. (21a) and (21b).

Equation (19) for the planar case appears as Eq. (A10) in Ref. 26 if that result is simplified by setting $|p1|=0$ and using Eqs. (5) and (6) to eliminate $dp1/dx$. If the solid has high enough thermal conductivity and/or specific heat to enforce $T1ws=0$ despite the liquid's motion, then $\u03f5s=0$ and Eq. (19) becomes Eq. (4.67) in Ref. 22. Table I shows how some thermal properties of water near room temperature compare with those of three common solids, suggesting why $T1ws=0$ would be a poor approximation for water and these solids.

. | $\u2009k/ks$ . | $\u2009\rho cp/\rho scs$ . | $\u2009\delta \kappa /\delta s$ . | $\u2009\phi s$ . |
---|---|---|---|---|

Kapton | 2.8 | 2.7 | 1.0 | 2.8 |

Quartz | 0.4 | 2.7 | 0.4 | 1.0 |

S.steel | 0.04 | 1.2 | 0.2 | 0.2 |

. | $\u2009k/ks$ . | $\u2009\rho cp/\rho scs$ . | $\u2009\delta \kappa /\delta s$ . | $\u2009\phi s$ . |
---|---|---|---|---|

Kapton | 2.8 | 2.7 | 1.0 | 2.8 |

Quartz | 0.4 | 2.7 | 0.4 | 1.0 |

S.steel | 0.04 | 1.2 | 0.2 | 0.2 |

Figures 2(a) and 2(b), respectively, show the real and imaginary parts of *u*_{1} and *T*_{1} for water adjacent to a Kapton (polyimide) wall near room temperature in boundary-layer approximation. These curves represent Eqs. (6), (13), (18), and (19) with Eq. (7) for *h* and *f*. The velocity grows from its $u1=0$ boundary condition at *y *=* *0 toward $u1=\u27e8u1\u27e9$, where the scale length is $\delta \nu =\sigma \delta \kappa =2.65\delta \kappa $ such that most of the variation occurs for $y<6\delta \kappa $. The $T1(y)$ curves qualitatively resemble those of $u1(y)$ but with important differences. Most obviously, there is a $\pi /2$ phase shift arising from the *i* in the denominator in Eq. (19). Extremes of temperature coincide with extremes of displacement, $u1/i\omega $, far from the wall. In the absence of thermal conductivity, the normalized temperature and velocity curves in Fig. 2 would be identical, except for the factor of *i*. The actual temperature curves are more complicated because of nonzero conductivity and because we chose solid properties that imperfectly enforce the $T1=0$ boundary condition at the wall. The cosh dependence in Eq. (13) is visible for *y *<* *0, and the discontinuity in slope at *y *=* *0 reflects continuity of heat flux with the discontinuity in thermal conductivity there.

As shown in Sec. 5.2 in Ref. 22, enthalpy is carried down the temperature gradient by ordinary conduction and the time-averaged product of oscillating velocity and temperature given here in Eqs. (6) and (19), respectively. With more generality, if a nonzero time-averaged molar flow, $N\u0307tot$, also flows through the channel, the discussion near Eq. (7.105) in Ref. 22 gives the additional term that accounts for its enthalpy flux with the formal caveat that any such time-averaged flow must not overwhelm the other terms. The result for the time-averaged thermal power carried by the mixture-separation process here is

where *A* is the cross-sectional area of the liquid in the channel, *A _{s}* is the cross-sectional area of the solid channel wall, $mavg$ is the average molar mass, and

*h*is the mean enthalpy per unit mass. With $N\u0307tot=0$, this is the enthalpy flux given by Eq. (A30) in Ref. 26, which is equivalent to Eq. (4.21) in Ref. 25 in the boundary-layer limit.

_{m}## III. MOLE FRACTION AND MOLE FLUX

Next, consider mass diffusion and thermal diffusion in the binary liquid mixture in the channel, whose velocity profile, *u*_{1}, is given by Eq. (6) and temperature profile, *T*_{1}, is given by Eq. (19). The mass-diffusion equation is

where *n* is the mole fraction of the solute, *D*_{12} is the binary mass-diffusion coefficient, and *k _{T}* is the thermal diffusion ratio. ( Appendix B compares our notation with that of the thermophoresis and other thermal diffusion literature.) Equation (23) can be obtained as the acoustic approximation to the combination of Eqs. (57.3) and (58.11) in Landau and Lifshitz,

^{30}converted from their mass-based notation to our molar notation. At the heart of this paper, the thermal diffusion term in Eq. (23) accounts for the fact that a temperature gradient, $\u2207T$, causes a mole-fraction diffusion flux density, $(D12kT/T)\u2207T$, and the divergence of that flux density causes the local mole fraction,

*n*, to change in time.

The boundary condition that the solute's flux perpendicular to *x* must be zero at the channel walls is expressed as

Noting the similarity of Eq. (23) to the velocity and temperature equations above, we expect the solution to be of the form

Freely using the identities in Appendix A, substitution of Eqs. (6), (19), and (25) into Eq. (23) determines the particular-solution coefficients $E\u2032,\u2009\u2009F\u2032$ and $G\u2032$, and confirms that *h _{D}* is the general solution if $\delta D=2D12/\omega $. The boundary condition, Eq. (24), then determines $H\u2032$. This yields the full solution for the complex amplitude of the oscillating mole fraction,

where $L=k/\rho mcpD12$ is the Lewis number of the liquid mixture. The $dnm/dx$ term in Eq. (26) is the same as that in Eq. (68) in Watson^{14} for a two-dimensional channel. (We write the factors $\sigma \u22121$, *L* – 1, and $\sigma L\u22121$ for minus-sign convenience for liquids, which typically have $\sigma >1$ and $L>1.$)

It is helpful to think about the $dnm/dx$ and $dTm/dx$ terms in Eq. (26) separately. These *y*-dependent parts of *n*_{1} are displayed in Figs. 2(c) and 2(d) in boundary-layer approximation for the same conditions as in Figs. 2(a) and 2(b). In the absence of any mass diffusion or thermal diffusion, $L\u2192\u221e$ and *k _{T}* = 0, and Eq. (26) shows that

*n*

_{1}would be simply

as the velocity simply carries the mean mole-fraction gradient back and forth along the channel. This dominant relationship between *u*_{1} and the $dnm/dx$ term in *n*_{1} is apparent in Fig. 2. Allowing finite *L* lets the mole fraction diffuse along *y*, altering the $dnm/dx$ term in *n*_{1} in Fig. 2(c) slightly, just as thermal diffusivity altered the shape of *T*_{1} in Fig. 2(b).

Further allowing nonzero *k _{T}* then brings the $dTm/dx$ term in Eq. (26), shown in Fig. 2(d), into consideration. The complexity of its graph reflects the complexity of the term itself in Eq. (26). A little insight can be gained by taking $L\u226b1$ and $\sigma L\u226b1$. Then, in boundary-layer approximation for fluid and solid, the part of the $dTm/dx$ term in square brackets simplifies to the sum of three exponential terms,

where

is the boundary-layer limit of *ϵ _{s}*, approximately 2.8 for water with Kapton and 0.2 for water with stainless steel, as shown in Table I. The sharp features near

*y*=

*0 in the $dTm/dx$ term in Fig. 2(d) arise from the*

*h*term here, boosted in magnitude by $L$ but decaying exponentially over a distance of only $y\u2243\delta \kappa /L\u2243\delta \kappa /10$. At larger

_{D}*y*, the more leisurely dependences of $h\kappa $ and $h\nu $ unfold. For large

*L*, the $h\kappa $ and $h\nu $ terms in Eq. (28) arise directly from $i\omega n1=(D12kT/Tm)\u2207\tau 2T1$ in Eq. (23), and this relationship is visible in Fig. 2.

Another noteworthy feature of the $dTm/dx$ term in Eq. (26) is its transverse spatial average, which must be zero, whether in boundary-layer approximation or not. This is easily shown algebraically and is at least approximately apparent to the eye in Fig. 2(d).

We now have all of the ingredients for the most important step of the derivation. The time-averaged mole flux of the solute is given by

where $N\u0307tot$ is the bulk molar flow rate (if any) and $mavg$ is its average molar mass (therefore, $\rho m/mavg$ is the total number of moles of fluid per unit volume). Using Eqs. (6) and (26) for the velocity and mole fraction, respectively, and with frequent use of the equations in Appendix A, the challenging part of Eq. (30) becomes

where

Setting $\epsilon =0$ in Eq. (A12) in Ref. 21 (being attentive to the typographical error described in the erratum to Ref. 21) yields Eq. (32). Setting $\epsilon =0$ in Eq. (39) in Ref. 20 gives the same result as setting $\u03f5s=0$ in Eq. (33).

The $dTm/dx$ term in Eq. (31) is the principal focus of this paper, expressing the ability of oscillating motion of a fluid mixture along a temperature gradient to cause time-averaged separation flux at a rate proportional to the square of the velocity amplitude.

As was shown by Watson,^{14} the $dnm/dx$ term in Eq. (31) is always negative, creating a mole flux down the mole-fraction gradient, in agreement with human intuition and the second law of thermodynamics: Simple oscillatory motion of a fluid mixture along a channel must tend to mix the mixture, not to separate it. In his Eqs. (73) and (96), Watson cast this in terms of the ratio of the oscillation-induced effective diffusivity along the channel to the ordinary steady diffusivity along the channel. In our notation, this ratio is

This can easily be large if the average fluid displacement amplitude, $|\u27e8\xi 1\u27e9|=|\u27e8u1\u27e9|/\omega $, is on the order of centimeters and $\delta \nu |1\u2212f\nu |\u2243rh2/\delta \nu $ [obtained using Eq. (10)] is on the order of 1 mm or less as long as the Im[-] factor is not too small. Defining $\u211c\u2207T$ as the similar ratio of the oscillating to steady *k _{T}* terms yields an expression with a similar, potentially large prefactor,

For a 50–50 He–Ar mixture at atmospheric pressure and room temperature in a circular steel tube, $F\u2207T$ peaks at about 0.2 when $\delta \kappa \u223crh$; and indeed Ref. 20 saw a small but measurable effect of $dTm/dx$ in He–Ar.

For a liquid solution, the $dTm/dx$ term's plot in Fig. 2(d) suggests that a channel with $rh\u2273\delta \kappa $ might take advantage of the broad features in *n*_{1}, or a channel with $rh\u223c\delta D\u223c\delta \kappa /L$ might take advantage of the sharp features, although it is difficult to visualize how the spatial averages of the products of *u*_{1} and *n*_{1} could turn out. To examine these regimes more clearly, Fig. 3 shows $Fgrad$ and $F\u2207T$ over a range of *r _{h}*. The peak in $|F\u2207T|$ near $rh\u2245\delta D$ is disappointingly small, right where $|Fgrad|$ is largest—an unfavorable combination for separation. On the other hand, $F\u2207T$ and $Fgrad$ are comparable for $rh\u2273\delta \kappa $, the region we will focus on henceforth as most promising for high separation flux.

The $|\u27e8\xi 1\u27e9|2$ dependence in Eq. (35) can reduce equilibration times and increase throughput significantly compared with steady thermal diffusion along the same axial distance. If the only limitation on $|\u27e8\xi 1\u27e9|$ is the transition to turbulence in the channel, then keeping the Reynolds-number amplitude,^{31} $4rh|\u27e8u1\u27e9|\rho /\mu $, less than about 2000 is equivalent to

which could be as large as $\u223c106$ for aqueous solutions near $rh/\delta \kappa \u223c1$.

The thermoacoustic remixing effect easily dominates the steady remixing effect because $\u211cWatson\u226b1$ for easily achievable fluid motion. A similar argument allows neglect of the steady thermal diffusion term compared with the $F\u2207T$ term in these circumstances. Then, separation saturates and stops when the thermoacoustic flux proportional to Eq. (31) is zero, i.e., when

which can be rewritten using Eq. (B11) as

In the range where Fig. 3 shows that $F\u2207T$ and $Fgrad$ are comparable, thermoacoustic Soret separation's limiting $|dnm/dTm|$ can be comparable to that of thermal diffusion without oscillating flow. Figure 4 shows their ratio, $RLim=Fgrad/F\u2207T$. For aqueous copper sulfate,^{27} $ST\u22430.009$ °C^{−1}. Thus, a temperature difference of only a few tens of °C can create a signficant difference in mole fraction along a channel.

Consider two volumes, *V*, connected by a separation tube of length $\Delta x$ with a uniform $dTm/dx=\Delta Tm/\Delta x$, all initially at uniform *n _{m}*. The exponential approach of this system toward its final state, which has $\Delta nm=\u2212F\u2207TkT\Delta Tm/FgradTm$ according to Eq. (37), occurs with a time constant $t\u2217$ equal to the number of moles of solute to be moved through the tube, $(\rho mV/mavg)(\Delta nm/2)$, divided by the initial flow rate of the solute given by Eqs. (30) and (31) with $dnm/dx=0,\u2009N\u0307tot=0$, and neglecting the steady-diffusion term. The result is

Had the same system been allowed to approach a final state with $|\u27e8u1\u27e9|=0$, the final $\Delta nm$ would have been $kT\Delta Tm/Tm$ and the time constant would have been $\omega t**=\omega V\Delta x/2AD12$. Hence, thermoacoustics speeds up the separation process by

For aqueous CuSO_{4} near room temperature in a 20-cm-long, 1-mm inside diameter tube joining 1-cm^{3} volumes with $|\u27e8\xi 1\u27e9|=1$ cm, choosing $\delta \kappa =rh$ yields $\omega /2\pi =0.7$ Hz, and Eq. (39) shows that $\omega t*/2\pi \u223c50\u2009\u2009000$ oscillations (14 h) would create a significant change in $nm(x)$. Under these conditions, $\u211cWatson=1500$.

## IV. ENERGY EFFICIENCY OF THERMOACOUSTIC SORET SEPARATION

In this section, consider a very dilute solution and assume that its bulk thermodynamic properties are dominated by only one parameter: the solvent's heat capacity per unit mass, $csolvent$. Let *T*_{0} be the reference temperature for thermodynamic energies, and let this also be the temperature at which heat has no value when considering the second law of thermodynamics. Then, integrating $dh=csolvent\u2009dT$ and $T\u2009ds=csolvent\u2009dT$ yields the molar flow exergy^{32} of the solvent,

For a very dilute solution, this is also the partial molar flow exergy of the solvent.

For a sufficiently dilute solution, the exergy change of mixing or separating is dominated by the entropy term associated with the solute. For example, with NaCl in water, the enthalpy term is only 20% of the entropy term at $nm=0.001$. Therefore, the partial molar flow exergy of the solute can be taken to be approximately

where *n*_{0} is the mole fraction where the solution has no thermodynamic value, such as an initial solution from which one might want to enrich or deplete the solute, and $Runiv$ is the universal gas constant.

With reference to Fig. 5 and following an approach like that of Ref. 33, the increase in the total flow exergy, $dX\u0307m$, associated with time-averaged mole fluxes that a length *dx* of separation channel accomplishes is the difference between the corresponding outgoing and incoming total flow exergies:

Combining logarithms and using $N\u0307solventmsolventcsolvent$$\u2243N\u0307totmavgcp$ in the first term quickly yields

Equation (6.27) in Ref. 22,

gives the additional contribution to the exergy change per unit length due to acoustic-power dissipation, thermoacoustic enthalpy transport, and ordinary thermal conductivity along the channel. For the approximations of this paper, $E\u03072$ can be neglected compared with $H\u03072$ in the last term of Eq. (45), and $dE\u03072/dx$ and $H\u03072$ are given by Eqs. (11) and (22b), respectively. $N\u0307solute$ is given by Eqs. (30) and (31). The total $dX\u0307/dx$ is the sum of Eqs. (44) and (45).

The only term in Eqs. (44) and (45) depending on $dnm/dx$ is the $N\u0307solute$ term, and that term is quadratic in $dnm/dx$ [once explicitly in Eq. (44) and, again, in Eq. (31)] with a maximum at

This shows that the maximum exergetically beneficial separation occurs halfway between zero mole-fraction gradient and the gradient at which there is no useful flow of the solute.

There are four terms in the sum of Eqs. (44) and (45), but we will neglect two of them. The ratio of the $dE\u03072/dx$ and $H\u03072$ terms in Eq. (45) is

where

neglecting the last two terms in Eq. (22b). The prefactor is $6\xd710\u22124$ for room-temperature water at 0.6 Hz and 100 K/m. The second factor, $Im[\u2009f\nu ]/\Lambda $, decreases from 10 at $rh/\delta \kappa =0.5$ to 1.5 at $rh/\delta \kappa =4$, the region of interest here. Hence, the $dE\u03072/dx$ term in Eq. (45) is negligible. We can also set $N\u0307tot\u22430$ for a situation with negligible flow of the solvent. Thus, we can define a second-law efficiency of the separation process at the best value of $dnm/dx$ as

Onsager^{34} derived an expression for the maximum second-law efficiency of any separation process based on thermal diffusion. Like ours, his derivation relies on the conclusion that the best local $\u2207n$ must always be half of the limiting value, which is $(kT/Tm)\u2207Tm$ for steady thermal diffusion. In our notation and for small *n _{m}*, Onsager's result is

This leads to the simple result that

Figure 6 shows this result for aqueous CuSO_{4} with stainless steel, quartz, and Kapton. For large enough $rh/\delta \kappa $, the efficiency of thermoacoustic Soret separation is comparable to the best possible efficiency for any separation based on thermal diffusion. This shows that the ratio of the microscopic, transverse, time-dependent mole-fraction gradients and temperature gradients in the oscillating process do not, on average, differ dramatically from $\u2207nm=(kT/2Tm)\u2207Tm$ in that range of $rh/\delta \kappa $ and at $dnm/dx|best$, although both transverse gradients are much larger than the corresponding axial gradients.

For salt water, $\eta 2,Onsager\u223c10\u22123nm$, which is much smaller than the efficiencies of the industrial-workhorse separation methods such as distillation and semi-permeable membranes. Thermoacoustic Soret separation does not overcome this handicap.

## V. CONCLUSIONS

High-amplitude laminar oscillating flow increases the rate of Soret separation in a mixture-filled channel with a temperature gradient by a factor that scales as $(|\u27e8\xi 1\u27e9|/rh)2$. If $rh\u223c\delta \kappa $, the factor could be as large as 10^{6}. The increase occurs without such dramatic changes in either the limiting mole-fraction gradient, $dnm/dTm$, or the best possible second-law efficiency.

For aqueous solutions, $\delta \kappa \u2272rh\u2272\delta \nu $ is the most favorable region. Smaller *r _{h}* suffers from smaller $F\u2207T$ and larger $Fgrad$; therefore, the achievable concentration gradient is reduced. Larger

*r*simply leaves a larger core of the channel uselessly isolated from the boundary layers.

_{h}If advanced experiments with this technique are pursued, it will be interesting to learn whether multiple, parallel, nominally identical channels can easily increase the separation flux in a single apparatus or if they will suffer from remixing caused by different streaming flows appearing spontaneously in different channels. It is also interesting to speculate how purity might be increased by cascading such sets of channels in series, using something like gently mixed thermal buffer tubes^{22} to reset the temperature from the end of one set to the beginning of the next set without resetting the concentration.

## ACKNOWLEDGMENTS

We thank Kim Williams for the idea at the heart of this paper. This work has been enabled by the support of our group-level management at Los Alamos National Laboratory. We are still grateful for the financial support of the U.S. Department of Energy's Office of Basic Energy Sciences many years ago, which launched this line of our research.

### APPENDIX A: HELPFUL EXPRESSIONS

The first half of this appendix is based on the insights of Arnott *et al.*,^{23} regarding general properties of solutions to the complex Helmholtz equation as applicable to thermoacoustics. The second half summarizes useful identities gathered from our own previous thermoacoustic mixture-separation publications.

If *h* satisfies Helmholtz's equation, then

identifies the associated penetration depth, *δ*. By convention, we normalize *h* such that

which is convenient when applying the boundary conditions for *u*_{1} and *T*_{1}, and we define *f* to be the transverse spatial average of *h*,

Application of the gradient boundary conditions for *T*_{1} and *n*_{1} is facilitated by using the planar divergence theorem on the transverse gradient of *h*,

where $\tau \u0302$ is the normal unit vector pointing out of the channel at the channel boundary and $\Pi $ is the boundary's perimeter, together with Helmholtz's equation,

to show that

where $rh=A/\Pi $ is the hydraulic radius. Equation (A6) works for channels such as circular tubes ($rh=R/2$) and parallel planes ($rh=y0$) having the same gradient at all locations on the perimeter, and the sign is straightforward for those geometries in which *y* or *r* increases in the fluid *toward* the surface. Note, however, that $\u2207\tau h|wall=\u2212(dh/dy)wall$ for boundary-layer approximation because *y* increases into the fluid *away from* the surface in Eq. (7); thus, Eq. (A6) needs a minus sign in that case.

The time-averaged products that form $H\u03072$ and $N\u0307solute$ involve spatial averages of products of the *h* functions of the form given in Eq. (A9) in Ref. 21,

which is based on Eq. (45) in Ref. 23. The time-averaged products also use $\u27e8h\u27e9=f$, introduced as Eq. (A3) above.

The definitions of the penetration depths,

lead to the compact results,

where $\sigma =\mu cp/k$ is the Prandtl number and $L=k/\rho mcpD12$ is the Lewis number. (Our use of the subscripts *ν* and *κ* for the viscous and thermal penetration depths, respectively, traces back to the use of *ν* as kinematic viscosity and *κ* as thermal diffusivity in much of the literature.)

In gas mixtures, the thermal and mass-diffusion penetration depths get blended by a nonzero $\u2202T/\u2202n$ at constant entropy as encountered in Appendix C. References 11, 13, and 20 show that the blended penetration depths are

where

and *m _{H}*,

*m*, and $mavg=nmmH+(1\u2212nm)mL$ are the molar masses of the heavy and light components, and the average molar mass, respectively. Useful identities involving these blended penetration depths include

_{L}Also, note that $\delta \kappa D\u2192\delta \kappa $ and $\delta D\kappa \u2192\delta D$ as $\epsilon \u21920$.

### APPENDIX B: MIXTURE DIFFUSION NOTATION

History has led to different vocabularies and notations in different communities who study and use mass diffusion in binary mixtures.

Cussler^{29} explains why, when both diffusion and convection are present, there is no unique or best choice for how to define the convective velocity. “It might be the mass average velocity that is basic to the equation of motion. It might be the velocity of the solvent, because that species is usually present in excess. … We only know that we should choose…so that [this velocity] is zero as frequently as possible…so…we are left with a substantially easier problem.” He mentions two other useful averages: the molar average velocity and volume average velocity. Here, we focus on the molar and mass perspectives.

The mole fraction or number fraction, *n*, and the mass fraction, *c*, are related by

where *n* and *c* are taken to be the local fractions of the heavier component, and *m _{H}*,

*m*, and $mavg$ are the molar masses of the heavier component, lighter component, and mixture, respectively. This quickly yields

_{L}(Notations vary widely among authors. Some use *c* as mole fraction and *n* as number density or mole density. Some call a variable “concentration” without being explicit about whether that means number per unit volume or a number fraction. When in doubt, study authors' equations to understand their notations.)

Landau and Lifshitz^{30} focus on mass because they want to use the average velocity, $vmass$, “the total momentum of unit mass of the fluid,” i.e., they want $vmass$ to be what is used in the Navier-Stokes equation. Previous Los Alamos thermoacoustic mixture-separation papers and Appendix C, here, share this approach because momentum is important in acoustics. The diffusive mass flux density, $jmass$, of the heavy component, relative to $vmass$, and its coefficients are defined by Eq. (58.11) of Landau and Lifshitz to be

so that the total mass flux density of the heavy component is given in Eq. (57.2) of Landau and Lifshitz as $\rho cvmass+jmass$, and the total mass flux density of the light component is $\rho (1\u2212c)vmass\u2212jmass$. These sum to $\rho vmass$ as they must. The average velocities of the two types of molecules are then

Hirschfelder *et al.*^{35} use molar notation (as do we in this paper, except in Appendix C). Converting their *n*_{1}, *n*_{2}, and *n*, which are moles per unit volume, to our *n* and $(1\u2212n)$, their Eq. (8.1−13) becomes

Jones and Furry^{4} also use molar notation, and their Eqs. (1)–(4) can be straightforwardly combined to yield Eq. (B7). To compare this with the mass-based Eq. (B4), use Eqs. (B5) and (B6) on the left-hand side and Eqs. (B2) and (B3) on the right-hand side to show that

Thus, the diffusion coefficients are the same in either basis, and we have written this variable as *D*_{12} in this paper. However, the thermal diffusion ratios are not the same.

The thermal diffusion *constant*, *α*, is related to the thermal diffusion *ratios* via

and *α* is indeed roughly constant—roughly independent of *n*—for many gas mixtures and is the same in the mass basis and mole basis. The second equality in Eq. (B10) is obtained from Eqs. (B2) and (B9).

Molar notation is most common in publications about gas mixtures. In publications about liquids, the Soret coefficient,

### APPENDIX C: CLARIFYING A PREVIOUS EQUATION

In previous Los Alamos publications about gas mixtures,^{11,13,20} the variable *ε* was needed to account for $\u2202T/\u2202n$. We originally expected to simply set $p1=0$ and $\epsilon =0$ in the results of Ref. 20 to check the $\u03f5s=0$ limit of Eq. (26). However, the results in Ref. 20 seem to rely on the assumption that $\epsilon \u22600$ in the step between Eqs. (15) and (16) therein, leading to subtle $\epsilon /\epsilon $ factors in some subsequent expressions. The same $\epsilon \u22600$ assumption was made in Refs. 11 and 13. Here, we outline a derivation for the conditions of Ref. 20 that does not require $\epsilon \u22600$ to confirm that this result is consistent with the results in those previous papers and the present paper.

The coupled differential equations for temperature and mole fraction in Ref. 20 are

The derivations in Ref. 20 and this appendix are expressed in terms of mass fraction, *c*, instead of mole fraction, *n*, and, thus, rely on the mass-based thermal diffusion ratio, $kT\u2032$. Otherwise, the notation is the same as in this paper.

To avoid dividing by *ε*, solve Eq. (C1) for $\u2207\tau 2T1$ and substitute it into Eq. (C2) to obtain

Then, apply $\u2207\tau 2$ to Eq. (C3) and, again, substitute $\u2207\tau 2T1$. Use Eq. (6) for *u*_{1}. This yields a fourth-order differential equation for *c*_{1},

The particular solution has one term that is independent of the transverse coordinates and a second term proportional to $h\nu $. The general solution allows $h\kappa D$ and $hD\kappa $ with $\delta \kappa D$ and $\delta D\kappa $ given by Eqs. (A13) and (A14).

As in Ref. 20, the coefficients of those two terms in the general solution are obtained from two boundary conditions,

using Eq. (C3) for *T*_{1} and the results from Appendix A many times. This yields

where

For nonzero *ε*, this is the same as Eq. (33) in Ref. 20, and this way of writing *c*_{1} avoids $\epsilon /\epsilon $ ambiguities when $\epsilon =0$. Setting $p1=0$ and $\epsilon =0$ produces an expression equivalent to Eq. (26) in this paper but with $kT\u2032$ here instead of the mole-fraction-based *k _{T}* in Eq. (26). They are related as shown in Eq. (B9).

Moving forward with Eq. (C7) to obtain the second-order separation flux does indeed yield Eq. (39) for $F\u2207T$ in Ref. 20, which had no $\epsilon /\epsilon $ ambiguity.

We have not attempted a solution for $\epsilon \u22600$ and $\u03f5s\u22600$ simultaneously.

## References

_{4}, CoSO

_{4}, and mixed salt aqueous solutions using an improved design of a Soret cell