Ind. Eng. Chem. Rocess Des. Dev. 1903, 22, 42-49
42
functional groups forming carbonates which are the most stable compounds under these conditions. Above 800 K, carbonates in close proximity to coal will decompose (Franklin et al., 1981) releasing COP Schafer (1979a) also reported that calcium-form coals pyrolyzed at low heating rates produced more C02 than could be accounted for by the carboxyl oxygen present, while stoichiometric quantities of C02 were evolved by acid-form coals. In his study, though, the excess C02came almost entirely at the expense of H20 production, while CO yields and char and tar oxygen contents were relatively unaffected by cation addition. It is possible that the carbonate intermediate mechanism proposed here applies to Schafer’s experiments as well. The differences in results can be ascribed to different decomposition pathways of noncarboxyl oxygen groups in the acid-form coal, which can in turn be caused by the radically different experimental conditions and time-temperature histories used in the two studies. Thus certain noncarboxyl groups which decompose yielding H 2 0 during slow heatup pyrolysis of acid-form coal may yield CO and/or char and tar oxygen during rapid pyrolysis. These same functional groups will react with cations to form carbonates during pyrolysis of cation-form coals under either heatup condition. At temperatures above 1100 K both C02 (Figure 5 ) and H20 (Figure 7) yields decrease with increasing temperature for cation-form coals. This behavior is probably due to secondary gasification of the coal char by these volatile products. Support for this supposition may be found in the plots of CO (Figure 6) and H2 (Figure 8) yields, which show that production rates of these gases from cation-form coals are greater than those from acid-form coals at temperatures above 1100 K, although some of this extra CO and H2 may be due to cation effects on primary pyrolysis rather than on secondary gasification. Catalysis of H20
and C 0 2 gasification by cations has been reported by Walker et al. (1979). Conclusions Exchanging cations onto the carboxyl groups of low rank coals reduces the yields of hydrocarbon volatile products from pyrolysis of these samples. Cation treated samples evolve more C02 than can be accounted for by carboxyl group decomposition, probably reflecting conversion of noncarboxyl to carbonates as an intermediate step. Registry No. Ca, 7440-70-2; Na, 7440-23-5; K, 7440-09-7; Ba, 7440-39-3; COB,124-38-9; CO, 630-08-0; H2, 1333-74-0;methane, 74-82-8.
Literature Cited Cosway, R. G. M.S. The&, Department of chemical Engineering, Masachusetto InstlhJte of Tedwrdogy, Cambridge, MA, 1981. Franklln, H. D.; Peters, W. A.; Carldlo, F.; Howard, J. B. fnd, €ng. Chem. procesS Des. Dev. 1981, 20, 670. Frankh, H. D.; Peters, W. A.: Howard, J. B. fuel 19824, 61, 155. Franklin, H. D.: Peters, W. A.; Howard, J. B. fuel 19826, in press. Hlppo, E. J.; Jenklns, R. G.; Walker, P. L., Jr. fuel 1979, 58, 338. Johnson, J. L. Am. Chem. Soc. Dlv. fuelchem. Prepr. 1975, 20(4), 85. Morgan, M. E.; Jenkins, R. 0.; Walker, P. L., Jr. fuel 1981, 60, 189. Nandi, S. P.; Johnson, J. L. Am. C h m . Soc.Dlv. FueIChem. Prep. 1979, 24(3), 17. Schafer, H. N. S. fuel 19790, 58, 667. Schafer, H. N. S. fud197ob. 58, 673. Schafer, H. N. S. fwl198Oa, 50, 295. Schafer, H. N. S. fuel19M)b, 59, 302. Tykr, R. J., Schafer, H. N. S. fuel 1980, 50, 487. Walker. P. L.. Jr.: Mahajan, 0. P.; Komatsu, M.. Am. Chem. SOC.Dlv. fuel Chem. Prepr. 1979. 24(3), 10.
Received for review July 24, 1981 Accepted June 1,1982
Financial support was provided by the United States Department of Energy under Contract EX-76-A-01-2295,Task Order 26. Presented in part at the InternationalConference on Coal Science, Dusseldorf, West Germany, Sept 1981.
Some R e s u b on Dynamic Interactton Analysis of Complex Control Systems Thomas J. McAvoy Department of Chemkai and Nuclear Engheering, Universiiy of Maryland,
college Park, Maryland
20742
The problem of analyzing dynamic interactions in complex control schemes is addressed. Several practical examples are chosen to illustrate when dynamic interaction analysls is requled. The methodology of analyzing interactlons involves calculating a frequencydependmt relative gain array (RGA) based on process dynamic considerations only. It is shown how the dynamic RGA can be used to indicate adverse loop interaction at the design stage of the process, and thls allows for design changes to improve control.
Introduction The relative gain array (RGA) was proposed by Bristol in 1966 as a means of measuring control loop interaction. For a number of years after its publication the RGA sat in the literature and was not used. Gradually industry began to use it and its importance and utility are well established today. As originally formulated, the RGA gave a steady-state measure of loop interaction. For a number of practical problems a steady-state RGA analysis is sufficient. However, for some it is necessary to extend the traditional RGA approach to consider dynamics. Recently, several
-
authors (Witcher and McAvoy, 1977; Bristol, 1978; Tung and Edgar, 1981; Gapnepaign and Seborg, 1982; and Jensen et al., 1982) have discussed dynamic extensions of the RGA. These authors also give a reasonably complete bibliography on the RGA. The purpose of this paper is severalfold. First, two practical examples are discussed which give insight into when a steady-state RGA analysis is not sufficient. Secondly, the question of how to analyze interactions in systems with purely integrating elements is examined. It is shown that a dynamic RGA analysis gives the same result as Woolverton’s (1980) heuristic RGA calculation
0 1984305/83/1122-Q042$0l.50/00 1982 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 43
method which is based on using the derivative of a controlled variable. Thirdly, one of the questions which arises in extending RGA analysis to large systems is examined. The 5 X 5 control problem which can occur in distillation columns is used for illustrative purposes. As pointed out by Shinskey (1977), the traditational RGA calculation which involves having all manipulative variables in manual is not appropriate for this 5 X 5 problem. Rather, to get meaningful results it is necessary to look at interactions when the reflux accumulator and reboiler level loops are closed. Shinskey’s analysis is extened to cover the case where level loops are loosely tuned. Closing level loops appears to be necessary in analyzing interaction in other large systems as well. One of the important potential uses of the RGA is to integrate the areas of process design and control. If simple approximate analytical dynamic models can be set up, then control loop interactions can be analyzed at the design stage of a process to indicate potential problems. With the resulta of the 5 X 5 distillation problem, this procedure for integrating design and control is illustrated on a 3 X 3 model of a light ends de-ethanizer published by Tyreus (1979). The operational problems which were actually experienced on the de-ethanizer could have been anticipated at the design stage of the deethanizer if the model were available. Cases Where Dynamic RGA Analysis Is Required The dynamic RGA definition that is used in this paper is based on the open loop matrix transfer function model for a process x = Gm (1) By calculating (G-’)T and multiplying corresponding elements, one can calculate a dynamic RGA. By setting s = io the frequency response of this dynamic RGA can be determined. For 2 X 2 systems the 1, 1 element of this array is given as (Witcher and McAvoy, 1977)
Note that the definition given by eq 1and 2 differs from that presented by Tung and Edgar (1981). An important question to be considered is when is dynamic interaction analysis necessary. Two conditions are required. First, the RGA should change substantially with frequency. Secondly, the RGA near the natural frequency of a loop should differ substantially from the steady-state RGA (w 0). Two 2 X 2 systems where dynamic interactions are important will be treated. Case 1. Head Box Control. The head box control system is shown in Figure 1. This example is given by Shinskey (1979), but he does not present a detailed dynamic model of it. A transfer function model is derived in the Appendix assuming that the accumulation of air takes place much faster than the accumulation of liquid. The open loop model is
Air
\e
Total Head
h Wire
Figure 1. Head box.
-
At steady state, All = 0 and the traditional RGA analysis calls for pairing h with m2. At high frequencies X,I 1.0 and the reverse pairing of h with ml is called for. As Shinskey (1979) points out, the nontraditional dynamic pairing of h with m, is used in actual practice. This is an interesting pairing because it has a steady-state RGA of 0. Pairing on a 0 in the RGA means that the h-ml loop can function only if the I-mz loop is closed. The reason that h-m, pairing works is because the ml variable has a much faster effect on h than m2has and both m, and m2 affect I slowly. Thus, the loop involving ml-h will be fast, Le., operate at high frequencies, and it can be tightly tuned. Since the h-m, loop has an RGA approaching 1.0 at high frequencies, it exhibits excellent dynamic performance even though ita gain approaches 0 at low a’s. Case 2. Distillation Column Example. McAvoy and Weischedel(l981) recently compared material balance and conventional dual composition control. They presented results for three different towers: A (low purity), B (moderate purity), and C (high purity). The resulta for column B will be considered here. Open-loop transfer function models for column B were determined from step testing a nonlinear column simulation. These models are shown in eq 5 and 6. Analyzer dynamics were modeled conventional control
-
-s
1
+ de
where a, b, d , and e are constants. Substitution of eq 3 into eq 2 gives S
A11
=s
+ ed
(4)
I
-0.516e-0*5s
0.562
(7.74s
+ 1)2
(7.1s
0.344e-0*5s
(15.88 + 1)(0.5s
+ 1)
+ 1)* -0.394
(13.8s
+ 1)(0.4s +
1) (5)
as a pure dead-time and perfect level control in the reflux accumulator and reboiler was assumed. Two PI feedback composition controllers were used with settings which were the best that would be found from simultaneous tuning. Additional details are given by McAvoy and Weischedel (1981). Figures 2 and 3 give resulta for both a 1-and 5min analyzer dead time. By comparing Figures 2 and 3 it can be seen that the magnitude of the dead time has an important effect on
44 r-
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 material balance control
Y
-0.805 (18.3s
+ 1)(5.6s +
1)
(5.76s + 1)(1.25s t 1 )
1 "
;I^ i
-0.465e-0*3S (28.3s
+
-0.055
1)(0.6% + 1 ) (3.3s
+
,0361
1)
determining which control strategy is best. For T = 1, Figure 2 shows that conventional control is superior to material balance control. Conventional control gives rise to smaller deviations from steady state and a faster response. For T = 5, Figure 3 shows that material balance control becomes more favorable. The conventional scheme exhibits a very sluggish response in this case. The simulations were run up to 200 min and terminated. At 200 min, XD for the conventional scheme was 0.9785 and X B was 0.0210 and both compoeitions were responding slowly. By contrast, the material balance scheme returns to steady state in approximately 150 min. The dynamic RGA can be used to explain the results in Figures 2 and 3. If eq 5 and 6 are substituted into eq 2 and s is set equal to io,the frequency response results shown in Figure 4 are obtained. It is important to note that Figure 4 is independent of analyzer dead time since it cancels in the calculation. As can be seen, the magnitude of the RGA for material balance control remains relatively constant as a function of frequency. By contrast, the magnitude of the RGA for conventional control changes substantially with frequency. To use Figure 4 it is necessary to estimate the frequency a t which the bottom and top loops operate. A well-designed control system will generally exhibit a decaying, oscillatory response. A good estimate of the frequency of this oscillation is the ultimate or natural fiequency of the loop with the other loop off. For conventional control these ultimate frequencies are shown in Table I together with the corresponding RGA values determined from Figure 4. For T = 1 the lower loop is essentially decoupled from the upper loop (A = 0.93) while the upper loop exhibits some interaction (A = 1.99). However, for T = 5 the magnitude of the interaction increases sharply and this is the reason that conventional control becomes inferior to material balance control. For T = 5, the magnitude of the RGA for the upper loop is 3.75 while for the lower loop it is 2.88. Both of these RGA's indicate a sluggish response which the results shown in Figure 3 confirm. If a still slower analyzer were used then the magnitude of the RGA's would increase further and, as Figure 4 shows, it would approach the steady-state value of 5.04. For a very slow analyzer the steady-state RGA would give an accurate measure of the amount of interaction present. In analyzing interactions in this example it was found that the results could be explained by considering only the magnitude of the RGA. Clearly the phase angle may be important in some cases. Further work on the significance of the angle of the RGA and its relationship to the question of when interaction can be favorable is required. Interaction in Systems Containing Integrating Elements One of the most common process variables that is controlled is level. It is well known that level can often re-
LEGEND
\
___
Material Balance
-
'I,
I
/
i
l
10
20
Conventional
40
30
50
I,min.
Figure 2. Plots of XD and XB va. t; T = 1. I
LEGEND
___
MATERIAL BALANCE
- CONVENTIONAL
c-
,'.--*' O I O L - L _ L ' 40
___--------
I'
'
'
80
'
'
120
' 160
'
'
*
200
t, min. Figure 3. Plots of XD and XB va. t ; T = 5.
Table I
1 5
0.5 0.215
1.99 3.75
1.19 0.33
0.930 2.88
spond as a non-self-regulatingpurely integrating element. If a level system is at steady state and the inlet or outlet flow is changed, then the level will not come to a new steady state until the vessel either overflows or runs dry. This behavior presents a problem in the traditional method of calculating the RGA where all manipulative variables are put into manual and one is step forced. This calculation procedure results in a ' s in the open loop gain matrix.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 45
5 = Fz - FIR
COLUMN B
1
°
,
0
Equations 7-9 and 11 form a set of algebraic equations for ~ which the RGA can be calculated by standard techniques. This RGA is m2
? .IO
I
.01
-__
-
MATERIAL BALANCE CONVENTIONAL I
,025
.05
.IO
.25
,50
LO
2.5
5.0
10.0
FREQUENCY ( r u d h n i n . )
Figure 4. Frequency response of dynamic RGA for tower B.
60 gpm 30 psi
m3
While this method avoids the calculation problems involved with using I , it can be questioned whether the interactions involving I and d l f d t will be the same. As an alternative a dynamic RGA can be calculated for this example in a straightforward, rigorous manner. If eq 7 to 10 are linearized, then the dynamic model in eq 13
[ [ =
180 p s i
(11)
r.]
~~~~~] -
l " 2 5 : 01s
145.71s -4.3041s
(13)
m3
-
results. As can be seen that last row of the G matrix does blow up as w 0. However, l f s can be factored from the matrix to give eq 14, from which (G-')T can be calculated G=
22.27 102.28 1.596 -3.421 0 145.7
0 1 0
102.28 -3.421 -4.304
1
(14)
as eq 15. If corresponding elements of G and (G-l)T are
r'
Ol
(G-')T= 0 1 O 0
LO 30 p s i
Figure 5. Woolverton example.
In a recent article, Woolverton (1980) presented an ad hoc approach to get around this problem. The system that he considered is shown in Figure 5. A stream containing two immiscible components flows into a vessel at a rate Fl. In addition to controlling Fl it is desired to control the pressure in the vessel, P1,and the interface between the two components, I. There are three control valves which can be manipulated to achieve control, ml, m2, and m3. The material balances which Woolverton used to describe the system are Fi = mi(Po - Pi) (7) J'2 = mz(P1- Pz) (8) F3 = m@i - P2) (9) dl/dt = F2 - FIR (10) Steady-state values are given in Figure 5. The difficulty in applying standard RGA analysis to this model comes from the fact that I drops out of the model at steady state. Kim and McAvoy (1981) have noted this same effect in distillation control analysis where the reflux accumulator and reboiler levels are involved. If m2 is step-changed holding m3 constant or vice versa, it is straightforward to show that I fm. To overcome this problem Woolverton proposed calculating the RGA based on dI/dt rather than I. If 6 is defined as d l l d t then
-
X
0 s J 0.01429 0.000191 0.4272 -0.002669 [0 0.006666
0.006475 -0.009035 -0.006666
1
(15)
multiplied, the RGA given by eq 12 results if 5 is replaced with 1. Thus, in this example the RGA for control of I is the same as the RGA for control of dI/dt. Also, even though the G matrix blows up as o 0, the RGA matrix is well defined. By treating the system shown in Figure 5 dynamically, the computational problems with traditional RGA analysis disappear. In the following section a 5 X 5 distillation example is treated which illustrates that the standard RGA approach can lead to another difficulty when level loops are involved.
-
A 5 X 5 Distillation Problem In a distillation column, shown in Figure 6, often there are five variables that one wishes to control. These are xD, xB, P, lA, and l p There are also five manipulative variables: D, B, L,S, and Qc. It is straightforward to write down and linearize a dynamic model relating all the manipulative and controlled variables. Of particular interest is the way in which D and B enter into the model. These two manipulated variables only appear in the equations describing lA and 1R, respectively (Kim and McAvoy, 1981)
Equations 16 and 17 hold under the assumptions that q = 1 and constant molal overflow exists. These two assumptions can easily be relaxed. If one calculates transfer
48
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
I 7 COOLANT-Qc -
&
li
STEAM-S
Figure 7. Recycle system.
BOTTOMS-B
Figure 6. A 5
X 5
distillation control problem.
functions from eq 16 and 17, then the well-known result that level responds as a purely integrating element to changes in flow is obtained. Additionally, the only place where D and B appear in the dynamic column model is in eq 16 and 17. Thus, if L, V , and Qc are held constant and either D or B is changed, xD, xB, and P are not affected. These controlled variables will be affected eventually when the reflux accumulator or reboiler overflows or runs dry. However, in the linear region near steady state, D only affects lA and B only affects la. Taking the above effects into account gives an open loop transfer function matrix with the form of eq 18. A dy-
c
1
1 0 PAAA~
+-
1
-
O
D
PAAA~
1
1
--
PRARS
0
O
0 0 L
AI-
B -
L-
v Qc
(1Sj
-1
namic RGA can be calculated as discussed earlier. Because of the 0’s in eq 18 this RGA has the form shown in eq 19.
iA i, E , .rg P 1
0
0
0
0
0
0
0 (19)
A=[:
0 0
0 0
feasible only pairings .
Equation 19 illustrates the problem with the standard relative gain approach. This standard approach indicates that D and B should only be used for level control and not for composition control. This result is in conflict with current industrial practice where material balance control (Shinskey, 1977) is used. With a material balance scheme, D controls X D and V controls xg. Shinskey (1977) was the
first to recognize this conflict and he proposed the following resolution of it. In calculating the RGA all manipulated variables are put into manual, and one variable at a time is step-forced to generate G. The RGA is then calculated by multiplying corresponding elements of G and With the standard RGA approach when D and B are changed, L and V are fixed, and therefore D and B never affect any other variable but 1A and le In practice, D and B affect xD, xB, and P by changing ZA and 1% Thus, it is through the level controllers that D and B have their effects. This leads to the conclusion that to properly analyze interactions in the system shown in Figure 6 the level loops have to be operational. In Shinskey’s analysis he noted that the speed of the pressure and level loops is significantly faster than the speed of the composition loops in most towers. Thus, it should be possible to tune the pressure and level loops very tightly. Therefore, Shinskey assumed perfect pressure and level control and reduced the 5 X 5 problem to a 2 X 2 problem involving composition loop interactions. The fact that level loops should be closed in analyzing interactions in large systems is probably a general result. Consider the system shown in Figure 7 and suppose that one is interested in interactions between the bottom loop and other loops, e.g., the reactor control loops. Unless the level controller is operational on the surge tank,changing B will have no effect on the feed flowrate to the reactor. Although at this time it appears that all level loops in a process should be closed, there may be some cases where some should be open. Further research and experience will help to provide guidance on this point. Since Shinskey’s approach to analyzing interactions when level loops are present is a steady-state approach, it was necessary for him to assume perfect level control. It is possible to extend his approach to the case of averaging level control (McAvoy, 1978) by using a dynamic RGA analysis. With averaging level control a surge tank is used to filter flow disturbances and the controller is tuned loosely. If proportional averaging level control is used on the surge tank shown in Figure 7, then R and B are related as
R / B = l/(Os
+ 1)
(20)
where 0 is the time constant of the fiiter (McAvoy, 1978). In carrying out the RGA analysis, R and the surge tank level would be dropped as variables under consideration and the dimenstion of the problem reduced by one. Where R appeared it would be replaced with B/(Os + 1). An advantage of this approach is that the size of the RGA to be considered is smaller. Also, using dynamic RGA analysis one can vary 0 and therefore the sue of the surge tank in the system and study its effect on interactions.
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983 47 0.3 3 3e- dE 20s+ 1 0.038(182s
S
+
1)
(27s t 1)(1Os + 1)(6.5s + 1) 0.313 k(29s
-0.1
+ 1)(17s + 1 )
S
0.36
-0.12~3~
-
+ 1)(8s + 1 1.563
s(14s
S
-2.222
s ( l 0 s + 1)(6.7s
For cases where tight level control is used, e.g., in reflux accumulators, Shinskey's analysis calls for dropping level as a variable in the RGA analysis. A manipulative variable, e.g., reflux, would be dedicated to controlling level and also removed from the RGA analysis. This dedicated variable would change in response to changes in other manipulative variables such that the level stays constant. The transfer function models given by eq 5 and 6 were determined in this manner. In the nonlinear dynamic column simulations the levels in the reflux accumulator and reboiler were held constant as step testa in manipulative variables were made. For eq 5, D controlled the reflux accumulator level and changes in D were equal and opposite to changes in L and V. For eq 6,L controlled the reflux accumulator level and changes in L were equal and opposite to changes in D and V. When averaging level control is used, the appropriate closed loop transfer functions between the manipulated variable dedicated to level control and the other manipulated variables in the system can be written down. These transfer functions would involve closing only the level loop, and by using them the variable dedicated to level control can be eliminated from the analysis. This procedure is illustrated below. Integrating Process Design and Process Control One of the most important potential uses of the RGA is to indicate control problems at the design stage of a process. If simple, approximate, dynamic models can be set up, then an RGA analysis can be used to assess potential interaction problems. With existing simulation packages it should be possible to estimate accurately the steady-state gains for the process models although some care may have to be taken (Kim and McAvoy, 1982). If simple estimates for proteas dead times and time constanta can be gotten, then it is straightforward to carry out an RGA analysis. To illustrate this procedure, a model of a de-ethanizer will be used. The de-ethanizer is shown in Figure 8. As can be seen, there are three controlled variables, temperature, pressure, and accumulator level, and three manipulative variables, heat input, distillate flow, and reflux flow. The actual variable pairings that were implemented are shown. The tower was experiencing severe operational problems. To analyze these problems Tyreus (1979)experimentally forced the tower and determined the model shown in eq 21. This model clearly was not available at the design stage of the de-ethanizer. If one goes to the trouble of developing a rigorous dynamic model, either through simulation of experimental testing, then control approaches such as the inverse or direct Nyquist array would be more appropriate to analyze interactions in the system. However, in principle it should be possible to develop approximate models which would be available at the design stage and which would yield the same information as eq 21. If such approximate models can be developed, then the RGA could prove to be a valuable tool for integrating process design and process control. To analyze interactions in the de-ethanizer, the accu-
+ 1) s(17s + 1)
--I
Figure 8. De-ethanizer example.
mulator level loop will be closed. Tight level control will be assumed so that 0.313 i A = o = 429s 1)(17s 1)QB 2.222 1.563 s(l0s 1)(6.7s l)D - s(17s 1)L (22)
+
+
+
+
+
In the deethanizer control system L controlled 1A. Solving eq 22 for L and substituting it into eq 21 gives the resulting model for perfect lA control as eq 23,where the G,'s are
the original elements of eq 21. Equation 23 can be substituted into eq 2 and a dynamic RGA can be calculated. However, for this particular example a low-frequency analysis will suffice. For small values of s eq 23 can be approximated as P
T
For small values of s the relative gain calculated from eq 24 is
-
-
-57.9
+58.9
+58.9
-57.9
A = [
P
]5
QB
48
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 1, 1983
The large negative All element indicates that the actual pairings that were used on the de-ethanizer are not feasible. It is straightforward to show (Gagnepaign and Seborg, 1982) that if the T-QB,P-D pairings are used, the resulting system will be unstable. Thus, the RGA gives an accurate assessment of the control problems that were actually experienced. This RGA information would have provided the control system designer important insight into potential problems had it been available when the tower was designed. Tyreus used inverse Nyquist array techniques to reconfigure the control system to solve the problem. If averaging level contrpl were used in the 2,-L loop, then .L would be related to QB and D as
.L= QBG31Gc
+
DG32GC
(26) 1 + G33Gc 1 + G33GC where G, is the transfer function of the averaging level controller. G, includes the gain of the level transmitter which is assumed to be very fast. If we know G33, G, can be chosen to achieve a desired averagjng level response. Equation 26 can be used to eliminate L from the dynamic RGA analysis. Conclusions Several aspects of dynamic interaction analysis have been treated. It has been shown that dynamic RGA analysis is required when the RGA changes substantially with frequency, and ita magnitude at the natural loop frequency differs substantially from ita steady-state value. It has been shown that dynamic RGA analysis can be used in a straightforward manner to treat integrating systems where the steady-state gain matrices blow up. A 5 X 5 distillation problem has been used to discuss the problem of directly extending RGA analysis to complex systems. Rather than having all manipulative variables in manual when an RGA analysis is carried out, level loops in the process should be closed. In the present analysis either tight or averaging level control can be assumed. Lastly, one example showing how the RGA can be used to integrate the areas of process control and process design has been treated. With the RGA it should be possible to uncover control problems at the design stage of a process and thereby take steps to avoid such problems. Acknowledgment This work was supported by the National Science Foundation under NSF Grants ENG 76-17382 and CPE8025301. Appendix The head box is shown in Figure 1. To derive a dynamic model for this process, linearized flow relationships will be used. For example, the air inlet flow is
F,,
= mls(P0 - Pis) + mAPo - p13 + m1,(Po - P1)
(A-1) where the subscript s indicates steady state. Similar relationships are used for all flows through valves. Dynamic mass balances on air and liquid are air balance dn M.W.- = Q(PO - PI,) + mls(P0 - PI) + dt liquid balance
The total head in the head box is given as h = Pl + pgl
(-4-4)
Deviation variables can be defined as ‘6 = h - h,
(A-5)
1=1-1, (A-6) The accumulation of air is assumed to be fast and dnldt is taken as 0 in eq A-2. Substitution of eq A-4 to A-6 in eq A-2 and A-3 gives di/dt = bfiz - d h (A-7) h = a f i , + el (A-8) where (A-9)
Taking the Laplace transform of eq A-7 and A-8 gives eq 3. Nomenclature a = constant given by eq A-9 A = area of head box A A = area of reflux accumulator AR = area of reboiler b = constant given by eq A-10 B = bottoms flowrate d = constant given by eq A-11 D = distillate flowrate e = constant given by eq A-12 F = feed flowrate FqiD= inlet air flowrate to head box g = acceleration due to gravity G = open loop matrix of process transfer functions G, = averaging level controller transfer function including gain of level transmitter h = total - head in head box i = d-1 ki = constant I = interface in Woolverton example 1 = level in head box 1A = level in reflux accumulator 1R = level in reboiler L = reflux flowrate mi = manipulated variables n = moles of air in head box Pi = pressure QB = reboiler heat duty Qc = condenser heat duty R = F a / F l s for Woolverton’s example and flowrate out of surge tank (Figure 5) s = Laplace variable S = steam flowrate t = time T = temperature V = vapor boilup x = controlled variable X B = bottoms product composition X D = distillate product composition Greek Letters B = averaging level control time constant hi, = RGA element of A A = RGA matrix p = density P A = density in reflux accumulator pR = density in reboiler
Ind. Eng. Chetn. Process Des. Dev. 1883, 22,49-53 T
= dead time
w = frequency w, = ultimate frequency - = Laplace transformation
-= deviation variable
Literature Cited Brlstol. E. “Recent Results On Interactions In Mukivarieble Process Control”; A I C M 71st Annual Meeting, Mlaml. FL, Nov 1978. Gagnepalgn, J.; Seborg, D. Ind. Eng. Chem. Process Des. Dev. 1982, 27, 5. Jensen. N.; Fisher, D. G.; Shah, S. L., submltted for publication in AIChE J . , 1982. Kim, Y. S.; McAvoy, 1.J. Ind. Eng. Chem. Fundam. 1981, 20, 381. Klm, Y. S.; McAvoy. 1. J. “Computing The Relethe @In for Pressure and Composition Control of a Single lower”; Proceedings of Automatic Con-
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trol Conference, Arlington, VA, June 1982. McAvoy, T. J. “Process Dynamics and Control”; Adlo Course, American Chemical Society: Washington, DC, 1978 Unk 4. McAvoy, 1.J.; Welschedel, K. “A Dynamic Comparison Of Materiel Balance vs. Conventbnal Control Of Dlstlktbn Columns”; Paper 107.2, International Federation of Automatic Control Congress, Kyoto, Japan, Aug 1981. Shinskey, F. G. “Distlliatlon Control for Productivity and Energy Conservation”; McQraw-Hill: New York, 1077; Chapters 1.2. Shlnskey, F. 0. “Process Control Systems”. 2nd ed.; McGraw-Hill: New York, 1970; Chapter 8. Tung, L.; Edgar, 1.AIChEJ. 1981, 27, 690. lyreus. B. Ind. Eng. Chem. Process Des. Dew. 1979, 78, 177. Wkcher, M.; McAvoy. 1.J. ISA Trans. 1977, 18, 35. Woolverton, P. Intech. 1980. 27(9), 63.
Receiued for review August 10, 1981 Accepted June 18,1982
Representation of Petroleum Fractions by Group Contribution Vlastlmll RuilEka, Jr.,’ Aage Fredendund,’ and Peter Rasmwsen Instkuttet for Kemlteknik, Danmerks Tekniske H0)skok, DK-2800 Lyngby, Denmark
Most oil and gas processing operations require estimation of phase equilibria. Proper characterization of the complex mixtures encountered in petroleum fractions is a major problem. I n this work, the UNIFAC groupcontribution methcd for predicting vapor-liquid equilibria has been used as a basis for describing complex petroleum fractions in terms of model compounds. Standard procedures may be used to estimate critical properties, acentric factors, and molecular weights for the model compounds. This allows the inclusion of complex petroleum fractions in already available, generalized methods for phase equillbrlum calculations, based on equations of state or the UNIFAC group-contribution method. Good results are obtained for lower and medium molecular weight petroleum fractions at temperatures up to 600 K. At higher temperatures, the method may fail due to present limitations of the UNIFAC method.
Reservoir oil and gas fluids are complex mixtures of mainly paraffins, naphthenes, and aromatic compounds. For oil fractions of molecular weight higher than about 100, it is unpractical to list all of the compounds present. Hence one of the major problems in phase equilibrium calculations involving such fractions is the representation of the many different hydrocarbons in terms of a few properly averaged characteristic parameters. This work describes a new method of characterizing heavy petroleum fractions. The method is based on the UNIFAC group-contribution model for predicting vaporliquid equilibria (see, e.g., Fredenslund et al., 1977) and purecomponent vapor pressures (Jensen et al., 1981). The model is described in the Appendix. In additidn, a true boiling point analysis (TBP) and, if available, paraffinnaphthene-aromatic (PNA) analysis are used. A TBP curve for a complex petroleum mixture (here a lean absorber oil) is shown in Figure 1. The method suggested in this work entails the following steps: (1) division of the TBP curve into a number of subfractions; (2) definition of model components for each subfraction in terms of UNIFAC groups such as -CH3, -CH2-, and ACH (aromatic hydrocarbon group); (3) adjustment of the number of groups in each model compound so as to match the midvolume boiling point for each subfraction. The result of this procedure is a set of well-defined model compounds which represent the complex petroleum ‘Onleave from Prague Institute of Chemical Technology, 166
28 Prague 6,Czechoslovakia.
mixture. Molecular weights, acentric factors, and critical properties of the model compounds may be readily established by use of standard procedures. As indicated in Figure 1, usually 5-10 subfractions are required, each containing three model compounds: one paraffinic, one naphthenic, and one aromatic. The approach has been found to yield reliable results for temperatures up to 600
K. Data Requirements The method requires the following data: a complete TBP-analysis (boiling point temperature vs. liquid volume percent boil-off), a PNA (paraffin-naphthene-aromatic) analysis, preferably for each subfraction, and density, preferably for each subfraction. Often, all of these data are not available for complex hydrocarbon mixtures. Various procedures for transforming incomplete information on C,+ fractions into satisfactory TBP analyses are outlined by Erbar (1977). Although primarily developed for absorber oil C6+ fractions, the procedures can also be used for other petroleum fractions. A review of different categories of basic data available for C,+ fractions is given by Wilson et al. (1978). Procedure The TBP curve is broken into five to ten subfractions as indicated in Figure 1. The number of subfractions must be kept as low as possible so as to keep the computer requirements reasonably low. For the PNA analysis, no distinction is made between volume, weight, or mole percent. This is a reasonable 0 1982 American Chemical Society
