A GRAPHICAL SOLUTION OF
CYCLIC EXTRACTION JORGEN LOWLAND Department of Chemical Engineering, Norwegian Institute of Technology, Trondheim, Norway
Both cyclic-flow and continuous-flow operation of multistage countercurrent extraction may be described b y a series of linear material balances and (generally) nonlinear equilibrium relationships, making the same assumptions in both cases. For cyclic operation, some of the equations have to be solved b y trial and error calculations. A graphical method, eliminating trial and error calculations, is presented for the simplest case of the transfer of a single solute between two immiscible phases. The method is similar to the McCabeThiele method used for continuous-flow operation. If the total holdup of each phase i s transferred during each cycle, cyclic operation gives the same extraction in N stages as continuous-flow operation gives in 2N stages. ELTER
and Speaker (1967) have given a mathematical
B model of a multistage countercurrent liquid-liquid extraction process operated by controlled cycling and have solved the resulting equations analytically. Among their assumptions was a linear equilibrium relationship. A graphical solution of the same problem is presented here, but for the more general case of an arbitrary equilibrium relationship. Otherwise, the assumptions are the same in both cases. T h e graphical method is similar to the well-known McCabe-Thiele method used for continuous flow under similar assumptions. The system with S stages is shown in Figure 1. The controlled cycle operation is simulated by the following sequence (Belter and Speaker, 1967) : Transfer fraction p of the holdup, H, of heavy phase on each stage to the adjacent stage to the right. Equilibrate and let phases separate. The concentration of solute on stage n then becomes x,’ in the heavy phase and y, in the light phase. Transfer fraction q of the holdup, L, of light phase on each stage to the adjacent stage to the left. Equilibrate and let phases separate. T h e concentration of solute on stage n now becomes x, in the heavy phase and yn’ in the light phase. The process is described by a series of material balances and equilibrium relationships. The equilibrium relationships holding alternately on stage n are
T h e most convenient material balances depend upon the end of the process a t which the calculation is started. I n the first case, the conditions at the left end of the processL i g h t p h a s e flow
Heavy p h a s e flow
Le., the point ( X O , yJ-are considered known. A total solute balance over stages 1 u p to and including stage n then yields
PH
xo
+ qLYn+,
= PH x n
+ qLyl
(3)
or
(4) Equation 4 represents a straight line with slope PH - between the
qL points ( X O , y l ) and (x,, y n + l ) , and extended to stage N it becomes the same as the operating line for continuous-flow extraction. One more equation is needed, and it is obtained by taking a material balance around stage n after the heavy phase flow, equating the amount of solute before and after equilibration:
pH xn-1
+ (1 - P ) H + L xn
~ n = ’
H x,’
+L
(5)
or
Starting from the point ( x , - ~ , y,), use of Equations 1, 2, 4, and 6 allows one to advance the calculations one stage: Combined solution of Equations 6 and Equation 1 yields x,’. 2 then yields x , and yn’, and Equation 4 finally yields Y,+~. The simultaneous solution of Equations 6 and 2 will involve some kind of trial and error calculation. This is in contrast to the calculations for continuous-flow operation, where a material balance and an equilibrium relationship are solved sequentially. A graphical solution of the equations for stage n is shown in Figure 2, where it is assumed that the solute is extracted from the light phase to the heavy phase. The starting point ( x ~ - ~y,), lies on the operating line. A horizontal line from this point to the equilibrium curve gives xn’. Equation 6 is a linear relationship between y,’ and x , which intercepts the H y-axis a t y n - (xn’ p x n J . T h e lines below the abscissa
+L
-
with slopes -1 and -p are used to determine
Figure 1. Schematic diagram of cyclicflow operation of N extraction stages
yn
PX,-~.
The
H line with slope - from ( P X , - ~ , yn) then intersects the vertical L H line through x,’ at a point with ordinate y n - (xnl P X , - ~ ) . L
+
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JANUARY 1968
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urz Q
d
SOLUTE CONCENTRATION IN HEAVY P H A S E
Figure 2. Graphical determination of from xn-1 and Yn
xn and yn+l
Figure 3. Graphical determination of x.-~ and y. from xn and Y n t l
From the point on the y-axis with this ordinate, Equation 6 is then drawn as a straight line with slope -(1
H and L
- p) -,
this line intersects the equilibrium curve a t (xn, yn’). Finally, a vertical line through x,, to the operating line determines T h e graphical solution eliminates the trial and error calculation necessary in a numerical solution. If, instead, the calculations start from the point ( X N , Y N + I ) the operating line remains the same, with the constant term now changed toy.V+l
-P -- H x , ~ . ?L
The material balance around
xn’
+ qL
-
n
.c
-P c v;
5 e
+ (1 - 4 ) L
~ n + l
Yn =
Hxn
LY,’
(7)
C
v
2
H (1
- 4)L
xn’
-
4 1 - 4 . -
-~
y
+
1 Yn’
+ (1 -H4 ) L xn
Solute concentrotion in heavy phose
(8)
The calculations around stage n now start from ( x n , yn+l). Equation 2 yields y,,‘ and solution of Equations 1 and 8 yields 2% and x n ’ , whereupon Equation 7 yields x,-]. T h e graphical solution is shown in Figure 3, where it again is assumed that the solute is extracted from the light phase to the heavy phase. Equation 8 is a straight line in j n and xn’ which intercepts the L abscissa at x,’ = x n - (jn’ - q j n + ] ) . The intercept is de-
+H
termined first, and a straight line with slope
-
Operating line
aJ
In K
,“Y5 K
-
-P 6
-2e c
Y,
-
c
” c 0
H ___
rontinuous flow
-
(1 4) L is drawn from the intercept to the equilibrium curve which is intersected a t the point ( x n ’ , y n ) . If the solute is extracted from the heavy phase to the light phase, the operating line lies below the equilibrium curve. However, the same equations are still used, and the graphical method will also remain unchanged. Figure 4 shows the method applied to an extraction process with three stages. The calculations have been started at 66
y2
m 0
Yn+l
1 - q
Y,’
al
2
or Yn=--
Y3
L
stage n is now taken after the light phase flow, equating the amount of solute before and after equilibration as before :
H
pi
.c
I&EC PROCESS DESIGN A N D DEVELOPMENT
xo
xi
x2
x3
x,
x5
Solute concentrotion in heavy phose
Figure 4. Comparison of cyclic-flow and continuous-flow operation for the same total extraction Starting point is (xg, y1); determination of p x not shown
( x ~ yl), , and the graphical determination ofpx,-l is not shown in Figure 4. Continuous-flow operation to obtain the same total extraction is also shown in Figure 4, illustrating the advantage obtained by cyclic operation. I n the special casep = q = l--i.e., the total holdup of each
phase is transferred-Figure
H
- (x,'
L
1
- pa-,)now
[
2 shows that the point x,',y
falls on the operating line, while the
line representing Equation 6 becomes horizontal. in Figure 3, the point x,
[
,+
+
L
- (y,'
H
- q Y,+~),y,'
Similarly,
1
falls on the
operating line, while the line representing Equation 8 becomes vertical. I n both cases, the various concentrations are now
found by drawing steps between the operating line and the equilibrium curve, just as for the continuous-flow case. For cyclic operation, however, there are two steps between the two curves for each stage. Hence, cyclic-flow operation of N stages gives the same separation as continuous-flow operation of 2N stages. This result was shown analytically by Belter and Speaker (1967) for the case of linear equilibrium relationship, but is seen to hold for the more general case treated here. literature Cited
Belter, P. A., Speaker, S. M., IND.END.CHEM.PROCESS DESIGN DEVELOP. 6, 36 (1967). RECEIVED for review May 8, 1967 ACCEPTEDAugust 31, 1967 Supported by the Norsk Hydro Co.
COKE FORMATION KINETICS ON S I L I C A - A L U M I N A C A T A L Y S T Basic Experimental Data Y U l C H l O Z A W A ' A N D K E N N E T H B. B l S C H O F F l The University of Texas, Austin, T e x . The kinetics of coke formation reactions during the cracking of ethylene over a silica-alumina catalyst was studied at atmospheric pressure and from 350' to 500". The coke had no significant effect on the surface area of the catalyst and the effective diffusivity through the catalyst pores in the range of less than 1 weight % coke. Evaluation of the effectiveness factor for the pertinent catalyst and reaction indicated that pore diffusion offered negligible resistance. A thermogravimetric system was used for continuous measurement of the weight of the coke formed. Plots of the weight per cent of coke on catalyst vs. process time on a loglog scale showed two straight lines, indicating an initial rapid coke formation. HE activity of catalysts declines rapidly because of the Taccumulation of carbonaceous deposits in most high temperature organic reactions using heterogeneous catalysts. This phenomenon is typically observed in the cracking of hydrocarbons over a silica-alumha catalyst. I t is important to characterize the nature of this coke formation process because of its commercial interest and the effect on the catalyst activity and selectivity. Several empirical equations have been presented to relate the weight of coke formed on the catalyst with the process time. Voorhies (1945) showed, for example, that the' carbon weight per cent on catalyst, C, approximately followed a relation C, = A . P . where t is the process time and A and n are constants. The values of exponent n were evaluated for different reactions by Rudershausen and Watson (1955), Tyuryaev (1939), and Wilson and Den Herder (1958), who found values ranging from 0.4 to 1.0. Voorhies reasoned that the rate of coke formation was controlled by a diffusion process; the diffusion rate could be taken to be inversely proportional to the weight per cent of carbon, since the coke itself became the diffusion barrier in the catalyst pellet. Similar semiempirical equations were presented by Blue and Engle (1951) and Panchenkov and Lolesnikov (1959). A recent extensive study by Eberly et al. (1966) used regression techniques to relate the weight per cent coke on catalyst as a function of process time and space velocity in the cracking of n-hexadecane.
Present address, Mobil Oil Co., Paulsboro, N. J. Present address, Department of Chemical Engineering, University of Maryland, College Park, Md. 20742
The present study was concerned with the over-all kinetics of coke formation reactions on spherical silica-alumina cracking catalyst. The surface areas of the fresh and coked catalysts were measured in order to determine the effect of coke deposition on the area. The effect of the coke on diffusivity was also determined to see whether the diffusivity remained constant during coke formation. With these values, the effective diffusivity of the reaction gas through the catalyst pellet was estimated to learn the importance of the pore diffusion process. The weight of coke was measured continuously by a thermogravimetric system. Most of the previous workers determined the weight of coke by burning the coke in a furnace and measuring the COn content of the combustion gas absorbed in a caustic solution (Appleby et al., 1962; Blue and Engle, 1951; Eberly et al., 1966; Greensfelder et a/., 1945; Smitkons, 1949 ; Voorhies, 1945). When the thermogravimetric system is used, the weight of coke can be instantaneously determined without interrupting the experiment. T h e effect of process time, feed rate, and temperature on the coke formation was thus studied using ethylene as a reaction gas in order to have a fairly simple reaction. Experimental
The Mobil Oil Co. Durabead 1 catalyst used contained 90 weight % of silica and 9.7 weight 7 0 of alumina, and had a surface area of approximately 200 sq. meters per gram. The catalyst was spherical and divided into four groups: VOL. 7
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