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Modeling, simulation and control of liquid-liquid extraction columns
Pergamon PII:
Chemical Engineering Science, Vol. 53, No. 2, pp. 325-339, 1998 ~c~,1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009-2509(97)00310-2 ooo9 2509/98 $19.00 + o.oo
Modeling, simulation and control of liquid-liquid extraction columns O. Weinstein, R. Semiat and D. R. Lewin* Department of Chemical Engineering, Technion I.I.T., Haifa 32000, Israel (First received 21 May 1997; accepted 12 June 1997)
Abstract A model describing the hydrodynamics and mass transfer of countercurrent liquidliquid extraction columns is developed and solved. The hydrodynamic model assumes that the dispersed-phase drops behave as spheres of uniform diameter. The model has been found to be qualitatively in agreement with experimental results published in the literature. It is shown that conventional dispersion interface level control using the continuous-phase effluent flow rate as the manipulated variable causes unavoidable overshoots and oscillations in the mean holdup and outlet concentrations, while an alternative scheme using the continuous-phase feed flow rate leads to a significant improvement in the dynamic response. A model-based decentralized M I M O control scheme is designed and tested by simulation, and is shown to provide excellent servo and regulatory performance. © 1997 Elsevier Science Ltd Keywords:
Liquid-liquid extraction column; mathematical models; dynamic simulation and
control.
l. INTRODUCTION
Solvent extraction in continuous columns is one of the most important separation processes in chemical engineering. However, the control of these units can often be problematic, partially due to their multiphase nature, and partially due to the difficulty of on-line measurement of output variables. Thus, the reliable simulation of the transient behavior of these columns is extremely valuable. A review of the progress on the dynamic modeling and simulation of extraction columns was published by Steiner and Hartland (1983). Most previous studies have dealt with mass-transfer transients while assuming hydrodynamic steady state. In contrast, the detailed experimental results published by Hufnagl et al. (1991) imply that the dynamic response of outlets, holdup and concentrations are determined largely by the dynamic behavior of dispersed phase. Recently, several papers (e.g. Casamatta and Vonderpohl, 1985; Dimitrova A1 Khani et al., 1989; Tsoris et al., 1994) have developed models relying on a drop population. These models have the potential of explaining the nonuniformity of holdup profiles along the contactor, and thus they allow a more realistic description of flooding conditions. On the other hand, this approach leads to complicated mathematical formulations. Aside from this, drop breakage and
*Corresponding author. E-mail: [email protected].
coalescence models, on which the approach relies, have not yet been developed for all types of columns. Perhaps for these reasons, this approach has not been used for control purposes. For example, the holdup overshoot and oscillations in response to input disturbances shown analytically by Brownstein and Schegolev (1988), and demonstrated experimentally by Hufnagl et al. (1991), is an important characteristic of extraction column dynamical behavior. Nevertheless, this phenomenon has not been studied in the works dealing with drop population models. This paper will show that important aspects of extraction-column dynamics may be investigated with sufficient accuracy using a simpler approach, based on a hydrodynamic model relying on the Sauter mean diameter, which is an accepted measure of the average diameter of a dispersed-phase drop. This has been widely used in relationships for parameters of extraction processes and for the estimation of flooding conditions (Bair and Lane, 1973; Baird and Shen, 1984). The model assumes a uniform drop diameter and thus, the number of equations that need to be solved is much smaller than that for drop population models. The model for mass-transfer developed accounts for changes in the continuous and dispersed flow rates caused by the holdup profile transient. The model is intended to be used for control purposes, and is used to simulate conditions both for a Karr column (Karr et al., 1987), in which the dispersion layer is supported on a stack of perforated, vibrating plates, and for a Kuhni column, in which it is supported on rotating plates (Hufnagl et al., 1991). The novel 325
326
O. Weinstein et al.
dispersion-level control scheme suggested by Hufnagl et al. (1991) is simulated using the proposed model, and compared to the conventional control commonly adopted. Finally, a multivariable controller for a Karr column is developed, relying on a linear model approximating the dynamics of the column. This is tested by simulation on the detailed nonlinear model developed in this study. 2. M A T H E M A T I C A L
liquid-liquid extraction equipment is discussed by Godfrey and Slater (1991). The relationships for Vs and other parameters used in the model are discussed below. Since the cross-sectional area is constant, it is possible to introduce new variables Ue = Qe/S, Uc = Qjs,
Ue.in = Qe.~./S, Uc.,n = 9_c.,#S, Ue .... = Qe.out/S
and U.... t = Q .... t/S. Thus, eqs (1) to (3) can be rewritten as
MODELING
2.1. Assumptions The mathematical model for a liquid-liquid extraction column relies on the following main assumptions:
Oe(h) Ua(h) = e(h) Ve(h) - De(h) - Oh U c - - U d = U c , in -
Ud,ou t =
U .... t --
(5)
U d . in = A U
(6) (1) The radial variations of the mean velocity and the concentrations of each phase are negligible. (2) The phase volume flow rates are independent of mass-transfer and solute concentration. (3) The dispersed-phase drops behave as spheres of a uniform diameter. (4) The model describes the processes occurring in the dispersion layer of an extraction column. The length of this dispersion layer is fixed. 2.2. Modeling of fluid dynamics Since the dispersed phase is assumed to consist of spherical drops of uniform diameter, the mean drop size in a dispersion is expressed as the Sauter mean diameter: d32 = Z n f l ~ / Z n i d 2 where di is the individual drop diameter and n~ is the number of drops with this diameter. The direction of movement of the dispersed phase coincides with the positive vertical direction. The flow rate of the dispersed phase, Qa, through the crosssection S is expressed by:
0~(h)
Qe(h) = St(h)Ve(h) - SDa(h) - Oh
=
Qc,out
--
Qa.i, = AQ
O~_
dUe_
O(Va~)
where Q~,in and Qa, in are continuous- and dispersedphase feed flow rates, Q.... t and Qa, o,t are continuous and dispersed phase effluent flow rates, and Vc is the velocity of the continuous phase:
O/
&\
(8)
Substituting eq. (5) in eq. (6) gives O~ U,. = AU + Ua = AU + e V e - De ~
(9)
and by using eqs (4), (7) and (9), Ve is obtained as: Va = (1 - e)V,(e) - AU + De -~.
(10)
The boundary conditions are & Ua, i n = r ~ V a - D e ~
(1)
(2)
(7)
A balance of dispersed phase in an elementary volume of the column leads to the following dynamic equation:
h=0;
where De(h) is dispersion coefficient, Vd(h) is velocity of dispersed phase, e(h) is the volume fraction holdup of dispersed phase at the height h of the dispersion layer in the column. The flow rate of continuous phase, Q~, can be obtained by assuming incompressible flow in the material balance between crosssections h = 0 and h = h Q c - - Q~d = Q c , in - - Q a , o u t
U~ Vc = 1 - e"
(11)
Oe
h=l;
T~=0.
(12)
These boundary conditions arise from material balances across the system's boundary planes, with the assumption that the column is a "closed system", i.e. dispersion occurs only between h = 0 + and h = l - . The significance of these conditions is discussed by Wehner and Willem (1956). Miyauchi and Vermeulen (1963) applied similar boundary conditions in their steady-state solutions of an axial dispersion model. The outlet flow, Ue,out, is expressed by
h=l,
Ue,o~t=eVe=e(V~- V~)=~ V~- 1-~]"
Qc V~
S(1 - e)'
(3)
For Vn and V~ the relationship
ve + v~ = v~(~)
(4)
is valid. An appropriate expression relating the slip velocity, V~, and the dispersed phase holdup, E, in
(13t The variation of the input parameter U~,i~ will cause an immediate change to the flow of the continuous phase on the whole column height. This follows from physical properties of the system. The variation of the input parameter Ud, i. induces the flow change that
327
Liquidqiquid extraction columns spreads with limited velocity, so that U .... t is obtained from material balance:
g .... t =
Uc, in "~ Ud, in - -
Ud. out"
(14)
2.3. Modelin9 of mass transfer The following flow rates of solute passes through the cross-section S of the column: for the continuous phase: Qy = S V c Y (1 - `g) + SDc
8[(I
e)Y]
-
Oh
(15)
tions. The column is considered as a series of n perfectly mixed stages with countercurrent flows. Subscripts 0 and n + 1 refer to the disengagement sections h = 0 and h = l, respectively, as shown in Fig. 1. In order to simplify matters, it has been assumed that the axial dispersion coefficients are independent of height. 3.1. Holdup dynamics Applying eq. (8) to an interior point i using a finite difference approximation gives ~'`gi
&
for the dispersed phase: Qx = SVdX`g + SDa
Ere,V e i l -- [F,V d ] i - 1
--
Ah
+
De
el-1 - 2`gi + ei+ l Ah 2 (22)
a[ex]
(16)
O~
where Y = Y(h) and X = X(h) are the concentrations in the continuous phase and in the dispersed phase, respectively. The mass transfer in an arbitrary volume element of column can be expressed as S A h a K x (X* - X)
or S A h a K y ( Y - Y*)
(17)
where X * is the solute concentration in equilibrium with Y and Y* is the solute concentration in equilibrium with X. The interfacial area per unit volume, a, is calculated by a = 6e/d32 , where d32 is the Sauter mean diameter. Using eqs (15~(17), a mass balance for the continuous and dispersed phase around an arbitrary volume element of column in the interval 0 < h < l leads to the following differential equations:
+ aKx(X* - X)
The boundary conditions (11) and (12) in discretized form are `g,+x - e, = 0;
Uda. = [eVd]o--Dd
el
-
,go
A--'--~ (23)
[ V J i : [-(1 - e)Vs(`g)]i - AU + Dd `gi+l - `gl Ah
Substituting the discretized boundary conditions (23) at i = 0 into eq. (22) for i = 1 gives
0el
(
(e2- 81)Ua.m)~
0t-= [eVe]x-De A ~
`g)r]
0`g.
[eVa]. -
(;t
[eVa].- ~
Ah
Ah 2
U c, in
Ud,out
(18)
`g.- ~ - `g.
t- De
O
Oh [V=(1 - `g)Y]
at
O/
+~tD<
O[(1
n
~E)Y])-aKx(X*-X)
(19)
n-I . . . . . . ........... Z ..........
with boundary conditions h = O,
D ispersion laver
Oh YinUc,in
`gVaX - De - Oh
-`g)Y]
O[(1
h=l,
I
O(ex)
X i n U d , in =
= (1 - e)V~Y + D~
O[ex]
- -
Oh
(20)
-0 8[(1
-e)Y] Oh
= 0.
' (21)
Ud, in
I
IUc,out
3. SOLUTION OF THE DYNAMIC EQUATIONS The dynamic equations derived in Section 2 are spatially discretized using finite-difference approxima-
(25)
The equation for section i = n accounts for the boundary condition at i = n + 1:
n+l
OE(1 -
(24)
Fig. l. The discrete grid scheme.
(26)
O. Weinstein et al.
328
Finally, eq. (13) in discretized form is expressed by
gd .... =[~gd],,+a =
e V~
0[(1 - DY]a &
1--e/J,,+1
[(1 - e) Y V ~ ] z - [(1 -- e) YVc]I
Ah
[ ( i - e) Y ] 2 - -
+ De 3.2. Mass transfer Finite-difference approximation of eqs (18) and (19) for an interior grid point i gives ~[~S]i
[eXVd]i -- [~XVd]i_ 1
8t
Ah
+ Da
[eX],_, - 2 1 8 X ] , - [eX],+~ Ah 2 + Ni
+
[(1 - c)YVJi+x - [(1 - ~)YV~]i Ah
D[(1 - e)Y]i+ a - 2[(1 - e)Y]i + [(1 - e)Y]i-1 Ah 2
-- N i
where (30)
The boundary conditions (20) and (21) in discretized form are
DX]o = D x ] I , [(1 - e ) Y ] .
Ud, oo,Xou, = DXVd].
(31)
= [(1 - e ) Y ] . + b
U ..... You, = [(1 - e) YVc]I.
(32)
The equations for sections i = 1 and i = n must include the effect of the boundary conditions at i = 0 and i=n+l:
8[eX]~ _ 8t
([eXVd]l - Dd [eX]z -- [eX]l \ Ah )1
- Ud,inXi. --~ + N1
8 Dx]. &
(33)
(35)
Uc, i.gi. - [(1 -- e) YVc].
[(1 - E ) Y ] . -
[(1 - e ) Y ] . - l ~ l / - -
ah
/ ah (36)
These equations are solved using a Runge-Kutta integrator. The solution method has been implemented using MATLAB/SIMULINK. This flexible platform also allows for the testing of alternative control schemes (see Section 6).
(29)
Ni = [aKx(X* - X ) ] , = [aKr(Y - Y*)],.
D~
(
-- N1
-- N . . (28)
3[(1 - e)Y], &
8[(1 -- 0 Y-]. c3t
I
[(1 - e) Y ] t
Ah2
4. OPEN-LOOPSIMULATIONS 4.1. Model parameters The developed model can describe any common type of liquid-liquid extraction column. The only additional requirements are relationships for parameters such as the Sauter drop diameter, the slip velocity, axial mixing coefficients and mass-transfer coefficients. In the following, calculations of model parameters are based on data from Hufnagl et al. (1991): the simulations results will therefore be compared to the experimental results which appear there. The experiments were performed using a Kuhni column: a liquid-liquid extraction column in which the dispersion layer is supported on rotating perforated plates. The liquid system was the EFCE-test system (toluene-acetone-water), for which physical data is available (Misek et al., 1984). The data for this column and physical properties of the employed system are given in Table. 1. The Sauter mean drop diameter is calculated from an equation of Fisher as reported by Hufnagel et al. (1991)
[eXVd]. - [ ~ x v A . _ 1
Ah [~x].-1
+ Dd
da2 = 0.19W~R0"61dR 1 + nzL22 / ~
R (1 + 2.8e).
- [Ex].
Ah 2
+ N.
(34)
(37)
Table 1. Physical properties and column data for the Kuhni column Parameter
Value
Units
Description
Pc Pd rlc qd I dr dR hz (o
998.2 866.7 1.003 x l 0- 3 0.586 X 10- 3 2.52 0.15 0.085 0.07 0.27
(kg/m 3) (kg/m 3) (kg/(m s)) (kg/(m s)) (m) (m) (m) (m) --
Continuous-phase density (water) Dispersed-phase density (toluene) Continuous-phase viscosity Dispersed-phase viscosity Column height Column diameter Rotor diameter Section height Free fractional area of plate
329
Liquid-liquid extraction columns
Mass-transfer outside the drops is determined using the film coefficient equation of Heertjes et al. (1954)
The slip velocity, V~, is given by the empirical equation of Kumar (1986) V, = 1.18 x
q00"g
10 - 3
(38)
~lcdRn2 "
k~ = 0.83 ~
(43)
where Dm.~ is the continuous phase molecular diffusivity coefficient. Mass-transfer inside the drops is determined by an equation of Handlos and Baron, as suggested by Blab et al. (1986)
Equation (38) is valid for holdups of less than 0.3. For higher values, this equation can be extended by a (1 - e) tenn. The dispersion coefficients for the continuous and the dispersed phase are computed using expressions which appear in Bibaud and Treybal (1966)
kd = 0.00375 V.~ e)] D,. = ~Vchz I - 0.171 + 0.02 dRn(1-~ I
/
\4.18
Dd=a2n 1.3x10-~We1'5~(
P--d~ ~
-1
The distribution coefficient, m, which depends on the effluent solute concentration, is calculated from an empirical expression presented by Misek et al. (1984):
1
.
d
3
In(m) = ~
(40)
10-3
i=1
with xp = 0.6367, al = 135.405, a2 = 79.72012. The overall mass-transfer coefficients Kx and Ky are calculated from the phase mass-transfer coefficients k~ and kd, and the distribution coefficient, m: 1
1
1
1
KG=k~ +~
and
I
m
(42)
KG=k~ +~"
0.2
w 0.1
/~'"
~3
/..'"
d0
~....=~
-0.l
I
I
i
I
I
i
1000
1500
2000
2500 t [sl
3000
,
g o.6 1 _
r."
500
0.8 / /
"~
1
/
.o 04. '~" ".. _ ~
-n ,21 v'-0
~
Ccl
-.......
Cc2
- - -
Cc3
......
Cc4
/
3500
i
i
4000
4500
5000
,
A \
~
C
(45)
4.2. Simulation results Figure 2 shows the simulated column dynamics resulting from an increase of 40% in the dispersedphase inlet flow. In the plots, the dispersed-phase variables, y(t), have been normalized with respect to their initial values: Kv = (y(t) - y(O))/y(O). Other symbols in the plot are Vca, the continuous-phase effluent flow rate; ed, the dispersed-phase mean holdup; and d32, the Sauter mean drop diameter. The simulations are based on the assumption of fixed interface level
(41)
ai(XA~ --Xp) i+l
bi(XAc- Xp) i
where xp = 0.6367, bo = - 5.0, b~ = - 2.711, be = - 9.356 and b3 = - 7.225. A grid size of n = 30 has been used in the simulations.
2
1 + 5XA~ ~
o
i:
The interfacial tension can be computed as a function of the acetone concentration, XA¢ (mass fraction), using an empirical correlation presented by Misek et al. (1984) O" m
(44)
(39)
Re°6~ I
k,G -- Pay
r/----L--~.
rl~ + qd
r~
v
Holdup
, "~." . . . . . . , 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 t lS]
Fig. 2. Simulated transient behavior in response to a step change of 40% in dispersed-phase feed flow rate. The initial values of the input variables at steady state just before the onset of the disturbance are Ud = Uc = 0.0024 m3/m z s, f = 2.9 l/s, y~, = 1 1 kmol/m 3 (Kuhni column). Co,, Col, Cc2, Cc3 and Cc4 represent concentrations in positions l, 6, 12, 18 and 24, for n = 30.
330
O. Weinstein et al.
between the two liquid phases in the upper separation chamber (i.e. perfect level control). This explains the slight difference between the simulated and experimental behavior of the continuous-phase effluent flow. The concentrations are presented as perturbations from the initial concentrations just before the onset of the disturbance. The simulated concentrations in Sections 1, 6, 12, 18 and 24 for n = 30 (see Fig. 1.) are denoted as Co,, C~1, Cc2, Cc3 and C~4, respectively. These positions approximately correspond to the locations of the concentration measurement detectors in the experimental system of Hufnagl et al. (1991). The simulated continuous-phase effluent concentration value (Cc, in Fig. 2), is related to the bottom cross-section of the dispersed layer, not to the bottom settler outlet pipe as in the experiments. Thus, the simulation does not account for the dispersion and time delay associated with the column settler. The simulation results compare well with the experimental transient values in response to a step of 40% in the dispersed-phase inlet flow, as shown in Fig. 3. In order to explain the resonance observed in the mean holdup in both the experimental and simulated transients, additional simulation results showing holdup profiles at various times are shown in Fig. 4. At the onset of the disturbance, the continuous-phase
outlet flow increases as a result of the displacement of the continuous phase by the dispersed-phase (segment OE in Fig, 2). As a result of the propagation of the local holdup to the top of the column (profiles from t = 149 276s in Fig. 4), the dispersed-phase outlet flow rate undergoes an incremental increase from about t = 250 s. As a result, there is a reduction in the effluent flow rate of the continuous phase at h/l = 0 due to mass balance (segment F G in Fig. 2). This in turn causes a reduction in the local holdup in the bottom cross-section (h/l = 0) due to the relationship between holdup and the flow rates [eqs (4), (7) and (10)], which then propagates up the column (see profiles t = 392-613 s in Fig. 4). This causes a decrease in the total holdup (segment AB in Fig. 2). When this holdup reduction disturbance arrives at the top of the column, the dispersed phase outlet flow decreases. Thus, the outlet of the continuous phase at h/l = 0 increases (segment GH in Fig. 2). This causes an increase in the local holdup in this section (see profiles t = 716-927 s in Fig. 4) and the described process is repeated with a lower amplitude (segments BCD and HIJ in Fig. 2). Thus, we see that the changes in the flows at h/l = 1 implies changes at h/1 = 0, with decaying oscillations in holdup. This phenomena has been demonstrated by Brownstein and Schegolev
0.4 0.3 E O
E ,x
0.2
o 0
0.1
"9
1
-
~
~'' " X,.r~
cc2
(3
O --O. 1 o
looo
2ooo
sooo
400o
5000
t Is] 1.O (?.8 0.6
.9
=~ o . 4
._E
.~
0.2
o
Vca
--0.2 looo
2000
50o0
4oo0
5ooo
t [sJ
Fig. 3. Experimental transient behavior in response to a step change of 40% in dispersed-phase feed flow rate (Kuhni column data from Hufnagl et al., 1991).
331
Liquid liquid extraction columns
0.38
w
0.3~
°
t = 276 s
0.34
t=392s ~
t=
. ~
" s
t = 613s . ~ . . .
0.32 ~
~
6
S
0.3
0.28 0.26 0.24 0.22 0.2 0.l~
0'.1 0'.2 0:3
0:4
01s 0'.6 017 018 019
Igl [dimensionless]
Fig. 4. Simulated transient behavior of the holdup profiles at various times in response to a step change of 40% in dispersed-phasefeed flow rate. The initial values of the input variables at steady state just before the onset of the disturbance are as in Fig. 2 (Kuhni column).
(1988) by an analytical solution of a system of equations describing the holdup dynamics in an extraction column using the method of small perturbations. This phenomenon is also observed in response to disturbances in continuous-phase flows and in rotor speed. Unfortunately, Hufnagl et al. (1991) did not report the actual values they used for parameters such as continuous- and dispersed-phase feed flow rates and rotor speed. Thus, the experimental and simulated results can only be compared qualitatively. Nevertheless, the model appears to accurately predict the main experimentally observed phenomena. 5. IMPROVING DISPERSION INTERFACE LEVEL CONTROL
The control of the dispersion interface level is crucial for the reliable operation of liquid-liquid extraction columns. If this level is not adequately stabilized, the dispersion layer could entrain out of the column, leading to loss of solvent and product. The resonance effects observed in the open-loop simulations are detrimental to good dispersion interface level control. Hufnagl et al. (1991) have suggested a novel method to solve this problem by manipulating the continuous-phase feed flow rather than the effluent flow, which is the industrially accepted practice. In this scheme, the continuous-phase effluent flow is left free to control the throughput. Applying these conditions to the model gives new relationships for the dispersed-phase effluent flow rate and the continuous- phase feed flow rate. Thus, the continuous-phase feed flow rate, Uc,in, is obtained from material balance Uc, in = U . . . . t -~- U d , o u t - Ud, in"
(46)
From eqs (13) and (46), the effluent flow, expressed by
Ua.....
is
h= h
(
Vd,ou t = ~. V s - -
Uc.out + Ua out - Ud.in) ~'~
= e[Vs(1 - e) - U..... t Jr- Ud,in]'
(47)
The simulated dynamic responses of the Kuhni column to step changes in the manipulated variables: dispersed-phase feed flow rate, continuous-phase feed/effluent flow rate and rotor speed, are plotted in Figs 5-7. These are the manipulated variables usually used to control the holdup and outlet concentrations. In the plots, the responses of the conventional levelcontrol scheme are shown by solid lines, while those of the improved scheme by dashed lines. The open-loop responses for the case of the improved level-control scheme exhibit monotonic behavior, while those for the conventional level-control show overshoots, oscillatory behavior and even inverse response (e.g. the effluent concentration plotted in Fig. 7). Clearly, the control of the column will be much easier if the improved level-control scheme is adopted. 6. DESIGN AND SIMULATION OF MULTIVARIABLE CONTROL
For the case of the conventional interface level controller, the manipulated variables available for control are the plate-stack oscillation frequency, the dispersed- and continuous-phase feed flow rates. When the modified interface level controller is used, the continuous effluent flow rate may be employed as the control variable instead of the continuous-phase
332
O. Weinstein et al. 0.4
~
0.3 0.2
6 0.1
lobo
' 2000
t [s]
3o;o
' 4000
5000
30;0
40;0
5000
~0.4 e~
~-o.3
~o.2
2o o
t [s]
Fig. 5. Comparison of the transient behavior in response to a step change of 40% in dispersed-phase inlet flow for the two level-control schemes: () conventional scheme; (---) novel scheme. The initial values of the input variables before the onset of the disturbance are Ud = Uc = 0.0024 m3/m 2 s , f = 2.9 l/s, y~. = 1 1 kmol/m 3 (Kuhni column).
0.5
~
0.4
~0.3
6 0.2
1000
2000
3000
4000
5000
30b0
40;0
5000
t [s] -~ 0.3 cO
"~ 0.28
.e
~0.26 ~0.24 0.22 Z~
10;0
20;0 t [sl
Fig. 6. Comparison of the transient behavior in response to a step change of 40% in continuous-phase inlet flow for the two level-control schemes: ( ) conventional scheme; (---) novel scheme. The initial values of the input variables before the onset of the disturbance are Ud = Uc = 0.0024 m3/m 2 s , f = 2.9 l/s, Yi, = 1 1 kmol/m 3 (Kuhni column).
feed flow rate. Both types of level controller were discussed in the previous section. The system outputs are effluent c o n c e n t r a t i o n s a n d dispersed-phase holdup. Multivariable control for a K a r r column will now be designed a n d tested by simulation. F o r this system, p h o s p h o r i c acid is fed in the c o n t i n u o u s phase (pentanol) a n d extracted in the dispersed phase (water). Physical d a t a for the w a t e r - p e n t a n o l - p h o s p h o r i c acid system appears in D a u b e r t a n d D a n n e r (1985).
D a t a for the c o l u m n a n d physical properties are given in Table 2.
6.1. Generatin9 an empirical model The responses of h o l d u p a n d b o t h phase outlet c o n c e n t r a t i o n s to 10% step changes in the control variables are plotted in Fig. 8 for the case of the c o n v e n t i o n a l level controller, a n d in Fig. 9 for the case
Liquid-liquid extraction columns
333
0.4
~
0.35
¢* 0.3 0"250
1000
2000
3000
4000
5000
40b0
50o0
t [sl
~o.3 g~ Q
~-o.2
0.1~
10bo
20b0
t Is]
Fig. 7. Comparison of the transient behavior in response to a step change of 31% in rotor speed for the two level-control schemes: ( ) conventional scheme; (---) novel scheme. The initial values of the input variables before the onset of the disturbance are Ud = Uc = 0.0015 m3/m 2 s,f = 2.8 l/s, Yin = 1 1 kmol/m 3 (Kuhni column).
Table 2. Physical properties and column data for the Karr column Parameter
Value
Units
Description
pc
814.4 998.2 0.4045 x 10-3 1.003 x 10 3 12.25 2.5 0.05 0.027 0.9 0.6
(kg/m 3) (kg/m 3) (kg/(m s)) (kg/(m s))
Continuous-phase density (pentanol) Dispersed-phase density (water) Continuous-phase viscosity Dispersed-phase viscosity Distribution coefficient Column height Column diameter Rotor diameter Orifice coefficient Fractional open area
Pa
r/c qa
m l dr ho Co So
(m) (m) (m) ---
of the modified level controller. In both figures, the process outputs are simulated for positive (solid line) and negative (dashed line) step changes in inputs. For the case of negative inputs, changes in the responses have been plotted with opposite sign. As with the K u h n i column, the Karr column responses for the case of the conventional levelcontrol scheme (Fig. 8) show overshoots and oscillations in the holdup in response to disturbances in all control variables as well as an inverse response and overshoot in outlet dispersed-phase concentration. The modified level-control scheme shows significant improvement of the system's dynamic response (Fig. 9), which should permit more effective control. An empirical uncertain linear model has been identified by fitting low-order transfer functions model using simulated step response data for the modified
level-controller case in the vicinity of the operating point
Y=P'U~
=
P21
P22
P23]
Pij =
Ii
].E,
(48)
Uc
(K + AK) [(zt + Azl)s + 1] 272 + A~2)2S 2 + 2(z 2 + Az2)(~ + A~)s + l J i j '
i = 1,2; j =
1,2,3.
(49)
In the above, ~ is the change in holdup, )( is the change in the outlet dispersed phase concentration, f is the change in the plate stack oscillation frequency, Ua is the change in the dispersed-phase feed flow rate, Uc is the change in the continuous effluent flow rate.
O. Weinstein et al.
334
Ud [m/s]
f [l/s]
-
x 10 "a Uc [m/s]
0.04 i
0.02 ~::O0~l
10~:)0
o,~t
2000
0
:°]A .
0
1000
2000 x 103
X 103
x 10~
.
IOO0
.
.
.
2000
0
1000
x 104
,olaf
O: -2
~'-2 0
IOO0
2000
~
0
t[sl
2000
2000
x 10~
xlO 3
1000
2000
0
1000 t (sl
t [s]
2000
Fig. 8. Simulated transient behavior in response to positive ( ) and negative (---) 10% step change in the control variables for the case of conventional level control. The initial values of the input variables at steady state just before the onset of disturbance are Ud = 0.006 m3/(m 2 s), Uc = 0.003 m3/m z s , f = 1.2 l/s, Y~n = 0.2 kmol/m 3 (Karr column).
f [l/s]
!
Ud Ira/s]
-
o.~i
~:~0.01
IY
x 10 3
UC [m/s]
4i
0.02i
o ~ .,~ x 10~
X 2
....
0u 0
, 1000
20o0
] J 2000
o0
1000
2000
x 10"3
0
1000
2000
,!y
2000
x l O "4
xlO ~
x l o -~
2000 xlo -3
O:
~'-2 0
1000 t [s]
2000
0
1000 t [s]
2000
i/o
2000
t [s]
Fig. 9. Simulated transient behavior in response to positive ( ) and negative (---) 10% step change in the control variables for the case of modified level control. The initial values of the input variables at steady state just before the onset of disturbance are U~=O.OO6m3/m2s, Uc=0.003 m3/m2s, f = 1.2 l/s, Yio = 0.2 kmol/m 3 (Karr column).
Liquid liquid extraction columns
335
Table 3. Process transfer function coefficients Element
K
AK
Z1
A'r I
1:2
A~C2
~
A~
Pit (eft) P12 ('g/Ud) P13 (,g/Uc) P21 (X/f)
0.264 0.439 0.055 0.016 -- 0.077 0.091
0.018 0.082 0.003 0.003 0.005 0.001
16.1 23.3 16.5 773 123 42.2
2.7 6.6 0.03 261 123 2.4
84.9 93.2 83.9 65.9 98.3 109.9
3.2 18.2 3.9 16.9 52.2 3.6
0.85 0.86 0.84 1.53 0.76 0.98
0.03 0.07 0.01 0.46 0.15 0.05
P22 (X/Ua) P23 (X/U~)
Equations (48) and (49) describe the response of the two-process output variables to the three-process inputs as a matrix of uncertain linear transfer functions. The transfer functions parameters are given in Table 3. The uncertainties AK, Art, At2 and A~ reflect the variation in model parameters required to enable a reasonable fit to the process trajectories in response to positive and negative step changes in the inputs in the vicinity of the operating point. The simple transfer functions (49) are not able to capture all the observed nonlinear features shown in Fig. 9, especially for the response of the dispersed-phase effluent concentration. This leads to additional unquantified highfrequency uncertainty in the parameters associated with this output. 6.2. Control system design Decentralized (i.e. diagonal) control has been chosen. From RGA analysis (Bristol, 1966), it is apparent that the most promising pairing is to control the outlet dispersed-phase concentration using the continuous feed flow rate and the holdup using the plate-stack oscillation frequency. The IMC synthesis procedure described by Morari and Zafiriou (1989) has been used for control system design. As a result, the two single-loop IMC controllers are determined as
[ q'(i/)?)
=
q2(g/CL) =
z2s2+2r2~s+l
l
LK(77s +~+lj],1 [ r2~s= + 2 r d s + I ]
(50)
L.K(G7s + 1)i;,e3s + 1) 23
The tuning of the IMC controllers was carried out using the Rijnsdorp Interaction Measure (RIM, Rijnsdorp, 1965): nominal stability is guaranteed in the face of process interaction, if f P2.ico)p13(i~)"]-1 ~/[P11(ko)P23(io)) // >
Ihis(i~)lV~
(51)
where the left-hand-side-term is the RIM, and his is the nominal complementary sensitivity function relating process output i to set-point change j. According to IMC design procedure, his = Paisfq. Since in our case PA~S= 1 (the process has no N M P components),
his(s) =f~(s) = (2iss + 1)- 1.
(52)
The RIM provides a sufficient stability condition for decentralized diagonal control. Clearly, if the offdiagonal elements of the MIMO system are small relative to the diagonal elements, then it will be relatively easy to satisfy eq. (51), since the complimentary sensitivity functions are of the order of unity. Frequency ranges where diagonal dominance does not hold will result in RIM values under unity and will limit achievable bandwidth. The RIM parameter therefore represents constraints on achievable performance caused by the interactions in the process. Thus, plotting RIM as a function of frequency indicates the bandwidth over which stabilizing control is guaranteed. The limiting time constant value, 2~s, of filter, f, can be determined by eq. (51). This procedure is illustrated in Fig. 10. for 2 = 15s. The filter time constant values, 211 and 223 used in closed-loop simulations have been chosen as 100s, providing responses free of oscillations. The increased filter time constant values reflect a desire for increased gain margin in the face of unmodeled uncertainty (especially in the response of X). The dispersed-phase holdup is an important system output variable because its overshoot may lead to flooding, aside from its effect on the concentration dynamics. For this reason, a dynamic decoupler and feedforward controller have been designed to stabilize the holdup. A block diagram for the control system is shown in Fig. 11. A dynamic decoupler has been designed in order to compensate for the effect of continuous-phase feed on holdup magnitude. The parameters of the decoupler are calculated using the nominal process model (indicated by the superscript ~) --/D13 [K(rls -[- 1)]13[-r2s 2 -1- 2Zz(S + 1111 DIs) = /311 = - [K(rlS + 1)]11[z22s2 + 2"c2~s + 1113"
(53) A feedforward controller has been introduced in order to compensate for the effect of dispersed-phase feed flow on holdup. The IMC design procedure has been used for the feedforward controller -- /~12
qf(g/Oa)-- Plx
[K(rls + 1)]12[r2s 2 --k 2%~s + 1111 [K(rxs + l)]11['T22s 2 + 2%¢s + 1112' (54)
336
O. Weinstein et al. 101
........
,
........
,
.......
,
........
,
.......
i
........
,
........
,
....
,
........
":....,
........
,
........
,
.......
Z 10°
10~10-5 i , ,,....i 104
........
t O "3
10 -2
1 0 -1
10 °
101
10 2
Frequency [rad/s]
Fig. 10. RIM test for tuning the IMC controller.
] £ ~p
•
+
{ ql
I
Fig. 11. The block diagram of the control scheme.
The designed control scheme was implemented using MATLAB/SIMULINK. 6.3. Closed-loop simulation results Closed-loop simulations were carried out both for decentralized IMC control alone and for an IMC controller augmented by a decoupler and a feedforward controller. Figure 12 shows the simulated control system performance for a 10% set-point change in outlet dispersed-phase concentration: the decoupled scheme [-Fig. 12(b)] performs significantly better than the scheme without the decoupler [Fig, 12(a)]. The regulatory closed-loop response to a 10% disturbance in continuous-phase feed concentration is
shown for the IMC scheme in Fig. 13(a) and for the decoupled IMC scheme in Fig. 13(b). The level is maintained perfectly by the decoupled scheme. The dynamic decoupler eliminates interaction between the continuous-phase feed flow rate and the holdup almost perfectly due to relatively low uncertainties in the linear approximation and the relatively low sensitivity of holdup to the continuous-phase feed flow rate. Finally, the regulatory performance of the decoupled IMC scheme augmented by feedforward control [-Fig. 14(b)] can be compared to that of the simple decentralized IMC scheme [Fig. 14(a)], for a disturbance in the dispersed-phase feed flow rate. Here,
Liquid-liquid extraction columns
337
(a) Decenla-allzedIMC 0.1
...... .........................................................................
Z ~. 0.05 "7.. f
0
I
I
I
I
100
200
300
400
I
I
500 600 time [s]
I
I
I
700
800
900
1000
Co) D c c o u p l e d I M C + F F
.."" ..............................................................................
0.1 Z
..'"
~
0.05
,--?
0
100
I
I
I
I
I
I
I
I
200
300
400
500
600
700
800
903
1000
time [sl
Fig. 12. Simulated closed-loop response to a set-point 10% step change in outlet dispersed-phase concentration: (a) decentralized IMC scheme; (b) decoupled IMC [ ( - - ) g/e(t = 0), (---) X / X ( t = 0)].
0
.
0
Z ore-
.."
+
• 0.01
~
:
3
[
:"
~
(a) DecentralizedIMC . . . .
'"' ............
""',,,..,..
O:
..................
-001 • 0
100
200
300
~
400
'
500
~
600
~
700
'
800
I
9OO
1000
time [sl
(b) DecoupledIMC + FF 0.03[
,
,,, . . . . . , . . .
,
°.°'t.
.............
~.ooi: • 0
100
200
300
400
500
.............................. 600
700
800
900
1000
time Is]
Fig. 13. Simulated closed-loop response to a 10% step change in feed continuous-phase concentration: (a) decentralized IMC scheme; (b) decoupled IMC [( ) g/~(t = 0), (---) X / X ( t = 0)].
the impact of augmenting feedforward and decoupler components for the holdup alone can be seen; the holdup level is almost perfectly stabilized in face of a - 5% step change in Ue, at the price of a slight deterioration in the response of the outlet dispersedphase concentration. A similar procedure was adopted to design decentralized control for the conventional level control configuration. The resulting performance is slightly worse than that obtained with the proposed modified level control strategy. It should be noted that this
conclusions based only on simulation results. It would be advisable to compare closed-loop performance for both level-control systems experimentally•
7. CONCLUSIONS A theoretical dynamic model has been developed for a countercurrent liquid-liquid extraction column based on the average dispersed-phase drop diameter. The dynamic simulations relying on this model have been found to be in agreement with experimental data
338
O. Weinstein et al. (a) Dccentrallz~IMC
".....
N -0.04 -0.%
100 '
200 '
300 '
4 0' 0
O0
i
i
i
700
800
900
7
8
time [sl fo) DecoupledIMC + FF
1000
Z 0.04
~. o.oz
"'.....
o
'-741.02 -0.04 -0.06
.
100
200
.
.
300
.
400
.
500
time [s]
600
1(300
Fig. 14. Simulated closed-loop response to a - 5% step change in feed dispersed-phase flow rate: (a) decentralized IMC scheme, (b) decoupled IMC augmented with feedforward action [( ) k/e(t = 0),
(---) 2/x(t and some published theoretical results. Thus, the model can be reliably applied to the control of an extraction column and for the study of the extraction process. Two control schemes for dispersion interface level have been investigated: conventional level control using the continuous-phase effluent flow rate as the manipulated variable, and level-control using the continuous-phase feed flow rate. It has been shown that the conventional level-control scheme leads to unavoidable overshoots and oscillations in the output variables, while the proposed alternative scheme leads to smoother, overshoot-free response. It has also been demonstrated by simulation that a model-based I M C controller, augmented by a dynamic decoupler and feedforward control, provides excellent servo and regulatory response for a Karr column.
a d32 dK dR D D,, f h hz K l n nz q t s
NOTATION specific interfacial area, m 1 Sauter mean drop diameter, m column diameter, m rotor diameter, m axial dispersion coefficient, m2/s molecular diffusivity coefficient, mZ/s IMC controller filter, dimensionless level, m section height, m overall mass transfer coefficient, m/s column height, m grid, dimensionless rotor speed, 1/s I M C controller transfer function, dimensionless time, s Laplace operator, i/s
= 0)].
S U
cross-section's area, dimensionless loading (Uc -- Qc/S and Ud = Qd/S), m3/m 2 s flow rate, m3/s velocity [V~ = U c / ( 1 - e) and Vd = Ud/E)], m/s continuous phase effluent flow rate, m3/s acetone mass fraction, dimensionless solute concentration in the dispersed phase, kmol/m 3 solute concentration in the continuous phase, kmol/m 3
Q V V~a Xac X Y
Greek symbols
volume fraction holdup of dispersed phase, dimensionless I M C controller filter time constant, s viscosity, kg/m s density, kg/m 3 interracial tension, N/m time constant, s frequency, rad/s free fractional area of plate, dimensionless
P a z co ~o
Subscripts c d
continuous phase dispersed phase
Dimensions numbers nzd~pc . ReR = - impeller Reynolds n u m b e r qc WeR
2 3 nzdRpc . - impeller Weber n u m b e r (7
Liquid-liquid extraction columns REFERENCES
Bait, M. H. I. and Lane, S. J. (1973) Drop size and holdup in a reciprocating plate extraction column. Chem. Engng Sci. 28, 947-957. Bair, M. H. I. and Shen, Z. J. (1984) Holdup and flooding in a reciprocating plate extraction column. Can. J. Chem. Engng 62, 218-227. Bibaud, R. E. and Treybal, R. E. (1966). Axial mixing and extraction in a mechanically agitated liquid extraction tower. A.I.Ch.E.J. 12, 472-477. BlaB, E., Goldmann, G., Hirschmann, K., Mihailowitsch, P. and Pietzsch, W. (1986) Progress in liquid-liquid extraction. Germ. Chem. Engng 9, 222-238. Bristol, E. H. (1966) On a new measure of interaction for multivariable processes. IEEE Trans. Automat. Control AC-11, 133-134. Brownstein, B. I. and Schegolev, V. V. (1988) Hidrodinamika, Masso i Teploobmen v Kolonnih Apparatah (Hydrodynamics, Mass and Heat Transfer in Column Apparatus). Khimiya, Leningrad, pp. 112 132. Casamatta, G. and Vonderpohl, A. (1985) Modelling of fluid dynamics and mass transfer characteristics of a Karr extraction column. Ger. Chem. Engng 8, 96 103. Daubert, T. E. and Danner, R. P. (1985) Data Compilation Tables of Properties of Pure Compounds, Design Institute for Physical Property Data, AIChE, New York. Dimitrova A1 Khani, S., Gourdon, C. and Casamatta, G. (1989) Dynamic and steady-state simulation of hydrodynamics and mass transfer in a liquidliquid extraction column. Chem. Engng Sci. 44, 1295-1305. Godfrey, J. C. and Slater, M. J. (1991) Slip velocity relationships for liquid-liquid extraction columns. Can. J. Chem. Engng 69, 130-141.
339
Heertjes, P. M., Holve, W. A. and Talsma, H. (1954) Mass transfer between isobutanol and water in a spray column. Chem. Engng Sci. 3, 122. Hufnagl, H., McIntyre, M. and BlaB, E. (1991) Dynamic behavior and simulation of a liquidliquid extraction column. Chem. Engng Technol. 14, 301-306. Karr, A. E., Ramanujam, S., Lo, T. C. and Baird, M. H. I. (1987). Axial mixing and scaleup of reciprocating plate columns. Can. J. Chem. Engng 65, 373 381. Kumar, A. and Hartland, S. (1986) Capacity and hydrodynamics of an agitated extraction column. Ind. Engng Process Des Dev. 25, 728-733. Misek, T., Berger, R. and Schroter, J. (1984) Standard test systems for liquid extraction. The Institution of Chemical Engineers, Rugby, Warwickshire. Miyauchi, T. and Vermeulen, T. (1963) Longitudinal dispersion in two-phase continuous-flow operations. I.&E.C. Fundamentals 2, 113-125. Morari, M. and Zafiriou, E. (1989) Robust Process Control. Prentice-Hall, Englewood Cliffs. Rijnsdorp, J. E. (1965) Interaction in two-variable control systems for distillation columns--I. Automatica 1, 15 18. Steiner, L. and Hartland, S. (1983) Unsteady-state extraction. In Handbook of Solvent Extraction, eds. T. C. Lo, M. H. I. Baird and C. Hanson, Chapter 6, Wiley, New York. Tsoris, S., Kirou, V. I. and Tavlarides, L. L. (1994) Drop size distribution and holdup profiles in a multistage extraction column. A.I.Ch.E.J. 40, 407-4 18. Wehner, C. R. and Wilhelm, R. H. (1956) Boundary conditions of flow reactor. Chem. Engng Sci. 6, 89 93.
Chemical Engineering Science, Vol. 53, No. 2, pp. 325-339, 1998 ~c~,1997 Elsevier Science Ltd. All rights reserved Printed in Great Britain S0009-2509(97)00310-2 ooo9 2509/98 $19.00 + o.oo
Modeling, simulation and control of liquid-liquid extraction columns O. Weinstein, R. Semiat and D. R. Lewin* Department of Chemical Engineering, Technion I.I.T., Haifa 32000, Israel (First received 21 May 1997; accepted 12 June 1997)
Abstract A model describing the hydrodynamics and mass transfer of countercurrent liquidliquid extraction columns is developed and solved. The hydrodynamic model assumes that the dispersed-phase drops behave as spheres of uniform diameter. The model has been found to be qualitatively in agreement with experimental results published in the literature. It is shown that conventional dispersion interface level control using the continuous-phase effluent flow rate as the manipulated variable causes unavoidable overshoots and oscillations in the mean holdup and outlet concentrations, while an alternative scheme using the continuous-phase feed flow rate leads to a significant improvement in the dynamic response. A model-based decentralized M I M O control scheme is designed and tested by simulation, and is shown to provide excellent servo and regulatory performance. © 1997 Elsevier Science Ltd Keywords:
Liquid-liquid extraction column; mathematical models; dynamic simulation and
control.
l. INTRODUCTION
Solvent extraction in continuous columns is one of the most important separation processes in chemical engineering. However, the control of these units can often be problematic, partially due to their multiphase nature, and partially due to the difficulty of on-line measurement of output variables. Thus, the reliable simulation of the transient behavior of these columns is extremely valuable. A review of the progress on the dynamic modeling and simulation of extraction columns was published by Steiner and Hartland (1983). Most previous studies have dealt with mass-transfer transients while assuming hydrodynamic steady state. In contrast, the detailed experimental results published by Hufnagl et al. (1991) imply that the dynamic response of outlets, holdup and concentrations are determined largely by the dynamic behavior of dispersed phase. Recently, several papers (e.g. Casamatta and Vonderpohl, 1985; Dimitrova A1 Khani et al., 1989; Tsoris et al., 1994) have developed models relying on a drop population. These models have the potential of explaining the nonuniformity of holdup profiles along the contactor, and thus they allow a more realistic description of flooding conditions. On the other hand, this approach leads to complicated mathematical formulations. Aside from this, drop breakage and
*Corresponding author. E-mail: [email protected].
coalescence models, on which the approach relies, have not yet been developed for all types of columns. Perhaps for these reasons, this approach has not been used for control purposes. For example, the holdup overshoot and oscillations in response to input disturbances shown analytically by Brownstein and Schegolev (1988), and demonstrated experimentally by Hufnagl et al. (1991), is an important characteristic of extraction column dynamical behavior. Nevertheless, this phenomenon has not been studied in the works dealing with drop population models. This paper will show that important aspects of extraction-column dynamics may be investigated with sufficient accuracy using a simpler approach, based on a hydrodynamic model relying on the Sauter mean diameter, which is an accepted measure of the average diameter of a dispersed-phase drop. This has been widely used in relationships for parameters of extraction processes and for the estimation of flooding conditions (Bair and Lane, 1973; Baird and Shen, 1984). The model assumes a uniform drop diameter and thus, the number of equations that need to be solved is much smaller than that for drop population models. The model for mass-transfer developed accounts for changes in the continuous and dispersed flow rates caused by the holdup profile transient. The model is intended to be used for control purposes, and is used to simulate conditions both for a Karr column (Karr et al., 1987), in which the dispersion layer is supported on a stack of perforated, vibrating plates, and for a Kuhni column, in which it is supported on rotating plates (Hufnagl et al., 1991). The novel 325
326
O. Weinstein et al.
dispersion-level control scheme suggested by Hufnagl et al. (1991) is simulated using the proposed model, and compared to the conventional control commonly adopted. Finally, a multivariable controller for a Karr column is developed, relying on a linear model approximating the dynamics of the column. This is tested by simulation on the detailed nonlinear model developed in this study. 2. M A T H E M A T I C A L
liquid-liquid extraction equipment is discussed by Godfrey and Slater (1991). The relationships for Vs and other parameters used in the model are discussed below. Since the cross-sectional area is constant, it is possible to introduce new variables Ue = Qe/S, Uc = Qjs,
Ue.in = Qe.~./S, Uc.,n = 9_c.,#S, Ue .... = Qe.out/S
and U.... t = Q .... t/S. Thus, eqs (1) to (3) can be rewritten as
MODELING
2.1. Assumptions The mathematical model for a liquid-liquid extraction column relies on the following main assumptions:
Oe(h) Ua(h) = e(h) Ve(h) - De(h) - Oh U c - - U d = U c , in -
Ud,ou t =
U .... t --
(5)
U d . in = A U
(6) (1) The radial variations of the mean velocity and the concentrations of each phase are negligible. (2) The phase volume flow rates are independent of mass-transfer and solute concentration. (3) The dispersed-phase drops behave as spheres of a uniform diameter. (4) The model describes the processes occurring in the dispersion layer of an extraction column. The length of this dispersion layer is fixed. 2.2. Modeling of fluid dynamics Since the dispersed phase is assumed to consist of spherical drops of uniform diameter, the mean drop size in a dispersion is expressed as the Sauter mean diameter: d32 = Z n f l ~ / Z n i d 2 where di is the individual drop diameter and n~ is the number of drops with this diameter. The direction of movement of the dispersed phase coincides with the positive vertical direction. The flow rate of the dispersed phase, Qa, through the crosssection S is expressed by:
0~(h)
Qe(h) = St(h)Ve(h) - SDa(h) - Oh
=
Qc,out
--
Qa.i, = AQ
O~_
dUe_
O(Va~)
where Q~,in and Qa, in are continuous- and dispersedphase feed flow rates, Q.... t and Qa, o,t are continuous and dispersed phase effluent flow rates, and Vc is the velocity of the continuous phase:
O/
&\
(8)
Substituting eq. (5) in eq. (6) gives O~ U,. = AU + Ua = AU + e V e - De ~
(9)
and by using eqs (4), (7) and (9), Ve is obtained as: Va = (1 - e)V,(e) - AU + De -~.
(10)
The boundary conditions are & Ua, i n = r ~ V a - D e ~
(1)
(2)
(7)
A balance of dispersed phase in an elementary volume of the column leads to the following dynamic equation:
h=0;
where De(h) is dispersion coefficient, Vd(h) is velocity of dispersed phase, e(h) is the volume fraction holdup of dispersed phase at the height h of the dispersion layer in the column. The flow rate of continuous phase, Q~, can be obtained by assuming incompressible flow in the material balance between crosssections h = 0 and h = h Q c - - Q~d = Q c , in - - Q a , o u t
U~ Vc = 1 - e"
(11)
Oe
h=l;
T~=0.
(12)
These boundary conditions arise from material balances across the system's boundary planes, with the assumption that the column is a "closed system", i.e. dispersion occurs only between h = 0 + and h = l - . The significance of these conditions is discussed by Wehner and Willem (1956). Miyauchi and Vermeulen (1963) applied similar boundary conditions in their steady-state solutions of an axial dispersion model. The outlet flow, Ue,out, is expressed by
h=l,
Ue,o~t=eVe=e(V~- V~)=~ V~- 1-~]"
Qc V~
S(1 - e)'
(3)
For Vn and V~ the relationship
ve + v~ = v~(~)
(4)
is valid. An appropriate expression relating the slip velocity, V~, and the dispersed phase holdup, E, in
(13t The variation of the input parameter U~,i~ will cause an immediate change to the flow of the continuous phase on the whole column height. This follows from physical properties of the system. The variation of the input parameter Ud, i. induces the flow change that
327
Liquidqiquid extraction columns spreads with limited velocity, so that U .... t is obtained from material balance:
g .... t =
Uc, in "~ Ud, in - -
Ud. out"
(14)
2.3. Modelin9 of mass transfer The following flow rates of solute passes through the cross-section S of the column: for the continuous phase: Qy = S V c Y (1 - `g) + SDc
8[(I
e)Y]
-
Oh
(15)
tions. The column is considered as a series of n perfectly mixed stages with countercurrent flows. Subscripts 0 and n + 1 refer to the disengagement sections h = 0 and h = l, respectively, as shown in Fig. 1. In order to simplify matters, it has been assumed that the axial dispersion coefficients are independent of height. 3.1. Holdup dynamics Applying eq. (8) to an interior point i using a finite difference approximation gives ~'`gi
&
for the dispersed phase: Qx = SVdX`g + SDa
Ere,V e i l -- [F,V d ] i - 1
--
Ah
+
De
el-1 - 2`gi + ei+ l Ah 2 (22)
a[ex]
(16)
O~
where Y = Y(h) and X = X(h) are the concentrations in the continuous phase and in the dispersed phase, respectively. The mass transfer in an arbitrary volume element of column can be expressed as S A h a K x (X* - X)
or S A h a K y ( Y - Y*)
(17)
where X * is the solute concentration in equilibrium with Y and Y* is the solute concentration in equilibrium with X. The interfacial area per unit volume, a, is calculated by a = 6e/d32 , where d32 is the Sauter mean diameter. Using eqs (15~(17), a mass balance for the continuous and dispersed phase around an arbitrary volume element of column in the interval 0 < h < l leads to the following differential equations:
+ aKx(X* - X)
The boundary conditions (11) and (12) in discretized form are `g,+x - e, = 0;
Uda. = [eVd]o--Dd
el
-
,go
A--'--~ (23)
[ V J i : [-(1 - e)Vs(`g)]i - AU + Dd `gi+l - `gl Ah
Substituting the discretized boundary conditions (23) at i = 0 into eq. (22) for i = 1 gives
0el
(
(e2- 81)Ua.m)~
0t-= [eVe]x-De A ~
`g)r]
0`g.
[eVa]. -
(;t
[eVa].- ~
Ah
Ah 2
U c, in
Ud,out
(18)
`g.- ~ - `g.
t- De
O
Oh [V=(1 - `g)Y]
at
O/
+~tD<
O[(1
n
~E)Y])-aKx(X*-X)
(19)
n-I . . . . . . ........... Z ..........
with boundary conditions h = O,
D ispersion laver
Oh YinUc,in
`gVaX - De - Oh
-`g)Y]
O[(1
h=l,
I
O(ex)
X i n U d , in =
= (1 - e)V~Y + D~
O[ex]
- -
Oh
(20)
-0 8[(1
-e)Y] Oh
= 0.
' (21)
Ud, in
I
IUc,out
3. SOLUTION OF THE DYNAMIC EQUATIONS The dynamic equations derived in Section 2 are spatially discretized using finite-difference approxima-
(25)
The equation for section i = n accounts for the boundary condition at i = n + 1:
n+l
OE(1 -
(24)
Fig. l. The discrete grid scheme.
(26)
O. Weinstein et al.
328
Finally, eq. (13) in discretized form is expressed by
gd .... =[~gd],,+a =
e V~
0[(1 - DY]a &
1--e/J,,+1
[(1 - e) Y V ~ ] z - [(1 -- e) YVc]I
Ah
[ ( i - e) Y ] 2 - -
+ De 3.2. Mass transfer Finite-difference approximation of eqs (18) and (19) for an interior grid point i gives ~[~S]i
[eXVd]i -- [~XVd]i_ 1
8t
Ah
+ Da
[eX],_, - 2 1 8 X ] , - [eX],+~ Ah 2 + Ni
+
[(1 - c)YVJi+x - [(1 - ~)YV~]i Ah
D[(1 - e)Y]i+ a - 2[(1 - e)Y]i + [(1 - e)Y]i-1 Ah 2
-- N i
where (30)
The boundary conditions (20) and (21) in discretized form are
DX]o = D x ] I , [(1 - e ) Y ] .
Ud, oo,Xou, = DXVd].
(31)
= [(1 - e ) Y ] . + b
U ..... You, = [(1 - e) YVc]I.
(32)
The equations for sections i = 1 and i = n must include the effect of the boundary conditions at i = 0 and i=n+l:
8[eX]~ _ 8t
([eXVd]l - Dd [eX]z -- [eX]l \ Ah )1
- Ud,inXi. --~ + N1
8 Dx]. &
(33)
(35)
Uc, i.gi. - [(1 -- e) YVc].
[(1 - E ) Y ] . -
[(1 - e ) Y ] . - l ~ l / - -
ah
/ ah (36)
These equations are solved using a Runge-Kutta integrator. The solution method has been implemented using MATLAB/SIMULINK. This flexible platform also allows for the testing of alternative control schemes (see Section 6).
(29)
Ni = [aKx(X* - X ) ] , = [aKr(Y - Y*)],.
D~
(
-- N1
-- N . . (28)
3[(1 - e)Y], &
8[(1 -- 0 Y-]. c3t
I
[(1 - e) Y ] t
Ah2
4. OPEN-LOOPSIMULATIONS 4.1. Model parameters The developed model can describe any common type of liquid-liquid extraction column. The only additional requirements are relationships for parameters such as the Sauter drop diameter, the slip velocity, axial mixing coefficients and mass-transfer coefficients. In the following, calculations of model parameters are based on data from Hufnagl et al. (1991): the simulations results will therefore be compared to the experimental results which appear there. The experiments were performed using a Kuhni column: a liquid-liquid extraction column in which the dispersion layer is supported on rotating perforated plates. The liquid system was the EFCE-test system (toluene-acetone-water), for which physical data is available (Misek et al., 1984). The data for this column and physical properties of the employed system are given in Table. 1. The Sauter mean drop diameter is calculated from an equation of Fisher as reported by Hufnagel et al. (1991)
[eXVd]. - [ ~ x v A . _ 1
Ah [~x].-1
+ Dd
da2 = 0.19W~R0"61dR 1 + nzL22 / ~
R (1 + 2.8e).
- [Ex].
Ah 2
+ N.
(34)
(37)
Table 1. Physical properties and column data for the Kuhni column Parameter
Value
Units
Description
Pc Pd rlc qd I dr dR hz (o
998.2 866.7 1.003 x l 0- 3 0.586 X 10- 3 2.52 0.15 0.085 0.07 0.27
(kg/m 3) (kg/m 3) (kg/(m s)) (kg/(m s)) (m) (m) (m) (m) --
Continuous-phase density (water) Dispersed-phase density (toluene) Continuous-phase viscosity Dispersed-phase viscosity Column height Column diameter Rotor diameter Section height Free fractional area of plate
329
Liquid-liquid extraction columns
Mass-transfer outside the drops is determined using the film coefficient equation of Heertjes et al. (1954)
The slip velocity, V~, is given by the empirical equation of Kumar (1986) V, = 1.18 x
q00"g
10 - 3
(38)
~lcdRn2 "
k~ = 0.83 ~
(43)
where Dm.~ is the continuous phase molecular diffusivity coefficient. Mass-transfer inside the drops is determined by an equation of Handlos and Baron, as suggested by Blab et al. (1986)
Equation (38) is valid for holdups of less than 0.3. For higher values, this equation can be extended by a (1 - e) tenn. The dispersion coefficients for the continuous and the dispersed phase are computed using expressions which appear in Bibaud and Treybal (1966)
kd = 0.00375 V.~ e)] D,. = ~Vchz I - 0.171 + 0.02 dRn(1-~ I
/
\4.18
Dd=a2n 1.3x10-~We1'5~(
P--d~ ~
-1
The distribution coefficient, m, which depends on the effluent solute concentration, is calculated from an empirical expression presented by Misek et al. (1984):
1
.
d
3
In(m) = ~
(40)
10-3
i=1
with xp = 0.6367, al = 135.405, a2 = 79.72012. The overall mass-transfer coefficients Kx and Ky are calculated from the phase mass-transfer coefficients k~ and kd, and the distribution coefficient, m: 1
1
1
1
KG=k~ +~
and
I
m
(42)
KG=k~ +~"
0.2
w 0.1
/~'"
~3
/..'"
d0
~....=~
-0.l
I
I
i
I
I
i
1000
1500
2000
2500 t [sl
3000
,
g o.6 1 _
r."
500
0.8 / /
"~
1
/
.o 04. '~" ".. _ ~
-n ,21 v'-0
~
Ccl
-.......
Cc2
- - -
Cc3
......
Cc4
/
3500
i
i
4000
4500
5000
,
A \
~
C
(45)
4.2. Simulation results Figure 2 shows the simulated column dynamics resulting from an increase of 40% in the dispersedphase inlet flow. In the plots, the dispersed-phase variables, y(t), have been normalized with respect to their initial values: Kv = (y(t) - y(O))/y(O). Other symbols in the plot are Vca, the continuous-phase effluent flow rate; ed, the dispersed-phase mean holdup; and d32, the Sauter mean drop diameter. The simulations are based on the assumption of fixed interface level
(41)
ai(XA~ --Xp) i+l
bi(XAc- Xp) i
where xp = 0.6367, bo = - 5.0, b~ = - 2.711, be = - 9.356 and b3 = - 7.225. A grid size of n = 30 has been used in the simulations.
2
1 + 5XA~ ~
o
i:
The interfacial tension can be computed as a function of the acetone concentration, XA¢ (mass fraction), using an empirical correlation presented by Misek et al. (1984) O" m
(44)
(39)
Re°6~ I
k,G -- Pay
r/----L--~.
rl~ + qd
r~
v
Holdup
, "~." . . . . . . , 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 t lS]
Fig. 2. Simulated transient behavior in response to a step change of 40% in dispersed-phase feed flow rate. The initial values of the input variables at steady state just before the onset of the disturbance are Ud = Uc = 0.0024 m3/m z s, f = 2.9 l/s, y~, = 1 1 kmol/m 3 (Kuhni column). Co,, Col, Cc2, Cc3 and Cc4 represent concentrations in positions l, 6, 12, 18 and 24, for n = 30.
330
O. Weinstein et al.
between the two liquid phases in the upper separation chamber (i.e. perfect level control). This explains the slight difference between the simulated and experimental behavior of the continuous-phase effluent flow. The concentrations are presented as perturbations from the initial concentrations just before the onset of the disturbance. The simulated concentrations in Sections 1, 6, 12, 18 and 24 for n = 30 (see Fig. 1.) are denoted as Co,, C~1, Cc2, Cc3 and C~4, respectively. These positions approximately correspond to the locations of the concentration measurement detectors in the experimental system of Hufnagl et al. (1991). The simulated continuous-phase effluent concentration value (Cc, in Fig. 2), is related to the bottom cross-section of the dispersed layer, not to the bottom settler outlet pipe as in the experiments. Thus, the simulation does not account for the dispersion and time delay associated with the column settler. The simulation results compare well with the experimental transient values in response to a step of 40% in the dispersed-phase inlet flow, as shown in Fig. 3. In order to explain the resonance observed in the mean holdup in both the experimental and simulated transients, additional simulation results showing holdup profiles at various times are shown in Fig. 4. At the onset of the disturbance, the continuous-phase
outlet flow increases as a result of the displacement of the continuous phase by the dispersed-phase (segment OE in Fig, 2). As a result of the propagation of the local holdup to the top of the column (profiles from t = 149 276s in Fig. 4), the dispersed-phase outlet flow rate undergoes an incremental increase from about t = 250 s. As a result, there is a reduction in the effluent flow rate of the continuous phase at h/l = 0 due to mass balance (segment F G in Fig. 2). This in turn causes a reduction in the local holdup in the bottom cross-section (h/l = 0) due to the relationship between holdup and the flow rates [eqs (4), (7) and (10)], which then propagates up the column (see profiles t = 392-613 s in Fig. 4). This causes a decrease in the total holdup (segment AB in Fig. 2). When this holdup reduction disturbance arrives at the top of the column, the dispersed phase outlet flow decreases. Thus, the outlet of the continuous phase at h/l = 0 increases (segment GH in Fig. 2). This causes an increase in the local holdup in this section (see profiles t = 716-927 s in Fig. 4) and the described process is repeated with a lower amplitude (segments BCD and HIJ in Fig. 2). Thus, we see that the changes in the flows at h/l = 1 implies changes at h/1 = 0, with decaying oscillations in holdup. This phenomena has been demonstrated by Brownstein and Schegolev
0.4 0.3 E O
E ,x
0.2
o 0
0.1
"9
1
-
~
~'' " X,.r~
cc2
(3
O --O. 1 o
looo
2ooo
sooo
400o
5000
t Is] 1.O (?.8 0.6
.9
=~ o . 4
._E
.~
0.2
o
Vca
--0.2 looo
2000
50o0
4oo0
5ooo
t [sJ
Fig. 3. Experimental transient behavior in response to a step change of 40% in dispersed-phase feed flow rate (Kuhni column data from Hufnagl et al., 1991).
331
Liquid liquid extraction columns
0.38
w
0.3~
°
t = 276 s
0.34
t=392s ~
t=
. ~
" s
t = 613s . ~ . . .
0.32 ~
~
6
S
0.3
0.28 0.26 0.24 0.22 0.2 0.l~
0'.1 0'.2 0:3
0:4
01s 0'.6 017 018 019
Igl [dimensionless]
Fig. 4. Simulated transient behavior of the holdup profiles at various times in response to a step change of 40% in dispersed-phasefeed flow rate. The initial values of the input variables at steady state just before the onset of the disturbance are as in Fig. 2 (Kuhni column).
(1988) by an analytical solution of a system of equations describing the holdup dynamics in an extraction column using the method of small perturbations. This phenomenon is also observed in response to disturbances in continuous-phase flows and in rotor speed. Unfortunately, Hufnagl et al. (1991) did not report the actual values they used for parameters such as continuous- and dispersed-phase feed flow rates and rotor speed. Thus, the experimental and simulated results can only be compared qualitatively. Nevertheless, the model appears to accurately predict the main experimentally observed phenomena. 5. IMPROVING DISPERSION INTERFACE LEVEL CONTROL
The control of the dispersion interface level is crucial for the reliable operation of liquid-liquid extraction columns. If this level is not adequately stabilized, the dispersion layer could entrain out of the column, leading to loss of solvent and product. The resonance effects observed in the open-loop simulations are detrimental to good dispersion interface level control. Hufnagl et al. (1991) have suggested a novel method to solve this problem by manipulating the continuous-phase feed flow rather than the effluent flow, which is the industrially accepted practice. In this scheme, the continuous-phase effluent flow is left free to control the throughput. Applying these conditions to the model gives new relationships for the dispersed-phase effluent flow rate and the continuous- phase feed flow rate. Thus, the continuous-phase feed flow rate, Uc,in, is obtained from material balance Uc, in = U . . . . t -~- U d , o u t - Ud, in"
(46)
From eqs (13) and (46), the effluent flow, expressed by
Ua.....
is
h= h
(
Vd,ou t = ~. V s - -
Uc.out + Ua out - Ud.in) ~'~
= e[Vs(1 - e) - U..... t Jr- Ud,in]'
(47)
The simulated dynamic responses of the Kuhni column to step changes in the manipulated variables: dispersed-phase feed flow rate, continuous-phase feed/effluent flow rate and rotor speed, are plotted in Figs 5-7. These are the manipulated variables usually used to control the holdup and outlet concentrations. In the plots, the responses of the conventional levelcontrol scheme are shown by solid lines, while those of the improved scheme by dashed lines. The open-loop responses for the case of the improved level-control scheme exhibit monotonic behavior, while those for the conventional level-control show overshoots, oscillatory behavior and even inverse response (e.g. the effluent concentration plotted in Fig. 7). Clearly, the control of the column will be much easier if the improved level-control scheme is adopted. 6. DESIGN AND SIMULATION OF MULTIVARIABLE CONTROL
For the case of the conventional interface level controller, the manipulated variables available for control are the plate-stack oscillation frequency, the dispersed- and continuous-phase feed flow rates. When the modified interface level controller is used, the continuous effluent flow rate may be employed as the control variable instead of the continuous-phase
332
O. Weinstein et al. 0.4
~
0.3 0.2
6 0.1
lobo
' 2000
t [s]
3o;o
' 4000
5000
30;0
40;0
5000
~0.4 e~
~-o.3
~o.2
2o o
t [s]
Fig. 5. Comparison of the transient behavior in response to a step change of 40% in dispersed-phase inlet flow for the two level-control schemes: () conventional scheme; (---) novel scheme. The initial values of the input variables before the onset of the disturbance are Ud = Uc = 0.0024 m3/m 2 s , f = 2.9 l/s, y~. = 1 1 kmol/m 3 (Kuhni column).
0.5
~
0.4
~0.3
6 0.2
1000
2000
3000
4000
5000
30b0
40;0
5000
t [s] -~ 0.3 cO
"~ 0.28
.e
~0.26 ~0.24 0.22 Z~
10;0
20;0 t [sl
Fig. 6. Comparison of the transient behavior in response to a step change of 40% in continuous-phase inlet flow for the two level-control schemes: ( ) conventional scheme; (---) novel scheme. The initial values of the input variables before the onset of the disturbance are Ud = Uc = 0.0024 m3/m 2 s , f = 2.9 l/s, Yi, = 1 1 kmol/m 3 (Kuhni column).
feed flow rate. Both types of level controller were discussed in the previous section. The system outputs are effluent c o n c e n t r a t i o n s a n d dispersed-phase holdup. Multivariable control for a K a r r column will now be designed a n d tested by simulation. F o r this system, p h o s p h o r i c acid is fed in the c o n t i n u o u s phase (pentanol) a n d extracted in the dispersed phase (water). Physical d a t a for the w a t e r - p e n t a n o l - p h o s p h o r i c acid system appears in D a u b e r t a n d D a n n e r (1985).
D a t a for the c o l u m n a n d physical properties are given in Table 2.
6.1. Generatin9 an empirical model The responses of h o l d u p a n d b o t h phase outlet c o n c e n t r a t i o n s to 10% step changes in the control variables are plotted in Fig. 8 for the case of the c o n v e n t i o n a l level controller, a n d in Fig. 9 for the case
Liquid-liquid extraction columns
333
0.4
~
0.35
¢* 0.3 0"250
1000
2000
3000
4000
5000
40b0
50o0
t [sl
~o.3 g~ Q
~-o.2
0.1~
10bo
20b0
t Is]
Fig. 7. Comparison of the transient behavior in response to a step change of 31% in rotor speed for the two level-control schemes: ( ) conventional scheme; (---) novel scheme. The initial values of the input variables before the onset of the disturbance are Ud = Uc = 0.0015 m3/m 2 s,f = 2.8 l/s, Yin = 1 1 kmol/m 3 (Kuhni column).
Table 2. Physical properties and column data for the Karr column Parameter
Value
Units
Description
pc
814.4 998.2 0.4045 x 10-3 1.003 x 10 3 12.25 2.5 0.05 0.027 0.9 0.6
(kg/m 3) (kg/m 3) (kg/(m s)) (kg/(m s))
Continuous-phase density (pentanol) Dispersed-phase density (water) Continuous-phase viscosity Dispersed-phase viscosity Distribution coefficient Column height Column diameter Rotor diameter Orifice coefficient Fractional open area
Pa
r/c qa
m l dr ho Co So
(m) (m) (m) ---
of the modified level controller. In both figures, the process outputs are simulated for positive (solid line) and negative (dashed line) step changes in inputs. For the case of negative inputs, changes in the responses have been plotted with opposite sign. As with the K u h n i column, the Karr column responses for the case of the conventional levelcontrol scheme (Fig. 8) show overshoots and oscillations in the holdup in response to disturbances in all control variables as well as an inverse response and overshoot in outlet dispersed-phase concentration. The modified level-control scheme shows significant improvement of the system's dynamic response (Fig. 9), which should permit more effective control. An empirical uncertain linear model has been identified by fitting low-order transfer functions model using simulated step response data for the modified
level-controller case in the vicinity of the operating point
Y=P'U~
=
P21
P22
P23]
Pij =
Ii
].E,
(48)
Uc
(K + AK) [(zt + Azl)s + 1] 272 + A~2)2S 2 + 2(z 2 + Az2)(~ + A~)s + l J i j '
i = 1,2; j =
1,2,3.
(49)
In the above, ~ is the change in holdup, )( is the change in the outlet dispersed phase concentration, f is the change in the plate stack oscillation frequency, Ua is the change in the dispersed-phase feed flow rate, Uc is the change in the continuous effluent flow rate.
O. Weinstein et al.
334
Ud [m/s]
f [l/s]
-
x 10 "a Uc [m/s]
0.04 i
0.02 ~::O0~l
10~:)0
o,~t
2000
0
:°]A .
0
1000
2000 x 103
X 103
x 10~
.
IOO0
.
.
.
2000
0
1000
x 104
,olaf
O: -2
~'-2 0
IOO0
2000
~
0
t[sl
2000
2000
x 10~
xlO 3
1000
2000
0
1000 t (sl
t [s]
2000
Fig. 8. Simulated transient behavior in response to positive ( ) and negative (---) 10% step change in the control variables for the case of conventional level control. The initial values of the input variables at steady state just before the onset of disturbance are Ud = 0.006 m3/(m 2 s), Uc = 0.003 m3/m z s , f = 1.2 l/s, Y~n = 0.2 kmol/m 3 (Karr column).
f [l/s]
!
Ud Ira/s]
-
o.~i
~:~0.01
IY
x 10 3
UC [m/s]
4i
0.02i
o ~ .,~ x 10~
X 2
....
0u 0
, 1000
20o0
] J 2000
o0
1000
2000
x 10"3
0
1000
2000
,!y
2000
x l O "4
xlO ~
x l o -~
2000 xlo -3
O:
~'-2 0
1000 t [s]
2000
0
1000 t [s]
2000
i/o
2000
t [s]
Fig. 9. Simulated transient behavior in response to positive ( ) and negative (---) 10% step change in the control variables for the case of modified level control. The initial values of the input variables at steady state just before the onset of disturbance are U~=O.OO6m3/m2s, Uc=0.003 m3/m2s, f = 1.2 l/s, Yio = 0.2 kmol/m 3 (Karr column).
Liquid liquid extraction columns
335
Table 3. Process transfer function coefficients Element
K
AK
Z1
A'r I
1:2
A~C2
~
A~
Pit (eft) P12 ('g/Ud) P13 (,g/Uc) P21 (X/f)
0.264 0.439 0.055 0.016 -- 0.077 0.091
0.018 0.082 0.003 0.003 0.005 0.001
16.1 23.3 16.5 773 123 42.2
2.7 6.6 0.03 261 123 2.4
84.9 93.2 83.9 65.9 98.3 109.9
3.2 18.2 3.9 16.9 52.2 3.6
0.85 0.86 0.84 1.53 0.76 0.98
0.03 0.07 0.01 0.46 0.15 0.05
P22 (X/Ua) P23 (X/U~)
Equations (48) and (49) describe the response of the two-process output variables to the three-process inputs as a matrix of uncertain linear transfer functions. The transfer functions parameters are given in Table 3. The uncertainties AK, Art, At2 and A~ reflect the variation in model parameters required to enable a reasonable fit to the process trajectories in response to positive and negative step changes in the inputs in the vicinity of the operating point. The simple transfer functions (49) are not able to capture all the observed nonlinear features shown in Fig. 9, especially for the response of the dispersed-phase effluent concentration. This leads to additional unquantified highfrequency uncertainty in the parameters associated with this output. 6.2. Control system design Decentralized (i.e. diagonal) control has been chosen. From RGA analysis (Bristol, 1966), it is apparent that the most promising pairing is to control the outlet dispersed-phase concentration using the continuous feed flow rate and the holdup using the plate-stack oscillation frequency. The IMC synthesis procedure described by Morari and Zafiriou (1989) has been used for control system design. As a result, the two single-loop IMC controllers are determined as
[ q'(i/)?)
=
q2(g/CL) =
z2s2+2r2~s+l
l
LK(77s +~+lj],1 [ r2~s= + 2 r d s + I ]
(50)
L.K(G7s + 1)i;,e3s + 1) 23
The tuning of the IMC controllers was carried out using the Rijnsdorp Interaction Measure (RIM, Rijnsdorp, 1965): nominal stability is guaranteed in the face of process interaction, if f P2.ico)p13(i~)"]-1 ~/[P11(ko)P23(io)) // >
Ihis(i~)lV~
(51)
where the left-hand-side-term is the RIM, and his is the nominal complementary sensitivity function relating process output i to set-point change j. According to IMC design procedure, his = Paisfq. Since in our case PA~S= 1 (the process has no N M P components),
his(s) =f~(s) = (2iss + 1)- 1.
(52)
The RIM provides a sufficient stability condition for decentralized diagonal control. Clearly, if the offdiagonal elements of the MIMO system are small relative to the diagonal elements, then it will be relatively easy to satisfy eq. (51), since the complimentary sensitivity functions are of the order of unity. Frequency ranges where diagonal dominance does not hold will result in RIM values under unity and will limit achievable bandwidth. The RIM parameter therefore represents constraints on achievable performance caused by the interactions in the process. Thus, plotting RIM as a function of frequency indicates the bandwidth over which stabilizing control is guaranteed. The limiting time constant value, 2~s, of filter, f, can be determined by eq. (51). This procedure is illustrated in Fig. 10. for 2 = 15s. The filter time constant values, 211 and 223 used in closed-loop simulations have been chosen as 100s, providing responses free of oscillations. The increased filter time constant values reflect a desire for increased gain margin in the face of unmodeled uncertainty (especially in the response of X). The dispersed-phase holdup is an important system output variable because its overshoot may lead to flooding, aside from its effect on the concentration dynamics. For this reason, a dynamic decoupler and feedforward controller have been designed to stabilize the holdup. A block diagram for the control system is shown in Fig. 11. A dynamic decoupler has been designed in order to compensate for the effect of continuous-phase feed on holdup magnitude. The parameters of the decoupler are calculated using the nominal process model (indicated by the superscript ~) --/D13 [K(rls -[- 1)]13[-r2s 2 -1- 2Zz(S + 1111 DIs) = /311 = - [K(rlS + 1)]11[z22s2 + 2"c2~s + 1113"
(53) A feedforward controller has been introduced in order to compensate for the effect of dispersed-phase feed flow on holdup. The IMC design procedure has been used for the feedforward controller -- /~12
qf(g/Oa)-- Plx
[K(rls + 1)]12[r2s 2 --k 2%~s + 1111 [K(rxs + l)]11['T22s 2 + 2%¢s + 1112' (54)
336
O. Weinstein et al. 101
........
,
........
,
.......
,
........
,
.......
i
........
,
........
,
....
,
........
":....,
........
,
........
,
.......
Z 10°
10~10-5 i , ,,....i 104
........
t O "3
10 -2
1 0 -1
10 °
101
10 2
Frequency [rad/s]
Fig. 10. RIM test for tuning the IMC controller.
] £ ~p
•
+
{ ql
I
Fig. 11. The block diagram of the control scheme.
The designed control scheme was implemented using MATLAB/SIMULINK. 6.3. Closed-loop simulation results Closed-loop simulations were carried out both for decentralized IMC control alone and for an IMC controller augmented by a decoupler and a feedforward controller. Figure 12 shows the simulated control system performance for a 10% set-point change in outlet dispersed-phase concentration: the decoupled scheme [-Fig. 12(b)] performs significantly better than the scheme without the decoupler [Fig, 12(a)]. The regulatory closed-loop response to a 10% disturbance in continuous-phase feed concentration is
shown for the IMC scheme in Fig. 13(a) and for the decoupled IMC scheme in Fig. 13(b). The level is maintained perfectly by the decoupled scheme. The dynamic decoupler eliminates interaction between the continuous-phase feed flow rate and the holdup almost perfectly due to relatively low uncertainties in the linear approximation and the relatively low sensitivity of holdup to the continuous-phase feed flow rate. Finally, the regulatory performance of the decoupled IMC scheme augmented by feedforward control [-Fig. 14(b)] can be compared to that of the simple decentralized IMC scheme [Fig. 14(a)], for a disturbance in the dispersed-phase feed flow rate. Here,
Liquid-liquid extraction columns
337
(a) Decenla-allzedIMC 0.1
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Fig. 12. Simulated closed-loop response to a set-point 10% step change in outlet dispersed-phase concentration: (a) decentralized IMC scheme; (b) decoupled IMC [ ( - - ) g/e(t = 0), (---) X / X ( t = 0)].
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Fig. 13. Simulated closed-loop response to a 10% step change in feed continuous-phase concentration: (a) decentralized IMC scheme; (b) decoupled IMC [( ) g/~(t = 0), (---) X / X ( t = 0)].
the impact of augmenting feedforward and decoupler components for the holdup alone can be seen; the holdup level is almost perfectly stabilized in face of a - 5% step change in Ue, at the price of a slight deterioration in the response of the outlet dispersedphase concentration. A similar procedure was adopted to design decentralized control for the conventional level control configuration. The resulting performance is slightly worse than that obtained with the proposed modified level control strategy. It should be noted that this
conclusions based only on simulation results. It would be advisable to compare closed-loop performance for both level-control systems experimentally•
7. CONCLUSIONS A theoretical dynamic model has been developed for a countercurrent liquid-liquid extraction column based on the average dispersed-phase drop diameter. The dynamic simulations relying on this model have been found to be in agreement with experimental data
338
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Fig. 14. Simulated closed-loop response to a - 5% step change in feed dispersed-phase flow rate: (a) decentralized IMC scheme, (b) decoupled IMC augmented with feedforward action [( ) k/e(t = 0),
(---) 2/x(t and some published theoretical results. Thus, the model can be reliably applied to the control of an extraction column and for the study of the extraction process. Two control schemes for dispersion interface level have been investigated: conventional level control using the continuous-phase effluent flow rate as the manipulated variable, and level-control using the continuous-phase feed flow rate. It has been shown that the conventional level-control scheme leads to unavoidable overshoots and oscillations in the output variables, while the proposed alternative scheme leads to smoother, overshoot-free response. It has also been demonstrated by simulation that a model-based I M C controller, augmented by a dynamic decoupler and feedforward control, provides excellent servo and regulatory response for a Karr column.
a d32 dK dR D D,, f h hz K l n nz q t s
NOTATION specific interfacial area, m 1 Sauter mean drop diameter, m column diameter, m rotor diameter, m axial dispersion coefficient, m2/s molecular diffusivity coefficient, mZ/s IMC controller filter, dimensionless level, m section height, m overall mass transfer coefficient, m/s column height, m grid, dimensionless rotor speed, 1/s I M C controller transfer function, dimensionless time, s Laplace operator, i/s
= 0)].
S U
cross-section's area, dimensionless loading (Uc -- Qc/S and Ud = Qd/S), m3/m 2 s flow rate, m3/s velocity [V~ = U c / ( 1 - e) and Vd = Ud/E)], m/s continuous phase effluent flow rate, m3/s acetone mass fraction, dimensionless solute concentration in the dispersed phase, kmol/m 3 solute concentration in the continuous phase, kmol/m 3
Q V V~a Xac X Y
Greek symbols
volume fraction holdup of dispersed phase, dimensionless I M C controller filter time constant, s viscosity, kg/m s density, kg/m 3 interracial tension, N/m time constant, s frequency, rad/s free fractional area of plate, dimensionless
P a z co ~o
Subscripts c d
continuous phase dispersed phase
Dimensions numbers nzd~pc . ReR = - impeller Reynolds n u m b e r qc WeR
2 3 nzdRpc . - impeller Weber n u m b e r (7
Liquid-liquid extraction columns REFERENCES
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Heertjes, P. M., Holve, W. A. and Talsma, H. (1954) Mass transfer between isobutanol and water in a spray column. Chem. Engng Sci. 3, 122. Hufnagl, H., McIntyre, M. and BlaB, E. (1991) Dynamic behavior and simulation of a liquidliquid extraction column. Chem. Engng Technol. 14, 301-306. Karr, A. E., Ramanujam, S., Lo, T. C. and Baird, M. H. I. (1987). Axial mixing and scaleup of reciprocating plate columns. Can. J. Chem. Engng 65, 373 381. Kumar, A. and Hartland, S. (1986) Capacity and hydrodynamics of an agitated extraction column. Ind. Engng Process Des Dev. 25, 728-733. Misek, T., Berger, R. and Schroter, J. (1984) Standard test systems for liquid extraction. The Institution of Chemical Engineers, Rugby, Warwickshire. Miyauchi, T. and Vermeulen, T. (1963) Longitudinal dispersion in two-phase continuous-flow operations. I.&E.C. Fundamentals 2, 113-125. Morari, M. and Zafiriou, E. (1989) Robust Process Control. Prentice-Hall, Englewood Cliffs. Rijnsdorp, J. E. (1965) Interaction in two-variable control systems for distillation columns--I. Automatica 1, 15 18. Steiner, L. and Hartland, S. (1983) Unsteady-state extraction. In Handbook of Solvent Extraction, eds. T. C. Lo, M. H. I. Baird and C. Hanson, Chapter 6, Wiley, New York. Tsoris, S., Kirou, V. I. and Tavlarides, L. L. (1994) Drop size distribution and holdup profiles in a multistage extraction column. A.I.Ch.E.J. 40, 407-4 18. Wehner, C. R. and Wilhelm, R. H. (1956) Boundary conditions of flow reactor. Chem. Engng Sci. 6, 89 93.