STUDIA UNIVERSITATIS BABEŞ-BOLYAI, PHYSICA, SPECIAL ISSUE, 2003
THE THEORY OF THE THERMAL-DIFFUSION COLUMN: A CRITICAL
REVIEW
Ilie HODOR
National Institute for R&D of Isotopic and Molecular
Technologies
P.O.Box 700, 400293 Cluj-Napoca, Romania
Abstract. It is critically reviewed the theoretical literature of the
thermal-diffusion column: the original and improved Furry-JonesOnsager theories, the forgotten effect, the involvement of the chemical
exchange reactions, and the application of separation-of-variables to
the linearized problems. A series of weaknesses are pointed out and
tentative plans for more rigorous theoretical approach are mentioned.
Introduction
Thermal-diffusion (TD) consists in the fact that a temperature gradient in
a mixture of fluids gives rise to a flow of one component relative to the mixture
as a whole. The phenomenon has a small separation effect, which is very small
for isotopic mixtures.
In 1938 Clusius and Dickel[1] found their thermal-diffusion column,
which can multiply considerably the TD elementary separation effect with a
countercurrent convection in the column. The mixture is placed between two
vertical walls, one hot, and one cold. There are two typical geometries: (a) the
plane case, when the two walls are plan-parallel, and (b) the cylindrical case,
when the two walls are coaxial cylinders. The horizontal temperature gradient
gives rise to a density gradient, which, at its turn, gives rise to the convective
countercurrent flow. The geometrical conditions are chosen so that the flow is
lamellar.
In 1938, Furry, Jones, and Onsager (FJO)[2-4] published their famous
theory of the TD column, which has played a central role in numerous studies
that have followed.
A few years ago I discovered an error in the FJO theory and decided to
re-examine the main theoretical TD literature. The purpose of this
communication is to show the present state of this undertaking.
FJO theory
The FJO theory is derived for a binary mixture and assumes the validity
of the following equations for the mass flux J 1* due to molecules of species 1
and for the molecular average velocity of all species v respectively,
ILIE HODOR
J1*   [vc1  D( c1   c1c2  ln T )]
(1)
r ( / r )r (v / r )  (dp / dz )   g
(2)
1
where  - mass density, c i - mole fraction of species i , D - coefficient of
ordinary diffusion, T - absolute temperature,  - thermal diffusion constant,
p - pressure,  - viscosity, g - acceleration of gravity, and (r, z) - cylindrical
coordinates. Eq. (2) involves the assumption that v is parallel to z-axis. The
main result of the column theory is the transport equation
 1*   *c1  H *c1 (1  c1 )  ( K c*  K d* )( dc1 / dz ) ,
(3)
where  1* stands for mass transport of the species 1 through the column,  *
stands for total mass transport through the column, and H * , K c* , K d* are some
definite integrals.
An error in the FJO theory
In 1984 I published an overall axiomatic theory of the separation
column[5] and, as an application, derived an independent theory of the TD
column. My result and that of FJO were similar and I thought, at that time, that
they coincide.
Kitamoto et al.[6] studied hydrogen isotope separation using thermaldiffusion column for hydrogen gas. When the mixture is binary, say H-D, three
molecular species D2-DH-H2 are present and the exchange reaction
H2+D2=2HD takes place at the hot wall. For such a system Kitamoto et al.[6]
developed a theory with 12 column constants instead of the three constants
( H * , K c* , K d* ) in Eq. (3).
From the theory in Ref. 5 it follows that if only two isotopes are present
and if the isotopic effects are small, then an equation of the type (3) can be
derived, with only three constants, indifferent of how many molecular types and
what kind of exchange reactions are involved. Thus, I derived an equation of the
type (3)[8] for the Kitamoto et al.’s problem. To verify my result I considered
the limit case when deuterium concentration is very small, in which case, my
coefficients (H, Kc, Kd) must coincide with those of FJO. After repeated
verification I concluded that the two results did not coincide. Then I analyzed
the original FJO papers and made an astonishing discovery: the discrepancy was
due to an error in the FJO theory. Namely, in the FJO theory, it is constantly
assumed that: the mass of a component in a binary mixture is given by the
product between the mass of the whole mixture and the molar concentration of
that component. This is strictly correct only if m2 / m1  1 , where m1 and m2
(with m2  m1 ) are the molecular masses of the two components. The same
error is included in Eq. (1), the corresponding correct equation of the flux being
THE THEORY OF THE THERMAL-DIFFUSION COLUMN: A CRITICAL REVIEW
J1  m1n[c1v  D(c1   c1c2 ln T )]
(4)
where n is the total molar density.
I corrected[9] the FJO theory using the original FJO method of derivation
though the more rigorous method from Ref. 5 could be applied. The result was
similar to Eq. 3,
 1  m1ˆc1  Hc1 (1  c1 )  ( K c  K d )( dc1 / dz )
(5)
where ˆ is the total molar transport through column. For distinction, the
quantities affected of the FJO error are marked with asterisk; that is, the
equations for (H, Kd, Kc) differ from those corresponding to ( H * , K c* , K d* ).
The correlation between corrected and original theory is given in Ref. 8.
The derivation procedure of the FJO theory is not rigorous; it uses a
series of intuitive simplifying assumptions. This circumstance has generated
discussions in literature and attempts have been made to improve the theory.
However, the FJO error was not noticed in an interval of about sixty years; there
are two explanations for that:
1. As shown in Ref. 8, the effect produced by FJO error is really large
only if the following two conditions are simultaneously fulfilled:
i. The TD-column is operated with extraction (not at total reflux);
ii. The ratio m2/m1 is sufficiently large (for a 3He-4He mixture, when
m2 / m1  1.25 , the error effect can be up to 33%; if m2 / m1  10
the error effect can be huge).
2. Usually, a paper is not analyzed in order to find elementary errors in it,
the more so when the authors are eminent (Onsager won Nobel Pries in 1968).
(It is to note that I discovered the error by verifying a new theory for which I
took FJO theory for granted.)
It seems that the two conditions (i and ii) have not been fulfilled in the
experimental research. For simplicity, the TD column was generally studied at
total reflux (closed column) and not many studies were made with a ratio m2/m1
sufficiently greater then unity.
One could assert that the curious FJO-assumption is not an error but
merely a simplifying assumption. That cannot be true as this assumption is not
necessary in the derivation of the theory and, on the other hand, FJO had in
view to apply their theory to helium isotope separation[4] where the ratio m2/m1
is substantially greater then unity.
Other comments on TD column theory
The comments that follow are based on a series of previous theoretical
studies[5,7-11].
1. Besides of the original[2-4] and corrected[9] FJO theory, a variant
derived by Rutherford[12] should be mentioned.
ILIE HODOR
Rutherford[12] affirmed that “the FJO theory was originally developed for
mixtures of heavy isotopes” even if Jones and Furry applied their theory also to
helium isotope separation (Ref. 4, pg. 210). He considered that “the theory can
be extended to include mixtures of light isotopes” merely by using mass
fraction wi instead of mole fractions ci and mass average velocity vm instead of
molar average velocity v . Doing this change, Rutherford re-derived the theory
using the flux equation
J1   w1vm   D(w1   w1w2  ln T ) .
(6)
One can demonstrate by algebraic manipulation that this flux is identical with
that given by Eq. (4). That is, Rutherford used a correct flux so that he derived a
correct theory, but he did not mention this essential quality as compared with
the FJO theory.
Both theories, Rutherford’s[12] and mine[9], are similar corrected FJO
theories, but strictly speaking they do not coincide: in the first[12] it is supposed
that vm, in the second[9] that v is parallel with z-axes; or it was shown[9] that the
two conditions generally cannot be fulfilled simultaneously. However, both
theories assume that  is small, which mathematically means   0 , and for
this limit (vm  v )  0 . It follows that the two theories coincide to the limit, or
that the difference between them is a term of the second order (proportional to
 2 ).
Rutherford tested his theory by separation experiments at total reflux with
mixtures of 3He and 4He. In spite of the fact that the experimental errors were
quite large, he considered that his theory is suitable for mixtures of light
isotopes. We know now that, at total reflux, all three theories FJO, ([9]), and ([12])
are equally suitable, the difference between them being only a term of the
second order which should be covered by experimental errors.
2. Forgotten effect. In the original[2-4] and corrected[9,12] FJO-theory, it is
assumed that the horizontal gradient of density is caused only of temperature
gradient. De Groot et al.[13] drew the attention upon a “forgotten effect” which
consists in the influence of the concentration gradient upon the density gradient.
Numerous attempts[14-19] have been made to derive the phenomenological theory
of the TD-column with forgotten effect included but to date there has no
rigorous derivation.
Prigogine et al.[20] published batch experiments on TD-column of some
binary organic solutions in which they obtained bizarre separation results: the
difference in concentration between top and bottom started in one direction,
slowed, and then went in the opposite direction, giving separations opposite
from the original direction. This behavior, which has been called concentration
reversal, was attributed by them to the influence of concentration on density,
that is to forgotten effect. Concentration reversal was also observed
experimentally by Jones and Milberger[21].
THE THEORY OF THE THERMAL-DIFFUSION COLUMN: A CRITICAL REVIEW
Using carefully constructed and operated columns, Korchinsky and
Emery[22] were unable to obtain concentration reversal under conditions similar
to those in which it has been reported in the literature[20,21]. They also solved
numerically the basic equations that describe the TD-column, including the
forgotten effect. They concluded that: “When density of the liquid in the
column varies with concentration, the forgotten effect is important in transient
batch operation of columns, increasingly so as the wall spacing decreases, but it
has no influence at the steady state.” This last part of the conclusion that the
forgotten effect “has no influence at the steady state” cannot be true. Indeed, let
us suppose a binary mixture with the property  / T  0 . In this case the
countercurrent convection is caused only of the concentration gradient. With
other words, the multiplication of the elementary separation effect on the TDcolumn is produced only of the forgotten effect. It means that the numerical
method used by Korchinsky and Emery, which is not presented in detail in Ref.
22, has weaknesses.
Horne and Bearman[23], observing that their results does not agree with
the new literature data[22,24], reanalyzed their theory and added a corrective term
to their forgotten effect. However, this correction was not sufficient to obtain an
agreement with Korchinsky and Emery’s result[22].
The conclusion is that in spite of the fact that many authors have studied
the forgotten effect; there is not a satisfactory theoretical treatment of this
subject.
3. The thermal diffusion term in Eq. (4) contains the product
c1c2  c1 (1  c1 ) , which is not linear. When c1 does not vary much along
column, this term can be linearized and the whole mathematical problem
became linear in c1. The linearized problem can be solved by separation-ofvariables method. Tsay and Yen[25] used this method but they did not arrived to
understand and overcome the encountered special difficulties. However, there
are studies on similar systems[7,10,11,26,27] so that linearized TD problems could be
correctly solved.
4. Spindel and Taylor[28] studied the concentration of 15N by chemical
exchange 14NO2+15NO2=15NO2+14NO in a TD column. No theory was derived
so far for such complex processes on a TD column.
The chemical exchange in the TD column represents a domain that is
insufficiently explored and it is possible to find in it processes of remarkable
practical interest. For this reason it would be worth to develop adequate
theoretical means.
Tentative plans for future theoretical studies
Much theoretical work on TD column have been done so far,
nevertheless, as shown above, many important problems are not
satisfactorily solved. My tentative plans are to complete the TD-column
theory by finalization and publication of the following series of
theoretical works:
ILIE HODOR
a. The rigorous theory of the TD column (without forgotten effect;
done[8]);
b. The rigorous theory of the TD column with forgotten effect included;
c. Theory of the TD column with chemical exchange at the wall
(done[8]);
d. Theory of the TD column with chemical exchange in inner fluid
mixture;
e. Theory of the TD column with misaligned cylindrical walls (problem
studied numerically by Sørensen et al.[29]);
f. Second order terms in the theory of the TD-column (the theories a—e
refer to the first order terms obtained in the supposition that  is
small);
g. Solving linearized problems of the TD-column by separation-ofvariables method.
These problems can be solved using the methods developed in Refs. 5 and 11.
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