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Transient behaviour of multicomponent distillation columns
Transient Behaviour of Multicomponent Distillation Columns H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY and J. A. CHALLIS Introduction The work described here grew out of an earlier investigation I, 3 into binary systems. A computer programme was developed to give the transient response of a column, and Armstrong and Wilkinson 2 compared the computed and measured results for a particular binary system. Extension of the computer programme to multicomponent systems was straightforward, and was carried out about two years ago. This paper describes the means used to obtain a mathematical solution and presents a comparison with experiment. Mathematical Theory As in the earlier work3, a number of assumptions were made to simplify the mathematics. In principle, none of these is enforced by the method of solution, since almost any problem which can be defined mathematically can be solved with a digital computer. The computing time, and even more the programming time, would however have been increased considerably in removing the assumptions, which were the following: (a) The vapour and liquid flow rates (including feeds and offtakes) are assumed to be known and constant during the transient. Vapour and liquid rates can be different on each plate, to suit a heat balance taken before the transient starts, but no arrangements are made for correcting the heat balance (and hence the flow rates) as the transient proceeds. (b) The vapour hold-up on each plate is neglected, and the liquid hold-up is assumed constant during the transient. The liquid hold-up can be different for every plate, but no arrangements are made to correct it for changing conditions during the transient. (c) The pressure in the column is equal at all points and does not change during the transient. (d) There is perfect mixing on each plate, so that each plate has a unique value for vapour and for liquid composition. (e) The calculations are made for' theoretical plates'. This procedure seems to be unavoidable at present, since there appears to be no extension to multicomponent systems of the plate efficiency which can be defined for binary systems. With these assumptions, a mass balance for thejth component around the rth plate gives d Hr d! X,j = Vr-IYr-l,j- VrYrj-(Lr+Pr)xrl +Lr+lxr+l,j
+ Frzrj
(1)
where plates are numbered from 1 to n starting from the bottom of the column, and : Hr is the liquid hold-up in moles on the rth plate. X,I is the proportion of the jth component in the liquid on plate r.
Yrj is the proportion of the jth component in the vapour above plate r. V, is the flow rate in moles/sec of vapour leaving the rth plate for the (r + l)th. L, is the flow rate in moles/sec of liquid leaving the rth plate for the (r-l)th. Pr is the flow rate in moles/sec of liquid product withdrawn from plate r. Fr is the flow rate in moles/sec of a feed to plate r . Z rj is the known and constant proportion of the jth component in the feed to the rth plate. Equation 1 can be taken to apply to the reboiler (r = 0) and to the condenser (r = n+ 1) if it is understood that V-I
=
Lo
=
Ln+2
= 0
and Vn+1 = 0 when no vapour product is taken from the column : otherwise Vn +1 is the flow rate of this vapour product in moles/sec. In addition to equations 1 there is a relationship, generally non-linear, betweenYrj and Xrj at the assumed column pressure: (2)
As shown, K for each component is a function of the temperature Tr on the appropriate plate and through this (as below) of Xrj . From the definition of Xrj it follows that
L X'I
= 1
(3)
j
The corresponding relationship for Yrj must also be true LYrj = L K;(Tr)X'1 = 1
(4)
j
and given equation 3, equation 4 will only be true when T, has its correct value. The method chosen for solving the equations was the simplest one that seemed likely to succeed. The temperature Tr on each plate was assumed constant during each time step, and equal to its known value at the beginning of the step. There is then one set of n + 2 equations, equations 1, for each component, and the sets can be solved independently to give Xrj at the end of the time step. A check is then made to see that equation 3 is true for each plate and an alarm is set if the discrepancy is more than a selected amount. The values of Xrj are corrected proportionately so that equation 3 is exactly true for each plate. Equation 4 is then solved by approximation to find each of the n + 2 values of T r • This procedure is repeated for the next time step. The method was checked by computing transients for a binary system and comparing the results with those obtained for the same problem by means of the binary programme described elsewhere 3 • To solve equations 1 the method described in reference 3 was 303
1816
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND J. A. CHALLIS
taken over without change. If the right-hand side of equations 1 is replaced for brevity by .p" then the method is to replace equations 1 by
= (I-().pr(tO)+().pr(tl)
Hr Xrj(t;)-;rj(t o)
(5)
1- 0
where () is a constant. When () = t the process is the one given by Crank and Nicolson4, and the error in the time step St is 0(St)3. For () = 1 the error is O(St)2, but the process is better for finding the steady-state values. If.p is a linear function of x, the process with () = 1 gives the steady state in one step if I/St is first put equal to zero. When.p is non-linear, the process (with () = 1 and liSt = 0) leads to a generalization 5 of the Newton-Raphson method for finding the steady-state values. Equation 5 leads to -()Vr- IK,-1. jSxr-1.j+ [()(Lr+Pr+ VrKrj )+ !]SXrl -()Lr+ISxr+l ,j
= .p,(to)
(6)
This has the form BrjSX,-hl+ CrjSXrj+ DrSX,+hl
=
Grl
(7)
and from the end conditions BOI
=
Dn+!
(8)
=0
Thus for each component the SXrj are to be found from a set of equations COjSXOI + DoSxlj = GOI BIjSXOj+C li SXIi+D ISX2j = Gl/ (9) BnjSXn_l,j+ CnjSXnl + DnSXn+I ,1 = Gnl Bn+!,jSXnl + Cn+1 ,jSXn+l,j = Gn+1.1
}
Equations 9 are reduced, by substitution, to the equivalent set COjSXOj + DOSXII Cg>SXIi + DISxzj
= GOj = GW = GW
CWSXnj+ D nSXn+1.1 q~\,ISxn + I,1
where C,j(l)
-
} (10)
= G~~I,I
Crj-cro B'I D ,-I
(11)
r-l.i
G(I) = G
'}
G(I) rj -~ C(l) ,-1,1
(12)
r-I,I
Then by back-substitution SXn +1,j
=
G~~I,;/C~~I,I
(13)
and in general SXrj =
~W {GW - D,SXr+l,j}
(14)
This procedure is repeated for each value of j in turn to give all the X,j at the end of the time interval St. The subsequent approximation to find Tr is carried out by linear interpolation. Let T, = K j(Tr(l)x rl and yW = K j(T,(2»x rj, where the X,j are known. A third temperature T,(3} is obtained from
1-2yg> T,(3) r -
T,(1)+ r
T"L.. y"
j
(2) _ "
j
(I)
L.. y'} i
(15)
Now T,
304
1817
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
The data must satisfy the following restrictions: (i) If A is an integer such that 32(A -1) < n
~
32A
(16)
then 8A+J+AJ
~ 203
(17)
This condition restricts the number of plates and components which can be dealt with simultaneously and arises from the limited storage (about 8,000 words) of the computer. For example, with 6 components the number of theoretical plates must not exceed 448. (ii) J ~ 32. This restricts the number of components. (iii) F ~ 1, L ~ 1, P ~ 1, VK ~ 1. These inequalities are achieved by scaling. (iv) s ~ 8. Results are punched out in binary form after p time steps. A separate short programme converts the results to decimal form. The time, and the temperature and compositions on each plate, can then be printed. Experimental Equipment The arrangement of the experimental equipment is shown in Figure 1. The column was a standard glass bubble-cap column 8 of nominal 6 in. diameter, with four plates above the feed and four below. Each plate had one bubble-cap, and the plates
were separated by conventional 6 in. bore distance pieces each 6 in. long as shown in Figure 2. Top and bottom products were controlled by regulating valves and measured by rota meters. The sum of the top and bottom products shown by their rotameters was always set to equal the feed rate indicated by a third rotameter. Feed was introduced into the column as liquid at the boiling point, this being secured by a feed pre-heater with a reflux condenser attached. A total reflux, glass spiral condenser was fitted above a reflux divider, returning condensate to the column at a temperature near to the vapour equilibrium temperature of the top plate. The reboiler had a capacity of about 45 litres, and was fitted with a sight glass so that the constancy of the level could be checked during a run. Two 3 kW electric heaters were fitted, the electrical input being continuously variable. Temperature measurements were made on each plate by gasfilled platinum resistance thermometers fitted through side arms in the column walls, with the measuring elements located about 1 in. above the dome of the bubble-cap. Each resistance element could be connected in turn through a selector switch to a five-decade Wheatstone bridge. K
Vent to atmosphe-re
A _ _~~~--B
Cooling water
C ----I--..,JII
D---.J
'----E
"'-t----J
Feed regulating valvE"
1/ kW fE'ed pre-heater
Vent
/Rolatamell"r
A - Downcomers B - Bubble cap C - Resistance thermometer bulb D - liqUid sample point E - Vapour sample pOint F - Feed point G - Vapour inlet H - Reflux return J - Riser K - Reflux timer section connection
Rotamele>r
G---* To draJn
H
Figure 1. General arrangement and flow diagram of the S-plate distillation columll 20-A.R.C.4
Figure 2. 305
1818
Column details
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND J. A. CHALLIS
Feed to the column was drawn from a tank, and the products were stored in two product tanks. A fourth tank was used for mixing the feed between experiments. The components used were selected to meet the following requirements: (a) A reasonable degree of separation with 8 plates. (b) Easy analysis of the mixture by means of an infra-red
spectrometer. (c) Small difference between the molal latent heats. The system chosen was toluene-iso-octane-paraxylene, which fitted the requirements satisfactorily. A temperature difference across the column of 20 could be achieved, and the molal latent heats 9 are 7'98, 8·63 and 7·40 kcal/mole respectively, at the boiling point of the pure component. Preliminary experiments were made to find the dynamic hold-up of the column. With the column 'operating at total reflux the heaters in the re-boiler were suddenly switched off and at the same time the valve in the liquid return line to the re-boiler was closed. The liquid in the column was allowed to drain out, and was measured. Heat storage in the electrical heaters in the re-boiler was not enough to affect the measurements. 0
In order to ascertain the overall efficiency of the column it was allowed to come to a steady state under total reflux and measurements were made of the compositions and temperatures. A set of five runs to the steady state was then carried out on the computer with 8, 7, 6, 5 and 4 theoretical plates respectively, starting in each case with the experimentally obtained compositions. A comparison of the five curves with the experimental values showed that eight theoretical plates gave the closest agreement. Results With a three-component system two measurements are needed to fix the composition. Owing to difficulties with the infra-red equipment which was to be used for analysis, two measurements could not be obtained and the comparison is made for temperature only. Figures 4--7 show computed transients and the corresponding experimental points obtained in two trials.
- I- The~ret i cia I
.U
o
110I - - fx First trial
o Second trial
..
109
11\
o
CII
-j....--- r--
~
E
Z 108
y
6
--
2
~
°
~:1 E 10 ., 7 jO ~
V
~V:
CII
a. CII I-
106
--
° °
. ~~
"f
104 0
,
~
·~o
105 : 100
I-'"
.~~
uu
ZUU
400
500
600
Time
o
Figure 5.
0·04
003
0·05
0·06
Boil-up
Figure 3.
moles/sec
Variation of hold-up with vapour rate
u
°0
111
~. V
x'
106
°
110 ~
:J
109
"§
it
I(
~
107 500
Time
Figure 4.
106 min
X
,
Ox
x
It
~.
1 300
306
1819
I 400
..: o
Second trial
I 500
Time
Figure 6.
[Plate 7)
'
I ! I
x First trial
- Theoretical [ I
200
--
I
;.-
100
~
j
~~
. o
y
F
x
~108 ° E
400
[Plate 5]
-~ ~
o
o 107 ~--~--~---+--~----+---~--- r~
JOO
min
The experimental points were obtained in the following way. The column was started under total reflux and allowed to run until changes of temperature became slow. A mixture of toluene, paraxylene and iso-octane in the molar proportions 0'368:0'323 :0'309 was then fed to the column at the rate of 0'0251 mole/sec. At the same time the top and bottom offtakes were set to 0'0061 and 0'0190 mole/sec, respectively.
u
200
700
0·08
007
Figure 3 shows the average hold-up on each plate as a function of vapour rate. The slope of the curve is not great in the region of interest, and when the vapour rate was varied a mean value of hold-up was used in the computer.
100
-
I--
f
I
,1h"/ '
~
00
[Plate 4 (feed»)
I 600
I 700 min
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
u
0
Appendix-Flow Chart of Computer Programme
115 -
Theoretical
114
,First trial OS~~
00 0 0
113 q, 112
j
:::J
-;;; q,
x x/ V xV 0 00
x
x x
L
8.'" E
!--" 000
/
xV / x51 0
0
00
~/
x ~o
I/x
1-110
fx
x 0
~~
0
x
><
109
x
x
)(,),;)'
0
1~
:lo
107
100
200
/V
,.,
-
300 400
500
Time
600
700 min
Figure 7. [Plate 3]
When the changes due to introduction of feed had slowed down, the reflux ratio was changed by adjusting the top and bottom off takes to 0·0152 and 0'0099 mole/sec, respectively. This is the point from which the transients in Figures 4-7 begin. After a convenient time (about 180 min) the top and bottom off takes were re-set to their first values, and finally after a further 140 min they were set to 0·0152 and 0'0099 mole/sec again. Calculations on the computer followed the same course. They were started from the measured compositions in the column at the time when the reflux ratio was first altered. Since subsequent changes of reflux ratio were made before the column had reached a steady state, the calculations were continued beyond the points at which a change was made in order to show the further course of the transient. Further calculations were then made, starting from the points at which the changes were made, for comparison with the experimental results. In Figures 4-7 the computed curves have been moved slightly along the temperature axis to achieve better agreement. Their scale has not been changed. The two important characteristics for purposes of control, namely the shape of the curves and the total amount of the changes produced, are therefore preserved. Conclusions The limited comparison given here indicates the same degree of correspondence between theory and experiment for multicomponent systems as has been found before 1,2 for binary systems. The method of calculation is sufficiently flexible to allow considerable relaxation in the assumptions. This may be important in dealing with practical columns, where some of the assumptions made here (such as perfect mixing on the plates) may not be appropriate.
No
Punch t, Tand x No
Are an loxl < Ex '? Yes
The authors are grateful to British Petroleum Company Research Centre at Sunbury-on-Thames for help with the infrared analysis ofliquid samples," to their colleague Mrs. J. M. Hi/ton, who wrote the computer programme and carried out the calculations described here, and to the Directors of Constructors John Brown Limited for permission to publish this paper.
No ~------~E-------~
307
1820
Put loxl max. on Output Staticizer
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND J. A. CHALLIS leferences ROSENBROCK, H. H. An investigation of the transient response of a distillation column. Part 1: Solution of the equations. Trans. lnstn chem. Engrs, Lond. 35 (1957) 347 ARMSTRONG, W. D. and WILKINSON, W. L. An investigation of the transient response of a distillation column. Part 11: Experimental work and comparison with theory. Trans. lnstn chem. Engrs, Lond. 35 (1957) 352 ROSENBROCK, H. H. Calculation of the transient behaviour of distillation columns. Brit. chem. Engng 3 (1958) 364, 432, 491 CRANK, J. and NICOLSON, P. A practical method for numerical evaluation of solutions of partial differential equations of the heatconduction type. Proc. Camb. phi!. Soc. 43 (1947) 50
5
6
7
8 9
BOOTH, A. D. Numerical Methods, p. 155. 1957. London; Butterworths HrLTON, J. M. A computer programme to calculate the transient response of a multicomponent mixture in a plate type distillation cplumn. C.1.B. Report No. R&D/133 (1959) (unpublished) HALEY, A. C. D. Deuce-a high-speed general-purpose computer. Proc. lnstn elec. Engrs 103, Pt. B, Suppl. No. 2 (1956) 165 SMITH, J. C. and KELM, E. F. An all-glass bubble-cap column. Chem. Engng 58 (1951) 155 PERRY, J. H. (Ed.) Chemical Engineers' Handbook, p. 215. 1953. New York; McGraw-Hill
Summary
he equations governing mass-transfer in a multicomponent 'stem have been solved using a digital ·computer. Experimental ark in an eight-plate column distilling a mixture of toluene, iso;tane and paraxylene has been carried out to check the calculations.
The computer programme and the experimental equipment are briefly described. A comparison is made between computed and experimental results.
Sommaire es equations qui regissent le transfert de masse dans un systeme it ements multiples ont ete resolues par l'emploi d'un calculateur Imenque. Un travail experimental, dans une colon ne it huit .ateaux, pour la distillation d'un melange de toluene, d'iso-octane de paraxylene, a ete effectue pour verifier les calculs.
On decrit brievement le programme du calculateur et I'equipement experimental. Une comparaison est faite entre les resultats de I'experience et ceux du calcul.
Zusammenfassung ie Konstruktion einer automatischen Regelanlage fUr Destillations)Ionnen setzt eine Kenntnis des Ansprechverhaltens der Kolonne If Anderungen in den Betriebsbedingungen voraus. Das hei13t so, wir mlissen in der Lage sein, das Ansprechverhalten der Kolonne If eine bestimmte Anderung des Rlicklaufverhaltnisses, der Auf:izung, der Zusammensetzung des zugeflihrten Materials, usw. zu :rechnen. Wenn die StOrungen entsprechend klein sind, verhalt :h die Kolonne linear und die gewlinschten Angaben kann man aus :m Ansprechverhalten bei einer plotzlichen sprunghaften Anderung :r Betriebsbedingungen erhalten. Daraus la13t sich das Ansprechrhalten auf eine sinusformige Anderung oder auf irgendeine rlderung berechnen. In einem frliheren Aufsatz (Trans. Insfn. Chem. Engrs, 1957, 35, 7-351) wurde ein Elektronenrechner-Programm beschrieben, mit :ssen Hilfe das zeitliche Ansprechverhalten einer Destillations)Ionne flir Zweistoffgemisch berechnet werden kann. Die leichungen wurden nicht linearisiert, soda13 man das Verhalten bei einen oder gro13en Anderungen erhielt. In einem weiteren Auftz von W. D. Armstrong und W. L Wilkinson (ibid. 352-361) lrden Versuchsarbeiten beschrieben, und ein Vergleich zwischen esen und den mit den Digitalrechnern erhaltenen Ergebnissen rrde vorgenommen. Das Elektronenrechner-Programm wurde verallgemeinert, soda13 aus mehrereren Komponenten bestehende Gemische behandeln ,nn. Die Voraussetzungen bleiben die gleichen wie diejenigen flir ,s Zweistoffgemisch. lnsbesondere sind die Gleichungen nicht learisiert. Es sind Versuchsarbeiten mit einem Dreistoffgemisch I Gange, bei denen eine Kolonne mit einem Durchmesser von 6 Zoll Id 8 Platten benutzt wird; die Materialzufuhr erfolgt an der 4. atte. Die mit dieser Destillationskolonne erhaltenen Ergebnisse ~rden mit dem aus dem Elektronenrechner-Programm erhaltenen
Ergebnissen verglichen, wodurch eine Dberprlifung der Ausgangsvoraussetzungen der theoretischen Arbeit und eine Erweiterung der frliheren Arbeiten mit Zweistoffgemischen moglich ist, Die flir die Anwendung dieses Programms zur Konstruktion von Regeisystemen empfohlenen Methoden werden ebenfalls beschrieben. Zunachst wird der Gleichgewichtszustand der Kolonne mit Hilfe des DigitaIrechners festgestellt, und danach wird eine der Betriebsbedingungen (zum Beispiel das Rlicklaufverhiiltnis) durch eine kleine sprunghafte Anderung verandert. Der Elektronenrechner berechnet die Anderung in der Zusammensetzung der Fllissigkeiten an jeder Platte als Funktion der Zeit wahrend die Kolonne ihrem neuen Gleichgewichtszustand zustrebt. Dieses Verfahren wird flir andere Arten von StOrungen wiederholt und aus den Ergebnissen wahlt man diejenige Platte oder Platten aus, die dann zur Ableitung eines Regelsignals verwendet werden. Jetzt kann man die Dbergangsfunktionen der Kolonne ftir kleine Storgro13en ermittein, und diese dienen aIs Daten flir eine Untersuchung mit einem Analogrechner. Die bei diesem Verfahren auftretenden Schwierigkeiten werden erklart. Es wurde festgestellt, da13 die Regelung der Kolonne entscheidend von bestimmten zweitrangigen Charakteristiken abhangt, zum Beispiel wie lange es dauert, bis sich eine Anderung in der Zuflu13geschwindigkeit in der gesamten Kolonne auswirkt. Auch mu13 die ganze Untersuchung ftir jeden beabsichtigten Betriebszustand der Kolonne wiederholt werden. Bestimmte, interessante, aus der Anwendung dieses Verfahrens hervorgegangene Ergebnisse werden beschrieben. Als Nebenergebnis dieser Arbeit ergibt sich eine Moglichkeit zur Untersuchung, wie eine Kolonne bei ihrer erstmaligen Inbetriebnahme den Gleichgewichtszustand erreicht. Auch erhalt man dadurch in vielen Fallen die Moglichkeit, den Gleichgewichtszustand der Kolonne schnell zu berechnen.
DISCUSSION le following questions and answers arose during the DisLssion. Q. Please give a detailed description of the method of necting Xrj. Is TO) determined after satisfying the condition rrj = 1 ? A. The correction to Xrj is made by the formula
where X'rj is the corrected and Xrj the uncorrected value. This operation is carried out only if 11- LXrjl is less than a small, j
pre-assigned number. If not, an alarm is set and the calculation must be repeated with smaller time steps in order to achieve a smaller initial error in LXrj' The approximation procedure to j
X'rj = X rj/2 Xrj j
find
TO)
308
1821
is started after the
Xrj
have been corrected.
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
lations to be used for routine design. The procedure is first to compute the steady state of the column in one operating condition. Sufficiently small changes are then made in the operating conditions so that the computed transient responses can be regarded as linear. Changes are made in feed composition, reflux ratio , boil-up, etc. -in all for six or more quantities. The transient responses are then approximated by relatively simple transfer functions, and these are used in an analogue computer study of the column with various control systems. This procedure will occupy one man for about six weeks, and will cost £500 or more. If the cost and time could be reduced to one-tenth, the method would be useful for routine design, but at present we cannot achieve such an improvement. Because of these difficulties, studies of optimal control, etc., have not been made. On the other hand, the chemical engineering design (i.e. steady-state design) of columns has been optimized by use of a computer. Q . Please write down the transfer function of the whole system. A. The question seems to show a misunderstanding of the approach which has been used. Computations are made, as just mentioned, to obtain the transient response on each plate to a small change in each of the operating conditions. If, for example, it is proposed to control the reflux and the boil-up by measuring two temperatures in the column (Figure A), then
Q. What is the duration of the following steps of the calculation: (a) determination of Xrj satisfying 2.xrj = 1, for one plate? j
(b) the same, but for a column of 100 plates? (c) the same, but for the column for the whole transient
process? A. Using the DEUCE computer, 0·25 sec is required per plate per component per time step for the calculation. Printing adds 2 sec to this time, but need not be carried out for each time step. The time for a column to reach a steady state depends (in linear conditions) roughly on n 2 , but the number of time steps need not increase so rapidly. As an example, a column of 100 plates with three components would require about 3 h for the calculation of a single transient. If the column was producing very pure product, so that conditions for the transient were non-linear, the time needed might be considerably increased. Q . What are the least changes in the feed composition which cause transient processes? A. In theory there will be a transient response to any change in the conditions of the column, however small. In practice, very small changes affect only the last few digits in the computed results, and the accuracy with which the transient is obtained therefore suffers . With 10 decimal digits available, as in DEUCE, and allowing two digits for rounding errors, a change of about 0·0001 in composition is about the least that can be dealt with satisfactorily. The initial steady state must be found with high accuracy (to the eighth decimal digit) if such small changes are to be used . Changes of 0·001 have been used in a number of calculations, and these are satisfactory unless the column is producing a very pure product. When this happens, very small changes of feed composition can produce large changes of composition in the column l . It is then difficult to make the change in feed composition small enough for the response to be considered linear. The result of such a linearization, if it is possible, is in any case of doubtful value, for it applies only to changes much smaller than will be met in practice. Q. Is it more worthwhile to use digital computers to calculate the dynamics of the column when the temperature difference between the plates is small (fractions of a degree) or great? A. Calculations have been made in both circumstances. The response of a heavy water distillation column with 350 plates was computed. This was useful in estimating the time needed after start-up before the column would be giving product. For such columns the linear analytical theory 2 can be used , but it is difficult to apply because of the large number of plates. On the other hand, when the temperature difference between plates is large, the problem is generally non-linear. Analytical methods can then be misleading, since they usually depend on a drastic linearization. Q . In equation 2 of the paper, Kj is a function of Tr and consequently of Xrj. Does K j depend on the concentration of other components on a given plate, for instance on Xr k> and if so why? A. There are two points here. First, Tr depends on all the Xrk (k = 1,2 ... , J) since it is calculated from equation 4 in which they all appear. Secondly, the assumed form of Kj , depending only on TT and not explicitly on the Xrk> is an approximation that is usually made 3 . The difficulty of using a more general form for Kj is not so much one of computation, but rather one of obtaining the necessary practical data. Q . In what state of development at the present time are the authors' investigations in the field of automation of distillation columns? Have they considered the problems of optimization of the process , extremal control, etc.? If they have, then in what direction ? A. At present we can compute the behaviour of a column, but the cost and time required are too great to allow the calcu-
Condenser
Top product
Feed
Steam Bottom product
Reboiler
Figure A
the two plates on which tempera tu res will be measured are selected at this stage. The system show n in Figure B is then
Reflux
BOil-up
O"---~
1---~T2
L...._--l
Figure B
set up on the analogue computer. Transfer functions All, A 22 , A 12 and A2\ are obtained by approximating to the computed results by means of simple functions. The transfer
309
1822
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND functions Bl and B2 of the controllers are selected according to the equipment which will be used. Q. Why were fresh changes made in the operating conditions of the experimental column before it had reached a steady state? A. A complete run lasted 12 h, including the initial approach to a steady state before the transients were measured. By starting one transient before the last had finished, three transients could be studied in this time instead of one. Q. Why were the resistance thermometers situated in the vapour? A. Although the resistance thermometers were above the tops of the bubble-caps, they were immersed in foam when the column was operating. Their position was fixed by the sidelegs on the standard column used. Q. What examples of practical application of computers for distillation processes do the authors know? A. Selected references up to 1958 are given in a paperl quoted above. Since then, the chief advances have been in on-line computer control, to which references are given by Grabbe 4 in a Conference paper. Q. How do the authors propose to apply their experimental results in practice? A. The experimental results serve as a limited confirmation of the computed results. The use made of these has been described above. Q. How must the controlled variable (temperature, pressure) of the column be changed, when the quality and quantity of the feed are altered, in order to ensure pure products? A. Many different control systems have been suggested for distillation columns 5 • The best system can, in theory, be selected by analysing each alternative as explained above. At present this is expensive and time-consuming, and the best system will certainly differ according to circumstances. No general rules can therefore be given at this stage. Q. What can the authors say about the optimal control of distillation columns? A. It would certainly be desirable to have a single controller for a column, which accepted information from measuring instruments and sent out control signals to the valves. Such a controller should be able to give better control than is obtained with independent loops such as are used at present. The design and adjustment of such a controller, however, require information about the behaviour of the column which at present (see above) we can only obtain with difficulty and expense. L. G. PLISKIN (U.S.S.R.)
The paper by Rosenbrock and his colleagues is of great interest. It contains results concerning a method of calculating distillation-column dynamics and the application thereto of digital computing machines (DCM). It should be noted, however, that this work is not the first on this topic. The authors mention this but their paper leaves obscure just what is the basic difference between their procedure and that in other papers, for example papers by Rose and others in 1958 6 . Even before 1958 there were a number of papers on this subject. At the Conference on Plant and Process Dynamic Characteristics (Cambridge, 1957) Wilkinson 7 and Voetter 8 each presented papers deriving a variety of differential equations for column dynamics, although as far as I can remember they were talking about two-component mixtures and the DCM was not used. On the other hand, the paper by Rose et al. is closely allied to the present one. They considered a multicomponent mixture, the basic equations for the dynamics were extremely similar, they took into account the non-linear relation between the concentrations of vapours and liquids over the plate and a DCM was used to solve the problem. The difference between the papers as far as initial conditions in the calculation are concerned is insignificant. (Rose's calculation starts with the column filled with feed material and with all plates at the same temperature.)
J.
A. CHALLIS
The second comment is of a more general nature. In the West during recent years, many papers have appeared, devoted to the calculation of the statics and dynamics of distillation columns with the aid of a DCM. This raises the question: are they due to a direct need to use a DCM with distillation columns or is there some kind of collateral reason for it? It seems to us that a partial explanation for this is advertisement for firms making DCMs. First, a distillation column, described by an unwieldy number of algebraic equations, not susceptible to manual calculation, is an exceedingly appropriate object for demonstrating the possibilities of a DCM. Secondly, it is typical that among the many papers on this topic there are almost none on the application of a DCM to control systems, which should be the ultimate aim of the paper. The last comment was intended to emphasize the importance of knowing the statics of the object with the purpose of optimization, as compared with the dynamics. Finally, it is impossible to neglect the dynamics, but two of its aspects-time-shift between parameters involved in the optimization criteria, and the stability of a closed system-can be estimated and allowed for rather more easily than the statistical relationships of the process described. THE AUTHORS, in reply. It is true, as Dr. Pliskin remarks, that much previous work has been done on the calculation of transient responses for distillation columns. Analytical work 9 dates back at least to 1947, analogue computer studies 10 to 1953, and digital computer work l l to 1951. For binary systems, a number of comparisons have been made between theory and experiment. The present paper is, however, the only one known to the authors in which calculated and measured transient responses have been compared for a multicomponent system. Indeed, in the paper by Rose et al. 6, the calculations were set out as a means for obtaining the steady state, and not the transient conditions. This follows the procedure adopted by one of the present authors l2 for binary systems. In addition, the numerical method used for integrating the equations in the present paper is new in its application to distillation. It will be found in practice to be many times (up to several hundred times) faster than the usual methods (i.e. Runge-Kutta and allied methods). It is possible that distillation has attracted more than its fair share of mathematical effort in recent years. It is, however, a convenient example of a class of operations (absorption, liquid extraction, diffusion, or, indeed, almost any stagewise physical operation carried out in a cascade) which are widespread in chemical engineering. The data, being physical, are reasonably well known. Results from the computations should lead to better control, not necessarily by an on-line digital computer. The authors' interest in the problem is that of a Company designing and constructing chemical plant. Simple quantitative methods of designing control systems for such plants are much needed. As stated before, the methods given in the paper, though they represent some progress in this direction, are still too cumbersome for routine use. References I ROSENBROCK, H . H . Use of digital computers in chemical engineering. Process Control and Automation, 5 (1958) 466--470 2 COHEN, K. and MURPHY, G. M. The Theory of Isotope Separation as Applied to the Large-scale Production of U.235.
3 4
5
6
1951.
New York; McGraw-Hill PERRY, J. H. (Ed.). Chemical Engineers' Handbook. 1950. New York; McGraw-Hill. p.568 GRABBE, E. M. Digital computer control systems: an annotated bibliography. l.F.A.C. ConJ, Moscow. Vo\. 2 (1960) 1075 BOYD, D. M. Fractionation instrumentation and control. Petrol. Refin. 27 (1948) 533- 536, 595-597 ROSE, A., SWEENY, R. F. and SCHRODT, V. N. Continuous distillation calculations by relaxation method. Ind. El/g. Chem. 50 (1958) 737-740
310
1823
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
7 WILKINSON, W. L. and ARMSTRONG, W. D. An investigation of the transient response of a distillation column. Plant and Process Dynamic Characteristics. 1957. London; Butterworths. pp. 56-72 8 VOETTER, H. Response of concentrations in a distillation column to disturbances in the feed composition. Plant and Process Dynamic Characteristics. 1957. London; Butterworths. pp. 73-96 9 MARSHALL, W. R. and PIGFORD, R. L. The Application of Differential Equations to Chemical Engineering. 1947. University of Delaware. pp. 144-158
ACRIVOS, A. and AMUNDSON, N. R. Solution of transient stagewise operations on an .analogue computer. Ind. Eng. Chem.45 (1953) 467-471 11 ROSE, A., JOHNSON, R. C. and WILLIAMS, T. J. Stepwise plateto-plate computation of batch distillation curves. Ind. Eng. Chem. 43 (1951) 2459-2464 12 ROSENBROCK, H. H. An investigation of the transient response of a distillation column. Pt. I: Solution of the equations. Trans. Instn. chem. Engrs. Lond. 35 (1957) 347-351
10
1824
+ Frzrj
(1)
where plates are numbered from 1 to n starting from the bottom of the column, and : Hr is the liquid hold-up in moles on the rth plate. X,I is the proportion of the jth component in the liquid on plate r.
Yrj is the proportion of the jth component in the vapour above plate r. V, is the flow rate in moles/sec of vapour leaving the rth plate for the (r + l)th. L, is the flow rate in moles/sec of liquid leaving the rth plate for the (r-l)th. Pr is the flow rate in moles/sec of liquid product withdrawn from plate r. Fr is the flow rate in moles/sec of a feed to plate r . Z rj is the known and constant proportion of the jth component in the feed to the rth plate. Equation 1 can be taken to apply to the reboiler (r = 0) and to the condenser (r = n+ 1) if it is understood that V-I
=
Lo
=
Ln+2
= 0
and Vn+1 = 0 when no vapour product is taken from the column : otherwise Vn +1 is the flow rate of this vapour product in moles/sec. In addition to equations 1 there is a relationship, generally non-linear, betweenYrj and Xrj at the assumed column pressure: (2)
As shown, K for each component is a function of the temperature Tr on the appropriate plate and through this (as below) of Xrj . From the definition of Xrj it follows that
L X'I
= 1
(3)
j
The corresponding relationship for Yrj must also be true LYrj = L K;(Tr)X'1 = 1
(4)
j
and given equation 3, equation 4 will only be true when T, has its correct value. The method chosen for solving the equations was the simplest one that seemed likely to succeed. The temperature Tr on each plate was assumed constant during each time step, and equal to its known value at the beginning of the step. There is then one set of n + 2 equations, equations 1, for each component, and the sets can be solved independently to give Xrj at the end of the time step. A check is then made to see that equation 3 is true for each plate and an alarm is set if the discrepancy is more than a selected amount. The values of Xrj are corrected proportionately so that equation 3 is exactly true for each plate. Equation 4 is then solved by approximation to find each of the n + 2 values of T r • This procedure is repeated for the next time step. The method was checked by computing transients for a binary system and comparing the results with those obtained for the same problem by means of the binary programme described elsewhere 3 • To solve equations 1 the method described in reference 3 was 303
1816
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND J. A. CHALLIS
taken over without change. If the right-hand side of equations 1 is replaced for brevity by .p" then the method is to replace equations 1 by
= (I-().pr(tO)+().pr(tl)
Hr Xrj(t;)-;rj(t o)
(5)
1- 0
where () is a constant. When () = t the process is the one given by Crank and Nicolson4, and the error in the time step St is 0(St)3. For () = 1 the error is O(St)2, but the process is better for finding the steady-state values. If.p is a linear function of x, the process with () = 1 gives the steady state in one step if I/St is first put equal to zero. When.p is non-linear, the process (with () = 1 and liSt = 0) leads to a generalization 5 of the Newton-Raphson method for finding the steady-state values. Equation 5 leads to -()Vr- IK,-1. jSxr-1.j+ [()(Lr+Pr+ VrKrj )+ !]SXrl -()Lr+ISxr+l ,j
= .p,(to)
(6)
This has the form BrjSX,-hl+ CrjSXrj+ DrSX,+hl
=
Grl
(7)
and from the end conditions BOI
=
Dn+!
(8)
=0
Thus for each component the SXrj are to be found from a set of equations COjSXOI + DoSxlj = GOI BIjSXOj+C li SXIi+D ISX2j = Gl/ (9) BnjSXn_l,j+ CnjSXnl + DnSXn+I ,1 = Gnl Bn+!,jSXnl + Cn+1 ,jSXn+l,j = Gn+1.1
}
Equations 9 are reduced, by substitution, to the equivalent set COjSXOj + DOSXII Cg>SXIi + DISxzj
= GOj = GW = GW
CWSXnj+ D nSXn+1.1 q~\,ISxn + I,1
where C,j(l)
-
} (10)
= G~~I,I
Crj-cro B'I D ,-I
(11)
r-l.i
G(I) = G
'}
G(I) rj -~ C(l) ,-1,1
(12)
r-I,I
Then by back-substitution SXn +1,j
=
G~~I,;/C~~I,I
(13)
and in general SXrj =
~W {GW - D,SXr+l,j}
(14)
This procedure is repeated for each value of j in turn to give all the X,j at the end of the time interval St. The subsequent approximation to find Tr is carried out by linear interpolation. Let T,
1-2yg> T,(3) r -
T,(1)+ r
T"L.. y"
j
(2) _ "
j
(I)
L.. y'} i
(15)
Now T,
304
1817
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
The data must satisfy the following restrictions: (i) If A is an integer such that 32(A -1) < n
~
32A
(16)
then 8A+J+AJ
~ 203
(17)
This condition restricts the number of plates and components which can be dealt with simultaneously and arises from the limited storage (about 8,000 words) of the computer. For example, with 6 components the number of theoretical plates must not exceed 448. (ii) J ~ 32. This restricts the number of components. (iii) F ~ 1, L ~ 1, P ~ 1, VK ~ 1. These inequalities are achieved by scaling. (iv) s ~ 8. Results are punched out in binary form after p time steps. A separate short programme converts the results to decimal form. The time, and the temperature and compositions on each plate, can then be printed. Experimental Equipment The arrangement of the experimental equipment is shown in Figure 1. The column was a standard glass bubble-cap column 8 of nominal 6 in. diameter, with four plates above the feed and four below. Each plate had one bubble-cap, and the plates
were separated by conventional 6 in. bore distance pieces each 6 in. long as shown in Figure 2. Top and bottom products were controlled by regulating valves and measured by rota meters. The sum of the top and bottom products shown by their rotameters was always set to equal the feed rate indicated by a third rotameter. Feed was introduced into the column as liquid at the boiling point, this being secured by a feed pre-heater with a reflux condenser attached. A total reflux, glass spiral condenser was fitted above a reflux divider, returning condensate to the column at a temperature near to the vapour equilibrium temperature of the top plate. The reboiler had a capacity of about 45 litres, and was fitted with a sight glass so that the constancy of the level could be checked during a run. Two 3 kW electric heaters were fitted, the electrical input being continuously variable. Temperature measurements were made on each plate by gasfilled platinum resistance thermometers fitted through side arms in the column walls, with the measuring elements located about 1 in. above the dome of the bubble-cap. Each resistance element could be connected in turn through a selector switch to a five-decade Wheatstone bridge. K
Vent to atmosphe-re
A _ _~~~--B
Cooling water
C ----I--..,JII
D---.J
'----E
"'-t----J
Feed regulating valvE"
1/ kW fE'ed pre-heater
Vent
/Rolatamell"r
A - Downcomers B - Bubble cap C - Resistance thermometer bulb D - liqUid sample point E - Vapour sample pOint F - Feed point G - Vapour inlet H - Reflux return J - Riser K - Reflux timer section connection
Rotamele>r
G---* To draJn
H
Figure 1. General arrangement and flow diagram of the S-plate distillation columll 20-A.R.C.4
Figure 2. 305
1818
Column details
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND J. A. CHALLIS
Feed to the column was drawn from a tank, and the products were stored in two product tanks. A fourth tank was used for mixing the feed between experiments. The components used were selected to meet the following requirements: (a) A reasonable degree of separation with 8 plates. (b) Easy analysis of the mixture by means of an infra-red
spectrometer. (c) Small difference between the molal latent heats. The system chosen was toluene-iso-octane-paraxylene, which fitted the requirements satisfactorily. A temperature difference across the column of 20 could be achieved, and the molal latent heats 9 are 7'98, 8·63 and 7·40 kcal/mole respectively, at the boiling point of the pure component. Preliminary experiments were made to find the dynamic hold-up of the column. With the column 'operating at total reflux the heaters in the re-boiler were suddenly switched off and at the same time the valve in the liquid return line to the re-boiler was closed. The liquid in the column was allowed to drain out, and was measured. Heat storage in the electrical heaters in the re-boiler was not enough to affect the measurements. 0
In order to ascertain the overall efficiency of the column it was allowed to come to a steady state under total reflux and measurements were made of the compositions and temperatures. A set of five runs to the steady state was then carried out on the computer with 8, 7, 6, 5 and 4 theoretical plates respectively, starting in each case with the experimentally obtained compositions. A comparison of the five curves with the experimental values showed that eight theoretical plates gave the closest agreement. Results With a three-component system two measurements are needed to fix the composition. Owing to difficulties with the infra-red equipment which was to be used for analysis, two measurements could not be obtained and the comparison is made for temperature only. Figures 4--7 show computed transients and the corresponding experimental points obtained in two trials.
- I- The~ret i cia I
.U
o
110I - - fx First trial
o Second trial
..
109
11\
o
CII
-j....--- r--
~
E
Z 108
y
6
--
2
~
°
~:1 E 10 ., 7 jO ~
V
~V:
CII
a. CII I-
106
--
° °
. ~~
"f
104 0
,
~
·~o
105 : 100
I-'"
.~~
uu
ZUU
400
500
600
Time
o
Figure 5.
0·04
003
0·05
0·06
Boil-up
Figure 3.
moles/sec
Variation of hold-up with vapour rate
u
°0
111
~. V
x'
106
°
110 ~
:J
109
"§
it
I(
~
107 500
Time
Figure 4.
106 min
X
,
Ox
x
It
~.
1 300
306
1819
I 400
..: o
Second trial
I 500
Time
Figure 6.
[Plate 7)
'
I ! I
x First trial
- Theoretical [ I
200
--
I
;.-
100
~
j
~~
. o
y
F
x
~108 ° E
400
[Plate 5]
-~ ~
o
o 107 ~--~--~---+--~----+---~--- r~
JOO
min
The experimental points were obtained in the following way. The column was started under total reflux and allowed to run until changes of temperature became slow. A mixture of toluene, paraxylene and iso-octane in the molar proportions 0'368:0'323 :0'309 was then fed to the column at the rate of 0'0251 mole/sec. At the same time the top and bottom offtakes were set to 0'0061 and 0'0190 mole/sec, respectively.
u
200
700
0·08
007
Figure 3 shows the average hold-up on each plate as a function of vapour rate. The slope of the curve is not great in the region of interest, and when the vapour rate was varied a mean value of hold-up was used in the computer.
100
-
I--
f
I
,1h"/ '
~
00
[Plate 4 (feed»)
I 600
I 700 min
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
u
0
Appendix-Flow Chart of Computer Programme
115 -
Theoretical
114
,First trial OS~~
00 0 0
113 q, 112
j
:::J
-;;; q,
x x/ V xV 0 00
x
x x
L
8.'" E
!--" 000
/
xV / x51 0
0
00
~/
x ~o
I/x
1-110
fx
x 0
~~
0
x
><
109
x
x
)(,),;)'
0
1~
:lo
107
100
200
/V
,.,
-
300 400
500
Time
600
700 min
Figure 7. [Plate 3]
When the changes due to introduction of feed had slowed down, the reflux ratio was changed by adjusting the top and bottom off takes to 0·0152 and 0'0099 mole/sec, respectively. This is the point from which the transients in Figures 4-7 begin. After a convenient time (about 180 min) the top and bottom off takes were re-set to their first values, and finally after a further 140 min they were set to 0·0152 and 0'0099 mole/sec again. Calculations on the computer followed the same course. They were started from the measured compositions in the column at the time when the reflux ratio was first altered. Since subsequent changes of reflux ratio were made before the column had reached a steady state, the calculations were continued beyond the points at which a change was made in order to show the further course of the transient. Further calculations were then made, starting from the points at which the changes were made, for comparison with the experimental results. In Figures 4-7 the computed curves have been moved slightly along the temperature axis to achieve better agreement. Their scale has not been changed. The two important characteristics for purposes of control, namely the shape of the curves and the total amount of the changes produced, are therefore preserved. Conclusions The limited comparison given here indicates the same degree of correspondence between theory and experiment for multicomponent systems as has been found before 1,2 for binary systems. The method of calculation is sufficiently flexible to allow considerable relaxation in the assumptions. This may be important in dealing with practical columns, where some of the assumptions made here (such as perfect mixing on the plates) may not be appropriate.
No
Punch t, Tand x No
Are an loxl < Ex '? Yes
The authors are grateful to British Petroleum Company Research Centre at Sunbury-on-Thames for help with the infrared analysis ofliquid samples," to their colleague Mrs. J. M. Hi/ton, who wrote the computer programme and carried out the calculations described here, and to the Directors of Constructors John Brown Limited for permission to publish this paper.
No ~------~E-------~
307
1820
Put loxl max. on Output Staticizer
H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND J. A. CHALLIS leferences ROSENBROCK, H. H. An investigation of the transient response of a distillation column. Part 1: Solution of the equations. Trans. lnstn chem. Engrs, Lond. 35 (1957) 347 ARMSTRONG, W. D. and WILKINSON, W. L. An investigation of the transient response of a distillation column. Part 11: Experimental work and comparison with theory. Trans. lnstn chem. Engrs, Lond. 35 (1957) 352 ROSENBROCK, H. H. Calculation of the transient behaviour of distillation columns. Brit. chem. Engng 3 (1958) 364, 432, 491 CRANK, J. and NICOLSON, P. A practical method for numerical evaluation of solutions of partial differential equations of the heatconduction type. Proc. Camb. phi!. Soc. 43 (1947) 50
5
6
7
8 9
BOOTH, A. D. Numerical Methods, p. 155. 1957. London; Butterworths HrLTON, J. M. A computer programme to calculate the transient response of a multicomponent mixture in a plate type distillation cplumn. C.1.B. Report No. R&D/133 (1959) (unpublished) HALEY, A. C. D. Deuce-a high-speed general-purpose computer. Proc. lnstn elec. Engrs 103, Pt. B, Suppl. No. 2 (1956) 165 SMITH, J. C. and KELM, E. F. An all-glass bubble-cap column. Chem. Engng 58 (1951) 155 PERRY, J. H. (Ed.) Chemical Engineers' Handbook, p. 215. 1953. New York; McGraw-Hill
Summary
he equations governing mass-transfer in a multicomponent 'stem have been solved using a digital ·computer. Experimental ark in an eight-plate column distilling a mixture of toluene, iso;tane and paraxylene has been carried out to check the calculations.
The computer programme and the experimental equipment are briefly described. A comparison is made between computed and experimental results.
Sommaire es equations qui regissent le transfert de masse dans un systeme it ements multiples ont ete resolues par l'emploi d'un calculateur Imenque. Un travail experimental, dans une colon ne it huit .ateaux, pour la distillation d'un melange de toluene, d'iso-octane de paraxylene, a ete effectue pour verifier les calculs.
On decrit brievement le programme du calculateur et I'equipement experimental. Une comparaison est faite entre les resultats de I'experience et ceux du calcul.
Zusammenfassung ie Konstruktion einer automatischen Regelanlage fUr Destillations)Ionnen setzt eine Kenntnis des Ansprechverhaltens der Kolonne If Anderungen in den Betriebsbedingungen voraus. Das hei13t so, wir mlissen in der Lage sein, das Ansprechverhalten der Kolonne If eine bestimmte Anderung des Rlicklaufverhaltnisses, der Auf:izung, der Zusammensetzung des zugeflihrten Materials, usw. zu :rechnen. Wenn die StOrungen entsprechend klein sind, verhalt :h die Kolonne linear und die gewlinschten Angaben kann man aus :m Ansprechverhalten bei einer plotzlichen sprunghaften Anderung :r Betriebsbedingungen erhalten. Daraus la13t sich das Ansprechrhalten auf eine sinusformige Anderung oder auf irgendeine rlderung berechnen. In einem frliheren Aufsatz (Trans. Insfn. Chem. Engrs, 1957, 35, 7-351) wurde ein Elektronenrechner-Programm beschrieben, mit :ssen Hilfe das zeitliche Ansprechverhalten einer Destillations)Ionne flir Zweistoffgemisch berechnet werden kann. Die leichungen wurden nicht linearisiert, soda13 man das Verhalten bei einen oder gro13en Anderungen erhielt. In einem weiteren Auftz von W. D. Armstrong und W. L Wilkinson (ibid. 352-361) lrden Versuchsarbeiten beschrieben, und ein Vergleich zwischen esen und den mit den Digitalrechnern erhaltenen Ergebnissen rrde vorgenommen. Das Elektronenrechner-Programm wurde verallgemeinert, soda13 aus mehrereren Komponenten bestehende Gemische behandeln ,nn. Die Voraussetzungen bleiben die gleichen wie diejenigen flir ,s Zweistoffgemisch. lnsbesondere sind die Gleichungen nicht learisiert. Es sind Versuchsarbeiten mit einem Dreistoffgemisch I Gange, bei denen eine Kolonne mit einem Durchmesser von 6 Zoll Id 8 Platten benutzt wird; die Materialzufuhr erfolgt an der 4. atte. Die mit dieser Destillationskolonne erhaltenen Ergebnisse ~rden mit dem aus dem Elektronenrechner-Programm erhaltenen
Ergebnissen verglichen, wodurch eine Dberprlifung der Ausgangsvoraussetzungen der theoretischen Arbeit und eine Erweiterung der frliheren Arbeiten mit Zweistoffgemischen moglich ist, Die flir die Anwendung dieses Programms zur Konstruktion von Regeisystemen empfohlenen Methoden werden ebenfalls beschrieben. Zunachst wird der Gleichgewichtszustand der Kolonne mit Hilfe des DigitaIrechners festgestellt, und danach wird eine der Betriebsbedingungen (zum Beispiel das Rlicklaufverhiiltnis) durch eine kleine sprunghafte Anderung verandert. Der Elektronenrechner berechnet die Anderung in der Zusammensetzung der Fllissigkeiten an jeder Platte als Funktion der Zeit wahrend die Kolonne ihrem neuen Gleichgewichtszustand zustrebt. Dieses Verfahren wird flir andere Arten von StOrungen wiederholt und aus den Ergebnissen wahlt man diejenige Platte oder Platten aus, die dann zur Ableitung eines Regelsignals verwendet werden. Jetzt kann man die Dbergangsfunktionen der Kolonne ftir kleine Storgro13en ermittein, und diese dienen aIs Daten flir eine Untersuchung mit einem Analogrechner. Die bei diesem Verfahren auftretenden Schwierigkeiten werden erklart. Es wurde festgestellt, da13 die Regelung der Kolonne entscheidend von bestimmten zweitrangigen Charakteristiken abhangt, zum Beispiel wie lange es dauert, bis sich eine Anderung in der Zuflu13geschwindigkeit in der gesamten Kolonne auswirkt. Auch mu13 die ganze Untersuchung ftir jeden beabsichtigten Betriebszustand der Kolonne wiederholt werden. Bestimmte, interessante, aus der Anwendung dieses Verfahrens hervorgegangene Ergebnisse werden beschrieben. Als Nebenergebnis dieser Arbeit ergibt sich eine Moglichkeit zur Untersuchung, wie eine Kolonne bei ihrer erstmaligen Inbetriebnahme den Gleichgewichtszustand erreicht. Auch erhalt man dadurch in vielen Fallen die Moglichkeit, den Gleichgewichtszustand der Kolonne schnell zu berechnen.
DISCUSSION le following questions and answers arose during the DisLssion. Q. Please give a detailed description of the method of necting Xrj. Is TO) determined after satisfying the condition rrj = 1 ? A. The correction to Xrj is made by the formula
where X'rj is the corrected and Xrj the uncorrected value. This operation is carried out only if 11- LXrjl is less than a small, j
pre-assigned number. If not, an alarm is set and the calculation must be repeated with smaller time steps in order to achieve a smaller initial error in LXrj' The approximation procedure to j
X'rj = X rj/2 Xrj j
find
TO)
308
1821
is started after the
Xrj
have been corrected.
TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
lations to be used for routine design. The procedure is first to compute the steady state of the column in one operating condition. Sufficiently small changes are then made in the operating conditions so that the computed transient responses can be regarded as linear. Changes are made in feed composition, reflux ratio , boil-up, etc. -in all for six or more quantities. The transient responses are then approximated by relatively simple transfer functions, and these are used in an analogue computer study of the column with various control systems. This procedure will occupy one man for about six weeks, and will cost £500 or more. If the cost and time could be reduced to one-tenth, the method would be useful for routine design, but at present we cannot achieve such an improvement. Because of these difficulties, studies of optimal control, etc., have not been made. On the other hand, the chemical engineering design (i.e. steady-state design) of columns has been optimized by use of a computer. Q . Please write down the transfer function of the whole system. A. The question seems to show a misunderstanding of the approach which has been used. Computations are made, as just mentioned, to obtain the transient response on each plate to a small change in each of the operating conditions. If, for example, it is proposed to control the reflux and the boil-up by measuring two temperatures in the column (Figure A), then
Q. What is the duration of the following steps of the calculation: (a) determination of Xrj satisfying 2.xrj = 1, for one plate? j
(b) the same, but for a column of 100 plates? (c) the same, but for the column for the whole transient
process? A. Using the DEUCE computer, 0·25 sec is required per plate per component per time step for the calculation. Printing adds 2 sec to this time, but need not be carried out for each time step. The time for a column to reach a steady state depends (in linear conditions) roughly on n 2 , but the number of time steps need not increase so rapidly. As an example, a column of 100 plates with three components would require about 3 h for the calculation of a single transient. If the column was producing very pure product, so that conditions for the transient were non-linear, the time needed might be considerably increased. Q . What are the least changes in the feed composition which cause transient processes? A. In theory there will be a transient response to any change in the conditions of the column, however small. In practice, very small changes affect only the last few digits in the computed results, and the accuracy with which the transient is obtained therefore suffers . With 10 decimal digits available, as in DEUCE, and allowing two digits for rounding errors, a change of about 0·0001 in composition is about the least that can be dealt with satisfactorily. The initial steady state must be found with high accuracy (to the eighth decimal digit) if such small changes are to be used . Changes of 0·001 have been used in a number of calculations, and these are satisfactory unless the column is producing a very pure product. When this happens, very small changes of feed composition can produce large changes of composition in the column l . It is then difficult to make the change in feed composition small enough for the response to be considered linear. The result of such a linearization, if it is possible, is in any case of doubtful value, for it applies only to changes much smaller than will be met in practice. Q. Is it more worthwhile to use digital computers to calculate the dynamics of the column when the temperature difference between the plates is small (fractions of a degree) or great? A. Calculations have been made in both circumstances. The response of a heavy water distillation column with 350 plates was computed. This was useful in estimating the time needed after start-up before the column would be giving product. For such columns the linear analytical theory 2 can be used , but it is difficult to apply because of the large number of plates. On the other hand, when the temperature difference between plates is large, the problem is generally non-linear. Analytical methods can then be misleading, since they usually depend on a drastic linearization. Q . In equation 2 of the paper, Kj is a function of Tr and consequently of Xrj. Does K j depend on the concentration of other components on a given plate, for instance on Xr k> and if so why? A. There are two points here. First, Tr depends on all the Xrk (k = 1,2 ... , J) since it is calculated from equation 4 in which they all appear. Secondly, the assumed form of Kj , depending only on TT and not explicitly on the Xrk> is an approximation that is usually made 3 . The difficulty of using a more general form for Kj is not so much one of computation, but rather one of obtaining the necessary practical data. Q . In what state of development at the present time are the authors' investigations in the field of automation of distillation columns? Have they considered the problems of optimization of the process , extremal control, etc.? If they have, then in what direction ? A. At present we can compute the behaviour of a column, but the cost and time required are too great to allow the calcu-
Condenser
Top product
Feed
Steam Bottom product
Reboiler
Figure A
the two plates on which tempera tu res will be measured are selected at this stage. The system show n in Figure B is then
Reflux
BOil-up
O"---~
1---~T2
L...._--l
Figure B
set up on the analogue computer. Transfer functions All, A 22 , A 12 and A2\ are obtained by approximating to the computed results by means of simple functions. The transfer
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H. H. ROSENBROCK, A. B. TAVENDALE, C. STOREY AND functions Bl and B2 of the controllers are selected according to the equipment which will be used. Q. Why were fresh changes made in the operating conditions of the experimental column before it had reached a steady state? A. A complete run lasted 12 h, including the initial approach to a steady state before the transients were measured. By starting one transient before the last had finished, three transients could be studied in this time instead of one. Q. Why were the resistance thermometers situated in the vapour? A. Although the resistance thermometers were above the tops of the bubble-caps, they were immersed in foam when the column was operating. Their position was fixed by the sidelegs on the standard column used. Q. What examples of practical application of computers for distillation processes do the authors know? A. Selected references up to 1958 are given in a paperl quoted above. Since then, the chief advances have been in on-line computer control, to which references are given by Grabbe 4 in a Conference paper. Q. How do the authors propose to apply their experimental results in practice? A. The experimental results serve as a limited confirmation of the computed results. The use made of these has been described above. Q. How must the controlled variable (temperature, pressure) of the column be changed, when the quality and quantity of the feed are altered, in order to ensure pure products? A. Many different control systems have been suggested for distillation columns 5 • The best system can, in theory, be selected by analysing each alternative as explained above. At present this is expensive and time-consuming, and the best system will certainly differ according to circumstances. No general rules can therefore be given at this stage. Q. What can the authors say about the optimal control of distillation columns? A. It would certainly be desirable to have a single controller for a column, which accepted information from measuring instruments and sent out control signals to the valves. Such a controller should be able to give better control than is obtained with independent loops such as are used at present. The design and adjustment of such a controller, however, require information about the behaviour of the column which at present (see above) we can only obtain with difficulty and expense. L. G. PLISKIN (U.S.S.R.)
The paper by Rosenbrock and his colleagues is of great interest. It contains results concerning a method of calculating distillation-column dynamics and the application thereto of digital computing machines (DCM). It should be noted, however, that this work is not the first on this topic. The authors mention this but their paper leaves obscure just what is the basic difference between their procedure and that in other papers, for example papers by Rose and others in 1958 6 . Even before 1958 there were a number of papers on this subject. At the Conference on Plant and Process Dynamic Characteristics (Cambridge, 1957) Wilkinson 7 and Voetter 8 each presented papers deriving a variety of differential equations for column dynamics, although as far as I can remember they were talking about two-component mixtures and the DCM was not used. On the other hand, the paper by Rose et al. is closely allied to the present one. They considered a multicomponent mixture, the basic equations for the dynamics were extremely similar, they took into account the non-linear relation between the concentrations of vapours and liquids over the plate and a DCM was used to solve the problem. The difference between the papers as far as initial conditions in the calculation are concerned is insignificant. (Rose's calculation starts with the column filled with feed material and with all plates at the same temperature.)
J.
A. CHALLIS
The second comment is of a more general nature. In the West during recent years, many papers have appeared, devoted to the calculation of the statics and dynamics of distillation columns with the aid of a DCM. This raises the question: are they due to a direct need to use a DCM with distillation columns or is there some kind of collateral reason for it? It seems to us that a partial explanation for this is advertisement for firms making DCMs. First, a distillation column, described by an unwieldy number of algebraic equations, not susceptible to manual calculation, is an exceedingly appropriate object for demonstrating the possibilities of a DCM. Secondly, it is typical that among the many papers on this topic there are almost none on the application of a DCM to control systems, which should be the ultimate aim of the paper. The last comment was intended to emphasize the importance of knowing the statics of the object with the purpose of optimization, as compared with the dynamics. Finally, it is impossible to neglect the dynamics, but two of its aspects-time-shift between parameters involved in the optimization criteria, and the stability of a closed system-can be estimated and allowed for rather more easily than the statistical relationships of the process described. THE AUTHORS, in reply. It is true, as Dr. Pliskin remarks, that much previous work has been done on the calculation of transient responses for distillation columns. Analytical work 9 dates back at least to 1947, analogue computer studies 10 to 1953, and digital computer work l l to 1951. For binary systems, a number of comparisons have been made between theory and experiment. The present paper is, however, the only one known to the authors in which calculated and measured transient responses have been compared for a multicomponent system. Indeed, in the paper by Rose et al. 6, the calculations were set out as a means for obtaining the steady state, and not the transient conditions. This follows the procedure adopted by one of the present authors l2 for binary systems. In addition, the numerical method used for integrating the equations in the present paper is new in its application to distillation. It will be found in practice to be many times (up to several hundred times) faster than the usual methods (i.e. Runge-Kutta and allied methods). It is possible that distillation has attracted more than its fair share of mathematical effort in recent years. It is, however, a convenient example of a class of operations (absorption, liquid extraction, diffusion, or, indeed, almost any stagewise physical operation carried out in a cascade) which are widespread in chemical engineering. The data, being physical, are reasonably well known. Results from the computations should lead to better control, not necessarily by an on-line digital computer. The authors' interest in the problem is that of a Company designing and constructing chemical plant. Simple quantitative methods of designing control systems for such plants are much needed. As stated before, the methods given in the paper, though they represent some progress in this direction, are still too cumbersome for routine use. References I ROSENBROCK, H . H . Use of digital computers in chemical engineering. Process Control and Automation, 5 (1958) 466--470 2 COHEN, K. and MURPHY, G. M. The Theory of Isotope Separation as Applied to the Large-scale Production of U.235.
3 4
5
6
1951.
New York; McGraw-Hill PERRY, J. H. (Ed.). Chemical Engineers' Handbook. 1950. New York; McGraw-Hill. p.568 GRABBE, E. M. Digital computer control systems: an annotated bibliography. l.F.A.C. ConJ, Moscow. Vo\. 2 (1960) 1075 BOYD, D. M. Fractionation instrumentation and control. Petrol. Refin. 27 (1948) 533- 536, 595-597 ROSE, A., SWEENY, R. F. and SCHRODT, V. N. Continuous distillation calculations by relaxation method. Ind. El/g. Chem. 50 (1958) 737-740
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TRANSIENT BEHAVIOUR OF MULTICOMPONENT DISTILLATION COLUMNS
7 WILKINSON, W. L. and ARMSTRONG, W. D. An investigation of the transient response of a distillation column. Plant and Process Dynamic Characteristics. 1957. London; Butterworths. pp. 56-72 8 VOETTER, H. Response of concentrations in a distillation column to disturbances in the feed composition. Plant and Process Dynamic Characteristics. 1957. London; Butterworths. pp. 73-96 9 MARSHALL, W. R. and PIGFORD, R. L. The Application of Differential Equations to Chemical Engineering. 1947. University of Delaware. pp. 144-158
ACRIVOS, A. and AMUNDSON, N. R. Solution of transient stagewise operations on an .analogue computer. Ind. Eng. Chem.45 (1953) 467-471 11 ROSE, A., JOHNSON, R. C. and WILLIAMS, T. J. Stepwise plateto-plate computation of batch distillation curves. Ind. Eng. Chem. 43 (1951) 2459-2464 12 ROSENBROCK, H. H. An investigation of the transient response of a distillation column. Pt. I: Solution of the equations. Trans. Instn. chem. Engrs. Lond. 35 (1957) 347-351
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