Ind. Eng. Chem. Res. 1990,29, 1859-1868
1859
PROCESS ENGINEERING AND DESIGN Tuning Controllers on Distillation Columns with the Distillate-Bottoms Structure Heleni S. Papastathopoulou and William L. Luyben* Department of Chemical Engineering, 11 1 Research Drive, Lehigh University, Bethlehem, Pennsylvania 18015 Up until recently, the distillate-bottoms (D-B) control structure was not considered to be a viable scheme for controlling a two-product distillation column. However, Finco et al. demonstrated by simulations and plant tests that it outperforms conventional structures when implemented on distillation columns separating a low relative volatility mixture into high-purity products. The D-B structure has indeterminant steady-state gains, and because of that, it is not possible to derive the transfer function matrix of the D-B control structure by using standard identification techniques. In this work, the D-B matrix is found by appropriately transforming the transfer function matrix of another control scheme. Most of the analysis is based on the assumption that the distillatevapor (D-V) matrix is available. If the reflux-vapor (R-V) matrix is known instead of the D-V matrix, the necessary modifications in the analysis are also discussed. The methodology is extended to a three-product, sidestream distillation column. The full transfer function matrices of the distillate-sidestream-bottoms (D-S-B) and the D-NS-B structures are derived by using the same ideas as for the D-B structure in the two-product column.
1. Introduction Consider a two-product distillation column in which the distillate flow (D) is used to control the top composition and the bottoms flow (B) controls the bottoms composition (Figure 1). The base level is controlled by heat input and the reflux drum level by reflux. This control structure is called D-B. Up until recently, the D-B control structure was considered to be impossible. Indeed, based on steady-state considerations, this is the case. The distillate and bottoms flows are not independent of each other at steady state but are related through the total material balance equation of the column. F=D+B (1) For constant feed flow, changes in the distillate or bottoms flow are accompanied by opposite, equal-size changes in the bottoms or distillate flow, respectively. For variable feed flow, a change in the distillate, for example, will cause a change in the bottoms flow such that eq 1 is satisfied. If dual-composition control is important, then the previous remarks show that the D-B structure is impossible at steady state because there is a lack of 1 degree of freedom. Finco (1987) tested the dynamic performance of the D-B control structure on a rigorous nonlinear simulator and in plant conditions, and he found that it outperforms conventional structures in rejecting feed disturbances. The study concentrated on the separation of low relative volatility mixtures (propylene-propane) into high-purity products. Such separations require very tall distillation columns (150-200 trays) and large reflux ratios. This large number of trays introduces a significant hydraulic lag into the *Author to whom correspondence concerning this paper should be addressed.
dynamics of the column, and this decouples the distillate and bottoms flows during the transients. Assuming that the tray time constant is 6 s (according to the old rule of thumb), we find that it takes 15-20 min for a change in the reflux to reach the column bottoms. It is because of that lag that the D-B control structure is feasible dynamically. Actually, the D-B control structure is equivalent to the RR-BR scheme (the reflux ratio (RR) is used for controlling the top composition and the boil-up ratio (BR) is adjusted according to bottoms composition changes) if the levels in the reflux drum and the column base are perfectly controlled. Tight level control is necessary for good performance in the D-B structure because outflows are used for composition control. The RR-BR structure has been used in practice for some time (Shinskey, 1984, 1988). Thus, the D-B structure is not as unusual as it might initially look. Recently, the RGA (Bristol, 1966; McAvoy, 1983) of the D-B control structure was calculated as a function of frequency (Skogestad et al., 1990). It was found that although at low frequencies the RGA for the D-B configuration is much worse (higher) than that for the R-V structure, it is significantly better (closer to 1)in the frequency range important for feedback control (0.01-1 m i d ) . This result supports the recommendationsof Finco et al. (1989). The steady-state gains of the D-B structure are indeterminant. Suppose that the xD-D gain needs to be found. A small increase in the distillate flow causes an equal and opposite change in the reflux. The top composition ( X D ) decreases. As the reflux flow change reaches the column base, it causes an equal reduction in the vapor boilup, which eventually will cause a further decrease in the reflux flow. Then this cycle repeats itself until reflux goes to zero. Thus, the xD-D gain is indeterminant. The inverse of the 0 1990 American Chemical Society
1860 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 -1/
F,z
___L
ro’
B.XIj
m P 1
Figure 1. D-B control structure.
steady-state gain matrix is singular. Therefore, it is not possible to derive the full transfer function matrix by using standard identification techniques (Ljung and Soderstrom, 1987; Unbehauen and Rao, 1987). Finco handled this problem by tuning the proportional-integral composition controllers empirically, by trial and error. In this paper, the problem of deriving transfer functions for the D-B structure is discussed. As pointed out, it is not possible to use standard identification techniques. Instead the transfer function matrix is derived by transforming appropriately the transfer functions of another more conventional scheme. In this work, the starting point is the D-V (Figure 2) or the R-V scheme. Later on, the transformation is extended to account for a sidestream column. In this latter case, the problem is how one can derive the full D-S-B matrix (Figure 3) by starting from the D-S-V structure (Figure 4) when the sidestream flow rate (S) is used to control the sidestream composition. Finally, the method to derive the D-NS-B transfer function is also discussed (Figure 5 ) when the location of the sidestream draw-off tray is used to control the sidestream composition. Once transfer functions are derived, controller design and tuning can be accomplished in a consistent way and comparison of the performance of the D-B structure with other more conventional schemes is possible because proper controller tuning is not an issue any more. Frequency domain analysis techniques can now be applied for the D-B structure as for any other scheme, including multivariable predictive controllers. 2. Derivation of the D-B Transfer Function
Matrix Transfer functions for the W B control structure (Figure
L 1
B3’B
Figure 2. D-V control structure. r
&
/
0
A ..N..T....
. ..NS ......
F,z
, ..NF ......
.........
T+ B.XB
m
I
Figure 3. D-S-B control structure.
1) are derived by appropriately transforming the D-V transfer function matrix (Figure 2). Suppose that the D-V transfer function matrix has the form (2)
Ind. Eng. Chem. Res., Vol. 29, No. 9,1990 1861 the reflux drum level is justified because these level controllers would typically be tuned very tightly. In a following section, the proper changes in the derivation are discussed for the case when the perfect level control assumption at the column base is not reasonable. In addition, it is assumed that the liquid flow dynamics at the bottom of the column can be expressed as a function of the liquid flow changes at the top of the column by the relationship
i I
1
D. xD t
g, = L l / R
(3)
where L1is the liquid flow from the first tray of the column (the column reboiler is considered as the zeroth tray), R is the reflux flow, and gLis the transfer function relating R with L1. In the next section, forms of gLare suggested for different column configurations. At the base of the column, because of the perfect level control assumption, we have
I
V=Ll-B (4) where V is the vapor boilup and B is the bottoms flow rate. Similarly, at the top of the column, we have R=V-D (5) where D is the distillate flow. Substituting eqs 3 and 5 into eq 4, we get
Figure 4. D-S-V control structure.
or in matrix form
If the [D, VIT vector in eq 7 is substituted into eq 2 and the appropriate multiplications are performed, the transfer function matrix for the D-B control structure is found
where gll, g12,g21,and g2, are the elements of the D-V transfer function matrix. These elements can be found by conventional identification methods. 2.1. The gLTransfer Function. The liquid flow dynamics of any particular tray of the column can usually be described by a first-order lag. The column hydraulics are then described by a series of first-order lags. Thus, for a single column, which is comprised of one shell, the following expression is suggested for the gL transfer function: I 1 T (9) gL = (1 TLS)NT
m ---r Figure 5. D-NS-B control structure.
By using a transformation, changes in the vapor boilup (V) are related to changes in the distillate (D)and bottoms ( B ) flows. The analysis is based on the assumptions of equimolal overflow, perfect level control in the reflux drum, and perfect level control at the column base. The equimolar overflow assumption is reasonable for close boiling mixtures such as propylene-propane. The perfect level control assumption for the column base and
+
where rLis the tray hydraulic time constant and NT is the column total number of trays. T~ can be calculated from the equation describing the liquid hydraulics of the column. A. Split Column Systems. Very tall columns are often comprised of two shells (split column design). Suppose that the bottom shell has NB trays and the top shell has the remaining NT- NB trays. In such a case, eq 9 must be modified to account for the intermediate inventory dynamics if it is not on perfect level control. The level
1862 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
changes at the bottom of the top shell are Ap dh/dt = LNB+l- Lht
(10)
where A is the column cross-sectional area, p is the liquid density at the intermediate inventory, h is the liquid height, LNB+lis the liquid flow from the first tray of the top shell, and Lht is the flow rate of liquid pumped to the top of the bottom shell. A PI controller is used to adjust the intermediate level by manipulating the pumpover flow
where K ttand T? are the proportional gain and the integral time of the intermediate level controller, respectively. Laplace transforming eq 10 and substituting into eq 11, we get
For columns comprised of two shells, the following gL expression is suggested instead of eq 9
Table I. Data for the Calculation of the Tray Time Constant d , ft 10.0 WI, ft 6.41 M,, Ib-mol 29.0 p , Ib-mol/ft3 33.7 Mw.
wh, i n .
42.2
TL, s
1.o
4
Substituting eqs 17 and 18 into eq 19, linearizing, and Laplace transforming, we obtain the expression for T L in min
Table I summarizes the necessary data for the calculation of T L (eq 20). These data are from a propylene-propane splitter, and they are also used in the simulations results that are presented in a following section. 2.2. Discussion. The D-B transfer function matrix can be used to predict the steady-state gains of this structure. In the Laplace domain, as the system approaches the steady state, s approaches zero. At the limit lim gL = 1 (21) S-4
Substituting the appropriate expressions into the right-hand side of eq 13 gives gL =
Ktt(7i"tS + 1)
1 (1
1
+ T L s ) ~ "Ap~Pts2 B + Kift(+ts + 1) (1 +
7 L ~ ) N ~
(14) Finally after performing the multiplication Ktt(TPtS
gL = A ~ + V
+ 1)
1
+ ~ t t ( 7 ~+t 1) s (1 +
(15)
TLS)NT
B. Calculation of Hydraulic Lag 7L. If the column hydraulics are described by the Francis weir formula, then T L is calculated as shown below. The Francis weir formula is 1, = 3.33~lh,"~
(16)
where 1, is the liquid flow rate from tray n in ft3/s, wl is the length of the weir in ft, h, is the height of liquid over the weir in ft, and n is the tray number. When the liquid flow rate from tray n is expressed in lb-mol/min (L,), eq 16 becomes 200p L, = - ~ l h , l . ~ (17) M W where M, is the molecular weight of the liquid on tray n and p is the liquid density in lb/ft3. The height of the liquid over the weir is calculated as the difference between the total liquid height on the tray and the weir height 4MnMw hw=-ad2p
wh
where Mn is the tray total holdup in lb-mol, d is the column diameter in ft, and wh is the weir height in ft. A mass balance on tray n (assuming equimolar overflow) yields (19) dMn/dt = L,+1 - L ,
Equation 21 is true when either eq 9 or eq 15 is used for the gL expression. The term 1 - gLappears in the denominator of all the elements of the D-B transfer function matrix (eq 8). Thus, the previous analysis predicts that the steady-state gains of the D-B control structure are indeterminant. However, as the process moves away from steady state, the magnitudes of the transfer function elements become N ~ finite. This enables analysis of the D-B control structure by using the same tools as for any other control structure (find ultimate gains and frequencies, tune controllers according to a desired procedure, etc.). From a controller tuning point of view, the ultimate gain and ultimate frequency of a system transfer function are important since they must be close to the actual values of the model or the plant. The propylene-propane separation was used as a test case for the analysis of the D-B control structure. An AutoTuning Variation (ATV) test (Luyben, 1987) was performed on both the xD-D and xg-B loops of the rigorous, nonlinear, dynamic simulator of a propylenepropane splitter. In Figure 6, the behavior of the xD-D loop is presented during the ATV test. The distillate flow changed every time the distillate composition crossed the setpoint (steady-state value). The ultimate gain (K,) and frequency (mu) were determined from the following relations: 40 K, = w, =
a7r 2*
-
(23)
P U
where P, is the period of the limit cycle, p is the height of the relay, and a is the amplitude of the primary harmonic of the output xD. The ultimate gains and frequencies for the two D-B transfer functions were also found by using eq 8. Since the C3 splitter was assumed to have two shells, the gL expression used was that of eq 15. The ultimate gains and frequencies determined by both methods are compared in Table 11. The gains are dimensionless numbers and the frequencies are reported in radfmin. The conclusion is that the D-B transfer function matrix (eq 8) describes the
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1863 Table 11. Ultimate Gain and Crossover Frequency for the D-B Structure eq 8 ATV Kt"P -40.9 -41.5 up 0.199 0.209 K Pm 75.1 101 0.141 0.121
x
L
E
n
n?
- 0
'0 X
s p'
? qP
400.0
0.0
100.0
2W.0
500.0
300.0
Time [minl
9
.-------_______ 0.0
100.0
5M.O
200.0
400.0
500.0
T i m e Iminl
Figure 6. ATV test on the XD-Dloop. 9
W
Figure 8. Bode plot of the (1,2)element for the D-V structure (solid line) and the D-B structure (dashed line).
W
=rp
L
10-
o
1
.V.*.I.,.''#d
10"
",J
.1.1.1.1.,#',1' ' - . I '
lo*
lo-] W
1O.l
loo
Figure 7. Bode plot of the (1,l)element for the D-V structure (solid line) and the D-B structure (dashed line).
behavior of the column very well a t the crossover point. This validates the effectiveness of the proposed approach to tuning the D-B structure. In Figures 7-10, Bode plots of the elements of the D-V and the D-B transfer function matrices are presented. As the frequency decreases, the magnitudes of the D-B elements increase asymptotically to infinity. The magnitudes of the corresponding D-V elements approach the steadystate gains. A t higher frequencies, the D-B elements be-
'lo*
10"
10-
lo=
W
lo+
lo-!
loo
Figure 9. Bode plot of the (2,l)element for the D-V structure (solid line) and the D-B structure (dashed line).
have like the D-V transfer function elements. Note that the D-B structure Bode plots look like a pure integrator a t low frequencies (log modulus with a constant slope of -20 dbldecade and -90° phase angle). This be-
1864
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
9
Table 111. Data for the Propylene-Propane Splitter F, lb-mol/h 1000 P, psia 110 z , mole fraction 0.70 MB, Ib-mol 1060 xD, mole fraction 0.9966 M D ,lb-mol 780 x B , mole fraction 0.0002 L,, lb-mol/h 7450 NT 190 RR 10.6 NY 61 Table IV. Controller Parameters for the D-V Structure
Ku 0"
Kp TF"
K iy,T TpT
Xn
XR
-40.8 0.199 -18.6 26.3 -7.38 66.3
-2.86 0.141 -1.30 37.1 -0.517 93.3
Table V. Controller Parameters for the D-B Structure
K,, WU
KEN 7 : "
KELT
W
Figure 10. Bode plot of the (2,2)element for the D-V structure (solid line) and the D-B structure (dashed line).
havior can be explained by examining the elements of the D-B matrix (eq 8). They have the general form
is the numerator polynomial of the transfer where function elements. Substituting eq 9 into eq 24, we get I
1(TLS
Expressing the term a power series yields
(TL
+ 1)NT
+ 1)"
of the denominator in
+ lIN~@(,, TLs(TL~T-'s~T-' + r L N r 2 s N r 2 + ... + TLS
XB
75.1 0.141 34.1 37.1 13.8 91.6
The R-V transfer function matrix is easily transformed into the D-V matrix if the vector [R, UTis substituted by the right-hand side of eq 30 and the appropriate multiplications are performed. Once the D-V matrix is available, the previous analysis can be used. 2.3. Closed-Loop Control. The simulation results presented in this section are based on a propylene-propane splitter with design specifications summarized in Tables I and 111. The D-V or equivalently the D-Q transfer function matrix is
E]=[
-37.1e* (7200s + 1)(2s i1)
(159s + 1)(2.86s +])l
0.647e*
-43.8e-2" (2338s + 1)(12.7s + 1)'
-1.71e* (26s + 1)(3.82s
E]
+ 1)2 131)
After performing the necessary calculation, G,,) becomes (TLS
G(s, =
TfLT
XD
-40.9 0.199 -18.6 26.3 -7.52 65.0
+ 1) ( 2 7 )
Finally, (29)
This transfer function contains an integrator. The analysis above started with the D-V control structure. Similar arguments are valid if, instead of the D-V, the R-V transfer function matrix was available. Because of the perfect level control assumption in the reflux drum, eq 5 still holds. In matrix form, we have
The time constants are reported in minutes and the steady-state gains in dimensionless numbers. The sampling time of the composition analyser is 6 min. The D-Q transfer function matrix was derived from simulation of a rigorous, nonlinear, dynamic model of the process. The D-B transfer function matrix was used for designing and tuning the distillate and bottoms composition loops. Controllers were tuned by using the Biggest log-modulus Tuning (BLT) criterion (Luyben, 1986). In Tables IV and V, the ultimate gain and frequency, the proportional gain, and the integral time according to the Ziegler-Nichols guidelines and the BLT method are summarized for the distillate and bottoms composition loops for the D-V and the D-B control structures. In Figure 11, the distillate and bottoms compositions along with the corresponding manipulated variables are presented for a feed flow disturbance to the column. The tuning parameters of the composition controllers are those of Tables IV and V. The behavior of the column in the presence of a feed composition disturbance is presented in Figure 12. These results confirm those of Finco (1987). The D-B structure performs better than other more conventional structures in the separation of a low relative volatility mixture into high-purity products. The overshoot of the
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1865
=r-----l
?
9
=~
I 0.0
2W.O
400.0
800.0
600.0
10W.O
1200.
0.0
200.0
400.0
Time Iminl
0.0
400.0
200.0
m.0
600.0
600.0
8M.0
1000.0
1200.
8M.0
IOW.0
12W.
Time Iminl
lOW.0
0.0
1200.
200.0
400.0
600.0
Time Iminl
Time [minl 9
I
0 9 0.0
too.0
200.0
1ow.o
m.0
600.0
7 10.0
1200.
"
200.0
"
400.0
6w.o
"
W.O
"
10w.o
'
I
1200.
Time Iminl
Time Iminl
.
"
a
9
I
.
y , , ,
0.0
200.0
,
'
,
.
,
,
,
,
,
,
,
,
,
,
400.0
600.0
800.0
10W.O
.
,I
0 0
0.0
1200.
200.0
400.0
6~0.0
600.0
1m.o
1 2 ~ .
Time Imin)
Time (minl ?
-
2 , 1 n 2
a?
0 0.0
200.0
400.0
€00.0
W.0
10W.O
1200.
"x
I
0.0
Time Iminl
Figure 11. Performance of the D-V (solid line) and the D-B (dashed line) control structures for a feed flow disturbance.
distillate composition is similar for both cases, but the bottoms composition remains almost unchanged in the D-B structure. The settling time for both product compositions is significantly smaller than that when the D-V structure is implemented. The advantage of the analysis presented in this paper is that a fair comparison of the performance of the D-B structure against any other structure can now be made. The proposed procedure eliminates the need for empirical tuning of the D-B structure. Quantitative multivariable controller tuning methods can be employed.
200.0
tw.0
m.0
em.0
1000.0
1200.
Time (minl
Figure 12. Performance of the D-V (solid line) and the D-B (dashed line) control structures for a feed composition disturbance.
2.4. Imperfect Level Control. The derivation of the D-B transfer function matrix is based on the assumption of perfect level control in the reflux drum level and the bottom of the column. In this section, the perfect level control assumption in the column base is relaxed. The liquid flow dynamics at the bottom of the column are still related to the reflux flow changes by eq 3. At the base of the column, the vapor boilup is used to control the level
v = gcM, (32) where g, is the transfer function of the level controller at
1866 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
the column base and MB is the bottoms holdup. If the level controller is proportional, then gc = Kc (33) or if a proportional-integral controller is used at the column base, then TIS + 1 g, = Kc(34) TIS
where Kc is the proportional gain and T~ is the integral time of the level controller. A material balance at the column base yields dMB/dt = L1 - - B (35)
In matrix form, we have
Substitution of eq 45 into eq 40 yields the D-S-B transfer function matrix
v
Laplace transforming eq 35 gives sMB = L, - V - B Substitution of eq 32 into eq 36 results in V = gimp(LI - B )
(36) (37)
where
In the case of perfect level control in the column base, gi, is equal to 1 and eq 37 reduces to eq 4. #inally, combining eq 37 with eqs 3 and 5, we get
For imperfect level control, eq 39 should be used in place of eq 6. All remaining analysis is the same. 3. Derivation of the D-S-B Transfer Function Matrix The methodology to derive the D-B transfer function matrix is extendable to a higher order system, a sidestream column. In this section, the derivation of the D-S-B transfer function (Figure 3) is presented. The D-S-V transfer function matrix (Figure 4) is appropriately transformed to the D-S-B matrix. Suppose that the D-S-V matrix has the form
pJ p; If 3[i]
(40)
For the sidestream column, the changes in the vapor boilup are related to changes in the distillate, sidestream, and bottoms flows. The analysis is based on the same assumptions as those of the two-product column: equimolal overflow and perfect level control. A t the base of the column, we have VzL1-B (41) It is assumed that the liquid flow rate leaving tray 1 can be expressed as a function of the liquid flow changes at the sidestream tray L1 = gLsLs (42) and the liquid flow changes at the sidestream tray are dynamically related to the liquid flow rate at the top of the column and the sidestream drawoff rate. (43) Ls = gLTR - s The specific forms of g, and gLT are discussed later in this section. Substituting eqs 5, 42, and 43 into eq 41, we get
(46)
where the gi,j elements (i = 1-3 and j = 1-3) are the corresponding elements of the D-S-V control structure. If the sidestream drawoff location is on tray Ns and the column has NT trays, the following expressions are suggested for gLs and gLT: (47) (48) For very tall columns comprised of two shells, the previous expression for gLs or gLT should be appropriately adjusted to account for the intermediate inventory dynamics. Equation 12 is also applicable for the sidestream column. Depending on the location of the sidestream tray relative to the intermediate inventory, either g u or gLT is adjusted. For example, if the sidestream drawoff location lies above the intermediate inventory, then the appropriate expressions are
The steady-state gains of the D-S-B structure are indeterminant. As the process approaches steady state, s goes to 0, gLsand gLT go to 1, and the difference 1- g< approaches 0. The difference 1 - g< appears in the denominator of all the transfer function matrix elements. In the transients, when the process is away from the steady state, the magnitudes of the transfer function elements become finite and eq 46 can be used to analyze and study the D-S-B structure using the same tools as for any other control structure. 4. Derivation of the D-NS-B Transfer Function
Matrix In a sidestream distillation column, apart from the obvious choice of manipulated variables distillate, sidestream,
Ind. Eng. Chem. Res., Vol. 29, No. 9,1990 1867 and bottoms flows, the location of the sidestream draw-off tray (Ns)is sometimes used for controlling the sidestream composition (Tyreus and Luyben, 1975; Doukas and Luyben, 1978). In this section, the derivation of the D-NS-B transfer function matrix is presented. Only the necessary changes in the analysis are discussed. In Figure 13, a possible configuration of the system is presented. It involves three draw-off trays. Sidestream flow can be withdrawn from one or two trays at any point in time. The choice of the required actions depends on the behavior of the sidestream composition. First, the case of sidestream withdrawal from trays NS1 and NS2is discussed. The changes in the liquid flow from the first tray (L,) are related to the liquid flow changes from tray NS, through Ll = gLSILNSl (51)
* ............
D
9LS,
aLs,
9LS3
9L
where 1 gLsl
=
(TLS
+ 1)NSl
The changes in the LNsl flow rate are related dynamically to changes in the flow from tray NS2 and the first sidestream flow rate (SI) (53) LNSl = gL&'NS2 - s1
Figure 13. Distillation column with variable sidestream draw-off trays.
where (54) See Figure 13 for a summary of the notation. Finally the changes in the LN, flow are related to the reflux flow changes and the sidestream flow from tray NS2 (55) LNS2 = k7LNT.$ - s 2 where
50
0
Substituting eqs 53 and 55 into eq 51, we get
L1 = g L R - g L s l s l
- gLS*S2
Controller output (%)
(57)
Figure 14. Output of the sidestream composition controller.
(59)
Substituting eqs 63 and 64 into eq 57, we finally get an expression for the changes in liquid flow from the first tray as a function of the changes in the reflux flow and the output of the sidestream composition controller (CO). This variable (CO) is equivalent to the sidestream draw-off tray location.
where
The two sidestream flows S1and S2mix together to form the total sidestream flow ( S ) s = s1 s2 (60) In deviation variables, eq 60 is equivalent to
+
s, = -s2 (61) In Figure 14, the behavior of the output of the composition controller is presented. When sidestream flow is withdrawn from trays NS1 and NS2, the controller output is related to S1 and S2. If eq 61 is also taken into account, then 0 ICO 5 50 (62) SI
=
s 2
co --s 50
=
LV -s 50
100
co
L1 = g L R - &LS2 - g , s 1 W z
(65)
Equation 65 needs modification if the sidestream draw-off trays are NS2 and NS3. The analysis is similar to that above. In this case, the output of the sidestream composition controller is 50 5 CO 5 100
(66)
and eq 65 is changed to
L1 = g,R - &LS3 - g where
(63) In general,
co
L s , ) s ~
(67)
1868 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
L1 = g,R - gSC0
(69)
P, = ultimate period Q = reboiler heat duty R = reflux flow RR = reflux ratio S = sidestream flow rate SI,S2,S3 = sidestream flow rate from the first, second, or third
(71)
V = vapor boilup xB = bottoms composition xD = distillate composition w , = weir length wh = weir height
where
gs = k L S 3 - gLSz'j0
if
50 ICO 5 100
Equation 69 is the expression for L1 when the location of the sidestream draw-off tray is used as one manipulated variable and it should be used in place of eqs 42 and 43. The remaining analysis is exactly the same as before. 5. Conclusion A methodology for deriving the transfer functions of the D-B, D-S-B, and D-NS-B control structures has been presented. It is based on appropriate transformations of the open-loop transfer function matrices of other more conventional control structures. Once the transfer functions are available, controller tuning can be done according t o any chosen method and a consistent comparison of the performance of the D-B structure with other commonly used schemes is feasible. Nomenclature A = column cross-sectional area B = bottoms flow BR = boil-up ratio CO = output of the sidestream composition controller D = distillate flow d = column diameter F = feed flow rate g, = controller transfer function gimp= imperfect base level control transfer function (eq 38) gL = transfer function for the column hydraulics gLs,,gut, gLs3= transfer functions for the liquid hydraulics between the column base and the first, second, or third sidestream draw-off tray, respectively gLa,, gLrp,s,gL, gL2-, = transfer functions for the liquid ydrau ICS of different sections of the column (Figure 13) h = liquid height in the column base hset = setpoint for the liquid holdup in the column base h, = height of liquid over the weir Kc = proportional gain of the controller K , = ultimate gain L, = liquid flow from tray n Lint = pumpover flow rate in the split column design MB = total holdup a t the column base MD = holdup of the reflux drum M, = total holdup on tray n M, = molecular weight of the liquid phase NB = number of trays in the bottom shell N , = location of the feed tray N s = location of the sidestream draw-off tray NT = column total number of trays P = operating pressure of the column
draw-off trays, respectively
z = feed composition
Greek Letters a = amplitude of the primary harmonic of the output (ATV test) /3 = height of the feedback relay (ATV test) p = density of the liquid phase T~ = integral time of the controller T~ = tray hydraulic time constant @.(s) = numerator polynomial of the transfer function elements W, = ultimate frequency
Literature Cited Bristol, E. On a New Measure of Interaction for Multivariable Process Control. IEEE Trans. Autom. Control 1966, AC-11, 133. Doukas, N.; Luyben, W. L. Control of Sidestream Columns Separating Ternary Mixtures. Instrum. Technol. 1978,25 (6), 43-48. Finco, M. V. The Modeling and Control of Low Relative Volatility Splitters. Master's Thesis, Department of Chemical Engineering Lehigh University, Bethlehem, PA, 1987. Finco, M. V.; Luyben, W. L.; Polleck, R. E. Control of Distillation Columns with Low Relative Volatilities. Ind. Eng. Chem. Res. 1989,28, 75-83. Ljung, L.; Soderstrom, T. Theory and Practice of Recursive Identification;MIT Press: Cambridge, 1987. Luyben, W. L. Simple Method fo; Tuning SISO Controllers in Multivariable Systems. Ind. Eng. Chem. Process Des. Dev. 1986, 25, 654-660. Luyben, W. L. Derivation of Transfer Functions for Highly Nonlinear Distillation Columns. Ind. Eng. Chem. Res. 1987, 26, 2490-2495. McAvoy, T. J. Interaction Analysis Principles and Applications; Instrument Society of America: Research Triangle Park, NC, 1983. Shinskey, F. G. Distillation Control for Productivity and Energy Conservation, 2nd ed.; McGraw-Hill: New York, 1984. Shinskey, F. G. Process Control Systems: Application, Design, and Tuning, 3rd ed.; McGraw-Hill: New York, 1988. Skogestad,S.;Jacobsen, E. W.; Morari, M. DB-Control of Distillation Columns. Ind. Eng. Chem. Res. 1990, submitted. Tyreus, B.; Luyben, W. L. Control of a Binary Distillation Column with Sidestream Drawoff. Ind. Eng. Chem. Process Des. Deu. 1975, 14, 391-398. Unbehauen, H.; Rao, G. P. Identification of Continuous Systems; North-Holland Systems and Control Series; Elsevier Science: Amsterdam, Holland, 1987; Vol. 10. Receiued for reoiew January 31, 1990 Revised manuscript received May 17, 1990 Accepted June 4, 1990
