I n d . E n g . C h e m . R e s . 1989, 28, 1202-1210
1202
Westerberg, A. W.; Hutchison, H. P.; Motard, R. L.; Winter, P. Process Flowsheeting; Cambridge University Press: Cambridge, MA, 1979. Yeh, N. C. C.; Reklaitis, G . V. Synthesis and Sizing of Batch/Semicontinuous Processes. Presented a t the AIChE Annual Meeting, Chicago, Nov 1985; paper 35a. Yeh, N. C. C.; Reklaitis, G. V. Synthesis and Sizing of Batchisem-
icontinuous Processes: Single Product Plants. Conzput. Chem.
Eng. 1987, 11, 639-654. Received for reuiew October 17, 1988 Reoised manuscript receiued April 13, 1989 Accepted May 15, 1989
Dynamics of the Flash Fermentor System with Recycle Shuji Ohbayashi and Kazuyuki Shimizu* Department
of
Chemical Engineering, Nagoya University, Chikusa, Nagoya 464, J a p a n
Masakazu Matsubara Department
of
Electrical Engineering, Daido Institute
of
Technology, M i n a m i - k u , Nagoya 457, J a p a n
T h e dynamic characteristics of the immobilized-cell flash fermentor system with recycle were investigated in terms of the frequency responses, the pole/zero locations, and the step responses. All eigenvalues are real in the left half of the complex plane for the case of a single-stage flash column, while N - 1 pairs of complex conjugates appear as the recycle ratio is increased for the case where the number of stages of the flash column, N , is greater than one. This causes the appearance of the resonant peak(s) in the frequency responses. It is also shown that the amplitude ratio becomes greater than the steady-state gain and that the phase lead occurs a t certain frequency ranges for the case where the recycle flow was considered to be the manipulated variable. Some physical interpretation on the dynamic behavior was made based on the step responses. 1. Introduction
In our previous study (Shimizu et al., 19881, we found the promising feature of the immobilized-cell type of flash fermentor system with recycle as compared to the conventional systems without a flash column. The performance evaluation was made in terms of a vector-valued objective function whose components are the productivity, the product concentration or the energy required for the product separation, and the substrate conversion. The next problem is the understanding of the dynamic characteristics of the flash fermentor system with recycle, which is essential to the control system design. Although very few papers have appeared so far on the dynamics of the system with recycle, it has been known that the system having recycle streams may show significantly different dynamic characteristics as compared with the system without any closed loops. Gilliland et al. (1964) have studied the effect of recycle structure on the process dynamics. Attire and Denn (1978) observed a significant change in the response time of an activated sludge plant with recycle. Kapoor et al. (1986) studied the effect of the recycle structure on the distillation tower time constants. Denn and Lavie (1982) showed that the dynamics of a plant with recycle are equivalent to those of the system with a feedback controller and that the general effect of recycle is to increase the steady-state gain and the dominant plant time constant. Papadourakis et al. (1987) have shown that the presence of recycle loops in a process can have a significant effect on the relative gain array (RGA) which is often used to assess the static interaction between the single-input/single-output (SISO) control loops for the control system design of multiinput/multioutput (MIMO) systems.
* Corresponding author. 0888-5885/89/2628-1202$01.50/0
In the present paper, we investigate the dynamic characteristics of the flash fermentor process with recycle from various points of view such as the frequency response, the pole/zero location, and the step responses in order to make clear the dynamic characteristics inherent in this very promising process. 2. Mathematical Model Equations
The following assumptions were made in deriving the mathematical equations that describe the dynamic behavior of the flash fermentor system as shown in Figure 1: (i) The dynamic behavior of the immobilized-cell concentration is slow enough to be negligible. (ii) The fermentor and the liquid holdups in the flash column are perfectly mixed. (iii) The vapor holdups are negligible. (iv) The hydraulic delay occurring in the liquid flows is negligible. (v) The liquid holups are constant with respect to time. (vi) The feed stream of the flash column is introduced at the top of the column with saturated liquid. (vii) The feed stream of the flash column is a ternary mixture of glucose, ethanol, and water. (viii) The vapor is in equilibrium with the liquid leaving the stage. (ix) No glucose is present in the vapor flow streams. (x) The immobilized-cell system is assumed to respond instantaneously to any perturbation in the substrate or ethanol concentration. Assumption iv will be dropped later, and the effect of the time delay on the dynamics will be discussed. Several other assumptions made are mentioned after the description of the mathematical equations. Numbering the stages of the flash column from the top as stage 1and the reboiler 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1203 Fermentor
Flash Column Product
Feed
SF
arizing the above system equations around the steady state as follows:
T
sn
Recycle
R
P* EF
> Waste
Figure 1. Schematic process configuration of the flash fermentor system.
6Pi =
f 6Pi-, P(1
- 17)
i(l
P 1-77
+ K ( l - E)
Table I. Parameter Values Used for the Simulation K S = 1.6 ka/m3 YQ= 0.06 Pi = 90 ki/m3 p i = 0.24 h-' Yp = 0.16
I
6Pi +
a=l
as stage N, the governing equations for the dynamic model of a flash fermentor system can be stated as follows:
P(S,P) dP V - = RPN - ( F + R ) P + XV dt YP dS V - = FSF dt
+ RSN-
(la)
P(SQ) -
( F + R ) S - -X V (lb) YS
Hi dPi/dt = ( F + R)(Pi-1- Pi) + ( F - B)K(Pi+l- Pi) (i = 1, 2,
...,N-1)
(IC)
where V is the fermentor volume and Hi (i = 1,2, ...,N ) are the liquid holdups of the flash column. X is the immobilized-cell concentration. P and S denote the ethanol and the substrate concentrations in the fermentor, respectively. Pi and Si denote the ethanol and the substrate concentrations a t the ith stage of the flash column, respectively, where Po = P and So = s. S F is the feed substrate concentration, and F and R are the feed and the recycled flow rates, respectively. B is the flow rate of the bottoms to be withdrawn as waste. Here, we should be careful about the term "waste", Since the ethanol contents above 10 ppm may not be acceptable to discard, the bottoms should be introduced to downstream processes for further purification. The yield coefficients, Y pand Ys, are assumed to be constant. Noting that the maximum ethanol concentration in the fermentor is about 90 kg/m3 (3.7 mol %), it may be justified to assume the vapor-liquid equilibrium constant, K , to be the same throughout the stripper-type flash column. The specific growth rate, p, is assumed to be of the form
(")(
(2)
P(S,P) = Ks + S 1 - E -J The parameter values used for the steady-state calculation are listed in Table I (Lee et al., 1983). 3. Linear System Equations The linear system equations can be obtained by line-
where each coefficient of the perturbed variable is evaluated a t the steady-state operating condition. Letting H = HI= H2 = ... = HN, f, 7, p, and e were defined as F R H - B f E 77"pEe=V F+R V F In eq 3, p p and ps denote the derivatives of p with respect to P and S, respectively. By the Laplace transformation of eq 3e and 3f, we have the following expression:
-
where s is the Laplace variable and indicates the Laplace transformed variable. In eq 4, p is defined as P ( 1 - 7)
@E-
f
By the Laplace transformation of eq 3c and 3d and by solving the difference equation (Edmister, 1943), it can be shown that the following expression is derived: 8PN
-aP- - GPNo(s)
X ( A , - A,)/(LUN-'[(AIN(0s + (Y + 77 + E (1 - q ) ] - (AIN-' - AZN-')])(5) E
where CY E
K ( l - e ) ( 1 - 7)
A,, A2 = b f (b2- 1 / ( ~ ) ' / ~
where b = s:(
+ 1 + $/2
1204 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989
P
- -20 -m
:I 01
-40
r(
-601
. . . . . ., . . -.-
N-
----
----
I 5 10 15 20
(13e:Il (153.21 (153.3) (153.31 (153.31
-60-
----
-80'
0.001
0.I
0.01
-
1
10
-0.001
\I 100
0.95 (152.1) 0.01
0.1 I w (rad/h)
IO
100
0.01
0.1
I
10
100
0.01
0.1 1 w (rad/h)
10
100
0.01
0.1
10
100
w (rad/hl
1
1
m
m
u
u
-120 -
-120
-
-iBO+ _-.
0.001
0.01
I
0.1
10
0.001
100
w (rad/h) (a)
-
0
-
0
-20
-
-n
-40
-
-m
n 0
-
0
- -20
m
R r(
0
-60-
-p-I.1.50 . .. . . . . . . ---~
----
2.0 2.5 3.0
CI
-
-40 -
r(
(153.3) (153.3) (153.3) (153.3) ii53.3)
. . . .. . . ..
0
-60-
_.-
----
0~001
f-1.0 0.8 0.6 0.4
(107.0) (126.1) (153.3) (193.4)
a
01
-60
m
u -I2O
t I
-180' 0.001
-120
\0.01
0.I I w (rad/h) (Cl
10
100
-100 0. I1
1
w (rad/h) (d)
Figure 2. Effect of design and operating parameters on the frequency responses of G p s ~(=6P/6SF).The parameter values other than the changed parameters are SF= 400 kg/m3, v = 0.9, t = 0.3, p = 1, f = 0.6 h-', and N = 10: (a) effect of N , (b) effect of 11, (c) effect of p , and (d) effect o f f .
Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 1205 from which we have
+
Im POLES
t 2
- 1
SP
-=
Sii
Gp&) =
a22b13
- a12b23 A
where A
t
-
L l
-
* I
v
-5
-4
v
-3
”
1
-2
-1
m
8 ........ ‘ >Re
i o
4. Analysis of the Dynamic Characteristics
1-1
Figure 3. Pole/zero loci of CpsFwhen 7 was changed from O+ to 1-. The symbols 0 and X indicate the locations corresponding to 7 = O+ and 7 = I-, respectively. The parameter values are SF= 400 kg/m3, c = 0.3, p = 1, f = 0.6 h-l, and N = 3.
By the Laplace transformation of eq 3a and 3b and using eq 4 and 5, we have
where
N- 1 (Y
C (A2 - A 2 ) ] / [ ( A l N- AzN){@+
i=l
(Y
- =12a21
It should be kept in mind that the system zeros of GpsF are the poles of GpNo and GSNO. The derivation of b13 is not straightforward and is briefly given in the Appendix section. Now, we want to study the effect of several design or operating parameters on the dynamics, mainly focusing on the frequency responses, where the transfer functions are normalized such as GpSF( jW)/GpSF(O), etc.
Im 4
ZEROS
%a22
+ q + ~ (-1q)] 1 (AIN-’- A 2N - l ) ]
Figure 2 shows the effect of design or operating parameter? on-the frequency responses of the normalized GpsF (=6P/6SF),where the corresponding steady-state value of the ethanol concentration of the overhead vapor of the flash column, P,, is shown in the same figure. It is interesting to note from Figure 2a that no resonant peak appears for the case where N = 1,while it tends to appear as the value of N increases. In particular, there exist two resonant peaks for the case where N = 20. Figure 2b shows that, as the recycle ratio, q, is increased, the resonant peak appears at higher frequency with a smaller amplitude ratio. Parts c and d of Figure 2 show that the resonant peak appears at higher frequency with almost the same amplitude ratio when the value of p or f is increased. It can be shown that the number of poles of GPsFis 2N + 2 and the number of zeros is 2N. Figure 3 shows the pole/zero loci of GpsF when q was changed from O+ to 1-. As stated before, the zeros are the poles of GpNo and GSNO, and they are not affected by the change in q. In Figure 3 , O and X indicate the pole/zero locations when q = O+ and q = 1-, respectively. As indicated by the arrow, some of the poles move toward the left in the complex plane as the value of 77 is increased. It can be seen from Figure 3 together with another figure (data not shown) that N - 1 pairs of complex conjugates appear as the value of q is increased, which is consistent with the Bode plot of parts a and b of Figure 2. Figure 4 shows the effect of design or operating parameters on the frequency responses of Gpf. Figure 4b shows that no resonant peak appears at low values of q, while the resonant peak becomes significant as the magnitude of q becomes large. The effects of the steady-state values of p and f turn out to be similar to parts c and d of Figure 2, respectively, and therefore are not shown in Figure 4. More or less, a similar discussion can be applied as for Figure 2. Figure 5 shows step responses for the change in f. The simulations were conducted by using the linear model as well as the nonlinear model. As can be seen, the linear model satisfactorily predicts the original nonlinear model in the sense that both the transient and the steady-state gains were close to those obtained by the nonlinear model, where the value of 77 was changed by 0.06. The physical interpretation of the transient is as follows: The increase in f causes the decrease in the ethanol con-
1206 Ind. Eng. Chem. Res., Vol. 28, No. 8, 1989 0
-10 -
-20
-
N......... -------
E
Pv
\
a
(138.1)
I
5 10 15
Y
(153.2)
(153.3) (153.3) (153.3)
20
h 9 d
P
I
-30.
-31
5
0
15
10 t i m e (h)
20
-
L i n e a r Model
E
\
0 Y
tP
vi
P
-9oL
I
0.001
0.1
0.01
1
10
1 100
t i m e (h)
w (rad/hl
(8)
(a)
I20
1
N o n l i n e a r Model
-
0
-?SI5
/As.?;
