ENGINEERING, DESIGN, AND EQUIPMENT
Automatic Control in Continuous Distillation ARTHUR ROSE The Pennsylvania S a f e University, University Park, Pa.
THEODORE J. WILLIAMS U.S.A.F. institute of Technology, Wrighf-Patterson Air Force Base, O h i o
T
H E advent of continuous processing in the chemical industries has brought with it a requirement for a much more stringent control of the process variables. I n general, continuous processes are more sensitive to operating conditions than batch processes (7). I n addition, because of their generally larger size, operation a t other than optimum conditions is more costly in plant returns; this further increases the requirement for close control. Where essentially steady-state conditions can be maintained, as in the distillation of petroleum fractions, human operators or simple on-off automatic controllers have in most cases been satisfactory. However, where operating conditions vary greatly, feed-back type controllers have proved necessary in order to maintain process products of the proper quality. This is especially true in distillations carried out in the organic process industries where control of distillation columns has proved difficult and sometimes impossible (2). Analysis of the problem to date has in general been empirical. A controller is connected to the column, and control is attempted at various settings. This is unsatisfactory, however, because the range of adjustments of any one controller is limited and the tests are time-consuming and expensive (6). The developing science of servomechanisms and automatic control offers a solution to this problem as an accurate mathematical representation of the interaction of the process and the controller under various operating conditions can be developed (a, 14). The proper control parameters can than be determined theoretically and the correct controller installed prior to operation of the equipment. Servomechanism analysis is complex, especially in a multistage process such as distillation, but automatic computers can be used effectively in solving the resulting equations. This article describes the derivation of the problem for choice of a controller for a five-plate distillation column and its solution on an analog type computer for variations in feed composition and quality.
As written here (IC),the feed plate is No. 3, as evidenced by Fx, which expresses the contribution of the feed to the column dynamics. Also, the presence of a total condenser is implied since no separation beyond the top plate of the column is taken into account. Plate No. 1 is considered to be the still pot. The liquid composition, x, and the vapor composition, y, on any one plate are also specified to be related a t all times by
where a is a constant. I n addition to the dynamic relations of the distillation process itself we must also consider the dynamics of gas and liquid f l o ~ in the column. Thus the terms V and L are not constant with time when unst,eady-state conditions exist because of changes in feed rates, vapor rates, or liquid rates caused by external circumstances or action of a controller in correcting effects arising externally. The transfer function notation as used in servomechanism analysis provides the simplest method of expressing these fluid flow relations (3, 4, 14). By this method the vapor rate leaving a plate is expressed as a function of the vapor rate entering the plate by
Vn- I
vn = 711)+1 This is equivalent to
dVn -= dt
-.dtd
since p =
71
Vn-1
- Vn 71
is a constant which represents the inherent lag in
the system-here the hydraulic flow of vapor through the liquid on plate n. Similarly the liquid rates are related by the expression
Ln+ 1
Mathematical model deflnes operation of the column under all conditions
The so-called finite difference equations of distillation accurately represent the dynamic behavior of the distillation process in the transient state (11-15). These equations may be expressed in dxerential form for a five-plate continuous column as follows:
(3)
Ln =
7
x
(4)
Therefore the vapor rates in the column are expressed by the relations
vs
=
V, TlP
+1
The quantity VI is the boilup rate of the stillpot. Thus fluctuations in this quantity would be propagated throughout the column by the above equations. The expression (1 - g) determines the effect of feed vapor rate on the column Tap
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INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 11
ENGINEERING, DESIGN, AND EQUIPMENT ~~~
vapor rate and would be inserted in the equation for the feed plate (No. 3). The column liquid rates are LD = f(V6)
-D
=
Lc
-D
(4a)
LD L, = rzp 1
=
(4b)
L6 -
(4c)
+1 L4 F __ + 1 * G-T L2 = rzp + 1 L,
Lj
+
=
72p
+ P(z/,
4 = 0.5
- 51)
(6)
where p is a function of a: which is specified later. Similarly the proper feed plate location may be represented as a function of a as shown later.
(a)
+
72p
I n addition, for complete mathematical representation of the column, the value of p and the proper feed plate must be specified. For a constant feed temperature the value of q over a range of x/ such that q varies from zero to one may be specified by the relation
I
I
I
I
I
I
/ I
I
L8
LC is the liquid stream leaving the condenser. Different time constants have been used for the feed relations since they would not necessarily have the same time effect on the over-all rates as would the main column flow relations. The plates in a distillation column are usually spaced sufficiently close that transit times of vapor and liquid transfer from plate t o plate can be ignored. However this is not true of the transit time involved in transferring vapor from the top plate t o the condenser, condensing same, and then returning the liquid to the top plate. For this reason Equation 4a is expressed as a function of Va. An equation is therefore needed which will express the mathematical equivalent of an event that reoccurs some finite time after its original occurrence. That is, a particular flow history must be reproduced at the outlet of the condenser and associated lines some finite time after it enters the inlet; thus Lc will reproduce the time-flow function of Vb but a t a later time. 14
I
1
1
I
I
I
I
I
I
I
I
I :I I
*fO
Figure 1.
Variation of ,B with xfs for a equals 5.0
By the real translation theorem of the Laplace transformation it can be shown that
f(t
- T) = e-r~[f(t)]
where 7 is the true time delay or the actual time required for material to traverse the condenser and associated lines. This equation serves only to delay a variable, such as a flow rate of composition, for a given definite time and does not alter its mathematical form as does occur with the other expressions such as Equations 3 and 4. Therefore
(5) While this expression, if carried out by hand, must be evaluated by using the methods of the Laplace transformation it can readily be evaluated on an analog computer by means of the Pad6 delay circuit (8, 9). Lc =
November 1955
f?76PVS
x'o Figure 2.
Designation o f optimum feed plate location for various values of xfa and X I
Equations la-e, 2, 3a-d, 4a-f, 5, and 6 and the empirical relations for p and feed plate location thus define the action of the distillation column under all conditions of operation. With the column defined, we can now turn our attention to the inechanics of controlling it.
Variation of reflux ratio held column compositions constant I n many distillations, both the bottoms and tops products are valuable and thus must be held to a defined purity. Because of the nature of most marketing arrangements, material of slightly better quality is permissible and indeed often desirable, but material of lower quality entails a penalty in price. Our problem therefore is to keep the distillate composition above a given composition of more volatile component and the bottoms below another established composition. Also this must be accomplished without gross overdesign of the column or the use of an excessively high reflux ratio. If a continuously operating analytical device is placed in the distillate product line and also in the bottoms product line, compositions as continuous functions of time can be obtained and compared with the desired values of distillate and bottoms product to determine the error in bottoms and top output. This resulting error is then used by the controller to apply the proper correction to the column flow parameters to bring the product compositions back to the desired value and to maintain them a t these values. A relation between error and correction which will permit the controller to accomplish these requirements must now be derived. For a given feed composition and rate a distillate composition which is lower than desired requires an increase in top reflux ratio-that is, a decrease in distillate takeoff rate and an increase in bottoms rate. Conversally, a bottom composition which is too high requires a decrease in bottoms rate and an increase in distillate rate. Let us now define the error as = -(xDdesired
if
XD
< XDdesired
INDUSTRIAL AND ENGINEERING CHEMISTRY
- xp) (7)
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ENGINEERING, DESIGN, AND EQUIPMENT Equations 4a-f show that for a constant feed rate and constant boilup rate the required correction is applied to W by material balance considerations if D is corrected. Therefore our controller need operate on the distillate product rate only and AD = f[E(x)I PLLTE 5
K’P
i 2 H
PLATE
(10)
There are three possible methods of conversion of the error into a correction by the controller ( 3 , 14). They are proportional, derivative, and integral control and may be used singly or in combination with each other. I n proportional control the correction is equal to some constant times the original error AD = K E ( x )
4
(loa)
I n derivative control the correction to distillate rate is equal to a constant times the rate of change of the error with time
Similarly integral control expresses the correction as another constant times the time integral of the original error
AD = K ”
E(z)dt =
K” - E(x) P
(10c)
Thus if the possibility of using all three types of controI in the controller is considered, Equation 10 can be written
L--&
‘TZP
AD = K E ( x )
+I
+ K’pE(2) + KE(x) P ”
(10d)
There is one other relation which while not directly applicable to controller operation gives a check on controller operation. Figure
3.
Distillation column automatic control dynamic relations
+W
DXD and if
XD
E 1 and
X =~F x f
(11)
xw E0 D EF x ~
and =
if
XU,
(XW
-
Computer capacity limits the parameters that may be investigated
XWdedrad)
> XWdeBired
(8)
Just as the plates will not respond instantly to a change in liquid or vapor rate, there is the possibility that the error detector may not respond instantaneously to a sudden change in distillate composition due to mixing in the test cell. Thus E ( x ) , the error transmitted to the controller, is related to C ( X ) the true error as
Any other two compositions besides those of distillate and bottoms, with a similar interrelationship to each other, could be used here to obtain the error if desirable. If it is required or desired to consider only one source of error-i.e., only one product stream or one plate composition is to be used to obtain the error for the controller-the true error equation would appear as E(X) =
(xfi
- xndesired)
(8a)
where xn is the composition variable being measured. The composition sampling arrangemente actually tried in this study were: 1. Two-point sampling using distillate and bottom plate
samplers 2. Two-point sampling using top plate and bottom plate sampling 3. Single-point sampling from distillate only 4. Single-point sampling from top plate only 5. Single-point sampling from an intermediate plate only 6 . Single-point sampling from bottom plate only
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The Systems Dynamics Analysis Branch of the Aeronautical Research Laboratory of Wright Air Development Center is equipped with a four-unit REAC (Reeves electronic analog computer). When this problem is applied to that machine up to five plates may be considered. Therefore the parameters to be investigated must be chosen with the computer capabilities in mind. The problems of commercial interest are mainly those giving separations 0.95 to 1.00 mole fraction purity a t reflux ratios (LID)less than 1O:l. A McCabe-Thiele type investigation (1) showed that a five-plate coIumn would give a separa-
=I 0.WO
0.022 0.m
=r O.SW
0.ss0 0.966
‘6
0.m
.@.?a6
Figure 4.
Response of column with controller set at 3.40,K’ = 0.0, and K ” = 0.0
K
=
Sornpling on distillate line ond on boftom plate
INDUSTRIAL AND ENGINEERING CHEMISTRY
Vol. 47, No. 11
ENGINEERING, DESIGN, AND EQUIPMENT -0.96
AD 0.0
Figure 5. K = 5.0, K' = 25.0, K" = 0
K = 0, K' = 0, 1.0
Figure 8.
K" =
4.96
AD 0.0
F i g u r e 9. Figure 6.
K = 5.0,
K = 5.0, K'
0, K"
K'=
=o
25.0,
K" = 0
0.660
0.955 5 0
675
0.795
-0.96
d 96
AD
AD
0.0
D O
fO
.(i.96
96
0.3m
F-igure 7 , ' : K = 0,K' = 25.0, ~ f = f o.5aa
Figurelo.
5.0,
K'
K = = 0,
'' o'5m
K" = 0
