Ind. Eng. Chem. Process Des. Dev., Val. 17, No. 4, 1978
If the gas phase is laminar, then 6 is proportional to u / a 2 where
(3)
485
than in terms of liquid holdup. This explains why the effect on holdup was not observed with respect to 1.8-mm packings even though it was significant in terms of pressure drop. Conclusions
By assuming ideal gas behavior, it follows that for given G (C - hJ2P(Ap/Z)'i3 7 = (4) N This is consistent with experimental observations and may be explained as follows. For a pre-wetted bed in which the gas flowrate is being increased from zero, the initial orientations of the liquid bridges are quite random. As gas flowrate is increased, bridges in directions transverse to the general flow of the gas tend to be broken down leaving those intact which are in the flow direction. The density of flowpaths N is increased and the length of the gas flowpaths, that is the tortuosity, is reduced. In view of the high surface tension effect in beds of small packings, this flow pattern is stable and the lower tortuosity and higher N are retained even when the gas flowrate is reduced. For a given G, A p l Z and P are observed t~ be lower which is consistent with (4). However, when the liquid flowrate is high and turbulence generates random changes in liquid and gas flowpaths any ordered pattern of liquid bridges is unstable, and the effect is no longer observed. Similarly, in beds of large packings, the surface tension effect is low and no large variation in tortuosity is possible. If surface tension were to play a major role in stabilizing various hydrodynamic states, then the addition of wetting agents should have the same effect as increasing packing sizes. That this was in fact not observed does not necessarily mean that the explanation provided above is invalid. It is possible that the decrease in surface tension induced by the surface active agent was not sufficient to cause a change in stability which is observable experimentally. It is also possible that the addition of wetting agent altered the flow regime so that film flow was more predominant rather than channel flow. This would certainly explain the observed increase in pressure drop. From eq 1to 4, it may be shown that the overall pressure drop Ap/Z is inversely proportional to ( t - hJ2.Thus, the effect is more easily observable in terms of pressure drop
For two-phase cocurrent flow in beds of small packings, a multiplicity of steady states is possible. Pressure drop is a function not only of the flowrate, properties of the phases and packing size, but also of the maximum gas flowrate experienced by the bed. Nomenclature
a = characteristic diameter of gas flow path, m G = gas mass velocity, kg m-2 s-l G,, = maximum gas mass velocity experienced by column,
kg m-2 s-l h, = total liquid holdup for G < G,, h,, = total liquid holdup for G = G,, (Le., path 0) L = liquid mass velocity, kg m-2 s-l N = number of flow paths per unit cross sectional area of packed bed P = absolute pressure, Pa ( A p / Z ) = pressure gradient in column, Pa m-l ReG = gas-phase Reynolds number based on packing diameter ReL = liquid-phase Reynolds number based on packing diameter u = real average gas velocity, m s-l Z = reactor length dimension, m Greek Letters 6 = pressure drop along flow path, Pa m-l t
= column voidage
p = 7
gas density, kg m-3
= tortuosity factor
Subscript 0 = refers to path 0 Literature Cited Charpentier, J. C., Chem. Eng. J . , 11, 161 (1976). Charpentier, J. C., Favier, M., AIChE J., 21, 1213 (1975). Hoffman, H., Int. Chem. Eng., 17, 19 (1977). Larkins, R. P., White, R . R., Jeffrey, D. W., AIChE J . , 7, 231 (1961). Satterfied, C. N., AIChE J . , 21, 209 (1975). Specchia, V., Baldi, G., Chem. Eng. Sci., 32, 515 (1977). Turpin, J. L., Huntington, R. L., AIChEJ., 13, 1196 (1967). Weekman, V. W., Myers, J. E., AIChE J., 10, 951 (1964).
Received for review October 17, 1977 Accepted June 8, 1978
Degeneracy of Decoupling in Distillation Columns Azmi Jafarey and Thomas J. McAvoy' Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts 0 1003
An interacting control system may be internally decoupled through the use of noninteracting manipulative variables. Application of this concept to dual composition control of a binary column (9= 1) leads to the unexpected result that the conditions for half-decoupling of either loop are effectively the same. This degeneracy very closely approximates the column material balance and makes lt impossible to completely decouple the system. The findings appear to be of general validity and afford insight into the success of the Shinskey material balance control scheme.
In recent years, an increasing significance has been attached to the steady-state control of industrial processes (Ellingsen, 1976; Lee and Weekman, 1976). Several tools are now available which aid in determining steady-state 0019-7882/78/ 1117-0485$01 .OO/O
control strategies for multivariable systems. These concepts have found a natural application in the design and implementation of control algorithms for distillation. The most successful of the syntheses has been the material @ 1978 American Chemical Society
486
Ind. Eng. Chem. Process Des. Dev., Vol. 17,No. 4, 1978
balance approach described by Shinskey (1967, 1977), wherein concepts such as the Bristol (1966) relative gain array have been used along with empirical correlations for the column, to design control systems. In this paper, the steady-state approach is retained, as is the premise that the interaction between distillation column control loops must be minimized. Attention is focused on dual composition control of a column with a saturated liquid feed, although it is shown that the conclusions drawn for this case may be expected to hold true for most distillation processes. The principal question which is addressed is: How can interaction between the composition loops be minimized or, preferably, removed? In general, there are two options available. In the first, any arbitrary pairing of manipulated and controlled variables is accepted, and the decoupling strategy is to employ external decouplers, which are designed on the basis of dynamic column information. Such schemes have been widely studied by Buckley (1967), Luyben (1972,1976), and Rijnsdorp (1965) among others. A major criticism of these schemes, apart from questions of cost, physical realizability, and integrity, has been their emphasis on the servo rather than the regulator problem in distillation (Niederlinski, 1971). The second approach is that of internal decoupling. Here, minimization of interaction is sought through the very act of choosing “correct” manipulative variables, and making proper pairings. This is the approach taken by Shinskey (1967, 1977), Nisenfeld and Stravinsky (1968), and others (Lloyd, 1973; Nisenfeld and Shultz, 1971; Rijnshdorp, 1965). The great advantage of internal decoupling is not only in its simplicity but also in obtaining a control scheme which is equally effective in disturbance filtering as it is in responding to set point changes. It is the second approach which is adopted here. The concept of noninteracting manipulative variables is first introduced by way of a simple blending example from Shinskey (1967, 1977). The procedure for finding noninteracting manipulative variables for dual composition control in distillation is then outlined. The preliminary work of McAvoy in this area (McAvoy, 1977) is extended, and it is shown that the conditions for half-decoupling degenerate to each other and also to the column material balance equation. As a consequence of this degeneracy, complete decoupling is shown to be impossible, from the point of view of its practical implementation for distillation. Although the column material balance very closely approximates the decoupling equations, material balance control schemes (Shinskey, 1967, 1977) show significant interaction (McAvoy, 1977). The reason for the success of material balance control, despite such interaction, is interpreted in the light of degeneracy. Other implications of the degeneracy result are also examined.
Noninteracting Manipulative Variables The most common application of Bristol’s relative gain array (Bristol, 1966) is in finding control pairs which give the least interaction for an arbitrarily chosen set of manipulative variables (Lloyd, 1973; Nisenfeld and Shultz, 1971; Nisenfeld and Stravinsky, 1968; Shinskey, 1967,1977; Witcher and McAvoy, 1977). As a logical extension of this approach, a more direct objective would be to find a set of manipulative variables whose very nature ensures complete or half-decoupling of the control system. McAvoy (1977) has applied this idea to dual composition control in binary distillation. Adopting the Shinskey concept of reversing the process model, McAvoy makes a direct numerical search for decoupling variables, using Smoker’s (1938) equation. A simple blending example
F. I.
Figure 1. Simple blending system.
from Shinskey (1967,1977) is used by McAvoy to illustrate his approach. This is briefly reproduced here, to provide a background against which McAvoy’s procedure for distillation can be described. These ideas are then applied to yield the degeneracy result. Consider the blending system of Figure 1,for which both the total flowrate, F , and composition, x , are to be regulated. It is decided that x is to be controlled by manipulating stream ml, and a noninteraction variable [ is postulated to control F. Then a half-decoupling condition may be imposed, which defines [: F is a function only of (. Thus if [ is constant, F is constant. The system material balance equations are overall: F = ml + m2 (1) component:
ml x = ___ ml
+ m2
ml --
F
(2)
A glance at eq 1 and 2 shows that if [ = ml + m2,then the decoupling condition is met. This can be confirmed mathematically, as follows. Consider the response of x to changes in ml, for two extreme cases: where the [ loop is open and where the [ loop is perfectly controlling. These situations are described respectively by the following equations (3)
(4) The relative gain term, A,,, for the x-m1 pair, is by definition the ratio of eq 3 and 4. Thus A,,, equals 1,and the complete RGA may be written as
;ii’ p, Thus the system clearly meets the desired objective of half-decoupling of the control loops. For complete decoupling, a second noninteraction variable 17 may be introduced, which controls x subject to analogous assumptions as those used for [. Then for the totally noninteracting system, one finds (5) ( = ml + m2
Graphs of [ = constant and 17 = constant are given in Figure 2. The reader is referred to the papers of Witcher and McAvoy (1977) and McAvoy (1977) for further details. Here, it is enough to note that the half-decoupling conditions exhibited by the constant ( and 17 curves of Figure 2 are clearly different. This distinction will be shown to break down in distillation. Development of the System Equations The analysis which leads to the degeneracy result closely follows McAvoy’s procedure for finding E and 17 variables
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978 487
column, regardless of whether it is disturbed or not. Once the column has been designed (NR and Ns fixed), then for constant a, eq 9 is simply an expression of the steady-state functional dependency between the column operating variables. It can therefore be called the column operating equation. Column Material Balance. In addition to the operating equation being valid, the column must satisfy its material balances at all times overall: F = D + W (10)
100
90
7
:CONSTANT
i
= t(ql = C O N S T A N T
80
TO
so :CONSTANT
50
E
component:
40
DESIGN VALUES
30
FxF
= DxD
+ WxW
(11)
These equations can be combined into the single expression
mi =70 m2 =30 20
F '100 x
=07
u1
40
IO
0
10
20
m L
50
60
70
__
Figure 2. Constant [ and q curves for blending.
for distillation (McAvoy, 1977). This requires development of the column operating equation, which must be used in conjunction with the column material balance. The simple case of binary distillation with a saturated feed ( q = l), for which the Smoker (1938) assumptions are retained, will be considered. Column Operating Equation. Consider a column for which the steady-state values of xF,xD, x W , CY, and R (reflux ratio) are specified. A column can then be designed using a version of Smoker's equation which has been suggested by Strangio and Treybal (1974)
(7)
NS
Here, kl and k2 refer to the two intersection points of the (respective) operating lines with the equilibrium curve y = a x / ( l + ( a - 1)~). These values are found algebraically by solving quadratic equations. For purpose of design, the matching composition x , which appears in eq 7 and 8 is the same as xF. Thus, NR and Ns are easily calculated, and the steady-state design of the column is now complete. Next, the column must be considered after a disturbance has entered it. The feed plate is now no longer in the optimum location for which the column had been designed. In other words, the design equality x , = xF can no longer be used in eq 7 and 8. However, x , has the same value in either equation and can therefore simply be eliminated by solving the two equations simultaneously. A single expression then results, which can be written as 0 = fl(xD, XWt XF, R , a , NR, NS) (9) The algebraic details of this equation may be found in the paper by McAvoy (1977). Note that eq 9 cannot be written explicitly for xD or xw. Equation 9 is no more than a restatement of the validity of eq 7 and 8, but in a form which is applicable to the
Equation 9, when used in conjunction with eq 12, determines the behavior of a specific column, according to the assumptions made in formulating the problem. Degeneracy of Decoupling With the previously developed column equations, it is now possible to find the noninteraction variables for distillation. The equations are developed for dual composition control, with the objective that the bottoms loop is to be decoupled from effects of the top loop. Exactly similar arguments would apply for half-decoupling in the other direction. By analogy to the blending example, it is first postulated that a noninteraction variable t exists, for controlling the bottoms composition xw. The reflux flowrate, L , is arbitrarily chosen to control xD. Then the half-decoupling condition may be expressed as: xW is a function only to E. Thus if 5 is constant, xW is constant. 5 is found as follows. Consider the general column operating equation, discussed previously 0 = fI(xD, XW, XF, R, a, NR,Ns) ( 9) For a particular column, the design is fixed by its steady-state specifications, and CY is constant by assumption. Since R can be expressed as L / ( V - L ) , eq 9 can be written as
L , V) (13) (NR,Ns, and a are not shown in the argument for eq 13 and subsequent equations since they are constants.) Next, the material balance for the column must be considered. For this purpose eq 12 is available and noting that D can be expressed as V - L, the material balance can be written as 0=
f I ( x D , XW, XF,
f&w, XF, F , L , V) (14) Since xF and F are fixed in deriving the decoupling equations, eq 13 and 14 become XD =
OP. eq:
mat. bal.:
0= XD
f l ( x D , XW, X F ~ L , ,
V) = fz(xw, xF0, F,, L, V)
(15) (16)
The decoupling condition is now imposed that xW remain constant a t its steady-state value. The above equations then reduce to OP. eq:
mat. bal.:
0 = XD
~ I ( X D xwo, , X F ~L , ,
V)
= f2(xWo,xF0, Fo, L ,
V)
(17)
(18)
From eq 17 and 18, it is clear that if any value of L is chosen, then the two equations can be solved simultaneously to yield a value of V , and a value of xD. Thus sets
488
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
Table I. A Few of the Cases Studied €or Degeneracy CASES FROM McAVOY’S STUDY
1
of values of ( V ,xD) vs. L can be generated. This data can be used to plot the curve
This curve represents a constant (steady state) value of xw,, while each point on it represents a distinct new value of xD. Thus eq 19 describes the algorithm which must be followed in order to decouple the xw loop from the XD loop. Since this must also be a statement of the constancy o f f , f can be defined as
If f is zero xw will be constant regardless of what happens to L, and the bottoms composition loop will be decoupled from the top composition loop. So far, the analysis has been in terms of the bottoms loop. A noninteracting manipulative variable q for control of xD can also be found. The derivation procedure is essentially the same as that for f . The half-decoupling condition in this case turns out to be
so that
Note that the same variables occur in eq 21 as those in eq 19, although a different functionality is expressed. In Figure 3, the graphs for f = 0 and q = 0 are plotted. Also shown in the figure is the steady-state material balance given by eq 12, where xw and XD are both at their constant steady-state values. Noting that D in eq 1 2 can be written as (V - L ) , this material balance plots simply as the linear relationship
2
3
4
HI-PURITY CASES
5
6
7
8
9
The case shown is for a column with the design specifications a = 1.5, XF = 0.5, XD = 0.98, xw = 0.02, and R c 6.65. This column showed the highest interaction in a 24-case study by McAvoy (1977). The graphs in Figure 3 represent computer iteration results, since eq 17 is complicated and implicit in XD (or xw). Convergence was assumed when XD (or xw) values given by eq 17 and 18 differed by less than It is clear from Figure 3 that the half-decoupling conditions for distillation are effectively the same, in the vicinity of the design point. The contrast with Figure 2 for blending is immediately obvious. Furthermore, both decoupling equations for the column are also closely approximated by the material balance equation. Thus for all practical purposes, the variables for noninteraction in the column may be represented by the material balance, so that
Since f z q , and since two manipulative variables are required to control XD and xw, eq 24 essentially shows that, in a practical sense, it is impossible to achieve complete decoupling in columns where V and L are manipulated. The decoupling problem is mathematically well posed and yields two distinct equations for f and q. Practically, however, one would not be able to implement the two equations simultaneously. Only one noninteracting manipulative variable is available for use, as reflected in the approximation given by eq 24. Thus at best only partial decoupling appears to be possible. Even partial decoupling may be prone to parameter sensitivity problems as discussed later on. The principal reason for graphing V / F ovs. LIFOin Figure 3 is simplicity, in that the material balance, eq 23, plots as a straight line in these variables. It should be emphasized that regardless of which variables are used for the analysis, e.g., D and V , V I L and L etc., the same type of degeneracy occurs. The two decoupling curves, f = 0 and q = 0 almost superimpose upon one another, and upon the material balance curve, as the region near the design point is approached. A plot involving D and V as manipulative variables is given later on, to illustrate Shinskey’s material balance scheme. Examination of Results The degeneracy result appears to be valid for a wide range of binary distillations. In addition to McAvoy’s most interacting case, more than 50 columns, a small sample of which are shown in Table I, were studied for a saturated liquid feed. Relative volatility values ranging from 1.02 to 2.5, symmetric and asymmetric product distributions, different reflux to minimum reflux ratios, and tower
Ind. Eng. Chem. Process Des. Dev., Vol. 17, No. 4, 1978
._._ (
---
32
34
36
30
4 0
= CONSTANT x w : f (
