- Email: [email protected]
Dynamic characteristic of binary srv distillation systems
Computers and Chemical EngineeringVol. 7, No, 2, pp. 105-122, 1983
0098-1354/83/020105-18503.00/0 Pergamon Press Lid,
Printed in Great Britain.
DYNAMIC CHARACTERISTICS OF BINARY SRV DISTILLATION SYSTEMS KAZUYIJKISHIMIZU'~and RICHARDS. H. MAH* Department of Chemical Engineering, Northwestern University, Evanston, IL 60201, U.S.A. (Received 27 July 1981; revision 5 April 1982)
Abstract--This paper reports a simulation study of SRV (secondary reflux and vaporization) distillation system using a perturbed linear model. The important factors which affect the dynamics are the heat transfer rate between the rectifying and the stripping sections, and the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler. The system becomes unstable if the mean temperature differences at the condenser and the reboiler are small, and the heat transfer rate between the rectifying and the stripping sections falls within a certain critical range of values. On either side of this range the system is insensitive to any kind of disturbances because of the inherent regulation occurring between them. Modal analysis shows that the external reflux rate activates the slowest mode which contains mainly the contributions of liquid concentrations on all stages. Scope--The drastic increase in energy costs over the last several years has stimulated the research on fractional distillation techniques, which are important energy consumers in the chemical and petroleum industries. The various schemes for reducing the energy requirements of distillation have been proposed under such categories as multiple effect, vapor recompression and heat pump[I]. Recently, Mah et al.[2] proposed a new type of distillation scheme which makes use of secondary reflux and vaporization (SRV). The advantages of SRV distillation has been evaluated from both the viewpoints of utility consumption[2] and of thermodynamic availability[3]. In addition to steady state performance and economic considerations, the dynamics and control characteristics of each new distillation scheme must be investigated as an integral part of the overall evaluation. It is dangerous to design control systems based on the knowledge of the conventional distillation, since the static and dynamic characteristics may be significantly modified as a result of energy integration. Tyreus & Luyben[4] presented the results of digital simulation studies of the dynamics and control of multi-effect, heat integrated distillation columns. Doukas & Luyben[5] showed the simulation studies of a prefractionator/liquid sidestream two column system separating a ternary mixture. In this paper we shall show how the dynamic characteristics of SRV distillation differ from the conventional distillation. Conclusions and Significance---The transient behavior of a 10-stage SRV distillation system was analyzed in terms of step responses based on linear equations, which were derived by perturbations about some steady state operating conditions with allowance for time variation of pressures. Dynamic characteristics of the system were interpreted for the changes of such variables as condenser cooling medium temperature, vapor flow from the stripping to the rectifying section, reboiler heating medium temperature, external reflux rate, feed composition, and feed flow rate. The study shows that the important factors which affect the dynamics are the heat transfer rate between the rectifying and the stripping sections, and the mean temperature differences at the condenser and the reboiler. The system becomes unstable or highly sensitive to the disturbances if the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler are small for a critical range of values of heat transfer rate between the rectifying and the stripping sections. On either side of this range the system is insensitive to any kind of disturbances because of the inherent regulation occurring in the system. Modal analysis shows that the slowest mode consists mainly of liquid composition elements and is activated by the external reflux rate. On the other hand, the two fastest modes are dominated by the pressure elements and activated by the condenser cooling medium temperature, the reboiler heating medium temperature and the vapor flow rate from the stripping to the rectifying section. This study underlines the significant difference in static and dynamic characteristics of SRV and conventional distillation systems, which must be taken into consideration in designing the control systems.
tKazuyuki Shimizu is now with the Chemical Engineering Department of NagoyaUniversity in Japan. *Author to whom correspondenceshould be addressed. 105
106
KAZUYUKISmMIZUand RICHARDS. H. MArt SRV DISTILLATION
A 10-stage SRV distillation scheme is shown in Fig. 1 with the condenser and the reboiler counted as stages 1 and 10, respectively. In order for the heat to be transferred from the rectifying section to the stripping section the former must be operated at a higher pressure than the latter. Compression and expansion devices are required between the two sections to sustain the required pressure differential. As a result of heat exchanges between the two sections, the liquid reflux rate at rectifying section steadily increases as the liquid proceeds down the column, and the vapor flow rate at stripping section steadily increases as the vapor proceeds up the column. Introduction of this secondary reflux and vaporization reduces the reboiler and condenser duties, and enhances the separation of products.
Numbering the stage from the top with the condenser as stage 1 and the reboiler as stage N, the governing equations for the dynamic model of an SRV distillation system can be stated as follows: Total mass balances: d M d d t = 0 = V 2 - ( L , + D)
(1)
dM~/dt = 0 = L~_, - (Vj + Wj) - (Lj + Uj) + Vj+, + Fi,
j=2,3 ..... N-1 dMN/dt = 0 = L N - ] - VN - LN.
(2) (3)
Component mass balances for the more volatile component: d x d d t = ~ , = { - ( L t + D)x, + V2y2}/M~
(4)
MATHEMATICALMODELING
The following assumptions were made in deriving the mathematical model for SRV distillation: (i) The columns are operated with a binary mixture. (ii) The vapor holdups are negligible. (iii) The liquid holdups are perfectly mixed. (iv) The vapor is in equilibrium with the liquid leaving the stage. (v) The heat transfer dynamics occurring between the rectifying and the stripping sections may be neglected. The same is assumed to be true for the condenser and the reboiler. (vi) A total condenser is used for the overhead vapor. (vii) The feed stream is introduced as a saturated liquid. (viii) Except for pressure drop across the valve the pressure drop across the column is negligible. (ix) The hydraulic delay occurring in the liquid flows is negligible (constant molar holdups). (x) Liquid molar holdup for each stage is constant. (xi) The overhead accumulator, the bottom of the rectifying section and the reboiler are under level control by the manipulation of the liquid outflows.
--y__ FC
dxjldt = ~ = { L j - , x j _ , - (Vj + W3Y~- (Lj + Uj)xj
+ Vj+,yj+, + ~Z~}/Mj
j = 2, 3. . . . . N - 1
(5)
dxNldt = ?PN = (LN-~XN--~- VNyN - LNXN)/MN. (6)
Energy balances: 0 = gJt = (142- h ~ ) V 2 - Q~
(7)
dhj/dt = ~j = {Lj-thj_~ - (V~ + Wj)Hj - (Lj + Uj)h~ + Vj+tHj+, + Fjhv i + UAj(Tj. - Tj)}/M~,
j=2,3 ..... N-1
(8)
dhN/dt = ~brq= ( L N _ ~ h ~ _ , - V~HN - LNhN + Qr)/Ms
(9) where Qc = U A c ( T 2 - % )
and Q, = U A r ( T , - TN)
ConcklflNr
and T'/is the temperature of the stage paired with stage j and h~ is the liquid enthalpy of the condenser outlet stream. Phase equilibrium relationships for the more volatile component:
L5
1~
yj = K,ixj, j = 2, 3. . . . . N
d v3
Dew point relationship for the condenser: y J K n + (1 - y2)/K2, = 1
--V, V-9
s
Reboiler
] LC
I0
(10)
Distillate
Or
Bubble point relationships: K~jxj + K2j(1 - xj) = 1, j = 2, 3. . . . . N.
Bottoms
(12)
Compression of vapor from stage m + 1 of the stripping section to stage m of the rectifying section: T*÷t
Fig. 1. Schematicrepresentation of a 10-stage SRV distillation.
(11)
=
T,,+,(pdp~) ("-t/")
(13)
where T~+, is the outlet temperature of the compressor and ~ is the polytropic coefficient. Note that the last term of Eq. (8) plays a characteristic role in the dynamic behavior of an SRV distillation system.
Dynamic characteristics of binary SRV distillation systems STEADY STATEBEHAVIOR
In contrast to the conventional distillation the vapor and liquid flow rates in SRV distillation change significantly throughout the column as a result of the heat exchanges between the rectifying and the stripping sections, and this will affect the dynamic behavior. Let us first examine the effect of the heat transfer between the two sections on the steady state behavior. For the following study, we chose a 10-stage (ethylene-ethane) C2-splitter. Thermophysical properties were evaluated using the Redlich-Kwong-Soave equation of state[6], and the steady state distillation calculations were carried out using a modified Wang-Henke method given earlier[2]. Steady state operating conditions chosen are listed in Table 1. A feed is introduced at the sixth stage and the value of # is taken as 1.3. For simplicity let us assume that the product of heat transfer area and overall heat transfer coefficient be the same for all stages. Notice that this product appears only in the energy balance, Eqn (8), for the SRV distillation. In Fig. 2, the effects of the heat transfer parameter UA on the liquid and vapor flow rates and condenser and reboiler duties are displayed for a fixed top and bottom product specification. As the heat transfer from the rectifying section to the stripping section increases, the liquid and vapor flow rates in the lower part of the rectifying section and the upper part of the stripping section ("middle sections") increase. Less external reflux and reboil are required to meet the same product specification, as shown in Fig. 2(c). These effects are illustrated by the positive slopes of L 3 - L 7 and the negative slopes of LL, L2, Ls, L9 in Fig. 2(a), and similar contrasting behavior of V3- V7 and V2, Vs- Vio in Fig. 2(b). In Fig. 3, the profiles of liquid and vapor flow rates and liquid compositions are shown for both UA = 0 J/s K and UA = 6x 103 J/s K. Figure 3(c) shows that as UA increases, the "middle sections" also assume a greater share of component separation in comparison with the "end sections." As pointed out by Wahl and Harriott[7], the liquid and
vapor flow rates may affect the time constants of the system. The liquid composition profile may affect the measurement locations for the control system design. Rademaker et at.[8] suggested that the control tray be chosen at the point where the temperature profile has a maximum gradient. In this sense, Fig. 3(c) shows that the control tray must be moved down at the rectifying section and moved up at the stripping section as the magnitude of the heat transfer increases in SRV distillation. PERTURBED LINEAREQUATIONS
Many papers dealing with the analysis of distillation columns have assumed the pressure to be constant and have consequently been unable to study certain phenomena associated with pressure variations. In 1959 Rijnsdorp & Maarleveld[9] developed a mathematical description of pressure variation for their analog simulation. Shortly afterwards, Davison[10] gave an approximate linearized mathematical model which also allowed for pressure variation. He applied modal analysis technique to find the control configuration which minimizes the dominant time constant of the controlled system. The linear state space representation of the system equations can be obtained quite directly by linearizing the nonlinear column model equations about a steady state operating condition. It requires less simplifications than the Davison formulation. The state equations for the pressures are derived from the following relationships: dhl/dt = ( ohd axOp/tj + ( ahd Op,)x#j, i = 2, 3 . . . . . N
(14) Although liquid enthalpy h~ is a function of Tj as well as p,, xj and t, T~ can be expressed as a function of xj and pj from the bubble point relationship. By virtue of assumption (viii), the state equations for the pressures at the rectifying and the stripping sections can be obtained from Eq. (14) using the energy balance equations. For
Table 1. Operatingconditions of 10-stage SRV distillation (ethylene-ethane system) Feed:
Flow Rate, F 6
0.126
Composition,
z6
0.5000
kg-mole/s
Flow Rate, D
0.63
Composition, x I
0.9000
Distillate: kg-mole/s
Column P r e s s u r e : Rectifier,
Pr
Stripper, Ps
4 . 6 2 x 105 Pa 3.08 x 105 Pa
Liquid holdup par stage: Overhead
107
Accumulator, M 1
50.0 kg-mole
Rectifying section, Mj
20.0 kg-mole
Stripping section, Hj
30.0 kg-mole
Reboller, Mlo
100.0 kg-mole
Condenser Temperature i T 1
202.0 K
Reboiler Temperature, TI0
206.0 K
108
KAZUYUrdSmmzu and RICHARDS. H. MAH I
~
St.age |
0.4
LJ- o,s 02 "o
0.1
I
2
3
4
5
6
7
Heat. transfer parameter, UA x I 0-,5 dis oK
Stage number
(b) 05
& 0.4 >-
0.3
0.2 ~ o
OI
I
I
I
I
2
I
3
4
I
5
I
6
Heat transfer parameter, UA x I0-,s d/s°K
~b
2~ °~
--
O~
I-0
I I
I
2
I
3
I
4
I
5
I
6
Heat transfer parameter, UA x I0-,s d / s °K Fig. 2. Effect of UA on the steady state operating conditions.
Dynamic characteristics of binary SRV distillation systems
z-
(a)
109
I
~x~
I
~X~
5-
E
2
~
6-CO
7-8--
J 9-IC 0
0.1 0.2 Liquid flow rotes,
L i,
03 0.4 kg-mole/s
0.5
(b) L
D
x~..
9
3 B
4-_o
E c
g CO
5 6
7 8P
91-IC 0
O. I
02
Vopor flow r o t e s ,
0.3
O.
0 .4
Vj, kg-mole/s
I
(c)
l
2 S
4 ..0
E
5 6--
o
7-8--
9-IC 0
f/
~¢ I
0.2
i
I
0 zl 06 L i q u i d composition,
i
08 xj
IO
Fig. 3. Effects of UA on the liquid and vapor flow rate profiles, and the liquid composition profile (©, UA =0J/s °K; x, UA = 6 x 10"~J/s °K).
110
KAZUYUm SrtImlzuand RICHARD S. H. MAH
the rectifying column pressure, we have
leaving only the slow modes which are important in determining the process dynamics. A knowledge of these modes can guide us in the selection of measurement locations for modal control[11]. Referring to Eq. (35) in the Appendix let us examine the dominant eigenvalues of A and their corresponding modes (right eigenvectors). The effect of heat transfer rate on system eigenvalues is shown in Fig. 4, where the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler are both taken to be 30 K. If the mean temperature differences are reduced to 20K, the dominant (numerically smallest) eigenvalue moves into the right half of the complex plane and the system becomes unstable at certain range of UA. This behavior will be further discussed later. The eigenvectors corresponding to the modes at UA = 4 x 105 J/s K are displayed as bar graphs in Fig. 5. Each bar represents the value of the corresponding eigenvector element. The elements are arranged from left to right in the order of the mole fractions of the more volatile component in the liquid phase (condenser through reboiler), rectifying and stripping pressures. Figure 5 shows that the first mode which corresponds to the largest time constant is mainly comprised of composition elements, the pressure elements being essentially negligible. As with the conventional distillation column[12], the influence of the first mode is spread fairly evenly throughout the two columns. The second mode contains a large contribution from the rectifying
dp./dt = [Llhl - V2H2- L,.h,. + V,.+~H,.+~ * m
+ ~{-
V~h. - WjHj + Fjh,:j + UAj(Tj. -
rj)
m
(15) where m is the bottom stage number of the rectifying column and H* is the outlet vapor enthalpy of the compressor. For the stripping column pressure, we have dpJdt = [L,.,hm - V,,,+lHm+l - LNh~ N
+ j.~+, { - Ujhj - WjHj + Fjhr,j + UAj(Tj. - Tj) -
(ahjlaxi),,,M#j
(ohjlap,),,j
.= I
(16)
I
where Try*= Th. The perturbed linear equations are given in the Appendix. In all subsequent figures, perturbations from the steady state conditions of Table 1 will be plotted. MODAL ANALYSIS
Many multi-stage processes have a wide spectrum of time constants. Typically, the fast modes decay rapidly
Heat. "Lransfer p a r a m e t e r , oo
I
2
3
U A x I0 -5, d / s ° K 4
5
6
X~
X2
i
X3 X4 X5
7 X6
_Io
~
f l X~
c~
-5
a~
ii
Xs
Fig. 4. Dominant eigenvalues of SRV distillation.
Dynamic characteristics of binary SRV distillation systems Eigenvect,or elemen't,s I 2 5 4 5 6 78
Eigenvect.or elements
9101112
Mode
,
[
I 2 34
5 6 78
9101112
.
I i
I
Mode
,lllllll,,..
I' ''II
I
II I
II,
III,
'
ii 'II'
II
i
II
,
I
,
w
I,
,
I.
I '1 I
o , I ,I
' I I ' ~
II
111
,
, .....
~
12
i
Ii I1
' I I
'
Fig. 5. Modes of a binary 10-stageSRV distillation UA = 4 x 105J/s K; ATm= 30 K.
pressure, while the third mode shows a large distillate composition element. On the other hand, the two fastest (1 lth and 12th) modes are comprised of the contributions from the rectifying and stripping pressures, the composition elements being almost totally absent. The effect of UA on the first three modes is displayed in Fig. 6. Figure 6(a) shows that pressure elements dominate the first mode at UA = 0 J/s K. But as UA increases, those elements become smaller, and for UA > 4x 105J/sK pressure elements are negligible in comparison with the other elements. Figure 6(b) shows that for the second mode the rectifying pressure element dominates at UA = 0 J/s K. But as UA increases, top and bottom composition elements also become numerically large. Figure 6(c) shows that the pressure elements are again dominant in the third mode at UA = 0 Jls K, but decreases rapidly as UA increases, while the reverse is true for the distillate composition. Another important factor in the modal analysis is the activation of the modes. The activation of mode i by the ]th input, uj, is defined by[ll] ~j = wiTbjuj
(17)
where b~ is the jth column of B in Eq. (35) and w~ is the ith left eigenvector of A. The activation is a measure of the extent to which an input affect a specific mode. The activation of modes caused by step changes in external
reflux rate, L~, condenser cooling medium temperature, T~, vapor flow rate from the stripping column to the rectifying column, 116, reboiler heating medium temperature, Th, feed composition, z6, and feed flow rate, F6, are given as bar graphs in Fig. 7. The length of each bar gives the value of the activation. Because of the differences in input step magnitudes, activations for a particular input cannot be directly compared with those for other inputs. Figure 7 shows that L~ activates the first mode. In other words, manipulation of LI affects the composition responses throughout the columns. On the other hand, the last two modes are activated mainly by To V6 and Th. Since the last two modes are dominated by pressure elements, the implication is that pressure responses may be modified by the manipulation of To, V6 and Th. SL~ULATIONR~ULTSANDVlSCVSSIONS Step response characteristics are important in understanding the transient behavior of the system. The main interest is to know the effect of the heat transfer between the rectifying and the stripping sections on the dynamic behavior. Step response curves were obtained, using the perturbed linear equations discussed in the previous section, for step increases in cooling medium temperature, vapor flow from the stripping to the rectifying section, reboiler heating medium temperature, external reflux rate, feed composition, and feed flow rate.
112
KAZUYUKISHIMIZUand RICHARDS. H. MAH i.Oi (a)
0
~"
I
2
HEAT TRANSFER
3
4
5
PARAMETER ,UA x 10-5
6
J/s
7
K
1 ~oo.~1-
~1,
/
~
~
I
oF
~
:~ 14~
0
"i
~ J
~
I
I
2
5
4
5
5
PARAMETER, UA x I0-
HEAT TRANSFER
6 d/s
7 K
~ 0.5
~
,o
~ o
-0"50I
~
.I
.
2.
HEAT TRANSFER
. 3 .
4
PARAMETER,
5 UA x I0
-5
J/s K
Fig. 6. Effect of UA on the modes of a binary 10-stage distillation systems, AT,. = 30 K (a) The first mode, (b) The second mode, (c) The third mode.
113
Dynamic characteristics of binary SRV distillationsystems
Cooling medium temperature Response curves of the rectifying column pressure and the top product composition to a unit-step increase in the condenser cooling medium temperature for different values of heat transfer parameter UA are given in Fig. 8. Much faster responses are observed for pressures than for compositions. This observation is consistent with the previous investigations for the conventional distillation columns [8]. The increase in the condenser cooling temperature decreases the condensation at the condenser, and therefore decreases the overhead vapor flow rate, which causes a rise in the rectifying column pressure. This tends to increase the temperatures in the rectifying column, which mitigates the primary effect on the heat transfer at the condenser. In contrast to the conventional distillation column, the step change also causes some secondary effects. The increase in the temperatures at the rectifying section increases the heat transfer between the rectifying and the stripping sections. This increase, in turn, causes a decrease in the rectifying column pressure, which partially nullifies the primary effect, and an increase in the stripping column pressure, which reinforces the primary effect. The increase in the stripping column pressure increases the stripping column temperatures and decreases the heat transfer between the two sections and
at the reboiler, which tends to decrease the stripping column pressure. Ultimately, the system settles down to a new steady state. In reality the situation is even more complex than the foregoing description because vapor and liquid compositions also play a role in these changes. Since the external reflux rate is kept constant, the decrease in the vapor flow in the rectifying section makes the operating line in the McCabe-Thiele diagram steeper, giving rise to better separation. Note that since the vapor flow through the compressor is kept constant, the vapor flow decreases more at the upper part than at the lower part of the rectifying column. Figure 8(b) shows the increase in the top product composition in all cases. In the case where there is no heat transfer between the rectifying and the stripping sections (UA = 0J/s K), this effect is suppressed by the pressure increase which reduces the relative volatility and results in less separation. For the case of UA greater than zero, namely, the case of SRV distillation, more enrichment in the top product composition might be expected for higher values of UA. But Fig. 8(b) shows that the top composition at UA = 2 x 105 J/s K increases more than at UA = 4 x 105J/s K or UA = 6 x 105 J/s K. If we plot the new steady state values after the unit-step increases in the coolant temperature, we notice that there
Modes
Modes I 2 3 4 56
7 5] 9101112
Input.
12 3 4 5 6 7 8 9101112 Input voriobles
variables
.
rc
I . ,.
V6
•
.
.
.
.
.i.
I
.
I
,If
]
,ll
ill..
]
Ill..
!
Th
"
'
I
"
. . . .
,,,11
Fig. 7. Activationsof the modes of a binary 10-stageSRV distillationsystem by input variables. CACE VoL 7 No. 2--D
114
KAZUYUK] SH[MIZOand R[CHAI~ S. H. MAH
exists a maximum at around UA = 1.5 x 105 J/s K (Fig. 9), and if the mean temperature differences at the condenser and the reboiler are 20°K or less, the system becomes unstable for values of UA between 1.5 and 3.5 x 105 J/s K. Figure 9 is based on the linear equations, and therefore, great care must be exercised in extrapolation, but it can be interpreted as indicating local sensitivities. More dependable values may be obtained by quasi-static calculations based on the nonlinear column model, but trialand-error calculations are necessary to find the new state of pressures. It may be supposed that the existence of a maximum is due to the fact that the unit-step increase in the cooling medium temperature does not cause the same amount of increase in the overhead vapor
flow at different values of UA, since the condenser duties are less at large values of UA than those at small values. The new steady state values after the step decrease in the overhead vapor flow rate fall almost on the curves given in Fig. 9, but the points tend to lie above the curves for higher UA values. They also exhibit a highly sensitive region. Let us look at the phenomena occurring in the stripping section. The new steady state values of the stripping column pressure are given in Fig. 9(c). The increase in the temperatures at the rectifying section will be transmitted either through the liquid which flows from the bottom of the rectifying column to the top of the stripping column, or through the pairing stages for the heat to be transferred between the two sections. Since there is
(a)
UA x 10-5 J / s * K 0
0
~-
0.15 f
x
0.10
2
a. c
E g oo5 C
4 6
u
n-
I0
O
o~5f
15 Time x I0-,3 s
I
I
20
25
(b) UA x io-SJ/s *K
O.IO
°°~
o
I
5
I
Io
1
15 Time x 10-3, s
1
2o
I
25
Fig. 8. Responses to a unit-step increase in the cooling medium temperature (a) Rectifying column pressure responses (b) Distillate composition responses.
Dynamic characteristics of binary SRV distillation systems
115
015
~n
010
r'lo~
E ~ 'o
~x 0.05
-
-
Q)
r~
0
2
3
4
H e o t t r o n s f e r porometer,
5
6
7
UA x 1 0 - S J / s ° K
(b)
0.08 x-
g
006
ATm, °K 0.04
2O 0
002
I
I
I
3
2
I
4
I
5
Heot tronsfer perometer,
6
7
UA x 1 0 - S J / s ° K
(c 015
c-
010
~o O.
._o.
005
if3
0
2
3
4
H e o t t r a n s f e r pororneter,
5
6
7
UA x 1 0 - s J / s °K
Fig. 9. Static gains for a unit-step increase in the condenser cooling medium temperature (a) Rectifying column pressure, (b) Distillate composition, (c) Stripping column pressure.
116
KAZUYUKISHIMIZUand RICHARDS. H. MAH
little change in the vapor and liquid flows in the stripping column, the pressure is the only factor which affects the separation and the bottom product becomes more volatile.
Vapor flow from the stripping section to the rectifying section Suppose there is no heat exchange between the rectifying and the stripping sections. The increase in the vapor flow from the top of the stripping section to the bottom of the rectifying section raises the rectifying column pressure almost instantaneously. This raises aU the temperatures in the rectifying section and increases the mean temperature difference in the condenser, which increases condensation and tends to decrease the rectifying column pressure until it settles at a new steady state as shown in Fig. 10. The inverse response is due to the right half-plane zeros of the transfer function in the complex plane and the system is non-minimum phase. This inverse response can also be seen in the conventional distillation columns as the inherent regulation when the reboiler heating is increased[8]. If there are heat exchanges between the rectifying and the stripping sections, the inverse response phenomenon tends to disappear, as for UA = 2 to 4x 10SJ/sK in Fig. 11. As noted by Davison[10], even for the conventional distillation columns, the inverse response phenomenon will disappear if the reflux rate instead of the top product flow is used for the liquid level control in the overhead accumulator. Let us now look at the phenomena occurring at the stripping section. The increase in the vapor flow from the
stripping to the rectifying section reduces the stripping column pressure (Fig. 12a). Since the external reflux rate is kept constant, the decrease in the stripping column pressure and the increase in the vapor flow in the section both favor the separation. As a result the bottoms composition becomes less volatile (Fig. 12b). Note that there exist inherent regulations to keep all the temperatures in the stripping section constant, since the decreases in the compositions tend to raise the temperatures, while the decrease in the stripping column pressure tends to lower them. If there are heat exchanges between the rectifying and the stripping sections, the increases in the temperatures at the rectifying section increase the heat transfer between them, and this tends to increase the stripping column pressure. As a consequence, the stripping column pressure is highly insensitive to the disturbance for IrA greater than 2 x 105 J/s K (see Fig. 12a).
Reboiler heating medium temperature The increase in the reboiler heating medium temperature causes an increase in the vapor flow at the stripping section, which makes the operating line in the McCabe-Thiele diagram less steep and the separation easier. However, since the vapor flow from the stripping to the rectifying section is kept constant, the stripping column pressure tends to increase and this affects the relative volatility in such a way as to cause less separation. If there is no heat exchange between the rectifying and the stripping sections, the pressure effect dominates and the bottoms composition becomes more volatile. On the other hand, if there exist heat exchanges between the two sections, the bottoms becomes less volatile (Fig. 13). The new steady state values of the stripping column
02
?0
~Tm, ° K
x
J
0 I 20
E 2
15
20
25
I
I
I
0 Time
x 10-,3 s
rr"
-0.1
40
Fig. 10. Responses to a 0.01kg-mole/s step increase in the vapor flow from the stripping to the rectifying section: UA = 0 J/s °K.
Dynamic characteristics of binary SRV distillation systems
117
0 O_
~_ o
~2
Cx 0 O~,.,~..r
u
1
I
I
15
20
25
Time x I0-,3 s
-Ot[ Fig. 11. Responses to a 0.01 kg/s step increase in the vapor flow from the stripping to the rectifying section: AT,. = 30 °K.
(a) Heat transfer parameter,
o
2
3
4
5
6
I
I
I
I
I
I
g3 o-
o
-I
"~
--
-2
c O_
Q_ C-. t/)
UA x I0-, s d / s °K
I
ATm,
--3
_
°K
\
(b)
o
7
Heat_ t r a n s f e r p a r a m e t e r ,
UA x 10-5,
d/s
°K
I
2
3
4
5
6
7
I
I
I
I
I
I
i
g ×
K
c
2
-0.O5
-O. lO
Fig. 12. Static gains for a 0.01 kg/s step increase in the vapor flow from the stripping to the rectifying section (a) Stripping column pressure, (b) Bottoms composition.
118
KAZUYUrJ SHI~ZU and RICHARD S. H. MAH
3
U A x IO-'~J/s*K
--
2--
%
Time x I0-~3 s
?
3
,?
25
1
8 2
-3
m
-4
5 Fig. 13. Responses to a unit-step increase in the rebofler heating medium temperature.
(a) ,.o
B r~
3 " 8 ° x
0.5
/
ATm,
I I
I 2
I 3
~
-
°K
I 5
Heat transfer parameter, UA x 10-5, d / s ° K
Cb)
H e a t t r a n s f e r parameter UA x I0~ 5 d / s °K x o
'0
I I
2 I
.3 1
4 I
6
5 I
1
7
I
x
tO
2 4
E
8
..9 ~o on
6 8
Tin,
°K
I0
Fig. 14. Static ains for a unit-step increase in the reboiler heating medium temperature (a) Stripping column pressure, (b) Bottoms composition.
119
Dynamic characteristics of binary SRV distillationsystems pressure and the bottom product composition after the unit-step increase in the heating medium temperature are plotted against the heat transfer parameter, UA in Fig. 14. The phenomena occurring in the stripping section affect the rectifying section through either the vapor flow from the stripping to the rectifying section or the heat transfer between the two sections. If there is no heat exchange between the two sections, the rectifying column pressure and the top product composition change very little. But if there are heat exchanges between them, the rectifying column pressure tends to increase and the top product composition tends to become heavier. The increase in the rectifying column pressure is caused by the fact that the increase in the stripping column pressure and the decreases in the compositions in the section both tend to raise the temperatures in the stripping section, and therefore decrease the heat transfer between the two sections.
External reflux ratio Since we are assuming that the hydraulic delay occurring in the liquid flows is negligible and that the bottoms of the rectifying and the stripping columns are under level control, an increase in the external reflux rate affects all stages instantaneously and increases the bottoms flow rate. The situation is almost the same as that of the conventional distillation. The increase in the external reflux rate leads to increases both in the top and the bottom product compositions, and, therefore, tends to decrease the temperatures, which, in turn, decreases the heat transfer at the condenser and increases the heat transfer at the reboiler. As a consequence, both of these increase the rectifying and the stripping column pressures. The increase in the pressures tends to increase all the stage temperatures, which counteract against the original effect. Finally, the states will be in balance at higher pressures and the contents at all stages will be more volatile than those before the disturbance.
Feed composition Consider the first instant after the feed composition is
suddenly increased. Note that the feed enthalpy is assumed to remain constant. The enthalpy of the vapor leaving the feed stage to the bottom of the rectifying column starts to decrease while the enthalpy of the liquid coming from the bottom of the rectifying column almost remains the same. The liquid enthalpy at the bottom of the stripping column also remains the same. These imply that the rectifying column pressure tends to decrease and the stripping column pressure tends to increase. If there exist heat exchanges between the two columns, the increase in the composition at the feed stage tends to increase the temperature difference between the paired stages, which causes the increase in the heat transfer between the two sections and results in the same situation as above. As the effect of the disturbance propagates to the condenser and the reboiler, the heat transfer rate in the condenser decreases while that in the reboiler increases, and these tend to raise the rectifying and the stripping column pressures. Consequently, the rectifying column pressure shows an inverse response in Fig. 15. The new steady state values of the rectifying column pressure and the top product composition after a 10% step increase in the feed composition are plotted in Fig. 16.
Feed flow rate Since the feed is assumed to be a saturated liquid, the increase in the feed flow rate causes the increase in the liquid flow rate in the stripping section. This tends to increase all the composition at the stripping section and decrease all the temperatures, which results in the similar situation as in the case of feed composition disturbance. The rectifying and the stripping column pressures increase and the former shows an inverse response. CONCLUDING REMARKS
The simulation results show that the system becomes unstable or very sensitive to the disturbances over a range of values of UA if the mean temperature differences at the condenser and the reboiler are small, which means that both UAc and UAr are large.
UAY
I 0 - ,5 J / s ° K
O O5
f
~
~-#_ o
i
0m ×
F~ (3d
5
10
15 T i m e × I 0 - ,3
i 2O
s
Fig. 15. Response to a 10% step increase in the feed composition:AT,. = 30°K.
25
120
KAZUYUKI SHIMIZU and RICHARD S. H. MAH
c t3._
o~ × C
ATm'
oK
L
'Z" ~
0
i
30 0
2
3
Heat transfer
4
parameter,
5
6
7
UA x I0 -5 d / s ° K
(b) 0.7
"
~ o
ATrn ,
0.4
K
20
0.3
i5
o.~
0
i
I
I
I
I
2
3
4
5
6
H e a l ~.ronsfer porome+.er,
7
UA x I0-,s d / s °K
Fig. 16. Static gains for a 10% step increase in the feed composition (a) Rectifying column pressure,
(b) Distillatecomposition.
Moreover, the range broadens, as the mean temperature differences decrease. The reason is as follows. Suppose the liquid composition at the top stage of the rectifying section is increased. Then the temperature tends to decrease and so does the overhead vapor flow rate, which causes a rise in the rectifying column pressure. This tends to decrease the top product composition, which counteracts against the original change. The system eventually settles down at a new steady state. This is the case where there is no heat exchange between the rectifying and the stripping sections. If there exist heat exchanges between the two sections, the pressure effect diminishes and the following effect becomes dominant. The decrease in the overhead vapor flow causes a decrease in the vapor flow rate in the rectifying section and this tends to increase the top product composition, which reinforces the original change, and if UA~ is large, this causes instability of the system. For the same product specifications as we increase the heat transfer rate between the two sections, we must reduce
the condenser duty, which means reducing UAo since 7'2- Tc remains the same. For high heat transfer rate the effect of disturbances tends to be reduced. This is why the system becomes sensitive to the disturbances over certain range of UA. A similar explanation applies to the stripping section and the reboiler.
Acknowledgement--This work
was supported by the National Science Foundation Grant CPE 764)6221. NOMENCLATURE
A (N + 2)x (N + 2) system matrix in the linear state space description, Eq. (35) B (N + 2)x 6 input matrix in the linear state space description, Eq. (35) b jth column of B Fi feed flow rate to stage j, kg-mole/s Hi vapor enthalpy at stage j, J/kg-mole H* outlet vapor enthalpy of the compressor, J/kg-mole h$ outlet liquid enthalpy of the condenser, J/kg-mole hi liquidenthaipy at stage j, J/kg-mole
Dynamic characteristics of binary SRV distillation systems hF~ Kis Ls Mr m N
UA~ UA~ UAr u Vs W~ wi x y z
feed enthalpy to stage j, J/kg-mole equilibrium constant of component i at stage j liquid flow rate leaving stage j, kg-mole/s liquid holdup on stage j, kg-mole bottom stage number of the rectifying column total number of stages including condenser and reboiler as stages pressure at stage j, Pa rectifying column pressure, Pa stripping column pressure, Pa condenser duty, W reboiler duty, W temperature at stage j, K temperature at the paired stage with stage j, K mean temperature difference at the condenser or the reboiler liquid side stream flow rate leaving stage j, kg-mole/s heat transfer parameter assumed constant throughout the columns, J/s K heat transfer parameter at stage j, J/s K heat transfer parameter at the condenser, J/s K heat transfer parameter at the reboiler, J/s K input vector in Eq. (35) vapor flow rate leaving stage j, kg-mole/s vapor side stream flow rate leaving stage j, kg-mole/s ith left eigenvector of A liquid composition at stage j vapor composition at stage j feed composition to stage j
Greek ~i~ ~j X~ /~ dp~ Os
symbols activation of mode i by input us (N 4- 2) x 1 state vector in Eq. (35) ith eigenvalue of A polytropic coefficient dxs]dt. See also Eqs. (4)-(6) function defined by the r.h.s, of Eqs. (7)-(9)
p~ p, p~ Q~ Q, Ts Tr AT,, Ui UA
121
APPENDIX
Derivation of perturbed linear equations for a binary S R V distillation system For small perturbations about a steady state Eqs. (4), (5) and (6) yield dgxddt = { - (L, + D)gxl - xl(gLi + 8D) + V28y2 + y28V2}/MI
(18)
dgxsldt = {Ls_, 8xj-t + xs-, 8Lj , - ( Vs + Ws)6yj - ys8Vs - (Lj + Ui)8xs - xigLj + Vi+,Syi+t+ ys.,SVj+,+ Fj,Szs + zs,SFj}/Mi, j=2,3 ..... N-1 (19) dgxN/dt = {LN-,gXN-t + X~-~gL~ , - V~6y~ - y~,SVs - L~,SX~ - X~,SLs}/M~.
(20)
Similarly, we obtain from Eqs. (1)-(3) and (7), 8D = 8 V z - 8LI ,SL~=gL~-,-gVs+6Vs+,+gF~,
(21) j=2,3 ..... N-I
(22)
6L~ = 8 L ~ - , - 8V~
(23)
8V~ = { U A A 6 T : - 6%) - V2(gH2 - 6h'~)}/(H2- h~).
(24)
Now Eqs. (22) and (23) can be re-expressed as 8L~=gL,-gV2+6Vs+,+~_jgFk,
j=2,3 ..... N-1
(25)
N-I
8Ls = 6 L I - 8V2+ ~
8Fk
(26)
k--2
and rearrangement of Eq. (14) yields dps/dt = {dhs/dt - (dxs/dt)( Oh~lcgx~)o~}/(Oh~/Ops)x~, REFERENCES 1. C. J. King, Separation Processes. McGraw-Hill, New York (1980). 2. R. S. H. Mah, J. J. Nicholas, Jr. & R. B. Wodnik, Distillation with secondary reflux and vaporization. A comparative evaluation. A.LCh.E.J. 23(5) 678 (1977). 3. R. E. Fitzmorris & R. S. H. Mah, Improving distillation column design using thermodynamic availability analysis. A.I.Ch.E.J. 26(2), 265 (1980). 4. B. D. Tyreus & W. L. Luyben, Controlling heat integrated distillation columns. Chem. Engng Prog. 72(9), 59 (1976). 5. N. P. Doukas & W. L. Luyben, Control of an energyconserving prefractionator/sidestream column distillation system. I&EC Proc. Des. Dev. 20(1), 147 (1981). 6. G. Soave, Equilibrium constants from a modified RedlichKwong equation of state. Chem. Engng Sci. 27, 1197 (1972). 7. F. W. Wahl & P. Harriott, Understanding and prediction of the dynamic behavior of distillation columns. I & E C Proc. Des. Dev. 9(3), 396 (1970). 8. O. Rademaker, J. E. Rijnsdorp & A. Maarleveld, Dynamics and Control of Continuous Distillation Units. Elsevier, New York (1975). 9. J. E. Rijnsdorp & A. Maarleveld, Use of electrical analogues in the study of the dynamic behavior and control of distillation columns. Proc. Joint Symp. on Instrumentation and Computation in Process Development and Design, Instn. Chem. Engrs. London, 135 (1959). 10. E. J. Davison, Control of a distillation column with pressure variation, Trans. Inst. Chem. Engng 45, %229 (1967). 11. H. H. Rosenbrock, Distinctive problems of process control. Chem. Engng Progr. 58(9), 43 (1%2). 12. R. E. Levy, A. S. Foss & E. A. Grens, II, Response modes of a binary distillation column. I&EC Fundtl. 8, 765 (1%9). CACEVoL 7 No. 2--E
j = 2, 3 . . . . . N.
(27)
If we substitute for dxjldt and dhsldt using Eqs. (5), (6) and (8), (9), respectively, and eliminate Lj using i
Lj = LL- V,.+ VS+I+ ~_~(Fk - Uk - Wk)
(28)
dps/dt = a t V2 +/3sVs 4- 3'jVs+,+ lrj
(29)
we obtain
where aj = { - hi , + h~ + (xj- , - xj)( ahjl axj)pj}/ $j (3j = {hi , - Hj - ( xj , - yj)( ahil axj)pj}l ~j ~,~= { - hs + Hj+ I + (xj - ys+l)( ahs/ axs')pj}l~j ~rj = [ { L,+ ~ ( F k - W k - U k ) } { h j - , - xj 4( ah,/ &rj)oj} i -Wj{Hs-yj(~ghj/c~xs)pj}-{Ll+ ~ (Fk - Wk - Uk) + Us} x {hi - xj(ahjl axj),,} + F~{hF~- zj(ahs/axs)o,} + UAs(Tj. - Tj)} ] / ~ s ii and
~j = {( ahjl apj)~ Uj}.
122
KAZUYUKI SH]~ZU and RICHARDS. H. MAH
Similarly, the perturbation of Eq. (16) yields
By virtue of assumption (viii) we have dp2/dt = dp3ldt . . . . .
dp,,+Jdt = dp.,+2ldt . . . . .
(30) (31)
dp,ddt dpNldt.
Substituting for dpj/dt using Eq. (29) and applying perturbations to each equation in turn we obtain
d~p~/dt = [L,a6h,. + h,.6L,,
-
Vm+16Hm+l -
H,,+t6Vm+l
-
L N S h N
-
hNSLN
N
+ ~__~+,{ - Uj~hi - Wi6Hj + h ~ F i + Fj6hF, + UAi(6Ti. - 6ri) N
~jaVj + ('/i -/3j+,)aVj+, - yj+~aVj+2
- (OhilOxi)
= - Vjaf~i- Vj+,(8~,j -af~j+,)+ Vj+:avj+~ + ( - aj + ai+,)~V2+ V2(- ~aj + ~ai+,)- 6~r~+ 87rj+,, j=2,3 . . . . . m - 2 a n d j = m + l , m + 2 . . . . . N - 2 .
(32)
Equation (32) may be solvedfor 6Vj,the coefficients6~j,8~j,and 61rjbeing obtained by perturbation. The perturbation of Eq. (15) yields dSpAdt = [Lt~hl + hlSLl - V26H2 - H26V2 - L,.6h,, - h,.SL,. m
j=
I
(34)
Equations (18)--(20), (32) and (33) constitute the set of perturbed linear equations. The perturbations of vapor and liquid enthalpies, 6/-/i and 6hi, may be evaluated numerically by perturbing xj and pj about the steady state conditions on the bubble point surface. Perturbations 8Lj and 8Vi are obtained from Eqs. (25), (26), (24) and (32), respectively. In summary, the perturbed linear equations may be written in the form of d.~/dt = A t + Bu
(35)
+ V~+~6H*~+,+ H*+t6V~+, + ,~={- U~ahi - WjSHj + hF~Fj m
+F~6ht:,+UAi(6T,.-,~,,-,O,..,,.,.,,,,]I(~,)
where ~ is a (12+ 1) state vector of liquid mole fractions (condenser through reboiler), rectifying and stripping pressures, and u is a (6 x 1) input vector of To, V6, Th, L1, z6 and F6.
0098-1354/83/020105-18503.00/0 Pergamon Press Lid,
Printed in Great Britain.
DYNAMIC CHARACTERISTICS OF BINARY SRV DISTILLATION SYSTEMS KAZUYIJKISHIMIZU'~and RICHARDS. H. MAH* Department of Chemical Engineering, Northwestern University, Evanston, IL 60201, U.S.A. (Received 27 July 1981; revision 5 April 1982)
Abstract--This paper reports a simulation study of SRV (secondary reflux and vaporization) distillation system using a perturbed linear model. The important factors which affect the dynamics are the heat transfer rate between the rectifying and the stripping sections, and the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler. The system becomes unstable if the mean temperature differences at the condenser and the reboiler are small, and the heat transfer rate between the rectifying and the stripping sections falls within a certain critical range of values. On either side of this range the system is insensitive to any kind of disturbances because of the inherent regulation occurring between them. Modal analysis shows that the external reflux rate activates the slowest mode which contains mainly the contributions of liquid concentrations on all stages. Scope--The drastic increase in energy costs over the last several years has stimulated the research on fractional distillation techniques, which are important energy consumers in the chemical and petroleum industries. The various schemes for reducing the energy requirements of distillation have been proposed under such categories as multiple effect, vapor recompression and heat pump[I]. Recently, Mah et al.[2] proposed a new type of distillation scheme which makes use of secondary reflux and vaporization (SRV). The advantages of SRV distillation has been evaluated from both the viewpoints of utility consumption[2] and of thermodynamic availability[3]. In addition to steady state performance and economic considerations, the dynamics and control characteristics of each new distillation scheme must be investigated as an integral part of the overall evaluation. It is dangerous to design control systems based on the knowledge of the conventional distillation, since the static and dynamic characteristics may be significantly modified as a result of energy integration. Tyreus & Luyben[4] presented the results of digital simulation studies of the dynamics and control of multi-effect, heat integrated distillation columns. Doukas & Luyben[5] showed the simulation studies of a prefractionator/liquid sidestream two column system separating a ternary mixture. In this paper we shall show how the dynamic characteristics of SRV distillation differ from the conventional distillation. Conclusions and Significance---The transient behavior of a 10-stage SRV distillation system was analyzed in terms of step responses based on linear equations, which were derived by perturbations about some steady state operating conditions with allowance for time variation of pressures. Dynamic characteristics of the system were interpreted for the changes of such variables as condenser cooling medium temperature, vapor flow from the stripping to the rectifying section, reboiler heating medium temperature, external reflux rate, feed composition, and feed flow rate. The study shows that the important factors which affect the dynamics are the heat transfer rate between the rectifying and the stripping sections, and the mean temperature differences at the condenser and the reboiler. The system becomes unstable or highly sensitive to the disturbances if the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler are small for a critical range of values of heat transfer rate between the rectifying and the stripping sections. On either side of this range the system is insensitive to any kind of disturbances because of the inherent regulation occurring in the system. Modal analysis shows that the slowest mode consists mainly of liquid composition elements and is activated by the external reflux rate. On the other hand, the two fastest modes are dominated by the pressure elements and activated by the condenser cooling medium temperature, the reboiler heating medium temperature and the vapor flow rate from the stripping to the rectifying section. This study underlines the significant difference in static and dynamic characteristics of SRV and conventional distillation systems, which must be taken into consideration in designing the control systems.
tKazuyuki Shimizu is now with the Chemical Engineering Department of NagoyaUniversity in Japan. *Author to whom correspondenceshould be addressed. 105
106
KAZUYUKISmMIZUand RICHARDS. H. MArt SRV DISTILLATION
A 10-stage SRV distillation scheme is shown in Fig. 1 with the condenser and the reboiler counted as stages 1 and 10, respectively. In order for the heat to be transferred from the rectifying section to the stripping section the former must be operated at a higher pressure than the latter. Compression and expansion devices are required between the two sections to sustain the required pressure differential. As a result of heat exchanges between the two sections, the liquid reflux rate at rectifying section steadily increases as the liquid proceeds down the column, and the vapor flow rate at stripping section steadily increases as the vapor proceeds up the column. Introduction of this secondary reflux and vaporization reduces the reboiler and condenser duties, and enhances the separation of products.
Numbering the stage from the top with the condenser as stage 1 and the reboiler as stage N, the governing equations for the dynamic model of an SRV distillation system can be stated as follows: Total mass balances: d M d d t = 0 = V 2 - ( L , + D)
(1)
dM~/dt = 0 = L~_, - (Vj + Wj) - (Lj + Uj) + Vj+, + Fi,
j=2,3 ..... N-1 dMN/dt = 0 = L N - ] - VN - LN.
(2) (3)
Component mass balances for the more volatile component: d x d d t = ~ , = { - ( L t + D)x, + V2y2}/M~
(4)
MATHEMATICALMODELING
The following assumptions were made in deriving the mathematical model for SRV distillation: (i) The columns are operated with a binary mixture. (ii) The vapor holdups are negligible. (iii) The liquid holdups are perfectly mixed. (iv) The vapor is in equilibrium with the liquid leaving the stage. (v) The heat transfer dynamics occurring between the rectifying and the stripping sections may be neglected. The same is assumed to be true for the condenser and the reboiler. (vi) A total condenser is used for the overhead vapor. (vii) The feed stream is introduced as a saturated liquid. (viii) Except for pressure drop across the valve the pressure drop across the column is negligible. (ix) The hydraulic delay occurring in the liquid flows is negligible (constant molar holdups). (x) Liquid molar holdup for each stage is constant. (xi) The overhead accumulator, the bottom of the rectifying section and the reboiler are under level control by the manipulation of the liquid outflows.
--y__ FC
dxjldt = ~ = { L j - , x j _ , - (Vj + W3Y~- (Lj + Uj)xj
+ Vj+,yj+, + ~Z~}/Mj
j = 2, 3. . . . . N - 1
(5)
dxNldt = ?PN = (LN-~XN--~- VNyN - LNXN)/MN. (6)
Energy balances: 0 = gJt = (142- h ~ ) V 2 - Q~
(7)
dhj/dt = ~j = {Lj-thj_~ - (V~ + Wj)Hj - (Lj + Uj)h~ + Vj+tHj+, + Fjhv i + UAj(Tj. - Tj)}/M~,
j=2,3 ..... N-1
(8)
dhN/dt = ~brq= ( L N _ ~ h ~ _ , - V~HN - LNhN + Qr)/Ms
(9) where Qc = U A c ( T 2 - % )
and Q, = U A r ( T , - TN)
ConcklflNr
and T'/is the temperature of the stage paired with stage j and h~ is the liquid enthalpy of the condenser outlet stream. Phase equilibrium relationships for the more volatile component:
L5
1~
yj = K,ixj, j = 2, 3. . . . . N
d v3
Dew point relationship for the condenser: y J K n + (1 - y2)/K2, = 1
--V, V-9
s
Reboiler
] LC
I0
(10)
Distillate
Or
Bubble point relationships: K~jxj + K2j(1 - xj) = 1, j = 2, 3. . . . . N.
Bottoms
(12)
Compression of vapor from stage m + 1 of the stripping section to stage m of the rectifying section: T*÷t
Fig. 1. Schematicrepresentation of a 10-stage SRV distillation.
(11)
=
T,,+,(pdp~) ("-t/")
(13)
where T~+, is the outlet temperature of the compressor and ~ is the polytropic coefficient. Note that the last term of Eq. (8) plays a characteristic role in the dynamic behavior of an SRV distillation system.
Dynamic characteristics of binary SRV distillation systems STEADY STATEBEHAVIOR
In contrast to the conventional distillation the vapor and liquid flow rates in SRV distillation change significantly throughout the column as a result of the heat exchanges between the rectifying and the stripping sections, and this will affect the dynamic behavior. Let us first examine the effect of the heat transfer between the two sections on the steady state behavior. For the following study, we chose a 10-stage (ethylene-ethane) C2-splitter. Thermophysical properties were evaluated using the Redlich-Kwong-Soave equation of state[6], and the steady state distillation calculations were carried out using a modified Wang-Henke method given earlier[2]. Steady state operating conditions chosen are listed in Table 1. A feed is introduced at the sixth stage and the value of # is taken as 1.3. For simplicity let us assume that the product of heat transfer area and overall heat transfer coefficient be the same for all stages. Notice that this product appears only in the energy balance, Eqn (8), for the SRV distillation. In Fig. 2, the effects of the heat transfer parameter UA on the liquid and vapor flow rates and condenser and reboiler duties are displayed for a fixed top and bottom product specification. As the heat transfer from the rectifying section to the stripping section increases, the liquid and vapor flow rates in the lower part of the rectifying section and the upper part of the stripping section ("middle sections") increase. Less external reflux and reboil are required to meet the same product specification, as shown in Fig. 2(c). These effects are illustrated by the positive slopes of L 3 - L 7 and the negative slopes of LL, L2, Ls, L9 in Fig. 2(a), and similar contrasting behavior of V3- V7 and V2, Vs- Vio in Fig. 2(b). In Fig. 3, the profiles of liquid and vapor flow rates and liquid compositions are shown for both UA = 0 J/s K and UA = 6x 103 J/s K. Figure 3(c) shows that as UA increases, the "middle sections" also assume a greater share of component separation in comparison with the "end sections." As pointed out by Wahl and Harriott[7], the liquid and
vapor flow rates may affect the time constants of the system. The liquid composition profile may affect the measurement locations for the control system design. Rademaker et at.[8] suggested that the control tray be chosen at the point where the temperature profile has a maximum gradient. In this sense, Fig. 3(c) shows that the control tray must be moved down at the rectifying section and moved up at the stripping section as the magnitude of the heat transfer increases in SRV distillation. PERTURBED LINEAREQUATIONS
Many papers dealing with the analysis of distillation columns have assumed the pressure to be constant and have consequently been unable to study certain phenomena associated with pressure variations. In 1959 Rijnsdorp & Maarleveld[9] developed a mathematical description of pressure variation for their analog simulation. Shortly afterwards, Davison[10] gave an approximate linearized mathematical model which also allowed for pressure variation. He applied modal analysis technique to find the control configuration which minimizes the dominant time constant of the controlled system. The linear state space representation of the system equations can be obtained quite directly by linearizing the nonlinear column model equations about a steady state operating condition. It requires less simplifications than the Davison formulation. The state equations for the pressures are derived from the following relationships: dhl/dt = ( ohd axOp/tj + ( ahd Op,)x#j, i = 2, 3 . . . . . N
(14) Although liquid enthalpy h~ is a function of Tj as well as p,, xj and t, T~ can be expressed as a function of xj and pj from the bubble point relationship. By virtue of assumption (viii), the state equations for the pressures at the rectifying and the stripping sections can be obtained from Eq. (14) using the energy balance equations. For
Table 1. Operatingconditions of 10-stage SRV distillation (ethylene-ethane system) Feed:
Flow Rate, F 6
0.126
Composition,
z6
0.5000
kg-mole/s
Flow Rate, D
0.63
Composition, x I
0.9000
Distillate: kg-mole/s
Column P r e s s u r e : Rectifier,
Pr
Stripper, Ps
4 . 6 2 x 105 Pa 3.08 x 105 Pa
Liquid holdup par stage: Overhead
107
Accumulator, M 1
50.0 kg-mole
Rectifying section, Mj
20.0 kg-mole
Stripping section, Hj
30.0 kg-mole
Reboller, Mlo
100.0 kg-mole
Condenser Temperature i T 1
202.0 K
Reboiler Temperature, TI0
206.0 K
108
KAZUYUrdSmmzu and RICHARDS. H. MAH I
~
St.age |
0.4
LJ- o,s 02 "o
0.1
I
2
3
4
5
6
7
Heat. transfer parameter, UA x I 0-,5 dis oK
Stage number
(b) 05
& 0.4 >-
0.3
0.2 ~ o
OI
I
I
I
I
2
I
3
4
I
5
I
6
Heat transfer parameter, UA x I0-,s d/s°K
~b
2~ °~
--
O~
I-0
I I
I
2
I
3
I
4
I
5
I
6
Heat transfer parameter, UA x I0-,s d / s °K Fig. 2. Effect of UA on the steady state operating conditions.
Dynamic characteristics of binary SRV distillation systems
z-
(a)
109
I
~x~
I
~X~
5-
E
2
~
6-CO
7-8--
J 9-IC 0
0.1 0.2 Liquid flow rotes,
L i,
03 0.4 kg-mole/s
0.5
(b) L
D
x~..
9
3 B
4-_o
E c
g CO
5 6
7 8P
91-IC 0
O. I
02
Vopor flow r o t e s ,
0.3
O.
0 .4
Vj, kg-mole/s
I
(c)
l
2 S
4 ..0
E
5 6--
o
7-8--
9-IC 0
f/
~¢ I
0.2
i
I
0 zl 06 L i q u i d composition,
i
08 xj
IO
Fig. 3. Effects of UA on the liquid and vapor flow rate profiles, and the liquid composition profile (©, UA =0J/s °K; x, UA = 6 x 10"~J/s °K).
110
KAZUYUm SrtImlzuand RICHARD S. H. MAH
the rectifying column pressure, we have
leaving only the slow modes which are important in determining the process dynamics. A knowledge of these modes can guide us in the selection of measurement locations for modal control[11]. Referring to Eq. (35) in the Appendix let us examine the dominant eigenvalues of A and their corresponding modes (right eigenvectors). The effect of heat transfer rate on system eigenvalues is shown in Fig. 4, where the mean temperature differences between the process fluid and the cooling or heating medium at the condenser and the reboiler are both taken to be 30 K. If the mean temperature differences are reduced to 20K, the dominant (numerically smallest) eigenvalue moves into the right half of the complex plane and the system becomes unstable at certain range of UA. This behavior will be further discussed later. The eigenvectors corresponding to the modes at UA = 4 x 105 J/s K are displayed as bar graphs in Fig. 5. Each bar represents the value of the corresponding eigenvector element. The elements are arranged from left to right in the order of the mole fractions of the more volatile component in the liquid phase (condenser through reboiler), rectifying and stripping pressures. Figure 5 shows that the first mode which corresponds to the largest time constant is mainly comprised of composition elements, the pressure elements being essentially negligible. As with the conventional distillation column[12], the influence of the first mode is spread fairly evenly throughout the two columns. The second mode contains a large contribution from the rectifying
dp./dt = [Llhl - V2H2- L,.h,. + V,.+~H,.+~ * m
+ ~{-
V~h. - WjHj + Fjh,:j + UAj(Tj. -
rj)
m
(15) where m is the bottom stage number of the rectifying column and H* is the outlet vapor enthalpy of the compressor. For the stripping column pressure, we have dpJdt = [L,.,hm - V,,,+lHm+l - LNh~ N
+ j.~+, { - Ujhj - WjHj + Fjhr,j + UAj(Tj. - Tj) -
(ahjlaxi),,,M#j
(ohjlap,),,j
.= I
(16)
I
where Try*= Th. The perturbed linear equations are given in the Appendix. In all subsequent figures, perturbations from the steady state conditions of Table 1 will be plotted. MODAL ANALYSIS
Many multi-stage processes have a wide spectrum of time constants. Typically, the fast modes decay rapidly
Heat. "Lransfer p a r a m e t e r , oo
I
2
3
U A x I0 -5, d / s ° K 4
5
6
X~
X2
i
X3 X4 X5
7 X6
_Io
~
f l X~
c~
-5
a~
ii
Xs
Fig. 4. Dominant eigenvalues of SRV distillation.
Dynamic characteristics of binary SRV distillation systems Eigenvect,or elemen't,s I 2 5 4 5 6 78
Eigenvect.or elements
9101112
Mode
,
[
I 2 34
5 6 78
9101112
.
I i
I
Mode
,lllllll,,..
I' ''II
I
II I
II,
III,
'
ii 'II'
II
i
II
,
I
,
w
I,
,
I.
I '1 I
o , I ,I
' I I ' ~
II
111
,
, .....
~
12
i
Ii I1
' I I
'
Fig. 5. Modes of a binary 10-stageSRV distillation UA = 4 x 105J/s K; ATm= 30 K.
pressure, while the third mode shows a large distillate composition element. On the other hand, the two fastest (1 lth and 12th) modes are comprised of the contributions from the rectifying and stripping pressures, the composition elements being almost totally absent. The effect of UA on the first three modes is displayed in Fig. 6. Figure 6(a) shows that pressure elements dominate the first mode at UA = 0 J/s K. But as UA increases, those elements become smaller, and for UA > 4x 105J/sK pressure elements are negligible in comparison with the other elements. Figure 6(b) shows that for the second mode the rectifying pressure element dominates at UA = 0 J/s K. But as UA increases, top and bottom composition elements also become numerically large. Figure 6(c) shows that the pressure elements are again dominant in the third mode at UA = 0 Jls K, but decreases rapidly as UA increases, while the reverse is true for the distillate composition. Another important factor in the modal analysis is the activation of the modes. The activation of mode i by the ]th input, uj, is defined by[ll] ~j = wiTbjuj
(17)
where b~ is the jth column of B in Eq. (35) and w~ is the ith left eigenvector of A. The activation is a measure of the extent to which an input affect a specific mode. The activation of modes caused by step changes in external
reflux rate, L~, condenser cooling medium temperature, T~, vapor flow rate from the stripping column to the rectifying column, 116, reboiler heating medium temperature, Th, feed composition, z6, and feed flow rate, F6, are given as bar graphs in Fig. 7. The length of each bar gives the value of the activation. Because of the differences in input step magnitudes, activations for a particular input cannot be directly compared with those for other inputs. Figure 7 shows that L~ activates the first mode. In other words, manipulation of LI affects the composition responses throughout the columns. On the other hand, the last two modes are activated mainly by To V6 and Th. Since the last two modes are dominated by pressure elements, the implication is that pressure responses may be modified by the manipulation of To, V6 and Th. SL~ULATIONR~ULTSANDVlSCVSSIONS Step response characteristics are important in understanding the transient behavior of the system. The main interest is to know the effect of the heat transfer between the rectifying and the stripping sections on the dynamic behavior. Step response curves were obtained, using the perturbed linear equations discussed in the previous section, for step increases in cooling medium temperature, vapor flow from the stripping to the rectifying section, reboiler heating medium temperature, external reflux rate, feed composition, and feed flow rate.
112
KAZUYUKISHIMIZUand RICHARDS. H. MAH i.Oi (a)
0
~"
I
2
HEAT TRANSFER
3
4
5
PARAMETER ,UA x 10-5
6
J/s
7
K
1 ~oo.~1-
~1,
/
~
~
I
oF
~
:~ 14~
0
"i
~ J
~
I
I
2
5
4
5
5
PARAMETER, UA x I0-
HEAT TRANSFER
6 d/s
7 K
~ 0.5
~
,o
~ o
-0"50I
~
.I
.
2.
HEAT TRANSFER
. 3 .
4
PARAMETER,
5 UA x I0
-5
J/s K
Fig. 6. Effect of UA on the modes of a binary 10-stage distillation systems, AT,. = 30 K (a) The first mode, (b) The second mode, (c) The third mode.
113
Dynamic characteristics of binary SRV distillationsystems
Cooling medium temperature Response curves of the rectifying column pressure and the top product composition to a unit-step increase in the condenser cooling medium temperature for different values of heat transfer parameter UA are given in Fig. 8. Much faster responses are observed for pressures than for compositions. This observation is consistent with the previous investigations for the conventional distillation columns [8]. The increase in the condenser cooling temperature decreases the condensation at the condenser, and therefore decreases the overhead vapor flow rate, which causes a rise in the rectifying column pressure. This tends to increase the temperatures in the rectifying column, which mitigates the primary effect on the heat transfer at the condenser. In contrast to the conventional distillation column, the step change also causes some secondary effects. The increase in the temperatures at the rectifying section increases the heat transfer between the rectifying and the stripping sections. This increase, in turn, causes a decrease in the rectifying column pressure, which partially nullifies the primary effect, and an increase in the stripping column pressure, which reinforces the primary effect. The increase in the stripping column pressure increases the stripping column temperatures and decreases the heat transfer between the two sections and
at the reboiler, which tends to decrease the stripping column pressure. Ultimately, the system settles down to a new steady state. In reality the situation is even more complex than the foregoing description because vapor and liquid compositions also play a role in these changes. Since the external reflux rate is kept constant, the decrease in the vapor flow in the rectifying section makes the operating line in the McCabe-Thiele diagram steeper, giving rise to better separation. Note that since the vapor flow through the compressor is kept constant, the vapor flow decreases more at the upper part than at the lower part of the rectifying column. Figure 8(b) shows the increase in the top product composition in all cases. In the case where there is no heat transfer between the rectifying and the stripping sections (UA = 0J/s K), this effect is suppressed by the pressure increase which reduces the relative volatility and results in less separation. For the case of UA greater than zero, namely, the case of SRV distillation, more enrichment in the top product composition might be expected for higher values of UA. But Fig. 8(b) shows that the top composition at UA = 2 x 105 J/s K increases more than at UA = 4 x 105J/s K or UA = 6 x 105 J/s K. If we plot the new steady state values after the unit-step increases in the coolant temperature, we notice that there
Modes
Modes I 2 3 4 56
7 5] 9101112
Input.
12 3 4 5 6 7 8 9101112 Input voriobles
variables
.
rc
I . ,.
V6
•
.
.
.
.
.i.
I
.
I
,If
]
,ll
ill..
]
Ill..
!
Th
"
'
I
"
. . . .
,,,11
Fig. 7. Activationsof the modes of a binary 10-stageSRV distillationsystem by input variables. CACE VoL 7 No. 2--D
114
KAZUYUK] SH[MIZOand R[CHAI~ S. H. MAH
exists a maximum at around UA = 1.5 x 105 J/s K (Fig. 9), and if the mean temperature differences at the condenser and the reboiler are 20°K or less, the system becomes unstable for values of UA between 1.5 and 3.5 x 105 J/s K. Figure 9 is based on the linear equations, and therefore, great care must be exercised in extrapolation, but it can be interpreted as indicating local sensitivities. More dependable values may be obtained by quasi-static calculations based on the nonlinear column model, but trialand-error calculations are necessary to find the new state of pressures. It may be supposed that the existence of a maximum is due to the fact that the unit-step increase in the cooling medium temperature does not cause the same amount of increase in the overhead vapor
flow at different values of UA, since the condenser duties are less at large values of UA than those at small values. The new steady state values after the step decrease in the overhead vapor flow rate fall almost on the curves given in Fig. 9, but the points tend to lie above the curves for higher UA values. They also exhibit a highly sensitive region. Let us look at the phenomena occurring in the stripping section. The new steady state values of the stripping column pressure are given in Fig. 9(c). The increase in the temperatures at the rectifying section will be transmitted either through the liquid which flows from the bottom of the rectifying column to the top of the stripping column, or through the pairing stages for the heat to be transferred between the two sections. Since there is
(a)
UA x 10-5 J / s * K 0
0
~-
0.15 f
x
0.10
2
a. c
E g oo5 C
4 6
u
n-
I0
O
o~5f
15 Time x I0-,3 s
I
I
20
25
(b) UA x io-SJ/s *K
O.IO
°°~
o
I
5
I
Io
1
15 Time x 10-3, s
1
2o
I
25
Fig. 8. Responses to a unit-step increase in the cooling medium temperature (a) Rectifying column pressure responses (b) Distillate composition responses.
Dynamic characteristics of binary SRV distillation systems
115
015
~n
010
r'lo~
E ~ 'o
~x 0.05
-
-
Q)
r~
0
2
3
4
H e o t t r o n s f e r porometer,
5
6
7
UA x 1 0 - S J / s ° K
(b)
0.08 x-
g
006
ATm, °K 0.04
2O 0
002
I
I
I
3
2
I
4
I
5
Heot tronsfer perometer,
6
7
UA x 1 0 - S J / s ° K
(c 015
c-
010
~o O.
._o.
005
if3
0
2
3
4
H e o t t r a n s f e r pororneter,
5
6
7
UA x 1 0 - s J / s °K
Fig. 9. Static gains for a unit-step increase in the condenser cooling medium temperature (a) Rectifying column pressure, (b) Distillate composition, (c) Stripping column pressure.
116
KAZUYUKISHIMIZUand RICHARDS. H. MAH
little change in the vapor and liquid flows in the stripping column, the pressure is the only factor which affects the separation and the bottom product becomes more volatile.
Vapor flow from the stripping section to the rectifying section Suppose there is no heat exchange between the rectifying and the stripping sections. The increase in the vapor flow from the top of the stripping section to the bottom of the rectifying section raises the rectifying column pressure almost instantaneously. This raises aU the temperatures in the rectifying section and increases the mean temperature difference in the condenser, which increases condensation and tends to decrease the rectifying column pressure until it settles at a new steady state as shown in Fig. 10. The inverse response is due to the right half-plane zeros of the transfer function in the complex plane and the system is non-minimum phase. This inverse response can also be seen in the conventional distillation columns as the inherent regulation when the reboiler heating is increased[8]. If there are heat exchanges between the rectifying and the stripping sections, the inverse response phenomenon tends to disappear, as for UA = 2 to 4x 10SJ/sK in Fig. 11. As noted by Davison[10], even for the conventional distillation columns, the inverse response phenomenon will disappear if the reflux rate instead of the top product flow is used for the liquid level control in the overhead accumulator. Let us now look at the phenomena occurring at the stripping section. The increase in the vapor flow from the
stripping to the rectifying section reduces the stripping column pressure (Fig. 12a). Since the external reflux rate is kept constant, the decrease in the stripping column pressure and the increase in the vapor flow in the section both favor the separation. As a result the bottoms composition becomes less volatile (Fig. 12b). Note that there exist inherent regulations to keep all the temperatures in the stripping section constant, since the decreases in the compositions tend to raise the temperatures, while the decrease in the stripping column pressure tends to lower them. If there are heat exchanges between the rectifying and the stripping sections, the increases in the temperatures at the rectifying section increase the heat transfer between them, and this tends to increase the stripping column pressure. As a consequence, the stripping column pressure is highly insensitive to the disturbance for IrA greater than 2 x 105 J/s K (see Fig. 12a).
Reboiler heating medium temperature The increase in the reboiler heating medium temperature causes an increase in the vapor flow at the stripping section, which makes the operating line in the McCabe-Thiele diagram less steep and the separation easier. However, since the vapor flow from the stripping to the rectifying section is kept constant, the stripping column pressure tends to increase and this affects the relative volatility in such a way as to cause less separation. If there is no heat exchange between the rectifying and the stripping sections, the pressure effect dominates and the bottoms composition becomes more volatile. On the other hand, if there exist heat exchanges between the two sections, the bottoms becomes less volatile (Fig. 13). The new steady state values of the stripping column
02
?0
~Tm, ° K
x
J
0 I 20
E 2
15
20
25
I
I
I
0 Time
x 10-,3 s
rr"
-0.1
40
Fig. 10. Responses to a 0.01kg-mole/s step increase in the vapor flow from the stripping to the rectifying section: UA = 0 J/s °K.
Dynamic characteristics of binary SRV distillation systems
117
0 O_
~_ o
~2
Cx 0 O~,.,~..r
u
1
I
I
15
20
25
Time x I0-,3 s
-Ot[ Fig. 11. Responses to a 0.01 kg/s step increase in the vapor flow from the stripping to the rectifying section: AT,. = 30 °K.
(a) Heat transfer parameter,
o
2
3
4
5
6
I
I
I
I
I
I
g3 o-
o
-I
"~
--
-2
c O_
Q_ C-. t/)
UA x I0-, s d / s °K
I
ATm,
--3
_
°K
\
(b)
o
7
Heat_ t r a n s f e r p a r a m e t e r ,
UA x 10-5,
d/s
°K
I
2
3
4
5
6
7
I
I
I
I
I
I
i
g ×
K
c
2
-0.O5
-O. lO
Fig. 12. Static gains for a 0.01 kg/s step increase in the vapor flow from the stripping to the rectifying section (a) Stripping column pressure, (b) Bottoms composition.
118
KAZUYUrJ SHI~ZU and RICHARD S. H. MAH
3
U A x IO-'~J/s*K
--
2--
%
Time x I0-~3 s
?
3
,?
25
1
8 2
-3
m
-4
5 Fig. 13. Responses to a unit-step increase in the rebofler heating medium temperature.
(a) ,.o
B r~
3 " 8 ° x
0.5
/
ATm,
I I
I 2
I 3
~
-
°K
I 5
Heat transfer parameter, UA x 10-5, d / s ° K
Cb)
H e a t t r a n s f e r parameter UA x I0~ 5 d / s °K x o
'0
I I
2 I
.3 1
4 I
6
5 I
1
7
I
x
tO
2 4
E
8
..9 ~o on
6 8
Tin,
°K
I0
Fig. 14. Static ains for a unit-step increase in the reboiler heating medium temperature (a) Stripping column pressure, (b) Bottoms composition.
119
Dynamic characteristics of binary SRV distillationsystems pressure and the bottom product composition after the unit-step increase in the heating medium temperature are plotted against the heat transfer parameter, UA in Fig. 14. The phenomena occurring in the stripping section affect the rectifying section through either the vapor flow from the stripping to the rectifying section or the heat transfer between the two sections. If there is no heat exchange between the two sections, the rectifying column pressure and the top product composition change very little. But if there are heat exchanges between them, the rectifying column pressure tends to increase and the top product composition tends to become heavier. The increase in the rectifying column pressure is caused by the fact that the increase in the stripping column pressure and the decreases in the compositions in the section both tend to raise the temperatures in the stripping section, and therefore decrease the heat transfer between the two sections.
External reflux ratio Since we are assuming that the hydraulic delay occurring in the liquid flows is negligible and that the bottoms of the rectifying and the stripping columns are under level control, an increase in the external reflux rate affects all stages instantaneously and increases the bottoms flow rate. The situation is almost the same as that of the conventional distillation. The increase in the external reflux rate leads to increases both in the top and the bottom product compositions, and, therefore, tends to decrease the temperatures, which, in turn, decreases the heat transfer at the condenser and increases the heat transfer at the reboiler. As a consequence, both of these increase the rectifying and the stripping column pressures. The increase in the pressures tends to increase all the stage temperatures, which counteract against the original effect. Finally, the states will be in balance at higher pressures and the contents at all stages will be more volatile than those before the disturbance.
Feed composition Consider the first instant after the feed composition is
suddenly increased. Note that the feed enthalpy is assumed to remain constant. The enthalpy of the vapor leaving the feed stage to the bottom of the rectifying column starts to decrease while the enthalpy of the liquid coming from the bottom of the rectifying column almost remains the same. The liquid enthalpy at the bottom of the stripping column also remains the same. These imply that the rectifying column pressure tends to decrease and the stripping column pressure tends to increase. If there exist heat exchanges between the two columns, the increase in the composition at the feed stage tends to increase the temperature difference between the paired stages, which causes the increase in the heat transfer between the two sections and results in the same situation as above. As the effect of the disturbance propagates to the condenser and the reboiler, the heat transfer rate in the condenser decreases while that in the reboiler increases, and these tend to raise the rectifying and the stripping column pressures. Consequently, the rectifying column pressure shows an inverse response in Fig. 15. The new steady state values of the rectifying column pressure and the top product composition after a 10% step increase in the feed composition are plotted in Fig. 16.
Feed flow rate Since the feed is assumed to be a saturated liquid, the increase in the feed flow rate causes the increase in the liquid flow rate in the stripping section. This tends to increase all the composition at the stripping section and decrease all the temperatures, which results in the similar situation as in the case of feed composition disturbance. The rectifying and the stripping column pressures increase and the former shows an inverse response. CONCLUDING REMARKS
The simulation results show that the system becomes unstable or very sensitive to the disturbances over a range of values of UA if the mean temperature differences at the condenser and the reboiler are small, which means that both UAc and UAr are large.
UAY
I 0 - ,5 J / s ° K
O O5
f
~
~-#_ o
i
0m ×
F~ (3d
5
10
15 T i m e × I 0 - ,3
i 2O
s
Fig. 15. Response to a 10% step increase in the feed composition:AT,. = 30°K.
25
120
KAZUYUKI SHIMIZU and RICHARD S. H. MAH
c t3._
o~ × C
ATm'
oK
L
'Z" ~
0
i
30 0
2
3
Heat transfer
4
parameter,
5
6
7
UA x I0 -5 d / s ° K
(b) 0.7
"
~ o
ATrn ,
0.4
K
20
0.3
i5
o.~
0
i
I
I
I
I
2
3
4
5
6
H e a l ~.ronsfer porome+.er,
7
UA x I0-,s d / s °K
Fig. 16. Static gains for a 10% step increase in the feed composition (a) Rectifying column pressure,
(b) Distillatecomposition.
Moreover, the range broadens, as the mean temperature differences decrease. The reason is as follows. Suppose the liquid composition at the top stage of the rectifying section is increased. Then the temperature tends to decrease and so does the overhead vapor flow rate, which causes a rise in the rectifying column pressure. This tends to decrease the top product composition, which counteracts against the original change. The system eventually settles down at a new steady state. This is the case where there is no heat exchange between the rectifying and the stripping sections. If there exist heat exchanges between the two sections, the pressure effect diminishes and the following effect becomes dominant. The decrease in the overhead vapor flow causes a decrease in the vapor flow rate in the rectifying section and this tends to increase the top product composition, which reinforces the original change, and if UA~ is large, this causes instability of the system. For the same product specifications as we increase the heat transfer rate between the two sections, we must reduce
the condenser duty, which means reducing UAo since 7'2- Tc remains the same. For high heat transfer rate the effect of disturbances tends to be reduced. This is why the system becomes sensitive to the disturbances over certain range of UA. A similar explanation applies to the stripping section and the reboiler.
Acknowledgement--This work
was supported by the National Science Foundation Grant CPE 764)6221. NOMENCLATURE
A (N + 2)x (N + 2) system matrix in the linear state space description, Eq. (35) B (N + 2)x 6 input matrix in the linear state space description, Eq. (35) b jth column of B Fi feed flow rate to stage j, kg-mole/s Hi vapor enthalpy at stage j, J/kg-mole H* outlet vapor enthalpy of the compressor, J/kg-mole h$ outlet liquid enthalpy of the condenser, J/kg-mole hi liquidenthaipy at stage j, J/kg-mole
Dynamic characteristics of binary SRV distillation systems hF~ Kis Ls Mr m N
UA~ UA~ UAr u Vs W~ wi x y z
feed enthalpy to stage j, J/kg-mole equilibrium constant of component i at stage j liquid flow rate leaving stage j, kg-mole/s liquid holdup on stage j, kg-mole bottom stage number of the rectifying column total number of stages including condenser and reboiler as stages pressure at stage j, Pa rectifying column pressure, Pa stripping column pressure, Pa condenser duty, W reboiler duty, W temperature at stage j, K temperature at the paired stage with stage j, K mean temperature difference at the condenser or the reboiler liquid side stream flow rate leaving stage j, kg-mole/s heat transfer parameter assumed constant throughout the columns, J/s K heat transfer parameter at stage j, J/s K heat transfer parameter at the condenser, J/s K heat transfer parameter at the reboiler, J/s K input vector in Eq. (35) vapor flow rate leaving stage j, kg-mole/s vapor side stream flow rate leaving stage j, kg-mole/s ith left eigenvector of A liquid composition at stage j vapor composition at stage j feed composition to stage j
Greek ~i~ ~j X~ /~ dp~ Os
symbols activation of mode i by input us (N 4- 2) x 1 state vector in Eq. (35) ith eigenvalue of A polytropic coefficient dxs]dt. See also Eqs. (4)-(6) function defined by the r.h.s, of Eqs. (7)-(9)
p~ p, p~ Q~ Q, Ts Tr AT,, Ui UA
121
APPENDIX
Derivation of perturbed linear equations for a binary S R V distillation system For small perturbations about a steady state Eqs. (4), (5) and (6) yield dgxddt = { - (L, + D)gxl - xl(gLi + 8D) + V28y2 + y28V2}/MI
(18)
dgxsldt = {Ls_, 8xj-t + xs-, 8Lj , - ( Vs + Ws)6yj - ys8Vs - (Lj + Ui)8xs - xigLj + Vi+,Syi+t+ ys.,SVj+,+ Fj,Szs + zs,SFj}/Mi, j=2,3 ..... N-1 (19) dgxN/dt = {LN-,gXN-t + X~-~gL~ , - V~6y~ - y~,SVs - L~,SX~ - X~,SLs}/M~.
(20)
Similarly, we obtain from Eqs. (1)-(3) and (7), 8D = 8 V z - 8LI ,SL~=gL~-,-gVs+6Vs+,+gF~,
(21) j=2,3 ..... N-I
(22)
6L~ = 8 L ~ - , - 8V~
(23)
8V~ = { U A A 6 T : - 6%) - V2(gH2 - 6h'~)}/(H2- h~).
(24)
Now Eqs. (22) and (23) can be re-expressed as 8L~=gL,-gV2+6Vs+,+~_jgFk,
j=2,3 ..... N-1
(25)
N-I
8Ls = 6 L I - 8V2+ ~
8Fk
(26)
k--2
and rearrangement of Eq. (14) yields dps/dt = {dhs/dt - (dxs/dt)( Oh~lcgx~)o~}/(Oh~/Ops)x~, REFERENCES 1. C. J. King, Separation Processes. McGraw-Hill, New York (1980). 2. R. S. H. Mah, J. J. Nicholas, Jr. & R. B. Wodnik, Distillation with secondary reflux and vaporization. A comparative evaluation. A.LCh.E.J. 23(5) 678 (1977). 3. R. E. Fitzmorris & R. S. H. Mah, Improving distillation column design using thermodynamic availability analysis. A.I.Ch.E.J. 26(2), 265 (1980). 4. B. D. Tyreus & W. L. Luyben, Controlling heat integrated distillation columns. Chem. Engng Prog. 72(9), 59 (1976). 5. N. P. Doukas & W. L. Luyben, Control of an energyconserving prefractionator/sidestream column distillation system. I&EC Proc. Des. Dev. 20(1), 147 (1981). 6. G. Soave, Equilibrium constants from a modified RedlichKwong equation of state. Chem. Engng Sci. 27, 1197 (1972). 7. F. W. Wahl & P. Harriott, Understanding and prediction of the dynamic behavior of distillation columns. I & E C Proc. Des. Dev. 9(3), 396 (1970). 8. O. Rademaker, J. E. Rijnsdorp & A. Maarleveld, Dynamics and Control of Continuous Distillation Units. Elsevier, New York (1975). 9. J. E. Rijnsdorp & A. Maarleveld, Use of electrical analogues in the study of the dynamic behavior and control of distillation columns. Proc. Joint Symp. on Instrumentation and Computation in Process Development and Design, Instn. Chem. Engrs. London, 135 (1959). 10. E. J. Davison, Control of a distillation column with pressure variation, Trans. Inst. Chem. Engng 45, %229 (1967). 11. H. H. Rosenbrock, Distinctive problems of process control. Chem. Engng Progr. 58(9), 43 (1%2). 12. R. E. Levy, A. S. Foss & E. A. Grens, II, Response modes of a binary distillation column. I&EC Fundtl. 8, 765 (1%9). CACEVoL 7 No. 2--E
j = 2, 3 . . . . . N.
(27)
If we substitute for dxjldt and dhsldt using Eqs. (5), (6) and (8), (9), respectively, and eliminate Lj using i
Lj = LL- V,.+ VS+I+ ~_~(Fk - Uk - Wk)
(28)
dps/dt = a t V2 +/3sVs 4- 3'jVs+,+ lrj
(29)
we obtain
where aj = { - hi , + h~ + (xj- , - xj)( ahjl axj)pj}/ $j (3j = {hi , - Hj - ( xj , - yj)( ahil axj)pj}l ~j ~,~= { - hs + Hj+ I + (xj - ys+l)( ahs/ axs')pj}l~j ~rj = [ { L,+ ~ ( F k - W k - U k ) } { h j - , - xj 4( ah,/ &rj)oj} i -Wj{Hs-yj(~ghj/c~xs)pj}-{Ll+ ~ (Fk - Wk - Uk) + Us} x {hi - xj(ahjl axj),,} + F~{hF~- zj(ahs/axs)o,} + UAs(Tj. - Tj)} ] / ~ s ii and
~j = {( ahjl apj)~ Uj}.
122
KAZUYUKI SH]~ZU and RICHARDS. H. MAH
Similarly, the perturbation of Eq. (16) yields
By virtue of assumption (viii) we have dp2/dt = dp3ldt . . . . .
dp,,+Jdt = dp.,+2ldt . . . . .
(30) (31)
dp,ddt dpNldt.
Substituting for dpj/dt using Eq. (29) and applying perturbations to each equation in turn we obtain
d~p~/dt = [L,a6h,. + h,.6L,,
-
Vm+16Hm+l -
H,,+t6Vm+l
-
L N S h N
-
hNSLN
N
+ ~__~+,{ - Uj~hi - Wi6Hj + h ~ F i + Fj6hF, + UAi(6Ti. - 6ri) N
~jaVj + ('/i -/3j+,)aVj+, - yj+~aVj+2
- (OhilOxi)
= - Vjaf~i- Vj+,(8~,j -af~j+,)+ Vj+:avj+~ + ( - aj + ai+,)~V2+ V2(- ~aj + ~ai+,)- 6~r~+ 87rj+,, j=2,3 . . . . . m - 2 a n d j = m + l , m + 2 . . . . . N - 2 .
(32)
Equation (32) may be solvedfor 6Vj,the coefficients6~j,8~j,and 61rjbeing obtained by perturbation. The perturbation of Eq. (15) yields dSpAdt = [Lt~hl + hlSLl - V26H2 - H26V2 - L,.6h,, - h,.SL,. m
j=
I
(34)
Equations (18)--(20), (32) and (33) constitute the set of perturbed linear equations. The perturbations of vapor and liquid enthalpies, 6/-/i and 6hi, may be evaluated numerically by perturbing xj and pj about the steady state conditions on the bubble point surface. Perturbations 8Lj and 8Vi are obtained from Eqs. (25), (26), (24) and (32), respectively. In summary, the perturbed linear equations may be written in the form of d.~/dt = A t + Bu
(35)
+ V~+~6H*~+,+ H*+t6V~+, + ,~={- U~ahi - WjSHj + hF~Fj m
+F~6ht:,+UAi(6T,.-,~,,-,O,..,,.,.,,,,]I(~,)
where ~ is a (12+ 1) state vector of liquid mole fractions (condenser through reboiler), rectifying and stripping pressures, and u is a (6 x 1) input vector of To, V6, Th, L1, z6 and F6.