Understanding and Prediction of the Dynamic Behavior of Distillation Columns Edward F. Wahll and Peter Harriott Cornell University, Ithuca, N . Y . 14850
A model of the dynamic behavior of a binary distillation column i s developed to obtain transfer functions for the system. Correlations allow the poles and zeros of these functions to be evaluated from a knowledge of steady-state conditions. The dynamic behavior is characterized by two parameters obtained from steady-state calculations. Equations and methods are presented for evaluating the gains of the transfer functions. The use of the model and correlations i s exemplified by the calculation of the transfer functions and time response of a feedback-controlled column. The general characteristics of such columns are discussed in terms of this example. The optimum sensor location i s determined mainly by steady-state error and controller sensitivity and to a lesser extent by dynamics of the control loop. The system's sensitivity to fluctuations and reflux rate i s discussed.
THIS
PAPER describes binary distillation columns, to elucidate their dynamic behavior and lead to a model for which the transfer functions are readily obtained. Where the mathematics become cumbersome, a correlation based on the model presented is used to predict the approximate transfer functions. The value of the model lies in its contribution to the understanding of the dynamics of a distillation column. It is to the dynamics as McCabe-Thiele is to the steady state. We are not discussing the solution to problems which require and can justify a detailed computer solution, taking into account all the very special characteristics of a particular column and a particular operation on that column.
plate piate 2I
i! F
molrs/mln.
Xf mole fraction A
I
Present address, 5284 S.R. 303, Richfield, Ohio 44286
396
Ind. Eng. Chern. Process Des. Develop., Vol. 9, No. 3, 1970
L = liquid rate molas/min.
-71 L1
X,= mole fraction on plate n H, = plate holdup, moles V = vapor rate moleslmin.
plate N
B
moleslmih. XR mole fraction A
The dynamic behavior of a distillation column depends primarily on two parameters, T, and L R ,which have the following physical significance:
Consider a binary distillation column operating a t some steady-state condition as shown in Figure 1. The plate composition varies with position in the column (Figure 2, left). Sometime later the column might be operating at some other condition, as shown by the dashed line. If the first condition is, considered the reference state, the difference between the two represents the perturbation in compositions, defined by x (Figure 2, right). The distillation column can be viewed as a black box
!
plate
Significant Parameters
T, is the time constant obtained, assuming that the product streams are a €unction of the average concentration in the column. L R is the reduced circulation rate and represents the extent to which the column is maintained at equilibrium and therefore the extent to which the above assumption in obtaining T, is valid.
?-
Figure 1. Schematic of N-plate distillation column 0 I 2 c
a
1
DEFINITION' X'X"
f
x
w
W
t
4
a N
h R
LIQUID PLATE COMPOSITION,
x
LIQUID PLATE COMP. CHANGE, X
Figure 2. Relation of column location to plate composition and perturbation on plate composition
which splits the incoming feed stream containing two components into two product streams, each enriched in one of the components (Figure 3). The black box contains a total amount of material which is about constant, but the average composition of which varies. The average composition (perturbation) for this black box column is calculated from the plate compositions as follows: R xa,=
R
C
Hnxn/ n = O
n = O
Hn
(1)
I n this calculation the holdups in pipelines and downcomers are included with the appropriate plate, the condenser, or the reboiler. The four loads or points a t which disturbances can enter this process are: (1) the feed composition, (2) the feed rate, (3) the boilup rate, and (4) the reflux rate. Change in the temperature of the feed corresponds to altering the reflux and vapor rate a t the feed plate, and so is a special case of 3 and 4 applied to part of the column and therefore is not listed separately. For this column, the dynamic response can be expressed in mathematical symbols as xn(t) =
C-’[.(”)
+$)
+ u(?)
+r ( 3 ]
Xf
where xn/nf, n.lf, x,Iu, and x,/r are the four transfer functions. An over-all component material balance on the column (black box) gives the desired relationship between input and output quantities for each of the four loads.
HTiav =
-Dx,- BxR + r ( X , - X R )
(2d)
The interior of the box is described as a set of N 2 well mixed tanks which are maintained at steady state (in equilibrium with one another) by a circulating flow and are upset by loads introduced into this system by the injection of material a t certain points and withdrawn at others. The circulating flow consists of the vapor stream in one direction and liquid stream in the other. The parameter which describes the ability of this system to maintain itself a t equilibrium is the ratio of the circulation rate to the holdup (Figure 4 ) . The reciprocal of this ratio can be considered the mixing or circulation time of the column. I t is a capacitance, H T , divided by a flow rate, L , through the capacitance. The choice of L is arbitrary here; V might just as well have been chosen. On the other hand, the rate at which the system responds to the upsets will be given by a characteristic time constant for the system. We define this as T,. I t is shown later that T, is approximately equal to the principal time constant of the system and can be calculated from the steadystate data for the column. Thus T, can be viewed as the total holdup divided by the rate a t which the load is introduced. A little thought will make it clear that the reciprocal of this is the loading rate per unit holdupthat is, it represents the rate at which the system is being forced away from the equilibrium (steady-state) condition. Because of the importance of the relationship of these two effects-namely, the circulation rate and the loading rate-we define the ratio of the two as the reduced circulation rate, LR-that is,
+
It is given this name because it represents the circulation (mixing) rate in the column relative to the rate the column is being forced away from steady state. At relatively high internal circulation rates such that H T / L is small compared with T,, the system will maintain itself close to equilibrium, and the composition at the top or bottom will depend only on the instantaneous state of the system. I n this case any single state property, such as xav, will completely define the composition, n, through the column. I n this case,
Unfortunately, no and X R may not depend only on naV. The solution of these equations as well as the prediction of the shape of the plate composition curve requires a knowledge of what occurs inside the black box.
TOP
FLOW RATE OF PRCOUCT
+t(”-
Dxo
BOTTOM
r , xo
FLOW RATE OF PROOUCT
B x
+
(f+r-v)
X
r
r
!
-*I I I
I I I
Figure 4. Essential interior of distillation column
Figure 3. Over-all view of binary distillation column ignoring interior, as a black box
Left. Low-holdup, high circulation rate
Righf. High-holdup, low circulation rate
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970 397
xo = fl(xav) XR
= f?(X,,)
and by the usual linearization of perturbation analysis, X" = XR
cs,,
= CRxav
where c, = d x o l d x a ,and C R= dxR/dx,,. I n this case, Equations 2a, b, c, and d can be solved, since they are all first-order differential equations of the form Load
B C R + DC,
(4)
where
Load = F q or X f f or uXR or r X , The constants C, may be evaluated by taking the derivative of x o with respect to the average concentration and rewriting so that
c, = d x J d x , ,
= (6x0/6u)(6u/6xa,)
Gou/.i6x,,/6u) refers to the load, xi, f, u , or r , and G,, is =
where u the gain on plate n due to load change u . Noting that 'V
-1
HnGnu/ Hr
(6xav/au) = n = O
the equations for C, and C Hbecome AV + 1
Co= HTG,,/
H,GnU n
=
0
Substituting these values in the definition of T,, the following relation is obtained: v
.+ 1
This equation is the same as that given by Rijnsdorp (1964) in the limit of infinitely small perturbations. The same value of T , should be obtained no matter which load is used to calculate T,. Calculations at typical values of L R show that the values of T , are within a fraction of a per cent of each other, except for feed composition loads for which 1 or 2% differences were obtained. No matter whether the load is reflux, vapor rate, feed rate, or feed composition, the distillation column will respond approximately as a first-order system with time constant T , given by Equation 5 . This becomes exactly correct in the limit as the reduced circulation rate, L R , approaches infinity. At the other limit of L X + 0, the 398
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970
behavior will depend on the load, as described below. Since this limit is not approached in any real column, it is a mathematical convenience rather than a reality. Feed Composition Load. The effect of a perturbation in feed composition for a column operating at very low circulation rate is most easily visualized by considering a case for which the feed is part vapor and part liquid. The feed stream then splits into two, one flowing up through the series of plates and the other down (Figure 4, right). This is equivalent to flow tnrough a series of well mixed tanks for which the transfer function is Gnu/ (T,s + 1)"'. Thus, the feed plate will respond as a firstorder lag, the plate on either side as a second-order lag, the third as a third-order lag, and so on. This means 0 that the column will respond at the limit of L H with a set of time constants with values determined by the holdups of the individual plates. Vapor Rate Load. Many writers have pointed out that the response time for rates of the vapor flow throughout the column is small compared to the response time for the composition changes, so that these two processes can be considered independently. As far as composition responses are concerned, a change in boilup rate causes a simultaneous change in vapor rates throughout the column. This vapor rate change is applied to each plate, and each plate responds therefore as a first-order lag to the vapor rate introduced at that plate. Since all plates are responding in the same manner, there is little effect on a plate due to the change in other plates as far as the response time (and frequency response) is concerned. There certainly is an effect on the final steady-state value to a step change, but this is expressed in the value of the gain for each plate. At the limit L H+ 0, each plate responds as a first-order lag with a time constant of a single plate. Feed Rate Load. The response time of the plates for composition changes is large compared to the time for changes in liquid level. Consequently, these two processes can be considered separately. Consider the case where the feed is all liquid. The feed plate and plates below the feed will nearly simultaneously receive the change in feed rate and will respond as first-order lags, just as for a vapor rate load. On the other hand, the plates above the feed plate will respond to the change in composition of the feed plate in the same manner as if the change were caused by a change in feed composition. Therefore, at the limit of L H 0, the lower portion of the column will respond as a first-order lag with a time constant of a single plate, whereas the top portion will respond as a series of first-order lags with transfer functions G,/ (T,s+ 1)"', Reflux Rate Load. Since the response time of composition change is large compared to the change in liquid flow rates, the response characteristics for reflux load will therefore be similar to that for a vapor load--that is, at the 0 each plate responds as a first-order lag with limit LH time constant equal to that of each plate. To complete the description of the dynamic behavior of the column on which the engineering prediction is to be based, we must consider what occurs between the limits discussed above. This will be the region of practical importance. Before describing this region, it is important to recognize that the time constants for all the plates for all types of loads are identical and that the differences that occur as discussed above must therefore result from -+
-
3
variations in the values of the zeros. The proof of this statement is as follows: The set of component material balances on each plate can be written in the matrix form:
ft = AX + BU
5
l
Y
where u is all the load variables. Taking the Laplace transform and solving for x results in
zra
x = [SI - A]-'Bu - [SI- A]* Bu [SI - A ]
8
bz
W
E
-
0
where F is the number of the feed plate. No relationship is implied between the time constants in this expression and the time constants of each plate-for example, the number of poles and zeros for each plate is shown in Table I for a nine-plate column. Since a t the limit of L R + m the response of all plates is identical with the time constant T,, the principal (largest) time constant must approach T , and all the other time constants (and also, therefore zeros) must be negligible in comparison. On the other hand, as LB + 0, the principal time constant must approach the same values as the others, and must have a value approaching that of the time constants of the plates. This trend is depicted in Figure 5, where
l
I
LR=IZ2
the values of the time constants are given for a nineplate column with reduced circulation rates of 0.65 and 17.2. The data were obtained by solving in the Laplace domain the matrix set of equations ir = Ax + Bu of a binary distillation column for the required transfer functions. At high LR the closest time constant is 5% of the principal time constant, whereas a t low L R it is about 70% of the principal time constant. Furthermore, the
2).
*
l
Figure 5. Typical time constant distribution for different reduced circulation rates
(s + . . . (s + 2,) I[ (s + ZF + . . .( S + ZN + = c [ ( s + 2,) ( T I S + l)(T*s+ 1) . . . . . . . . . . . ( T N + Z S + 1) 2 2 )
O
01 L, =.80
where * indicates the adjugate of [SI - A]. The time constants for all plate compositions, x., and all loads, ul, are the reciprocals of the poles which are the roots of the determinant of [SI - A]. These poles are the eigenvalues of A. Since A is independent of the type of load, the poles-i.e., time constants-are independent of the type of load. I n general, the number of time constants will be N + 2, one for each plate and one each for the condenser and reboiler. There may be more if some stages are divided to allow for imperfect mixing. Now the form of the transfer functions can be deduced from the limiting behavior. Consider first a feed composition load. I n view of the performance a t LR 0, the transfer function for each plate must contain a number of zeros such that the proper number of lags is obtained-that is, the transfer function for the nth plate of the enriching section is Xll xi
'a o ROOTS*'OF DEN.
2)
]
principal time constant is much closer to the value of the plate time constants for low LE. The ratio of T,/Ti is plotted for a large number of cases in Figure 6, which shows that T,/Tl approaches 1 as L R + m and that T, predicts T I in all cases very well, the worst deviation being 20%. This deviation can be predicted as a function of L R using Figure 6. The value of T , has been calculated for a wide range of binary distillation columns (Figure 7). Since T , is approximately equal to the principal time constant of the system, these IO
Table I. Number of Poles' and Zeros for Feed Composition Transfer Function of a Nine-Plate Column Plate
Condenser 1 2 3 4 Feed 6 7 8 9
Reboiler a
No. of
No. of
Zeros
Poles-Zeros
5 6 7 8 9
10 9 8
7 6 5
Number of poles is 11 for all plates.
6 5 4 3 2 1 2 3 4 5 6
09 t-"
.I
1.0 IO 100 REDUCED CIRCULATION RATE, L,
1000
Figure 6. Ratio of principal time constant to model time constant Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970
399
I
1.25
I
1.25
1.25
1000 -
IOOC
-
...
in
c” 3 z
100
W in
-
3
z
-
5
+*
+* 10
100
-
I
P
0.80 / 10
I
3
10
30
100
NUMBER OF PLATES
3
10 30 NUMBER OF PLATES
100
3
1
10 30 100 NUMBER O f PLATES
Figure 7. T, values for binary distillation columns
graphs can be used for estimating the response time of columns operating within the range covered. Consider now reflux and boilup loads. I n these cases the plates all react approximately as a single time constant, since the hydraulic lag is small compared to the composition response. Consequently, all but one of the time constants must be negated approximately by a zero. I n the limit as L R-, 03, this time constant is T,. Consequently, for flow rate changes the transfer function is approximately G,,/(T,s + 1). Transfer Functions
The dynamic terms in the transfer functions for the column may be estimated from the two parameters T , and L R and the other basic physical parameters of the column (Wahl, 1967). The authors have solved over 300 cases of binary distillation columns described by the matrix set i = Ax + Bu to determine the exact transfer functions
for the column for perturbation loads. These transfer functions were then approximated by appropriate simpler transfer functions, using the rationale developed. I n addition, cognizance was taken of the fact that the transfer functions for feed composition and feed rate loads do not normally enter into a feedback control loop, and consequently response a t high frequency is not required. What is required is good response in the time domain. A typical feedback control scheme is shown in Figure 8. The recommended transfer functions which were obtained are given in Figure 9. The flow rate changes respond as first-order lags, and furthermore, the time constant of the lag is equal to the principal time constant for the system, TI. For the feed composition load, adequate representation in the time domain was obtained by using the principal and third largest time constant, providing the proper zero is used. The first and third largest time constants were used because the second and fourth largest are effectively
Figure 8. Generalized schematic of top product control by reflux manipulation Control sensor on plate n A/D.Transfer function giving composition response of plate n due to load change
C/D. Tronsfer function giving composition response of condenser due to load change
GI&. Tronsfer function giving composition response of plate n due to load change r B/D2. Transfer function giving composition response of condenser due to load change r
+
H / s . Controller transfer function, H = K/TR Ks E / F . Transfer function for elements of control loop-measurement loop x , or f. Load function, feed composition, or feed rote r. Manipulated load function, reflux rate Summation, difference of two large numbers. Small difference should be examined for correctness
400
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970
For Respmse
r-
I
(T&i + 1)"'
A ; G , * 7;s + I
High Frequency
xn
Figure
Response
Bottom
of Column
For
of
9. Recommended transfer
;r""l Tls
+I
~
xn
Gf(T$ + I)
Top Column
functions
cancelled by zeros. For a feed rate load, however, the first and second time constants should be used for the top of the column, since the third and not the second time constant was effectively cancelled by a zero. Correlation graphs for estimating these time constants are given in Figures 10 to 14. The correlations are based on binary cases spanning the following range of variables: Relative volatility Feed composition, mole fraction Top product composition, mole fraction Recovery of light component, % Murphree vapor plate efficiency Number of plates
1.25 to 2.50 0.30 to 0.60 0.80 to 0.995 80.00 to 99.5 0.60 to 1.0 3 to 32
The time constants are relatively constant, for a large perturbation in the loads-for example, a 10% change in feed composition will change the principal time constant by 10%. On the other hand, the gain may vary severalfold. If the gain is assumed constant, the transfer functions will be limited to smaller perturbations than if the gain is taken as a function of column composition. The extent to which the linear transfer function can be assumed correct in a given case may be ascertained by calculating the transfer functions at the extreme operating points.
.I
I.o
IO
100
1000
" I I IO 100 REDUCED MIXING RATE, L, Figure 1 1 . Prediction of second time constant
NUMBER OF PLATES
REDUCED CIRCULATION RATE, L, Figure 10. Prediction of principal time constant
Figure 12. Prediction of third time constant for reflux ratios and plate efficiencies Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970
401
with change in liquid composition on each plate-that is,
K, = (dY/dX),
(6)
Values of K. can be obtained in a number of ways: From the slope of the equilibrium line or the pseudoequilibrium line in a McCabe-Thiele diagram From the equation for constant relative volatility and constant plate efficiency
From published K values for multicomponent systems.
1
~
:
,
2
4
I
:::
:
40
20
610
NUMBER
60
PLATES
OF
Figure 13. Prediction of fourth time constant
The equations in Table I1 can be used for multicomponent systems if there is a total condenser, equimolal overflow, a single feed stream, and top and bottom product streams only. The procedure in the appendix can be used to relax these restrictions by appropriate calculations. The values of K , are also needed in Equation 5 to get T,. Example
-1.0
As an example of the procedure given in this paper, the response of a feedback-controlled binary distillation column to a change in feed composition is presented. The general control scheme is given in Figure 8, and the following values of the principal parameters are used:
-0.5
-
Relative volatility Number of plates Feed plate (from top) Murphree vapor plate efficiency Feed rate, moles/minute Reflux rate, molesiminute Vapor rate, molesiminute Top product composition Feed composition Control Sensor on plate Reflux ratio Plate holdup, moles/plate Reboiler holdup, moles Condenser holdup, moles Total holdup, moles Refluximinimum reflux Hydraulic lag per plate, minutes Controller gain, K/K,, Reset rate, T,w,
I-*
+
t;" \ c
P I
FN
v
0.5
1.c
I
I
I
'
I
I
,
'
'
FRACTIONAL HOLDUP
Figure 14. Prediction of reciprocal zero Tz The remaining parameter that is required is the gain for each transfer function. Since the gain is in some cases sensitive to changes in the column, the limitation in predicting dynamic response may be the ability to make adequate steady-state calculations rather than the ability to predict the dynamic parameters-the poles and zeros. Two methods for calculating these gains are described below. The gains can be calculated from the steady-state compositions for two values of any load parameter. This method requires sufficient accuracy in the two steadystate calculations, better than can be done by simple graphing techniques. Alternatively the gains, G,,, can be determined from the linearized equations in Table I1 or following the computational procedure in the appendix, which explains the derivation of Table 11. This procedure requires the value of the change in vapor composition 402
Ind. Eng. Chern. Process Des. Develop., Vol. 9,No. 3, 1970
2.0 25.0 14.0 1.o 1.0 1.48 1.98 0.995 0.500 4.0 2.96 0.791 7.912 3.956 31.65 1.5 0.08 0.45 7.5
A steady-state calculation. gives the following values of the plate compositions: Plate No.
Vapor
Liquid
0 1 2 3 4 5 6 7 8 9 10 11 12 13
0.995016 0.991329 0.985893 0.977952 0.966501 0.950301 0.927984 0.898347 0.860843 0.816183 0.766691 0.716033 0.668224
0.9950 16 0.990082 0.982807 0.972179 0.956854 0.935174 0.905307 0.865644 0.815453 0.755684 0.689450 0.621654 0.557672 0.501754
Plate No.
Vapor
Liquid
14 15 16 17 18 19 20 21 22 23 24 25 26
0.626441 0.570098 0.498220 0.414354 0.326111 0.242810 0.171850 0.116504 0.076231 0.048383 0.029798 0.017688 0.009918
0.456072 0.398697 0.331753 0.261315 0.194822 0.138181 0.094002 0.061855 0.039626 0.024791 0.015124 0.008923 0.004984
The reflux and feed composition gains are then calculated by using Equation 7 and Table I1 to yield the following gains:
Table II. Calculation of Plate Composition Gains
Step 1. Determination of a, and b, n
b,
0 1
1 1iKi
[ vKbL,+ "'1 b,
2,. . .N F
- 1
-
VK, L b,
.. 2
N F+ 1 N F + 2,. . . N R
on for Lood Shown
n
Feed compn.
Feed rate
Reflux rate
Boilup rote
Equal reflux and boilup
0 1
0
0
0
0
0
0 0
0 0
2,. . .N F
0
0
NP+ 1
NF
-
F -_
VK,
+ 2,. . . N R
X n - 1 - X n__ 2 VK,
x, I - C' E + ____VK,
E
VK,
x,.1-xn-2
(+
+ ___---
VK, 2
E+
' + x,
Yn
I -xn-2
--
1 - Yn VK, Y"- I - Yn -__VK,
Yn
'+
xn-1-xnE
0
E+
VK,
~
1 - Yn VK,
E+-
x"-,-x"-2+ Y,
VK, Xn-l-Xn E + _____ VK,
2
+
x,-l-xn +
'+
2
1 - Y n
VK, Y"G1Y" VK, Yn-2 VK,
Yn-1-
VK,
Step 2. Determination of condenser gain Gm
F - DaR -~ D + BbR
For feed comp. load. G 'I-
GOF=
Xt - XR- BaR D + BbR
For reflux rate load. GOL=
X, - XR- BaR D + Bbx
For boilup rate load. GOR=
--
For feed rate load.
- BaR
D
+ BbR
Step 3. Determination of gain on the plates For all loads: G,,, = a, b,G,, where a, is the a, for appropriate load u as given above
+
Condenser mole fraction gain Plate 4 mole fraction gain
Per Mole Fraction Change Feed Compn.
Per Mole/ Min. Change in Reflux
Go,= 1.032 G,; = 8.535
G,. = 1.033 G,, = 8.492
I n addition, the gains are calculated for a reflux load for all plates. so that T, can be calculated from Equation 5 . The value of T, is 376 minutes. Alternatively, Figure 7 (center) could have been used to obtain the same value for T,. The reduced circulation rate is calculated according to Equation 3 to yield L R = 17.6. Using the above data, the values of the time constants are obtained following the procedure of Figures 10 to 14. ( H p /VI, minutes LH T,, minutes TI/ T , from Figure 10
0.40 17.6 376.0 0.975
T,; T 2from Figure 11 TI/(&/ V) from Figure 12 T,(H,/ V) from Figure 13 TI,minutes T,, minutes T?,minutes Ta,minutes To,minutes
28.0 15.1 11.9 365.0 13.4 6.03 4.76 2.00
The numerator time constant parameter corresponding t o the zero for plate 4 is obtained with the use of Figure 14. T o use Figure 14, the position of plate 4 expressed as fractional holdup is calculated:
H , + H p ( N- 1%) - 3.96 + (0.791)(3.5) = 0.45 3.96 + (0.791)(14) H, + H a / From Figure 14 is obtained the parameter
T, _ - T:! _ _ -- -0.86 T, + T, from which, together with the previously obtained values I d . Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
403
TZ and T4, the numerator time constant parameter, T,,is calculated to be -2.2 minutes. Similarly, T, for
of
TIME,
the condenser is found to be -4.3. When these parameters are inserted in the recommended transfer functions and in turn inserted in the block diagram for the single point control scheme of Figure 8, the schematic of Figure 15 is obtained. The transfer function shown for obtaining plate liquid rate, A, from reflux rate is obtained by assuming that the response to a change in reflux results in a change in flow rate at the fourth plate given by the transfer function.
MINUTES 2000
1000
0
0.0001 product plate 4
0
1 A4 _ -r (0.08s + 1)3
+
This may be approximated by 1/(0.08s 1)(0.16s + 1) and completes the statement of the transfer functions given in Figure 15. I n this way, the transfer functions which determine the dynamic response of a controlled distillation column system are obtained from basic column characteristics and relatively simple calculations. The time response of this system can then be determined by simple analog or digital computer techniques. The time response' for the example obtained from a digital computer solution is given in Figure 16 for a feed composition load.
- 0.06 0I
product composition, product composition plus the change due reflux change ordered
xf
&-+-K(+ x4
IO0
MINUTES
e,, is equal to the change in top due to the feed composition load to the change in reflux times the by the control loop:
e, = G,,x, + G,r The value of r at steady state is that which results in a zero error in composition on the control plate, n ,
so that e, is given by
e, = G o x ( l- R ) xf
I
-
I
I
Figure 16. Time response of controlled distillation system shown in Figure 15 to step change in feed composition of +0.05 mole fraction
The basic elements of feedback reflux control of a binary distillation column consist of a composition sensor located a t some position in the column and a control loop which manipulates the reflux so as to maintain constant composition a t that location. The location of the sensor should be chosen after considering both the dynamic response and the steady-state error in product composition. Steady-State Error. Since the adjustment of reflux necessary to maintain constant composition on a plate is not necessarily the same as that required for constant top product composition, control on any plate other than the condenser will result in a steady-state error of the top product, even though the control plate composition is maintained a t zero error through reset action. This steadystate error, which is different from the conventional definition, is an important effect which must be considered in any control scheme of this nature. Referring to the control system of Figure 8, the steady-state error in top
8.535 (-2.28+1) (365s .e I l ( 6 s + I)
I
50
TIME,
Characteristics of Feedback Control by Reflux Manipulation
8.492 (365s + I)
I
.
-_)b
1 k (0.16s + 1)(0.08st I)
I +I)+(0.050 I)(O.I t I)
+
r
L
1.033 (365s+l)(2s+I)
-I
1
1
1.032 (-4.3s
+
I)
(365s + 1 ] ( 6 s + I )
404
Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970
i_ _ - - --- - 4-l, z'+ * *e
Xo
where
Since the gains may be calculated as shown in Table 11, the steady-state error due to control on plate n is easily computed by Equation 8. The ratio of gain of plate n to the gain of the condenser for feed composition changes divided by the same ratio for reflux changes is about 1 or somewhat greater, reaching in some cases at the center of the column a value of 3 or more. Where R is close to 1, the steady-state error of top product composition is small. If a perfect controller were available, the best location for the sensor to control top product would be in the top product itself--the condenser, for example-at least as far as steady-state error is concerned. If it is assumed that the controller can determine composition to = t c mole fraction, the worst steady-state result will be when the actual composition is c units from the set point value. The contribution in overhead product composition error due to this effect will be the error, e , diminished by the ratio of the gain of top product response to reflux, G,,,to the gain of the control plate to reflux, Gnr. The net steady-state error will be the sum of this effect and that due to offset:
The trade-off between these two effects is shown in Figure 17 for a 25-plate column with feed on plate 14. The difference between the total error and the steady-state error is due to the sensitivity term, and the minimum error for the cases cited occurs when the control plate is located from 0.1 to 0.3 of the way from the top to the feed plate. Transient Response of Example-Controlled System. The control loop may be considered apart from the rest of
0.04
IOC
F
u 4 a
\
U
W
the column to evaluate the maximum gain and critical frequency. Figure 18 shows a plot of critical frequency and maximum gain as a function of control plate number, assuming that each additional plate adds one more hydraulic lag of 0.08 minute. The critical frequency for top plate control depends just on the measurement and valve lags, since the large column time constant contributes almost 90" phase lag a t very low frequencies. However, the magnitude of this time constant affects the maximum gain and the magnitude ratio is scaled accordingly in Figure 18. The transient response curves in Figure 3 6 were obtained using KIK,,, = 0.45 and W,TR = 7.5 as recommended settings for the control loop. Control on plate 4 gives a typical underdamped response curve which levels out with no noticeable error at plate 4 after several minutes. However, the error in product composition continues to increase for several hours, reflecting the slow change in column compositions with a 365-minute time constant. For control a t plate 1, the critical frequency is higher but the initial correction is made a little later, since it takes more time for the disturbance to reach the top than to reach plate 4. There was practically no error in product composition for top plate control, but the calculations were made assuming perfect sensitivity of composition measurement and perfect manipulation of the reflux flow rate. T o control composition to within 0.0001 mole fraction, as shown in Figure 16, the reflux flow must be regulated to better than 0.01%, which may not be possible in practice. The reflux oscillates only a little and the top product composition not at all for the step load shown in Figure 16. This might a t first seem to be in conflict with the usual oscillations obtained for step response of a feedback control system with controller settings as in this example. However, the system block diagram (Figure 15) shows that a step load in feed Composition is filtered through the response transfer function containing T 1 and TBtime lags before entering the feedback control loop. Consequently, the oscillations in reflux will be much less severe than if the load were introduced directly to the control plate.
0.03
I
Sensitivity
a*
iO.1 mole f r a c t i o n
0
[r a 0.02
!A
+ 0
10.01 c3
Sensitivity
i O . 0 1 mole t r a c t i o n
0
Steady state error (perfect seniitivlty)
CK
a 2
I
I
I
4
6
8
SENSOR
1.c I
PLATE NO.
Figure 17. Maximum steady-state top product error for various control system sensing capabilities
2
4 6810
2C
CONTROL PLATE NUMBER Figure 18. Critical frequency and open loop magnitude ratio for columns with hydraulic time constant of 0.08 minute as a function of control plate number Control loop lags, 0.05 and 0.1 minute
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970 405
If a perfect composition sensor is assumed, the best location of the sensor is at the top of the column for a step load. This is not necessarily true for other types of feed composition loads. Using controller settings of KIK,,, = 0.45 and TRc= 7.5 for the example case, the time response for various load disturbances was evaluated to determine optimum sensor location. The results are as follows: Type of Load Short pulses with time periods of a few minutes Pulses with time periods about Yi to 'h column time constant Pulses with time periods about same as column time constant Long pulses Step
Optimum Control Plate Plates 4 t o 6 Plates 2 to 4 Plate 1 Plate 1 Plate 1
The optimum location is plate 1,unless the loads consist of very short pulses with periods less than !/z the principal time constant of the column. I n that case the optimum sensor location will be closer to the feed plates. These optimum locations are a result of the reduction in the initial rise in composition due to earlier reflux response, without causing too large an overcorrection. Appendix
The gain in liquid composition for each plate per unit change in load is computed in the following manner. 1. Compute the gain of each plate x,/u by writing a steady-state component balance around the condenser and then each plate, working to the bottom of the column. These equations are evaluated in terms of x , / u and are all of the form (xn/U)=
an + bn(xo/U)
where u is the load and a and b are functions of n only. I n the above, advantage was taken of the fact that at steady state, the gain, G,,, is identical with the change in plate composition, xn, per unit change in load. 2. Write an over-all steady-state component balance to obtain an equation relating the top product gain and the bottom product gain. Use this equation and the equation for bottom product gain from the first step to solve for top product gain ( x o / u ) . 3. Evaluate the gain by inserting the value of ( x , / u ) from step 2 in the equation obtained in step 1. This procedure is demonstrated formally for feed composition loads as follows: Step 1. A steady-state component balance is written around each plate, beginning with the condenser. Each equation is rearranged to obtain the following set of equations. For the condenser
For n = 1,2 . . . . (nj - 1)
For n, = (nj + l ) , . . . . N (xn i
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 3, 1970
=
(VKn+ L e ) ( X n / U ) - Le(xn -- I / U ) (A VK"- i
14
The successive substitution of the solution of each equation into the following will transform this set of equations to a set of the form (xn/U)
= an
+ bn(xo/u)
(A2)
Therefore, the gain of each plate may be determined directly from this equation and ( x , / u ) . Step 2. The over-all steady-state component balance for the perturbed column is
B(XR+ XR) + D(X,+ x,) = F(X/+ xi) which (after noting that to
DX,+ B X R = FXj) is rearranged
D ( x ~ / u=) F - B(xR/u)
(-43)
Equations 3 and 2 for the reboiler gain, ( x R / u ) are , solved simultaneously for x,/ u :
Step 3. The gain for each plate is then determined from Equation A2, using the value of ( x , / u ) calculated by Equation A4. For computer or routine calculation, the equations for determining a, and 6 , are presented in Table 11. Table I1 shows that 6 , is the same for all load changes, and that a, is either zero or else the first two terms of a, are the same with different constant terms following, depending on the particular load case. Also shown in Table I1 are the equations for determining condenser gain and the other plate gains. Thus, Table I1 presents all the equations necessary for determining routinely the place composition gains. Nomenclature
A = matrix as defined in text = parameter as defined in text (Figure 15) = bottom product flow rate, moles/minute = parameter as defined in text (Figure 15) = matrix as defined in test c = controller sensitivity, mole fraction c = a constant as defined in text D = top product flow rate, moles/minute e, = error in plate n composition; equal to xn if reference state is desired value E = Murphree vapor plate efficiency f = perturbation in feed rate, molesiminute f ( x ) = function of x in time domain F = feed rate, moles/minute Gnu = gain for plate n composition response to a load u ; G,, to feed composition; G,, to feed rate, G,, to reflux rate; G , to vapor rate H = holdup, moles; T Ttotal column holdup, H , holdup of plate n, HR reboiler holdup, H, condenser holdup, H p plate holdup per plate I = unit matrix
a, B 6, B
K, K LR L
406
l/U)
= (dYldX), = controller gain = reduced circulation rate = liquid flow rate in enriching section, moles/ minute
Ls = flow rate in stripping section n = plate number N = total number of plates in column r = perturbation in reflux rate, molesiminute R = ratio of gains, (GorlGnr) 1 (GoxiGnJ T = time constant, minutes T , = simplified model time constant T , = time constant of plate i u = matrix (vector) of load variables, u u = load variable, f , q,v, or r v = perturbation in vapor rate, molesiminute vapor rate, molesiminute x = perturbation in composition of liquid, mole fraction x, = plate n composition x/ = feed composition composition of liquid, mole fraction zero, minutes-’
v =
x = z=
GREEKLETTERS OL = relative volatility = intermediate parameter used in Figure 15 A, = perturbation in liquid flow rate, molesiminute on plate n wC = critical frequency, radiansiminute
SUBSCRIPTS a av f h
= = = = n =
o =
r = R = T = u =
V = x =
approximating average feed hydraulic plate number condenser reflux rate load reboiler, same as N + 1 total load variable vapor rate load feed composition load
literature Cited
Rijnsdorp, J. E., discussion on “Approximation Models for the Dynamic Response of Large Distillation Columns,” J. S. Moczek, R. E. Otto, T. J. Williams, p. 246, “Automatic and Remote Control,” Proceedings of Second Congress of the I.F.A.C., Basel, Switzerland, 1963, Butterworths, London, 1964. Wahl, E. F., “Practical Prediction of Binary Distillation Column Transfer Functions and an Analysis of the Single Point Control System,” Ph.D. thesis, Cornel1 University, 1967. RECEIVED for review April 16, 1969 ACCEPTED April 9, 1970
Integrated Theory of Separation for Bulk Centrifuges Jan R. Schnittger Alfa-Lava1 A B , Tumba, Sweden An integrated theory i s suggested for separation of solids in bulk centrifuges, such as decanters in the food and chemical processing industries. The governing separation process takes place in a laminar flow boundary layer, 3 to 5 mm thick, with a free surface. The gradual, hindered settling of solid particles out of this layer, moving under the influence of centrifugal force and viscous friction, has been computerized.
This has allowed, for the first time, calculations of absolute values of the sedimentation efficiency. The results are presented from the viewpoint of the plant engineer. Flow rate, concentration of solids, revolutions per minute, viscosity of liquid, particle size, and hindered settling influence the operation of a bulk centrifuge. A series of field tests is simulated on the computer. The values of efficiency vs. flow rate are in very good agreement with the experimental ones until an excessive flow destroys the close validity of the model assumptions.
Bum
centrifuges offer many advantages in processes handling large amounts of solids. The decanter is especially suited for high tonnage applications with very different characteristics, from slimy sludges down to the 5-micron range or coarse, fast-draining crystals. The decanter is equipped with a screw conveyor, mounted in the rotating bowl. The moderate relative speed of the screw in the bowl moves the separated solids toward their exit (Figure 1). The clarified effluent leaves the bowl a t the opposite
end, usually through a number of exit holes, the radial location of which essentially determines the free surface radius, R,, of the rotating water volume in the bowl. The sedimentation distance, L , and the angular speed, W, of the bowl are other essential characteristics, as well as the differential speed between bowl and screw, the particle-liquid properties, and the general geometrical flow arrangements in the bowl. Any analytical model should be carefully designed to Ind. Eng. Chem. Process Des. Develop., Vol. 9,No. 3, 1970 407
