ANALYSIS OF TRANSVERSE-FLOW THERMAL DIFFUSION DA V I D FRA 2 I E R
,
The Standaid Oil Co. ( O h i o ) , Cleveland, Ohio
Described here are classes of thermal diffusion systems for continuous prcduction in which feed and take-off are at the ends of the thermal slit, and in which horizontal as well as vertical and thermal concentration gradients can exist. These are named “transverse-flow’’ systems. Analytical expressions based on a highly simplified model are derived for rate-separation curves of such transverse-flow systems and for comparable center-feed end take-off systems. Among parameters specifically included are “figures of merit” for performances of the feed and take-off channels.
(72) showed that a thermal gradient can cause a partial separation of a liquid mixture. Clusius and Dickel ( 7 ) constructed a n apparatus in which this effect (now called the ”Soret effect”) together with the natural thermal circulation of a fluid between vertical hot and cold walls brought about large changes in composition between reservoirs at the top and bottom of the “thermal slit.” This process has come to be called “thermal diffusion.” Addition of a central manifold and take-off ports at the ends by Jones and Hughes (9, 70. 7 7 ) made the process continuous. In this paper we discuss another flow scheme (2, 3. 4 ) for continuous thermal diffusion. in which feed is introduced a t diagonally opposite corners of a thermal slit, and top and bottom products are removed from the other two corners. Figure 1 illustrates the Clusius and Dickel apparatus, the Jones-Hughes apparatus. and the apparatus suggested here, which has come to be called the “transverse-flo\s system.” The primary tool for evaluation of continuous thermal diffusion columns has been rhe “rate-separation curve,” which is a plot of separation in terms of whatever property of the feed is of interest against the volumetric rate of production. \\’e shall derive expressions for the rate-separation curves for various embodiments of the apparatus of Figure 1. To do this it is necessary t o make certain assumptions about the process. \-cry broadly, these assumptions are slanted toward, or motivated by. the production of automobile lubricating oil from a partially refined crude oil distillate. Experience in this laboratory has shown (7) that such a distillate can give about a 507, yield of superior motor oil, and the separation required is ”small” compared to what can ultimately be done UDM’IG
Assumptions
1. Diffusion. M-e d o not concern ourselves with the mechanism of the Soret separation. \Ve merely assume that if the fluid under examination is confined in two well mixed chambers which are a t two different but uniform temperatures and are separated by a hypothetical porous membrane, a transfer of the property of interest Mill take place until Soret equilibrium is reached, and will take place at a rate proportional to the distance from Soret equilibrium. If y and z are the concentrations of the property! which is assumed to be conservative, additive, and volumeless. in the two cells, and if 2K is the Soret equilibrium separation, the flux, J ?of the property from one cell to the other \vi11 be: for all? and z : J
=
(-y
- 2
+ 2K)D
FEED
?AKE-OFF
’
BOTTOM
Figure 1.
Apparatus
Left. Clusius Center. Jones-Hughes Right. Transverse flow
mKE-OFF
(1)
Here D : the proportionality constant. can be thought of as a kind of diffusion coefficient, though its dimensions are It-’ and it is not to be confused \vith diffusion coefficients as usually defined. Dimensions of J are also It-‘. and y , z , and K are dimensionless. Compositions are measured from the feed, which is defined to have composition equal to zero.
2. Flow. .%. In the slit we assume, as did Furry and Jones (5)?that the rising and falling streams are of equal thickness, and that each stream is uniform in velocity and cornposition over its whole thickness. B. I t is desirable to construct the transverse-flow apparatus conceptually (and sometimes practically) as a series of smaller units connected in series. I t is necessary to make some assumption about the floiv from unit to unit. \Ye assume the flow pattern of Figure 2. The fraction (1 A ) / 2 of the flow to the next stage is ivithdrawn before the entering thermal stream is diluted with flow from the preceding stage, and the fraction (1 - A ) / 2 is withdrawn after dilution. The quantity -4(- 1 < -4 < 1) is a figure of merit for the channel or manifold system. C . Top and bottom take-offs, where they occur, are assumed equal in volume. D . Top and bottom feeds, where they occur, are assumed equal in volume.
BEED ‘ID4b BOTTOM
COLO-
to the oil by a really powerful column. This situation is different from that of Furry, Jones, and Onsager ( 6 ) :who were interested in separation of small amounts of UZ3jFs from a lot of nuclearly inactive material by separations that were “large” indeed. Our assumptions are simpler than those of Powers (73), but our center-feed equation is similar to his.
+
VOL.
1
NO. 4
OCTOBER 1 9 6 2
237
INPUT, COMPOSITION: !#u
OUTPUT,
RISING THERMAL STREAM
t
FI
i
I
\ ENTERING STREAM COYWSITK)” Cu
FLOW F A
t
I
FLOW: FJ.
I //A*, Figure 2.
FALLING THERMAL STREAM
Figure 3. Assumption about flow in center of center-feed apparatus
Top of mixed-end feed column
E. Some degree of mixing or diversion of the thermal streams may take place in the feed channel of the center-feed apparatus. IVe assume the pattern of Figure 3. Here the fraction (1 - B ) / 2 of each thermal stream entering the center channel is diverted into the other thermal stream, and the B)/2passes through the channel without diverfraction (1 sion. The quantity 6(-1 < B < 1) is thus a figure of merit for the center channel system. F. In the center-feed apparatus all feed is in the center and all take-off at the ends; there must therefore be a net flow through the thermal slit. Clearly the rising and falling streams are not independent: If a net upward flow is imposed, the downward flow must be reduced. We assume that an increase in flow in one direction reduces the flow in the opposite direction by the same amount-that is, if F is the rate of normal thermal circulation in each direction per unit width and F I and F2 are the rising and falling flows, then :
and solving for C, we have :
(4)
+
Ft
+ Fz
=
2F
(2)
Dimensions of F, FI, and F? are Pt-’.
But the dimensionless group. (DLF ) . can be seen to be the ratio of the total height of the column, L. to the height necessary for the leaving streams to be in Sorer equilibrium as defined in Equation 1. I t may therefoie be thought of as the number of “theoretical plates” in the column, to use distillation terminology. Therefore define :
P
D L (dimensionless) . F
(5)
Making this substitution, we reivrite Equation 4, and a similarly derived expression for C,:
Derivations
Clusius and Dickel Apparatus. Now consider the Clusius and Dickel apparatus sketched in Figure 4. I t will be convenient to develop expressions for C, and C,, the compositions of the streams entering the upper and lower reservoirs, respectively, in terms of K , D,and F as previously defined and I,,the height of the apparatus (dimensions = I ) . In general, in any small vertical section of slit, say the one of thickness dx a t x , there will be a flux of property of interest from one stream to the other as defined by Equation 1. Since the property is assumed conservative and additive, we have a t once :
These are equations for compositions of the streams entering the reservoirs in terms of the reservoir contents and column parameters. If we assume the column has come to steady state, the compositions of the entering stream will be equal to the composition:
c1 = c* =
If the reservoirs are of equal volume, and were loaded initially with material of zero composition, we also have:
(3)
Differentiating, omitting the t derivative :
-cc CD
This integrates on sight:
Y
= a1x
+ a2
Imposing these boundary conditions : At x = 0, y = C L , and z = CI At x = L , y = Cp, and z = Cu 238
l & E C PROCESS D E S I G N AND D E V E L O P M E N T
= CD
Solving, we find:
e, But the first derivatives are equal, by Equation 3. Hence:
CD
C”
=
=
KP -KP
This is the completely reasonable result that steady-state separation is just single cell separation. E;, times the number of “plates,” P. Mixed-End Feed Apparatus. Though the Clusius and Dickel apparatus is essentially a batch apparatus, it is practically as well as conceptually possible to operate it for continuous production by withdrawing product from the top and bottom reservoirs and simultaneously adding equal volumes of feed or partially separated mixture. .4n assumption about the mixing of the input with the circulating stream is necessary;
this has been stated as Assumption 2B. I t will now be convenient to derive a n expression for the steady-state output quality of such a system as a function of the volume rate of output. T h e top end of a mixed end feed apparams as assumed here is sketched in Figure 2 A flow, Z (dimensions = 6 t - I ) , of composition $u (dimensionless) is being continuously added. and the same amount \zithdra\vn. Of the material withdrawn, the fraction [(l A ) ’21 (dimensionless) is withdrawn from the circulating stream before addition of the input, and the fraction [(l - A ) , 21 is \vithdra\z.n after input. Acorresponding flow of composition $ n in and d D - A $ out is assumed through the lower reservoir The apparatus is X (dimensions = I ) wide; thus thermal circulation in and out of the reservoir is FA (P-’). Now define :
+
I=-
2
- FA
Here I , which can be called the “standardized take-off rate?” is the ratio of column output a t one end as a fraction of the thermal circulation rate. Taking into account conservation of flow and property, we have for the composition of the streams entering the thermal slit a t the top and bottom, in quantities previously defined:
c.’ =
[2 - ( I +A)rlCc+2r$v [2 (1 -
+
(11)
directly into the reservoirs from which products are removed ; a multistage, countercurrent array like that of Figure 5 might well be better. Here untreated feed (composition defined = zero) is brought in a t opposite “corners” of the array, and top and bottom products are withdrawn from the other corners. For generality, we include the possibility of “recycling” the fraction (1 - T ) of each take-off stream into its corresponding feed ( T i s dimensionless, 0 < T < 1). I t is convenient in such a n array to measure the throughput, Z , in terms of the total thermal circulation for all N columns, not in terms of a single one. Define:
Nothing but this definition (Equation 15), conservation of flow and property, and elementary algebra are required to derive the equation for the output composition, $. of the array in Figure 5. \$‘e find : 2 .VKP $‘ = ( 2 - T)[,VR(P - A )
2 KP
These equations. toTether with Equations 6 and 7, suffice to solve for A$. the composition change in each stream, as a function of r? the rake-off rate. and column parameters: A* =
2KP - ( $ L - - $ L ) [ r ( P - -4) 21
+
Clearly, this equation is meaningful only when:
(--2 1-)+ A
r
and grouping for convenience as before, we find : r
?RP
7
(24)
Useful Cases. T h e purpose of this work mas to develop a rational basis for comparing the newly suggested transverseflow and feed systems of thermal diffusion with the previously
240
UPPER TAKE-OFF FLOW = t
, . I
I & E C PROCESS D E S I G N A N D D E V E L O P M E N T
(7) Jones, A. L., Petrol. Process. 6 , 132 (1951). (8) Jones, A , L., Brown, G. R., Adzan. Pefrol. Chem. Refining 3, 43 (1960). (9) Jones, .4. L., Hughes, E. C.?U. S. Patent 2,541,069 (Feb. 13, 1951). (10) Ibid., 2,541,070 (Feb. 13, 1951). (11) Ibid.: 2,541,071 (Feb. 13: 1951). (12) Ludwig, C., Wien. Akad. Bey. 20, 539 (1956). (13) Powers, J. E., dissertation, University of California, Berkeley, August 1954. RECEIVED for review h?ay 19. 1961 . ~ C C E P T E D J u n e 20, 1962
American Institute of Chemical Enginrers. Yew Orleans, La., February 1961.
