A Simplified Analytical Design Method for Differential Extractors with Backmixing. II. Curved Equilibrium Line H. R. C. Pratt University of Melbourne, Victoria,Australia
Two alternative methods are presented for the extension of the closed solutions of the backmixing equation given in Part I to the case of a nonlinear equilibrium relationship. Both involve subdivision of the column into two or three sections, in each of which the equilibrium relation is approximated by a straight line. In the first method the length of each section is calculated using suitable approximate boundary conditions between the sections; in the second, overall lengths are first obtained for the full concentration range, and the fractions of these corresponding to each equilibrium curve segment are then calculated. Both methods are illustrated by a worked example which indicates that, when applicable, the second method is somewhat more convenient to use. Cases can arise, however, where it would be necessary to use the first method, either entirely or over a particular section of the column.
Introduction In Part I (Pratt, 1975) approximate closed solutions were presented to Miyauchi's (1957) backmixing model of differential extractors which enable the length to be calculated with considerable accuracy for the case of a linear equilibrium relation. These are not directly applicable, however, when the equilibrium curve is nonlinear. For such cases Mecklenburgh and Hartland (1967) have described a form of their own approximate method, based on a subdivision of the column into three sections in each of which one of the roots of the characteristic equation is dominant, but since this method is tedious, requiring both trial and error calculations together with graphical integration, a modification of the method of Part I was sought. Two alternative methods have, in fact, been devised and are presented below. Both involve subdivision of the column into two or three sections, in each of which the equilibrium curve segment can be approximated by a straight line; this subdivision is more arbitrary than in Mecklenburgh and Hartland's method. The two methods are described in turn below.
dition, the artifice is adopted of specifying that dncl/dzn = 0 a t the junctions of the sections, where n > 2. It will be shown later that a value of n = 3 gives the most satisfactory results, and this value is used below in the meantime. Equations are derived for the top, middle, and bottom sections of the column, although in fact the use of a middle section is optional, depending upon the degree of curvature of the equilibrium line and the concentration range involved. General (4-Parameter) Case (a) Top Section. The column is assumed to be divided at a Y-phase concentration cyl,t corresponding to an effectively straight section of equilibrium line AB (Figure 1) of slope rnt with an intercept q t on the c, axis. The coordinates are transformed to the Xt-Yt system using eq 7 and 8 of Part I, when, as usual, X t o = 1.0 and Yl,t = 0; also Ytois
is to be degiven by the specified exit value of cy, and termined. The solutions to the differential equation are given by eq 13 and 14 of Part I, in which the X i are the roots of the characteristic equation, Le., eq 15 of Part I. The boundary conditions for the top section are
First Method
In principle, this method involves subdividing the column into lengths, in each of which the equilibrium relation can be approximated by straight line segments such as AB, BC, and CD in Figure 1. The concentrations in each section are transformed into dimensionless units and the differential equation (Le., eq 9 of Part I) solved with suitable boundary conditions, to give the length of each section. In subdividing the column at a given value of, say cq, it is not possible, unlike the piston flow case, to calculate c, a t the same point by a material balance around one end since this contains two additional terms involving the unknown gradients d X l d Z and dYldZ (cf. eq A1 of Part I). The problem therefore arises, if the column is subdivided, of defining suitable boundary conditions at the junctions. A rigorous solution would require both compositions and gradients to be equated a t the junctions of each pair of sections; however, since the method involves the transformation of the individual compositions in each section into dimensionless units, the complexity of the resulting system of equations would defy simple analytical solution (cf. Wilburn and Nicholson, 1965). The correct boundary conditions are, in fact, continuity of the gradients, but since there is no suitable mathematical way of defining this con34
Ind. Eng. Chem., Process Des. Dev., Voi. 15,No. 1, 1976
2, = 1.0: Y, = 2, = 1.0:
Y1,t
= 0
(3 )
[3] Zt=l.O
= 0
Equations 1 and 2 are identical with (17) and (18) of Part I, but (3) and (4) relate specifically t o the present case. Following the procedure of Part I, the coefficients A, are obtained in determinantal form from the boundary condition equations and used to write out the full solutions for Xt and Y t , which are then simplified by disregarding terms in and ex3. The resulting expressions for X t and Yt are exactly analogous to eq 42 and 43 of Part I for F # 1.0, and to eq 52 and 53 for F = 1.0. The length of the section is obtained by substituting Zt = 0, Y t = Ytoin the expression for Y t and disregarding the term containing e-xz(l-zt). Substituting A4 = X4'Lt and solving for Lt then gives
7
I
z c; 0 e m 0
FLOW
-
m 0
FEED
z
YO.m (1
-
Y,,m)Hox
/Ip I
A OPERATING L I N E S ( T Y P I C A L ) @=PISTON
L,
( F = 1.0) (14)
(F
f
1.0)
(15)
( c ) Bottom Section. In this case all four terminal concentrations are known, and are transformed into Xb-Yh coordinates. The boundary conditions are X-JUMP-
2 , = 0, XO,, = 1.0
(17)
PHASE COMPOSITION, C X
Figure 1. Subdivision of column into three sections, approximating equilibrium curve by straight line segments, AB, BC, and CD.
(20)
5)
( 6)
The disappearance of exponent n when F = 1.0 indicates that the profiles in this case are linear and that the corresponding boundary condition is exact. Similarly, XI,^, the value of Xt a t the bottom of the section, is obtained by substituting Zt = 1.0, X = in the solution for Xt and disregarding the term containing exSz, giving
Having obtained and simplified the profile equations, the procedure is somewhat different since the exit concentration Xbl is specified, or obtained from the overall material balance. Consequently this value is substituted into the X-profile equation together with zb = 1.0, giving on solving for Lb
(F
+
1.0)
(21)
( F = 1.0) (22)
This completes the solution, but as a check on accuracy it is desirable to calculate Yo,b, and hence CyO,b, to compare it , ~is .obwith the previously assumed value, i.e., ~ ~ 1This tained from the 2 profile by inserting zb = 0 and disregarding the terms in e-x2(1-z), giving The value of X1,t is obtained from eq 7 or 8 using the value of Lt given by eq 5 or 6; transforming the coordinates by means of eq 7 of Part I then gives c x l , t ( = CxO,m). (b) Middle Section. As before, the column is divided a t a suitable value of the inlet Y-phase composition designated ~ ~ 1Of ,the ~ remaining . terminal concentrations, c ~is ~ the same as the assumed inlet value cyl,t to the top section, and c , , , ~ ( = cxl,J has been calculated. These are transformed into X,-Y, coordinates and the following boundary conditions used to evaluate the coefficients Ai,,,
z, = z,= 0 ,
0, X o , m= 1.0
[;%I
(9)
=o
(10)
z,=o
( F = 1.0) r ,
f
1
1
(23)
\l
~
L
( F = 1.0) (24)
(d) Terminal Gradients and “Jumps”. It is sometimes desirable to compute the terminal “jump” concentrations cXo and cylr particularly the latter, to enable the relevant part of the equilibrium curve to be ascertained. These are obtained via the gradients a t the terminals, obtained by differentiation of the profiles and substitution of 2 = 0 or 1.0, disregarding appropriate terms, giving
(12)
Following the same procedure as before the results are
(F
#
1.0)
(13)
(F
+ 1.0) ( 2 5 )
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
35
+ I G
I/
il
T
4a: +
C D 3
E
4 II
II
II
E.
11 E
4
36 Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
$
a
Table 11. Worked Examples: First Method (Note: All Examples below Are Based on a Linear Overall Equilibrium Relationship
1
0.025
0.100
0.50
2
0.025
0.100
0.50
3
0.025
0.100
1.00
4
0.025
0.100
2.00
5
0.025
50
0.50
6
0.025
m
1.00
7
m
0.025
0.50
8
50
0.025
1.00
Calcd Actual Calcd Actual Calcd Actual Calcd Actual Calcd Actual Calcd Actual Calcd Actual Calcd Actual
2.00 0.10 2.00 0.10 2.00
0.10 2.00
0.10
0
2.00 0.10 2.00 0.10 2.00 0.10 2.00 0.10
4.05 4.00 4.59 4.80 4.84 4.80 4.61 4.80 4.75 4.80 4.80 4.80 4.91 4.80 4.84 4.80
0.373 0.381 0.419 0.412 0.649 0.650 0.874 0.870 0.386 0.381 0.643 0.643 0.406 0.4 24 0.696 0.700
-
-
-
-
6.14 6.40 6.40 6.40 6.42 6.40 6.10 6.40 6.47 6.40 6.85 6.40 6.36 6.40
0.190 0.171 0.396 0.401 0.690 0.687 0.161 0.156 0.381 0.385 0.179 0.185 0.438 0.438
3.99 4.00 5.21 4.80 4.86 4.80 4.76 4.80 5.04 4.80 4.73 4.80 4.50 4.80 4.80 4.80
0.217 0.214 0.116 0.112 0.319 0.316 0.644 0.640 0.089 0.083 0.300 0.300 0.126 0.133 0.357 0.357
8.04 8.00 15.94 16.00 16.10 16.00 15.79 16.00 15.89 16.00 16.00 16.00 16.26 16.00 16.00 16.00
Overall column (not sectional) basis.
resulting equations are summarized in Table I, where the A, are given by eq 37 or 41 of Part I.
(F
#
1.0)
(27)
( F = 1.0) (28)
Substitution of the X-gradient from eq 25 or 26 into eq 1 then gives (29)
where Bt = Lt/d,. Finally transformation of the coordinates gives the inlet "jump" concentration cXo at the top of the column. The value of the inlet "jump" concentration cyl is obtained similarly via eq 19, viz.
Three Parameter Cases When backmixing occurs in only one phase, i.e., when P, or P, = m , the profile equations are less complex (Miyauchi, 1957) since only three boundary conditions are required. The present method is applicable to these cases, noting that boundary condition eq 2, 10, and 18 are redundant when P, = m , and eq 4, 12, and 20 when P, = m. The
Testing of Method To determine the best value of n to use in the boundary conditions of the type d"X (or Y)/dZ" = 0, the method was applied to a linear equilibrium relation case selected from the tabulation of McMullen et al. (1958), the parameter values for which are given in Table 11, example no. 1. Assuming the column to be divided into two equal sections, the calculated values of Lt were 4.23, 4.05, 4.08, and 4.07 ft for n = 2, 3, 4, and 5, respectively, with Xl,t = 0.344, 0.373, 0.368, and 0.369 a t the junction, the exact values being Lt = 4.00 ft, = 0.381. For the bottom half, using these X values, the calculated lengths were Lb = 3.99, 3.91, and 3.92 ft, respectively, for n = 3, 4, and 5 (exact value = 4.0 ft). It is therefore apparent that n = 3 is the most appropriate value to employ and this was used subsequently. The method was tested on the further seven examples listed in Table 111, using the McMullen tabulations for the four-parameter cases. For the three-parameter examples the exact profiles were calculated using the appropriate equations of Miyauchi (1957). In all cases (except no. 1)the column was assumed to be subdivided into three sections a t 2 = 0.3 and 0.7. It should be noted that the assumption of a straight equilibrium line relation simplified the computations since the values of A,' and a, were identical in all sections; this would not, of course, apply in the nonlinear case due to the variation in equilibrium line slope. The results in Table I11 indicate satisfactory agreement of the overall calculated length, all being well within 2% of the correct value. I t is interesting to note, however, that the error in the lengths of the individual sections-as well as the X values-was generally greater, but there was a close overall compensation of errors. Second Method In this method the straight line approximations to the equilibrium curve are extrapolated to cover the full specified concentration range; thus, in Figure 2, AB is extended to C and BD to G. For each, the overall length is calculated using the equations of Part I, giving obviously high results for the case shown, but low for an equilibrium line of oppoInd. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
37
Table 111. Design Equations (Second Method): Three-Parameter Systems
F
#
Term
1.0 L
Y
P,finite, P, =
P, =
m
P, finite
Eq 59 of Part I
Eq 5 8 of Part I
Multiply 2nd, 3rd, and 4th terms of numerators of X profiles by u 2 , a 3 , and u4 respectively. Equation 60 of Part 1
= 1.0 L
site curvature. The fraction 2 of the overall length is then calculated, and used to obtain the length corresponding to the equilibrium curve segment. To illustrate the procedure for the four-parameter case, eq 48 or 57 of Part I is used to calculate an overall length Lto for equilibrium line AC (Figure 2), using the appropriate transformed coordinates. One of the following rearranged forms of the simplified Y-profiles, eq 43 or 53 of Part I, is then used to calculate the fractional length Zt corresponding to concentration Yk at point B 1
2, = 7 In x
&’L,
(F
#
1.0)
(31)
( F = 1 . 0 ) (32)
These expressions are obtained by omitting the terms in exp [-X2’Lt0(l - Z ) ] and exp (X3’LtoZ) in the profile equa0.35 to 0.65. If tions, and are valid within the range Zt the calculated value of Zt lies tob far outside these limits, either the location of B on equilibrium line segment AC, or the line AC itself must be adjusted. Alternatively, the calculated values of Lto and Zt can be inserted in eq 43 or 53 of Part I, and a more exact value of Yk calculated, neglecting the terms in exp (X3’Lt0Z) when Zt > 0.6, and exp [-X2’Lto(l - Z ) ] when Zt < 0.4. Again some readjustment of the equilibrium line segments may be required. The length of the top section is then Lt = LtoZt,and the 38
m,
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
same procedure is applied to the bottom section, for which = Lbo(l - Z b ) . For a middle section, if used, values of Zml and Z2 , must be calculated at top and bottom, giving L m = LmO(Zm2 - z m 1 ) The “jump” concentrations can be calculated from the simplified profile equations at 2 = 0 and 1. Thus for the X-phase, eq 42 or 52 of Part I is used at Z = 0 neglecting the term containing exp [-X2’Lt(l - Z ) ] ; and for the Yphase, eq 43 or 53 of Part I is used at Z = 1, neglecting the term containing exp (X3’LG). The corresponding equations for Z for the three-parameter cases, and also the simplified profile equations, which were not given in Part I, are summarized in Table 111. Lb
Limitation of Method This method is subject to the limitation that, if either top or bottom equilibrium line segments on projection intersects the piston flow operating line, the column specification cannot be met so that it is inapplicable. In fact, it is desirable that they should not pass too close to this line, to avoid undue distortion of the profiles. In such circumstances it is necessary to revert to the first method, either overall, or for the particular section in question; in the latter case, the value of xk for the adjoining section, corresponding to the selected Yk, must be calculated from the profile equation, i.e. (42) or (52) of Part I. Discussion The worked example in the Appendix shows that, with careful selection of the equilibrium curve segments, both methods give results in close agreement with the exact value. The second method is somewhat easier to use since it avoids the need to calculate the Q 3 . I t might also be expected to be more accurate when applicable, at least for short columns, since the length is calculated initially over the whole conceetration range instead of a portion, thus avoiding the errors shown in Part I to occur with low values of X2’L and X 3 ’ L . However, such errors were not detectable
-
COMPOSITE FOR CURVE A E E
EQUILIBRIUM CURV
STON FLOW RATING LINE
01
02
0 3
05
04
FEED PHASE
07
06
08
09
10
COMPOSITION C x
Figure 2. Operating diagram for worked example.
in the example when solving by the first method even though the length of the bottom section was little more than 4.0 ft. The accuracy achieved with the column divided into only two sections can be attributed to the effect of the Y-phase “jump”, so that the portion OE (Figure 2) of the equilibrium line of greatest curvature is not involved. This indicates the desirability in accurate work of computing the “jump” concentration cJl, so that the slope of the segment BE’ can be adjusted if necessary. On the same basis it might be considered that the low concentration end of the equilibrium curve would need to be taken into account in the absence of backmixing in the Y-phase. However, further consideration in terms of the individual phase coefficients k , and k , shows that this is not necessarily the case. Thus there is evidence (cf. e.g. Gayler and Pratt, 1957) that these coefficients are of similar magnitude. Consequently a tie line of slope -k,lk, through the solvent inlet end of the operating line for no solvent phase backmixing, i.e., L in Figure 1, would intersect the equilibrium curve at a point such as T, with interfacial concentrations cxi,l and c y L , l .These are the lowest possible values of these concentrations in the column so that the driving force on an overall X-phase basis for the bottom section must be based on the equilibrium curve B T and not on BTO. With backmixing present in the solvent phase, the minimum interfacial concentrations cxr cyr would be higher still, e.g., a t S, the intersection of the line MS through the “jump” concentration M with the equilibrium curve. In a similar manner the tie line through the X-phase inlet end of the operating line (whether or not P, = a) would intersect the equilibrium curve at a concentration above the exit value, e.g., at R. Consequently the slope of the equilibrium curve above A must be taken into account in locating the straight line approximation to segment AB. More complex cases have been treated by Wilburn (1964), who allowed for mass transfer in the disengaging sections beyond the X and Y phase inlets, and by Wilburn and Nicholson (1965), who allowed also for curvature of the equilibrium relationship and for variations in holdup of the phases. In the latter case the column was divided into n sections and the profiles were obtained by numerical inversion of a 4n X 4n matrix. Such a method has disadvantages for general design purposes, however, and for the most part the present methods appear capable of giving a realistic aD-
proximation once reliable mass transfer coefficient data become available. Appendix Worked Example. Calculate the contactor length required to meet the following specification: c X o = 1.0 g-mol/ 1.; cxl = 0.512 g-mol/l.; cy1 = 0.0 g-mol/l.; U,/U, = 1.0; P, = P3 = 0.040; Ho, = 2.00 ft; d, = 0.10 ft. The equilibrium relationship is given by: cy = c X 2 .
First Method Equilibria. The equilibrium curve is shown in Figure 2, together with the piston flow operating line. The exit solvent phase composition, by overall material balance, is c y o = l.OO(1.000 - 0.512) = 0.488. Since P, is small the Y-jump will be considerable, and it will not be necessary to consider the lower portion of the equilibrium curve. Consequently, as a first trial the column will be divided a t Cy,k = 0.300 into two segments AB and BD. Top Section. Projecting AB to C, the intercept qt = 0.312. Also m, = F , = (0.6986 - 0.312)/0.488 = 0.7922. Transforming the coordinates through B ii/tC,,q
I’t 0
‘it =
-
0.7922
X
0.300
- 0.312
=
O,jjO
0.7922(0.488 - 0.300) = o.331 (1 - 0.550)
Solving the characteristic equation, Le., eq 28 of Part I, with P x / d , = Py/d, = 0.400, l / H o x = 0.500 and F t = 0.7922 gives A*‘ = A,’ = XI‘ =
0.7355546 4.7034252 4.0321294
=
0.3979649 -
=
“2 “3
4.00003316 ad
= 4.2340939 = -2.8808858 =
0.9305797
From eq 5, Lt = 5.781 f t . :. exp(X4’LJ = 0.830488. Hence from eq 7, using this value, X1,t = Xk = 0.1658. :. Cx,k = 0.1658(1 - 0.550) + 0.550 = 0.6246. Bottom Section. The line BD has intercept q b = 0.187 with slope mb = F b = 1.210. 1.210(0.300 - 0.0) (0.6246 - 0.187) = o’3295 (0.512 - 0.187) - o.7427 s,’- (0.6246 - 0.187) -
. . 17d,b =
Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
39
The roots, etc. are A ? ‘ = 0.7615375 A s ’ = -0.7894807 AJ’ =
“2
h f 3= -0.4920674
0.0279432
Xit3 =
0,0000218
From eq 21, Lb = 4.041 f t . From eq 23 YO,b = Yk = 0.8293.
. . Cyb
=
0.8293(0.6246 1.210
= -0.3766218
03
= -3.6953605
04
=
1.0519823
:.
exp(A4’Lb) = 1.119530.
-
0.187) = 0.300
This value of c Y k = 0.300 agrees with the assumed value at the junction of the sections and serves as a check on the result. ”Jump” Concentrations. From eq 27, dYb/dZb = -0.5922. .’.
f r o m eq. 30, Y I , =~ 0.5922/(0.40 Hence cy, =
x 4.041) = 0.3663
0.3633(0.6246 - 0.187) = o.1325 1.210
Similarly, using eq 25 and 29, c , ~ = 0.862. Total Length. L = 5.781 4.041 = 9.82 f t . Since the mass transfer rates are based on ko,, the driving force a t the solvent phase inlet is represented by EF, where F is the “jump” concentration cYl. Consequently the portion of the equilibrium curve below E is not required and a better approximation to the bottom section is given by straight line segment BE’D’. Recalculating the bottom section on this basis gives mb = 1.100, q b = 0.220, and Lb = 4.180, with the check value of Csl = 0.300 as before, and the “jump” concentration c y l = 0.127. The latter is sufficiently close not to warrant a further approximation. Hence: L = 5.781 4.180 = 9.96 f t . This agrees well with the exact value of 10.00 f t given by Mecklenburgh and Hartland (1967) for this example.
+
+
Second Method Approximating the equilibrium curve by lines AB and BD (Figure 2) gives the same values of mt, mb and q t , q b as before. Top Section. Transforming coordinates over the full concentration range AC gives
YO =
0.7922(0.488 - 0) = o.5619 (1 - 0.312)
Y i=
0.7922(0.300 - 0) = o.3454 ( 1 - 0.312)
The Xi’ and ai are the same as for the first method. From eq 48 of Part I, and (31) L t o = 10.540 ft
.’. eA4‘Lto=
2, = 0.5480
. * . LtoZt = 5.776 ft
0.712728
Bottom Section. Transforming the coordinates for the full concentration range GBD gives 1.210(0.488 - 0) = o,7263 YO = (1 - 0.187)
Y, =
1.210(0.300 - 0) ( 1 - 0.187) = 0‘4465
Again the Ai’ and ai values are the same as for the first method, hence as before Lbo = 11.131 f t
Zb = 0.6370 40
eAd’Lbo
= 1.364835
* * . L b o ( l- 2,) = 4.041f2
Ind. Eng. Chem., Process Des. Dev., Vol. 15,No. 1, 1976
“Jump” Concentrations. The “jump” concentrations are obtained by putting 2 = 0 and Z = 1.0 in the profile equations, i.e., eq 42 and 43 of Part I, for equilibrium lines AC and GD respectively, giving cXo = 0.8625 and c y l = 0.1325. Total Length. The total length is L = 5.776 4.041 = 9.82 f t . This is identical with the results of the first method, as are the “jump” concentrations. I t can be refined as before by recalculating the bottom section taking G’BE’D’ as the equilibrium line, again giving the same result. The operating lines in Figure 2 were calculated for equilibrium lines AC and G’D’ using eq 42 and 43 of Part I.
+
Composite Method Although the second method is completely satisfactory in the present case, it is of interest to consider its completion using the first method for the bottom section. Thus, having calculated the length of the upper section by the second method, it is found using the X-profile expression, i.e., eq 42 of Part I, that X1,t ( = Xk) = 0.4553. :. ck = 0.4553 X 0.688 0.312 = 0.6253. This agrees very closely with the value of 0.6246 calculated at the bottom of the top section using the first method. Completion by the latter method, as described earlier, would therefore give a virtually identical result for the bottom section, and hence for the total length.
+
Nomenclature
A, = coefficient in eq 13 and 14 of Part I, dimensionless less a, = defined by eq 16a and 16b of Part I, dimensionless B = L/d,, dimensionless cJ = concentration of solute in j phase, or ( m ~ l ) L - ~ cl, = interfacial value of c, d, = characteristic length, e.g., size of packing element, L E, = effective longitudinal diffusion coefficient in the j t h phase, L2T-l F = extraction (stripping) factor, mlJ,/lJ,, dimensionless Ho, = height of an overall (“true”) transfer unit based on X phase, L L = total length of contactor, L m = slope of equilibrium line, dc,/dc,, dimensionless No, = number of “true” overall transfer units based on X phase, dimensionless PI = Peclet number for the j t h phase, lJJd,/El, d’imensionless q = intercept of straight equilibrium line on c, axis, ML-3 or ( m ~ l ) L - ~ U, = superficial velocity of j t h phase, LT-I X = [c, - (mcyl. ?)]/[c,O - (mc,I q ) ] , generalized solute concentration in X (feed) phase, dimensionless Y = [m(cy - cY1)]/[c,O - (mcyl q ) ] , generalized solute concentration in Y (extractant) phase, dimensionless 2 = z/L, fractional length within column, dimensionless z = length within column measured from X phase inlet, L
+
+
+
Greek Letters A,, A’, = roots of characteristic eq 15 and 28 of Part I, respectively Superscripts 0 = feed inlet end, outside column (2= 0 ) 1 = extractant inlet end, outside column (2= 1.0) Subscripts b = bottom section of column i = number of root of characteristic equation, and of corresponding coefficient in solutions for X and Y, Le., eq. 13 and 14 of Part I j = X or Y phase k = intermediate concentration in column (second method) m = middle section of column
Mecklenburgh, J. E., Hartland, S., I. Chem. E. Symp. Ser., No. 26, 130 (1 967). Miyauchi, T., U S . Atomic Energy Commission Rept. UCRL-3911 (1957); see also Miyauchi. T., Vermeulen. T..Ind. Eng. Chem., Fundam., 2 , 113 (1963). Pratt, H.R. C., Ind. Eng. Chem., Proc. Des. Dev., 14. 74 (1975). Wilburn, N. P., Ind. Eng. Chem., Fundam., 3, 189 (1964). Wilburn, N. P., Nicholson, W. L., A.I.Ch.€.-l. Chem. E. Symp. Ser., No. 1, 105 (1965).
t = top section of column x = X phase (feed) y = Y phase (extractant) Literature Cited Gayler, R., Pratt, H. R . C., Trans. hst. Chem. Eng.. 35, 267 (1957). McMullen. A. R., Miyauchi, T., Vermeulen, T., U S . Atomic Energy Commission Rept. UCRL-3911-Suppl. (1958).
Received f o r review August 27,1974 Accepted June 20,1975
Derivative Decoupling Control Roger E. Palmenberg and Thomas J. Ward' Chemical Engineering Department, Clarkson CoNege Df Technology, Potsdam, New York 13676
Derivative decoupling control, a nonlinear noninteracting control approach, is discussed in terms of recent work in order to highlight some limitations and to provide a comparison with the linear structural analysis control approach.
Some time ago Liu (1967) presented an approach for noninteracting process control based on the decoupling of the state derivatives, rather than the state elements themselves. Since this has received new attention (Hutchinson and McAvoy, 1973; Rich et al., 1974), a detailed examination of this control method in terms of the recent work is warranted. Background Liu assumed that the uncontrolled process could be characterized by a nonlinear state-vector differential equation model of the form Y = F(Y,
x,t )
where Y = n X 1 column vector of state variables; X = n X 1 column vector of inputs; F = column vector of nonlinear functions f j ( Y, X,t ) ; t = time (the overdot represents differentiation with respect to t ) . The control object was the control of Y by the manipulation of X. With control, the manipulative input X became a function of the measurable state Y and the setpoint input vector R . As a result, the controlled process dynamics could be written as a function of the error E , where E = R-Y Y = G ( E ,t )
If G ( E , t ) were specified by design as a column. vector whose ith component was a function of only the corresponding error element ei and t , then the state derivatives would be decoupled. This also meant that the state elements themselves were decoupled. Such derivative decoupiing, which was analogous to the state decoupling used in early noninteraction approaches (Kavanagh, 1957), gave the noninteraction condition as a set of nonlinear algebraic equations
F(Y, X , t ) = G ( E , t )
(3)
where G ( E ,t ) was to be specified as above. Possible Difficulties There are three general difficulties that can be encoun-
tered in obtaining the xi controller equations from this system of equations. The first difficulty is the specification of G ( E , t ) . There are an infinite number of choices for each of the gi elements and there is no straightforward procedure for finding an effective, realizable choice. Liu arbitrarily assumed that a suitable form for all of the state derivatives was a type of proportional response 3.1 = a 1. e1. (4) where the a; are unspecified proportional coefficients. Since large values of the a; would provide faster response dynamics, Liu then proposed an algorithm that, starting from initial arbitrarily large values, would iteratively reduce the magnitudes until the input constraint limits were satisfied. This algorithm can be utilized as an off-line calculation that requires the complete solution of the nonlinear process equations or as an on-line approach involving integration through each time step. This calculation may not give any acceptable values of the coefficients in some cases without modification of the algorithm. The second difficulty is that it may not be possible to solve the equations for each of the x i as explicit functions of the state variables and setpoints. If eq 1 is linear in the xi, it is possible, as noted by Liu, to obtain an analytical solution. If an on-line numerical approach is required, there is no assurance of convergence to the correct xi values in a reasonable time. In order to overcome this general difficulty, Liu proposed a simplified procedure in which each equation of the set is solved separately for a single x i input as a function of the state variables, setpoints, and the other inputs. While this is an effective procedure for elementary systems, there is no general guide as to which input should be developed from a particular equation. Even more important, the procedure may fail in some cases. The third difficulty occurs in the common process case in which only a small subset of the inputs are manipulative and only a small subset of the state elements are actually controlled. It would often be impossible t o obtain the x i controller equations for this case without modification of the method or a careful reformulation of the process equations. Ind. Eng. Chem., Process Des. Dev., Vol. 15, No. 1, 1976
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