SOLUTION OF THE EQUATION OF THE THERMAL DIFFUSION COLUMN T H O M A S C . R U P P E L ' A N D
D~parfrnentof Chemical Engintering,
('nil
JAMES C O U L L
rrsitv of Piflsburgh, Pzttsburgh, Pa
The time-dependent partial differential equation of the batch-operated thermal diffusion column, closed at both ends, has been solved exactly using the Laplace transformation. The solution i s obtained in the form of a series of complementary error functions and i s suitable for treating thermogravitational data obtained from short experimental separation times over the full range of concentration. With this solution, it i s possible to solve explicitly for the thermal diffusion constant, CY.
a temperature gradient is placed across an initially homogeneous fluid mixture: a concentration gradient is established. This phenomenon is knojvn as thermal diffusion. I n the case of liquids: i t is called the Ludlvig-Soret (or simply Soret) effect after the two independent discoverers of the effect (25. 37). It has no special name in gases. Prior to 1938. the only kno\vn apparatus for thermal diffusion was the single-stage or convection-free unit-i.e.. two horizontal plates, sealed a t the perimeter, and separated b>- a slit ividth of kno\vn dimension, in Lvhich the fluid mixture was placed. .At that time Clusius and Dickel (8) introduced the thermogravitational column, a long vertical glass tube closed a t the ends and containing a gas mixture. A hot Lcire. serving as the hot ~vall.ran along the longitudinal axis. The resulting separation \vas many times that obtained with the convectionfree unit. Equal success was realized \\hen the thermogravitational method \cas applied to liquids (.9. 76, 23. 24). The basic mathematical theory of thermal diffusion in gases has been completely and independently \corked out by Chapman (6) and Enskog ( 7 2 ) . Theories have been put forth for the thermogravitational apparatus by- Furry. Jones. and Onsager ( 7 4 , de Groot (75).Debye (70). and LValdmann (32). Furry. Jones. and Onsager's theory \vas extended by Jones and Furry (20). A recent thermogravitational column theory is presented b>-Horne and Bearman ( 7 8 ) . These theories, with the exception' of Horne and Bearman's (78). lvhich is a steady-state theory. attempt to describe the concentration of one component in a binary fluid mixture as a function of time and height in the column. T o this end. all of the derivations proceed through the transport equation HEPI'
7 =
dc Hc(1 - c) - K - -
d2
(1)
In general. the nomenclature of Jones and Furry is used throughout this paper I n order to incorporate time into the transport equation. use is made of the equation of conservation of mass in thermal diffusion
1
Present address. Graduate School of Public Health. Cniversity
of Pittsburqh. Pittsburgh. Pa. 368
I&EC
FUNDAMENTALS
Taking the derivative of Equation 1 with respect to column length and equating to Equation 2 results i r
This is the equation of the thermogravitational column, Xvhich, kvhen solved in conjunction \vith prescribed boundary conditions. gives the Concentration at any point in the column as a function of time. The initial and boundary conditions. for a batch\vise-operated column closed a t both ends, to be applied to Equation 3 are T 7
= =
Oatz
=
Hr,jl -
Oandz = L c),
at t
=
0
(4)
(3
Integration of Equation of Thermal Diffusion Column
General Solution for Column Closed at Both Ends. Sirice Equation 3 is nonlinear: its solution poses some difficulty. Prior to 1951. only steady-state ( 7 = 0) and special case transient-state solutions \cere kno\vn. The derivation of the steady-state solution proceeds in a straightforLvard manner (271 and is not given here. Ho\vever: the result, in terms of the thermal diffusion constant. is
Bardeen ( 3 ) and Deb!-e (70) present transient-state solutions with the restriction that c
