JOHN H.
RAMSER
Research and Development Department, The Atlantic Refining Co., Philadelphia, Pa.
Theory of Thermal Diffusion under linear Fluid Shear
THE
J
mathematical theory of a thermal diffusion column in which fluid circulation is produced by convection has been given by Debye (2). He showed that the velocity distribution of the fluid in such a column depends on the properties of the fluid, the gap width, and the temperature difference across the column. I n order to accomplish more rapid separations, it is desirable to control fluid circulation in a manner which is independent of ‘these parameters. Thus, the suggestion has been made that fluid circulation be produced by a uniform motion of the walls of the column (7). This produces a linear shear of the fluid, if convection can be neglected either by making the shear rate sufficiently high or by placing the column in a horizontal position. In this discussion the mathematical theory for a plane-parallel thermal diffusion column is developed, in which the velocity distribution is linear. The mathematical treatment is based on the one used by Debye (2).
the order of to cm. sec.-l in practical cases. If the hot wall is made to move upward with constant velocity and the cold wall downaard with equal velocity, the fluid is subjected to a linear shear. For wall velocities of the order of 1 cm. sec.-l the velocity component due to natural convection may be neglected and the velocity distribution across the column becomes very nearly linear. Thus, ~ ( x assumes ) the form u = u (x
-
4)
(3)
where u is the rate of shear. If u is substituted from Equation 3 into 1 and dimensionless space coordinates are introduced as well as a dimensionless time coordinates, defined by the equations
After substitution of Equation 6 into 8, the boundary conditions for Equation 7 are found to be U‘
the following differential equation is obtained
in which U -
by
(1)
For a vertical column with stationary walls, the fluid velocity, v ( x ) , due to natural convection is small. The mean velocity of the fluid, obtained by averaging the velocity distribution, o ( ~ ) , as given by Debye, is
p
=
D‘/
and
w = u2u -
D
D
are dimensionless groups. Steady-State Solution The steady-state solution is obtained from Equation 4 by setting (5)
If u = cm., T = 60” C., p = poise, p = and p = 1, the mean velocity is about 3 X loF3 cm. sec.-l It may also be shown that the maximum velocity is only 54% higher than the mean velocity independent of the parameters in Equation 2. Thus, the fluid velocity in a thermal diffusion column with natural convection is of
{y
(9)
For small values ofp [the “Soret constant”
D
(3) is of the order of
to 10-2
7 is usually of the order of 100 to 10‘1 one may write
and
OL
dn
att’ =
+ . ..
(10)
where the U , are as yet undetermined functions of {’. Furthermore, one may set
D I=-t
a2
The basic partial differential equation ( 2 ) for a plane-parallel thermal diffusion column with fluid motion in the y direction has only the following form when expressed in ( x , y)-coordinates:
+$u = o
u = 1 +pu, +p2uz
Fundamental Equation
bn D‘-dT dx bx
The boundary condition for this equation may be derived from the fact that the total particle current, due to ordinary and thermal diffusion, must be zero a t both walls. This leads to the condition
The variables of the resulting partial differential equation may be separated by assuming that the solution has the form n = C,ew‘U(E’)
= pa1
+ p2az + . . .
(11)
where the 01, are as yet undetermined coefficients. For small values of p the value of U is nearly equal to 1 and the value of 01 is nearly equal to 0. Substitution of these values in Equation 6 yields the result that the particle concentration is nearly equaI to the as yet undefined constant, CO, a t every point in the column. Thus, the particular expansions, Equations 10 and 11, satisfy the initial condition. If U from Equation 10 and 01 from 11 are substituted in Equation 7, a polynomial in p is obtained which is zero for all values of ,$’only if the coefficients of the powers of p are identically zero. Thus, for the coefficient of$ in the polynomial
U;
+
wa1
(E‘
z)
- 1
= 0
(12)
and for the coefficient of p 2
(6)
This yields the ordinary differential equation in 4‘
Asp is assumed to be small, higher powers offi need not be considered. I n a similar VOL. 49, NO. 1
JANUARY 1957
155
way one obtains from Equation 3 the boundary conditions
After substituting n from Equation 18 in 19 and performing the double integration, it is found that
wall moves relative to the other with a velocity u,,
Asw+ The function lJ1 may be obtained by integrating Equation 12 tlvice and applying the first boundary condition Equation 14 for a determination of the first integration constant. The second integration constant may be set equal to zero without loss of generality, since Co still remains to be determined. Thus, it is found that
The coefficient al may be determined by substituting first C71 from Equation 15 in 13 and integrating once to obtain the function Ui. The result is
=
e
4/120
m,S-+l.
Nonsteady-State Solution w h Since p is very small and - 0,it is convenient to introduce the “separation factor,” S, which may be defined as the ratio of concentrations at
h
-
v’
=
0.
For small values of@ one obtains
a
to the concentration at 7 ’ =
The boundary condition is
Since
is very small, the eigen-values h K of Equation 26 are almost identica with the eigen-vaIues of the equation V“
The separation factor equals 1 for w = 0 and increases to a maximum a t w = I t follows that for any given value of the dimensionless group Fh -, the
+’=.
a
maximum separation is obtained if one
1 56
+?+P%-
_ = _
T h e steady-state solution in terms of symmetrical coordinates is given by Equation 36. I t is seen that the concentration a t equilibrium is a function of the dimensionless “shear group,” w. For zero rate of shear w = 0 and
INDUSTRIAL AND ENGINEERING CHEMISTRY
+ K2V = 0
(29)
A particular solution of Equation 29 satisfying the boundary condition 28 is V = A cos KE
with eigen-values ~ = 2 n n ( n= 0,1,2,. . .). Since K~ in Equation 27 is almost equal
to k2, it is clear that the only eigen-value of physical significance in 29 is the value K = 0, since the next higher value, K = 27r"k, would imply a n extremely fast approach to equilibrium according to Equation 24. Thus, a first approximation to the function V ( [ ) may be found by solving the equation y" = 0 which has the solution
v=A+Bq (30) The second approximation to V ( [ ) may be found by substituting V from Equation 30 into 26 and integrating. After adding to the resulting solution the function Equation 30, which is a particular solution of 26, one obtains
A more exact value of ~2 may now be found by differentiating V in Equation 31 and applying the boundary condition 28 to V ' . Thus, it is found that
Because the second term under the root is very small compared to 1, the root may be expanded by the binomial theorem. If the resulting value for ~2 is substituted into Equation 27 one obtains
A particular solution of Equation 4a
4 v=+m 7r2
*=o
sin ( 2 v (2v
+ 1)r 4 +
?I
1)2
Dt
if the following complex values are assigned to the arbitrary constants C:,
for which the value v = 0 is to be excluded. This leads to the following result
e-&
+
1)27r2
( g ( 1
ing $ may be neglected, irrespective of the value of w. After the 'dimensionless time
+ 2) I
{ has been replaced by - and constant a2 e introduced, defined by
Equation 38 may be written in the following form
(35) (40)
If the steady-state solution Equation 22 is expressed in symmetrical coordinates, it assumes the form
For w = 0, n = no for every 7. Again as w -+ 03, la -+ no, as is to be expected.
I n order to combine Equation 36 with the solution Equation 35, it will be necessary to replace q in Equation 36 by its Fourier series expansion in the h interval - -- 5 7 5 -.h 2a 2a'
Separation Factor The separation factor for nonsteady states could be obtained from Equation h 40 by substitutinginto it 7 = - - and 2 a h -, respectively, and dividing. and 7 = 2a However, the resulting expression for n,/nb would be too cumbersome for practical use. A close approximation of simpler form may be obtained by neglecting the very small variation of n with a t both ends of the column and making use of the following approximation for small values of E
+
After 7 from Equation 37 has been substituted in 36, a particular solution is obtained which may be added to the particular solution Equation 35 after
+
+
(1 - €)/(1
E )
= (1 -
€)2
=
1 - ,2c
With these provisions, one obtains for the separation factor
is therefore
t
(33) where V ( l ) is given by Equation 31. A solution, which satisfies the initial condition n = no a t { = 0 and converges to the steady-state solution as t ---t m, may be constructed in the following manner : As the variation of concentration n with E is negligibly small in comparison to its variation with 7 and {, it will be assumed that V is a constant which may be combined with C. Furthermore, as the partial differential equation (4a) is linear, a more general solution may be constructed by a linear combination of particular solutions, Equation 33, in which v may assume any integral value. Thus, one obtains:
the latter has been premultiplied b y the h constant - -. a
v==m
The complex exponential factor may be removed by making use of the identity
1
T h e result is Y
=o
m
-
- 7-r 2 8
The quantity 6, defined by Equation 39, has the dimension of time. The physical significance of 6 may be seen from the following argument: The first, and largest, term in the series in Equation 41 is e -tie 1
J
u - m
y =
In the derivation of this result use has been made of the identity
This solution satisfies the initial condition, n = no, since the term containing the series vanishes for { = 0 and p is assumed to be small compared to 1. As 4 m , the series term converges toward the Fourier series expansion of 7, so that tt converges toward the steadystate solution (Equation 36). For smaU values of p the term contain-
At t = 0 this term equals 1 ; if t = 6, the value of the term is l / e . For this reason, and in accordance with similar usage in physics, the quantity 6 may be called the "relaxation time" of the thermal diffusion column. Its value determines the rate at which the system passes from its original uniform concentration to its final state of equilibrium. This rate is small if 6 is large, and vice versa. Equation 39 shows how the relaxation time depends on the height of the column, h, the constant of ordinary VOL. 49, NO. 1
JANUARY 1957
157
diffusion, D, and the “linear shear group”
~
dimensionless
Table I.
Poise
Viscosity, p =
Per Cent Equilibrium Thermal Thermal Time, convec- Linear convec- Linear tion shear tion shear Days Separation
Factor _____
in which a is the gap width, u is the rate of shear, and D is the diffusion coefficient (see Equation 4). For a given height of column and gap width, the relaxation time may be decreased by increasing the rate of shear. The results obtained in this mathematical analysis can be used to determine the effect of different shear rates for any choice of apparatus constants and operating variables. I t is instructive to calculate the separations which can be achieved in a thermal diffusion column with externally applied linear shear and to compare them with results obtained in a thermal diffusion column of conventional design. Such a comparison may be made in a variety of ways, which may illustrate the advantages and disadvantages of either design under given conditions. In the numerical examples discussed below, a comparison is made on a basis of equal gap width and equilibrium separation. The gap width is assumed to be 0.01 cm. The equations derived by Debye (Z), as well as those derived above for linear shear, are valid only for small separations, because the exponential variation of concentration xvith height was replaced in either case by its series expansion, using only two terms for simplicity. Therefore, it is assvmed that the equilibrium separation, ( n , Inb)?> equals 0.791 in the numerical exan-ples discussed below. The purpose of these examples is to demonstrate that the rate of approach to equilibrium is considerably faster with linear shear when the viscosity of the fluid is sufficiently high. Numerical values were assigned as foiio~%7s: p = 10-3, = 103, = I, D = 10-5, 7 = 60’ c.,p = 5 x 10-4. (The value of p must be sufficiently small, so that the approximations used may hold.) Example 1. p = 10-2 poise. THERMAL CONVECTION.For the assigned numerical values of the constants the value of the dimensionless group, q, in Debye’s theory (2) becomes equal to 100. If h = 10, the equilibrium separation, ( n l / n b ) # ,calculated from Equation 23 in Debye’s paper, equals 0.791. The Debye relaxation time, OD, becomes 5.09 days and the average fluid cm. velocity V in the column, 3 X per second. LINEARSHEAR. If the velocity of either wall with respect to the central stationary plane at x =
is chosen to be 2 3 X lo-* cm. per second, the value of the dimensionless shear group, w , becomes equal to 60. I n order to obtain an equilibrium separation of 0.791, the
1 58
0.5 1 3 6 10 m
0.953 0.933 0.885 0.843 0.815 0.791
0.926 0.S95 0.830 0.800 0.792 0.791
22.5 32.1
55.0 75.1 88.5 100.0
35.4 50.2 81.3 95.7 99.5 100.0
height of the column as calculated from Equation 41 must be 25.9 cm. The corresponding relaxation time for linear shear calculated from Equation 39 is 2.05 days. Table I shows the values of the separation factor, ( n f / n b ) , as a function of time for thermal convection and linear shear, respectively. The fourth and fifth columns show the “per cent of equilibrium reached” as calculated from the expression
in xvhich ( n t / n J e = 0.791. I t is seen that a fivefold increase of the average fluid motion through externally applied linear shear results in a faster rate of approach to equilibrium. Example 2. p = 10-1 poise. THERMAL CONVECTIOK. As a result of the higher viscosity, q becomes equal to 10. In order to obtain an equilibrium separation factor of 0.791, the height of the column must now be 50.7 cm. With these values the Debye relaxation time becomes 297 days. -4s a result of the increased viscosity the average fluid cm. per second. velocity is only 3 X LINEARSHEAR. If the velocity of either wall is again chosen to be 3 X lo-* cm. per second, the results obtained with linear shear are the same as in Example 1, because the separation factor for linear shear is independent of viscosity. The separation factors for Example 2 are shown in Table I1 for thermal convection and linear shear, respec tively Columns four and five shows that equilibrium is reached at a considerably greater rate when the average fluid velocity is increased by a factor of 50 through externally applied linear shear. I t should not be concluded from these
.
Table II. Viscosity, p = 1 0 - I Poise Separation Per Cent Factor Equilibrium Thermal Thermal Time, convec- Linear convec- Linear shear tion shear Days tion
INDUSTRIAL AND ENGINEERING CHEMISTRY
1 10 100 m
0.991 0.972 0.913 0.791
0.895 0.792 0.791 0.791
4.3 13.4 41.6 100.0
50.2 99.5 100.0 100.0
examples that the advantage of linear shear over thermal convection, as far as rapidity of separation is concerned, is substantial under all conditions. Optimal conditions for good separation must be determined for each particular fluid, as both viscosity and the dimensionless group, p , have a large effect on the rate of separation. The significance of the thermal diffusion column with externally applied linear shear lies in the fact that the fluid motion in the column can be controlled independent of the properties of the fluid and of the geometry of the column. This should be of particular importance for fluids with high viscosity.
N o me nclat u re u
= linear fluid velocity, cm. per sec-
p
= coefficient of thermal expansion,
g
= gravitational constant, 981 cm. per
p
=
E
symmetrical horizontal position coordinate, dimensionless = symmetrical vertical position coordinate, dimensionless = height of column or length of gap, cm. = dimensionless “shear group” equal
ond reciprocal centigrade
secS2 fluid density, grams per cc. 7 = temperature difference, degrees centigrade p = absolute viscosity, poises a = gap Midth, cm. x = horizontal position coordinate, cm. x = 0 at cold wall 5 = average linear fluid velocity, cm. per second u = rate of shear, set.-' n = number of molecules per unit volume: ~ m . - ~ t = time, seconds y = vertical coordinate, cm. J = 0 at bottom D = constant of ordinary diffusion, sq. cm. per second D ’ = constant of thermal diffusion, s q . cm. per second;” C. T = temperature, degrees Kelvin E ’ = asymmetrical horizontal position coordinate, dimensionless 7 ’ = asymmetrical vertical position coordinate, dimensionless { = time variable, dimensionless D’ p = dimensionless group equal to - r
7
h (3
D
=
a26
to -D 0
= relaxation time, defined by Equa-
nt
=
no
=
tion 39, seconds concentration at top of column, cm.+ concentration at bottom of column. ~ m . - ~
Literature Cited (1) Beams, J. W., U . S. Patent 2,521,112 (Sept. 5, 1950). ( 2 ) Debye, P., Ann. Physik. 36, 284-94 (1939). (3) Prigogine, I., DeBrouck&re,L., Amand, R.,P h ~ ~ i c16, a 577-98 (1950). RECEIVED for review April 6, 1956 ACCEPTED October 4,1956
