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The stability of activated sludge reactors with substrate inhibition kinetics and solids recycle
War. Res. Vol. 24, No. 2, pp. 169-176, 1990 Printed in Great Britain. All rights reserved
0043-1354/90 $3.00+0.00 Copyright © 1990 Pergamon Press plc
THE STABILITY OF ACTIVATED SLUDGE REACTORS WITH SUBSTRATE INHIBITION KINETICS A N D SOLIDS RECYCLE ALBERTO BERTUCCOt*, PAOLA VOLPEj, HERBERT E. KLEI 2, THOMAS F. ANDERSON2 and DONALD W. SUNDSTROM2 qstituto di Impianti Chimici, Universita' di Padova, Via Marzolo, 9 35131 Padova, Italy and 2Department of Chemical Engineering U-139, University of Connecticut, Storrs, CT 06268, U.S.A. (First received November 1988; accepted in revised form July 1989) Abstract--The steady-state simulation of activated sludge reactors with settler and recycle of the concentrated biomass was considered for the case of substrate inhibition kinetics. Analytical relationships were developed for the continuous stirred tank reactor; numerical simulation was performed for other reactors. Occurrence of multiple solutions and hysteresis behavior are examined as a function of space time O, recycle ratio R, recycle solids concentration XR and sludge age O c. The sensitivity of the outlet substrate conversion with respect to the degree of substrate inhibition and of hydraulic mixing is discussed; criteria are suggested in order to operate such bioreactors and to prevent washout occurrence. All results are presented in dimensionless form. Key words--activated sludge reactors, substrate inhibition kinetics, stability, sludge age
NOMENCLATURE D z = axial dispersion coefficient in the reactor (area time - ~) k d =endogenous respiration constant (time -~) k o f m a x i m u m specific growth rate constant (time -t) Ka = substrate inhibition constant (mass, volume -~) KM= substrate saturation constant (masss volume-i) L = reactor length (length) q =specific biomass production and removal rate (mass~ volume-L time- ~) Q = volumetric flowrate (volume time-~) r = reaction rate (mass volume -~ time-~) R = recycle ratio S = limiting substrate concentration (mass volume -~) V = reactor volume (volume) X ffi biomass concentrations (free cells) (mass volume-~) Y, = yield coefficient (massx mass~- ~) : = axial reactor coordinate (length) E = substrate conversion O = fresh feed residence time (time) Oc = sludge age (time) v = axial velocity in the reactor (length time-~). Subscripts 0 = feed e = overflow settler stream i = reactor index I = reactor inlet stream n = net (referred to growth rate) R = recycle stream S = substrate U = reactor outlet stream X = biomass W = sludge withdrawal stream.
*Author to whom all correspondence should be addressed.
INTRODUCTION
The degradation of organic substrates in wastewater by means of microorganisms can be negatively affected by the concentration of the substrate itself, as it occurs, for example, in phenol degradation and in nitrification processes (Pawiowsky and Howell, 1973a; D ' A d a m o et al., 1984; Castens and Rozich, 1986), In general, substrate inhibition has been extensively investigated both in batch reactors and in once-through chemostats: we recall for example the works by Andrews (1968), Edwards (1970) and Pawlowsky and Howell (1973bj. Existence of multiple steady-states in continuous cultures has been discussed, a m o n g others, by Andrews (1968), Pawlowsky et al. (1973)--who also reported experimental results--and C h i e t al. (1974). DiBiasio et al. (1978, 1981) fully addressed the occurrence, stability and control of multiple steady-states in a continuous stirred tank bioreactor (CSTBR) without recycle, both theoretically and experimentally. A complete dynamic analysis and simulation of this same case is reported, for example, by Agrawal et al. (1982). The effects of substrate inhibited kinetics in an activated sludge reactor have been considered by Rozich and G a u d y (1984) and G a u d y and Rodney Sharp (1985). They showed the substrate conversion dependence of the dilution rate in a C S T B R with recycle of biomass, analyzed the influence o f R and XR as operating parameters and provided critical points curves for predicting how to avoid washout 169
170
ALBERTOBERTUCCOet al.
conditions. Rozich and Gaudy (1985) reported experimental results about the response to shock loadings of an activated sludge process using phenol as a substrate. Recently, Godrej and Sherrard (1988) addressed the problem of an activated sludge CSTBR for treatment of toxic organic wastewater by considering the sludge age as the reference operating variable. They also underlined the importance of quantifying the inhibitory effects of substrate in bioreactors. From all of these studies, it is apparent that substrate inhibition models are fundamental in predicting the results of wastewater systems containing toxic compounds. However, the theoretical models and experimental evidence proposed so far to demonstrate multilevel stability referred only to a one-CSTBR configuration and did not consider the effect of various types of mixing within the reactor. In two other papers (Klei et al., 1987; Bertucco et al., 1988), we have developed a general purpose computer program for the simulation of biochemical reactors which can be utilized to represent activated sludge configurations with recycle and substrate inhibition kinetics. By also using this program, we propose a different approach for characterizing the stability of steady-state conditions, for pointing out the importance of the sludge age as an independent variable, and for taking into account the effect of mixing within the reactor. These concepts and cases are discussed using dimensionless groups, according to the procedure presented by Sundstrom et al. (1973). The Haidane equation has been selected among the kinetic expressions available for describing substrate inhibition; according to Mulchandani and Luong (1989) it is accurate enough and requires only three adjustable parameters, therefore it is favourite with respect to more complicated equations. Although we do not present experimental results, we point out that, for all of the examples presented in our paper, values of the kinetic constants were investigated which are consistent with those calculated in kinetic studies for phenol degradation (Pawlowsky and Howell, 1973; D'Adamo et al., 1984; Rozich et al., 1985) and for nitrification process Oo SO' XO ~ PO
~ = - -s
y=
R . K"
~ = k2
KM
ko
i
Reactor
x
r,
rs = =
korsKM Oko
~ =
koKM y-------'~
vL Pe = - -
~c = Oc ko
Dz
(Castens and Rozich, 1986). Moreover, values of the operating variables were used, which are typical of current operations of activated sludge plants. MODEL BASED ON RECYCLE BIOMASS CONCENTRATION
The reference schematic of an activated sludge reactor with settler and recycle of biomass is reported in Fig. 1. General model equations for steady-state simulation and algorithms for their solutions have been developed in terms of dimensionless variables elsewhere (Bertucco et al., 1988). Table 1 summarizes the definitions used in the present paper. In developing our approach, the simplest model, i.e. a CSTBR with ideal settler is assumed as a first case. The material balances for substrate and biomass around the reactor are given by: S0 - S + Fs ~ = dS/dt"
+ (J:x- fl,~)~ = d,~'/df
(2)
with SY ex= - e , = 1 + g + S 2 / g
(3)
At steady-state, the right hand sides of equations (1) and (2) are zero. Under this hypothesis the combination of the two equations provides: So - S - ~f'(1 + R + f l O ) + :?0 + R-~'R = 0.
(2a)
and by introducing equations (2a) and (3) into equation (1), one can obtain a third-order equation A
Su, Xu ,Pu
O,
S e )Xe )
OR $R,×R,PR
(1)
,~'o + R-~R -- ~ ( 1 + R + fl~)
V, O R
St, Xz, Pz
I. Definition of dimensionless variables
Table
OW
Fig. I. Reference schematic of the activated sludge reactor with settler and recycle.
Stability of bioreactors
171
for E , at constant recycle biomass concentration )TR:
(4)
aE 3 + bE" + c,~ - 1 = 0 with
O.4
a
I/(S'oR)
=
0.2
[
b=
1/Eo-1//~
c = ~l/g° - 1
°
go(l+R+BO)
1
0
(l + R + ~ g ) + Eo(l + R + / t g ) J "
Equation (4) can be solved for g once g0, -~'0, R, )7,, and the kinetic constants have been assigned. Then the corresponding values for )7 can be calculated from equation (2a). Figure 2 shows E as a function of the space time for increasing dimensionless inhibition constant R. According to D ' A d a m o et al. (1984), the inhibition constant KH should always be greater than KM; we reported the curve at g = 1 as a limiting case of the actual situation. At very high Ks the expected Monod behavior is achieved, where for each ~ only one (g,)7) pair satisfies the material balance equations. On the other hand, at lower Ks a range of ~ arises where multiple solutions occur, namely a low and a high conversion state, which are each stable, and the intermediate unstable one. To better explain this matter, we consider a further representation of the solution, with )7 chosen as an independent variable and the conversion ~ as a dependent one. ~ is defined by:
E = (~o - g)lgo.
(5)
At a given ~ , the solutions for (E,)7") can be obtained also by solving the system: qx.p ----?x.n = E)7/(1 + E q- g : / R ) - / ~ ) 7
(6)
q,.,
(7)
=
[(R + 1))7 - R)Tr~- )7o1/~.
t6
12
g
~
30
40
~----~
1 + E0(l _ E) + E20(1 _ E)2/K + B/g0
2 = (2o + R)TR + Ego)/(R + 1 +/~).
I
3
I 4.
Fig. 2. CSTBR simulation with R--0.25, ~'R = 30t) and ~o = 20. S vs O profiles at different inhibition constants.
(9)
-]
g
+/~] - $(1 + E + E2/R)
D+R
2
(8)
In an ~ vs )7 diagram, equation (8) shows up as a sigmoid, while equation (9) appears as a straight line; both curves depend on ~ (see Fig. 3), but only the second one is a function of the recycle variables. It has to be pointed out that E can be rigorously used for expressing the production rate only when /~=0. The two curves in Fig. 3 intersect each other in three points (A, B, C), which are the solutions of the system of equations (8)--(9).Points A and C represent inherently stable operating situations. From a graphical point of view, itis apparent that any perturbation occurring on )7 inside the reactor will be annulled: in fact, on the right (or left) hand side of points A and C, the removal of biomass is higher (or lower) than its generation rate. Thus, the system is able to come back to its original steady-state. From an algebraic point of view, the conditions for stability can be applied to the system of equations (I)-(2), according to Liapunov's method (Morbidelli et al., 1987), in order to get:
L~
q
i
60
Equation (6) expresses the net production of biomass per unit reactor volume, and equation (7) the net removal of biomass from the reactor, in the same units. At steady-state, generation and withdrawal of biomass must equal each other, so that equations (6) and (7) are satisfied simultaneously. In terms of the selected variables, by recalling equations (2a) and (5), one can rearrange equations (6) and (7) to obtain:
33. (E2/R-I))7,>o
4
0
J
50
Fig. 3. Biomass production rate and biomass removal rate at different R and Ji'R= 300, O -- 1.05, ~o -- 20 and R = 3.
1 [I+R 8
10
2+R
(10)
E -¢ 1 + E + E:IR E a / R - 1))7
+(1 +E+E:/R) 2 /~>o.
(11)
172
ALBERTO BERTUCCO et al.
Condition (11) is satisfied for all three points while condition (10) is not for point B, which is then an unstable steady-state. The (E, ,~') plot is also helpful to inspect how the reactor conversion is affected by a change in the space time, due for example to a step change in the inlet flowrate, Q0. Starting from an initial steady-state [point A~ of Fig. 4(a)], an increase of Q0 makes the reactor move through states At, down to A2 where a limiting space time g * is reached. If ~ is further decreased, the solution jumps down to A~, the stable solution at low conversion, with a discontinuity for E. However, when operating at A3, an increase in does not shift the stable point back to high conversions until point A4 (at ~**) is attained. If ~ is further increased, the solution jumps to point A5 for the reasons previously discussed. This is the so-called hysteresis effect which is associated with substrate inhibition kinetics. For assigned R and avR, the values of ~*, g** can be calculated by equating equations (8) and (9) with E = Et and E = E2, where Et, E2are two solutions of the fourth-order equation obtained by imposing a tangency condition. The same phenomenon can be described by means of a ~ vs O plot, as is shown by the path evidenced in Fig. 4(b): here the shaded area represents the hysteresis loop, and the dotted part of the curve defines the range of instability, as it can be checked by applying equations (! 0)-(1 ! ) as well. Experimental evidence of hysteresis behavior has been provided by Pawiowsky et aL (1973) for a CSTBR without recycle. The effect of a step change in the recycle flowrate can be illustrated either by the ~ vs g plot of Fig. 5 or by the E vs ~' diagram of Fig. 3. It can be seen, for example, that starting from a high conversion steady-state situation (point A) at ~ = 1.05, R = 0.1, a decrease of R to a value of 0.05 at constant XR makes the system jump down to a low conversion steady-state (point D), which represents a totally
:oo,
_:_ _ x . . .
0.6 OA
.....
o.~ • / L /
2
0
3
4
5
6
7
Fig. 5. Effect of recycle ratio R in an ~ vs ~ plot. ~R = 200. S0=20and g = 3 . unacceptable performance. Very similar figures and conclusions could be derived if XR was varied and R kept constant. In Fig. 5 the curve at R = 0 is also reported; even though this case could be described by simplified relationships, it is easily simulated also as a limiting case of the general model developed above. It is remarkable to note that, without recycle and with -~'o= 0, washout conditions can be readily derived by differentiating [for equation (13)] or by setting S = So [for equation 04)], in the following: 1/O =
1 I/S + 1 + S / K
fl
which relates 0 and S at R = 0 and Xo = O. Washout occurs whenever 0 is less than its minimum O~n, given by the lower value between OM and OpR: 1 l/OM
=2/~-K_
i/erR =
1 /~ I
l/So + 1 + So/K
(13) -- #.
(b)
A ~
O.e
- .
1.14
A5
/~j' 0.6
0.4
0.4
0.2 ~ ' 1 /
~/
0.2 °A 3
O
20
40I
60I
(14)
When OpR > OM, OpR is an "activation" value of space time in the hysteresis loop: if OpR < O M no
leA2
0.6
(12)
0
0.5
1.0
1.5
Fig. 4. Hysteresis loop f o r a C S T B R according to the J/R-based model. ~' R = 200, R = 0.1, ~'0 = 20 and • = 3. (a) In an ~ vs ~' plot. (b) In an ~ vs O plot.
Stability of bioreactors hysteresis effect is possible because the inhibition is low enough to give results closer to Monod kinetics. It is also apparent from Fig. 5 that working with a recycle will lead to lower values of ~M, that is to less stringent operability limits for the reactor, and that complete washout is always avoided when ~R is kept constant.
173
Therefore, at assigned O and Q0, the sludge age depends upon Qw only, if R is kept constant by a recycle pump; in this way, Oc can be varied by simply opening or closing the biomass purging device. If equation (15), expressed in dimensionless form, is combined with equation (2), we also get: O c = ~x - fl;f' - £ 0
(18)
MODEL BASED ON SLUDGE AGE
Similar results for a CSTBR with substrate inhibition kinetics can be obtained on the basis of the variable called sludge age (Oc), defined as the ratio of biomass in the reactor to the net rate of biomass generation (Bertucco et al., 1988): VX VX OC= QoXo + rx.n = QwXR + Q~X .
(15)
Although Oc is not exactly the mean cell residence time, it is more commonly used because of the difficulty of determining the death constants of wastewater microbial populations; however, O c is usually an acceptable approximation of the biomass residence time. As it is known, Oc is related to the settleability of the sludge. In the above section we have seen that operating at constant recycle biomass concentration will never drive the system into total washout condition, so that a conversion of the substrate, even tiny, will always take place because the biomass concentration in the reactor inlet stream is never zero. Indeed, from a practical point of view, keeping XR constant means having a settling device capable of ensuring the desired recycle concentration at the bottom, no matter what is going on inside the reactor. This is a kind of attractive situation, but rather unrealistic when using a gravity settler, as usual in activated sludge plants. Therefore, it seems more appropriate in this case to adopt Oc as an independent variable. Moreover, Oc is related to parameters such as the flowrates, which can be predetermined and controlled easily by means of pumps and valves. To show the controllability of Oc, consider a (desirable) situation where no cells are wasted in the overflow stream Q,. Then, equation (I5) reduces to: Oc = ~
VX
On the other hand, by rearranging equation (15) and taking into account the material balance of biomass around the settler, one can get: O
R-4- I - R X R/X
(16)
that relates the concentration of biomass in the reactor to the recycle variables. A combination between equations (15a) and (16) finally provides: o~ = ~
1 llOc= ll~ + l +,~/R
ft.
(19)
When ~c is assigned, two solutions for ~ can be found, which are independent of ~; curves at constant ~ c are then given by straight lines on a ~ vs O plot, like the ones reported in Fig. 5. Equation (19) exhibits a minimum value OC.M and an "activation" value OC.PR: 1/~C,~a = -
-
1
2/x/~-
1
fl
(20)
1 I/~c'PR = 1/,~0 + 1 + ~'0/~ - fl"
(21)
Equations (19), (20) and (21) are the same as equations (12), (13) and (14) for R = 0, when ~ is replaced by ~c. Similar to previously considered values of ~, we now have to consider a washout value of ~ c given by ~c.~.., which is the minimum between ~C.M and ~C.PR. Due to the hysteresis effect, there are two washout conditions of ~ c to be determined: one is the ~c.mi. and the other an "activation" value, ~C.PR, if ~gC.PR > ~c.M.
The removal of biomass [equation (9)] is now expressed in terms of sludge age: = ~0~ + g0 .__~c
1 +fl~c
~
(9a)
whilst the biomass production is still given by equation (8). By looking at equation (9a) and recalling that for a CSTBR without recycle it holds:
(15a)
QwXR
Oc =
and, upon introducing the Haldane kinetics into equation (18), we have (for ~'0 = 0):
+ 1 .
(17)
1 + flO
(22)
lira Oc = ~
(23)
one can see that:
as would be expected from a correct definition of sludge age. As a consequence of the Oc-based model, we point out that the washout condition is always one of the stable solutions of the system given by equations (8) and (9a), as is evidenced in Fig. 6, in analogy with Fig. 3. Again, we have a sigmoid and a straight line, both depending on ~ (the straight line also depends
174
ALBERTOBERTUCCOet
on Oc); their intersections provide the possible steady-states A, B, C, but now the one marked by C is always at zero conversion. Point B represents an unstable steady-state. It can be seen that, if O is changed while Oc is kept constant, the system shifts to new possible operating points; increasing O at constant Oc reduces the cell concentration needed within the reactor, even though the substrate conversion remains unchanged. It is remarkable to note that complete washout occurs whenever Oc is lower than Oc.~,, since here a cell concentration in the recycle is not guaranteed, such as when operating at ,~'R= constant. Starting from Oc.min and increasing Oc with O, R is kept constant, a hysteresis behavior can be found, in analogy with the case at assigned ,I'R. Figure 6 shows details: the sigmoid not depending upon ~c, it is possible to follow the hysteresis loop by simply rotating the straight line. It is also apparent that above a value of dlc -OC.m~ (OC.~,~ = OC.PR if diC.PR> OC,M) only one solution is found, the high conversion one. It can be shown that for any value of O, the tangent point corresponding to OC,M provides a substrate conversion given by: Ev = I - x , ~ / ~ ' o
(24)
which is independent of O, being determined by the kinetic constants and the initial substrate concentration only. By imposing the two tangency conditions, equations (20) and (21) are obtained again, thus confirming that the Oc limiting values are not a function of O. The dependence of .~'t on Oc can be obtained analytically by combining equations (2a) and (16)-in dimensionless form--and (19). The solution is illustrated by means of Fig. 7: plots of Oc vs ,:I'R (bold lines for one CSTBR) exhibit minima and multiple solution ranges, which depend on the inhibition constant J~. It is remarkable to see that operating on substrate inhibited wastes requires values of ,f'R much higher than those used with Monod-like wastes: they can be achieved only by considerably raising the sludge age. All three curves are relative to cases with no cell death term in the kinetics; the introduction of a typical value for the factor ]~ results in a dramatic
o.
"21 0
20
40
60
80
I00
'120
'140
Fig. 6. Hysteresis loop for a CSTBR according to the Oc-based model. R = 0.25, ~ = 1.05. ~'0 -- 20 and /L'-- 3.
al.
2c ~
R, I
//
,
A.3 \ \\ '\ \\
~z
0
NON01~ 100
I 200
)¢Q-l_csrsR /fi~\scsrBR
I 300
t 40O
I
.
XR
Fig. 7. ~¢ vs "~'Rprofiles: for a CSTBR at different inhibition
constants and for different numbers of CSTBRs at /~ = 3 and ~"=0.1, R =0.25, (~= 1.26 and g0 = 20. raise of the sludge age required to ensure a desired .I'R or substrate conversion, as shown by the dashed curve for one CSTBR in Fig. 7. EFFECT OF MIXING
All considerations and remarks presented about the behavior of a CSTBR with substrate inhibited kinetics can be extended to cases when a different reactor configuration or a different hydraulic mixing model within the reactor are adopted for simulation. The definition of sludge age is the same, but the resulting expression in terms of recycle variables is slightly more complicated, since we now have: 0c =
;f'A~0 (R + l),t' - R~'R
(16a)
with
lff 3~ d V'.
~Av = ~
If a configuration with N CSTBR in series is considered, the material balances for ~, ~t' at the ith tank are now provided by:
o
S~_, - •, + ~s,, R + ! = 0
(25)
O .~_ ~ - ~ + r-~.,., R + I = 0
(26)
which have to be solved together with equation (16a). It is not possible to express these quantities by simple relationships as for a CSTBR and numerical solution is needed. The resulting Oc vs "]'R plot is somehow different, as indicated by the dotted lines in Fig. 7, which are calculated with the same/~ and operating conditions as for one CSTBR. Taking the number of CSTBR as a parameter (at constant total 0 ) and starting from a one tank situation, which is actually the lower limit, higher Ns lead to bigger values of Oc.mi, which are obtained at consistently higher ,I'a s. The upper limit is represented by the plugflow reactor (N --, oo). We note also the disappearance of intervals of multiple solutions, as N increases.
Stability of bioreactors The influence of the extent of mixing within a given reactor can be described by solving the pertinent material balance equations for S, .~ at different values of the Peclet number and constant space time and inlet concentrations ~0, -~'0:
175
O.8
0.6
1 d:~
dS
Pe dS:
d5
1 d2X
d.Y
Ped-'-
d5
,E -
-
R + 1
-
-
R + 1
~s = 0
(27)
e=OO
0.4
~x., = o
(28) 0.2 /
with the boundary conditions: d~ -~=Pe($'-.~0;
(a.~=0:
d..~
(~ 5=1:
dg
-
d J7 d.f
d.~ -~-=Pe(Y-J?l)
=0.
The system given by equations (27) and (28) can be solved by using an orthogonal collocation method, as outlined in a previous paper (Bertucco et al., 1988). An ~ vs z / L plot shows up, for example, as in Fig. 8, at constant XR = 300, and as in Fig. 9, at constant Oc = 9.1. We note a dramatic change in the profile when moving from a CSTBR situation (Pe = 0) to a plugflow one (Pe ~ oo). This also affects the outlet substrate concentration, especially at constant X R. An optimal mixing value can be found that maximizes the substrate conversion. In many cases, the pollutant removal gets worse as a plugflow situation is achieved, so that a certain extent of mixing is often ad~,isable when handling substrate inhibited wastes. It can be concluded that the degree of mixing could be used as an extra variable which affects the stability of the reactor. The influence of the degree of inhibition is finally illustrated by Fig. l0 at constant axial mixing (Pe = 3). Again, a critical range o f / ~ values can be identified, around which a sharp drop in the substrate conversion profiles takes place.
o. i •
0
/ /
~
I O.2
Pea oo
I 0.4
I 0.6
I 0.8
z/L
Fig. 8. Effect of mixing at -~'R= constant = 300, R = 0.l, (~ = 2.1, -~'o= 20 and g' = 3.
O
I
t
I
I
0.2
0.4
0.6
O.8
z/L Fig. 9. Effect of mixing at ~c = constant = 9.1, R = 0.1, = 2.1, ,go = 20 and E = 3.
0.8
0.6
Q4
0.2
I
I
I
I
0.2
0.4
0.6
0.8
z/L
Fig. 10, Effect of the degree of substrate inhibition in a reactor with constant axial dispersion: Pe = 3, R = 0.25, -~R = 200, ~ = 1.26, So = 20 and g = 3. CONCLUSIONS
The biological treatment of toxic wastewaters is of great interest for the future but also of great concern due to the problems arising from substrate inhibition effects. This is true either in designing and in operating activated sludge reactors dedicated to this scope. In the present paper the behavior of a bioreactor with a variable degree of internal mixing has been investigated taking into account a settler and a recycle stream in terms of dimensionless groups. Two different approaches have been considered, depending on which independent variable for the recycle is selected, namely the cell concentration ~R or the sludge age ~ c . From the reported results, four main conclusions can be derived. The first one is that special care should be taken about the possibility of kinetic effects due to substrate inhibition and to the death term, since the calculated steady-states of the system may deviate considerably from the common Monod-like ones. This is clearly shown, for example, by Figs 2 and 7, where the sensitivity of the system to relatively small changes in ~ or ~ c is large, if it is operated around critical conditions.
176
ALBERTOBERTUCCOet al.
As a second point, the occurrence of washout has to be considered as an actual danger. At A'R = constant, complete washout is never reached, but in the more common situation with ~ c assigned, any change in the space time or in the recycle ratio or in the sludge withdrawal flowrate will vary ~ c , thus requiring a careful check with respect to the desired performance of the plant. Also the degree of mixing can affect the critical point of operation. A third conclusion is related to the hysteresis effects: the dramatic changes of the overall substrate conversion occurring upon small perturbations of the system variables around their steady-state values are not reversible, as is represented by Figs 4 and 6. The "activation" value of ~ or ~ c , that is the threshold to overcome in startup or transient plant operation, is likely to be much higher than their minimum values at steady-state. Finally, we point out the remarkable effect of mixing, as is represented in Figs 8, 9 and 10: depending on it is the fact that the situation in any point within the reactor can be changed from a low to a high conversion one. All of these considerations are more relevant when the plant works at low values of recycle ratio. Therefore, both the design and the operation of activated sludge reactors for toxic wastes could take advantage of the capability of simulating their performance as a function of the inlet substrate and biomass concentration and the process variables, that is: space time, recycle ratio, biomass recycle concentration (or, preferably in our opinion, sludge age), kind of reactor configuration and degree of internal mixing. It is well known that for a given plant the flowrate and concentration of the feed tend to rise with time, so that the system could be led to operate in a "dangerous" region, and possibly fall into failure. On the other hand, even though the target values are not exceeded, the occurrence of random step changes in one of the above mentioned variables could be dangerous for the same reasons. All of these anomalous and unforeseen situations should be evaluated and their effects simulated, in order to design an effective control system and policy. We recognize that a lot of work has to be done by microbiologists so that proper microbial populations can be characterized and selected for processing toxic wastes; however, from the reaction engineering point of view, it also seems valuable to provide general equations and diagrams which allow one to predict and check the performance of a system where a substrate inhibited kinetic is dominant. Acknowledgements--The authors gratefully acknowledge Professor David DiBiasio from Worcester Polytechnic Institute, Worcester, Mass., for reviewing the manuscript and the Department of Higher Education (State of Connecticut) for financial assistance under High Technology Grant in developing computer models.
REFERENCES
Agrawal P., Lee C., Lim H. C. and Ramkrishna D. (1982) Theoretical investigations of dynamic behavior of isothermal continuous stirred tank biological reactors. Chem. Engng Sci. 3% 453-462. Andrews J. F. (1968) A mathematical model for the continuous culture of microorganisms utilizing inhibitory substrates. Biotechnol. Bioengng 10. 707-723, Bertucco A., Volpe P. and Klei H. E. (1988) Interactive biochemical reactor simulation on a microcomputer. Chem. biochem. Engng Q. 2, 127-136. Castens D. J. and Rozich A. F. (1986) Analysis of batch nitrification using substrate inhibition kinetics. Biotechnol. Bioengng 28, 461-465. Chi C. T., Howell J. A. and Pawlowsky U. (1974) The regions of multiple stable steady states of a biological reactor with wall growth, utilizing inhibitory substrates. Chem. Engng Sci. 29, 207-211, D'Adamo P. D., Rozich A. F. and Gaudy A. F. (1984) Analysis of growth data with inhibitory carbon sources. Biotechnol. Bioengng 26, 397-402. DiBiasio D., Lim H. C. and Weigand W. A. (1981) An experimental investigation of stability and multiplicity of steady-states in a biological reactor. A1ChE JI 27, 284. DiBiasio D., Lim H. C., Weigand W. A. and Tsao G. T. (1978) Phase-plane analysis of feed-back control of unstable steady-state in a biological reactor. AIChE JI 24, 686. Edwards V. H. (1970) The influence of high substrate concentration on microbial kinetics. Biotechnol. Bioengng 12, 679. Gaudy A. F. and Rodney Sharp H. (1985) Biological treatment of toxic wastes. Paper presented at Bioengineering Symposium, NJIT, Newark. Godrej A. N. and Sherrard J. H. (1988) Kinetics and stoichiometry of activated sludge treatment of a toxic organic wastewater. J. Wat. Pollut. Control Fed. 60, 221. Klei H. E., Baena R. D., Anderson T. F., Sundstrom D. W. and Bertucco A. (1987) Scaleup using a biochemical process simulator. In Biotechnology Processes (Edited by Ho C. S. and Oldshue J. Y.), pp. 185-199. AIChE, New York. Morbidelli M., Varma A. and Aris R. (1987) Reactor steady-state multiplicity and stability. In Chemical Reactions and Reactor Engineering (Edited by Carberry J. J. and Varma A.), pp. 973-1054. Dekker, New York. Mulchandani A. and Luong J. H. T. (1989) Microbial inhibitory kinetics resisted. Enzyme Microbiol. Technol. 11, 66-73. Pawlowsky U. and Howell J. A. (1973a) Mixed culture bioxidation of phenol. I. Determination of kinetic parameters. Biotechnol. Bioengng 15, 889-896. Pawlowsky U. and Howell J. A. (1973b) Mixed culture bioxidation of phenol. II. Steady state experiments in continuous cultures. Biotechnol. Bioengng 15, 887-903. Pawlowsky U., Howell J. A. and Chi C. T. (1973) Mixed culture bioxidation of phenol. III. Existence of multiple steady states in continuous cultures with wall growth. Biotechnol. Bioengng 15, 905-916. Rozich A. F. and Gaudy A. F. (1984) Critical point analysis for toxic waste treatment. J. envir. Engng Div., Am. Soc. cir. Engrs 110, 562-572. Rozich A. F. and Gaudy A. F. (1985) Response of phenolacclimated activated sludge process to quantitative shock loading. Wat. Pollut. Control Fed. 57, 795-804. Rozich A. F., Gaudy A. F. and D'Adamo P. C. (1985) Selection of growth rate model for activated sludge treating phenol. Wat. Res. 19, 481-490. Sundstrom D. W., Klei H. E. and Molvar A. E. (1973) The use of dimensionless groups in the design of activated sludge reactors. War. Res. 7, 1905-1913.
0043-1354/90 $3.00+0.00 Copyright © 1990 Pergamon Press plc
THE STABILITY OF ACTIVATED SLUDGE REACTORS WITH SUBSTRATE INHIBITION KINETICS A N D SOLIDS RECYCLE ALBERTO BERTUCCOt*, PAOLA VOLPEj, HERBERT E. KLEI 2, THOMAS F. ANDERSON2 and DONALD W. SUNDSTROM2 qstituto di Impianti Chimici, Universita' di Padova, Via Marzolo, 9 35131 Padova, Italy and 2Department of Chemical Engineering U-139, University of Connecticut, Storrs, CT 06268, U.S.A. (First received November 1988; accepted in revised form July 1989) Abstract--The steady-state simulation of activated sludge reactors with settler and recycle of the concentrated biomass was considered for the case of substrate inhibition kinetics. Analytical relationships were developed for the continuous stirred tank reactor; numerical simulation was performed for other reactors. Occurrence of multiple solutions and hysteresis behavior are examined as a function of space time O, recycle ratio R, recycle solids concentration XR and sludge age O c. The sensitivity of the outlet substrate conversion with respect to the degree of substrate inhibition and of hydraulic mixing is discussed; criteria are suggested in order to operate such bioreactors and to prevent washout occurrence. All results are presented in dimensionless form. Key words--activated sludge reactors, substrate inhibition kinetics, stability, sludge age
NOMENCLATURE D z = axial dispersion coefficient in the reactor (area time - ~) k d =endogenous respiration constant (time -~) k o f m a x i m u m specific growth rate constant (time -t) Ka = substrate inhibition constant (mass, volume -~) KM= substrate saturation constant (masss volume-i) L = reactor length (length) q =specific biomass production and removal rate (mass~ volume-L time- ~) Q = volumetric flowrate (volume time-~) r = reaction rate (mass volume -~ time-~) R = recycle ratio S = limiting substrate concentration (mass volume -~) V = reactor volume (volume) X ffi biomass concentrations (free cells) (mass volume-~) Y, = yield coefficient (massx mass~- ~) : = axial reactor coordinate (length) E = substrate conversion O = fresh feed residence time (time) Oc = sludge age (time) v = axial velocity in the reactor (length time-~). Subscripts 0 = feed e = overflow settler stream i = reactor index I = reactor inlet stream n = net (referred to growth rate) R = recycle stream S = substrate U = reactor outlet stream X = biomass W = sludge withdrawal stream.
*Author to whom all correspondence should be addressed.
INTRODUCTION
The degradation of organic substrates in wastewater by means of microorganisms can be negatively affected by the concentration of the substrate itself, as it occurs, for example, in phenol degradation and in nitrification processes (Pawiowsky and Howell, 1973a; D ' A d a m o et al., 1984; Castens and Rozich, 1986), In general, substrate inhibition has been extensively investigated both in batch reactors and in once-through chemostats: we recall for example the works by Andrews (1968), Edwards (1970) and Pawlowsky and Howell (1973bj. Existence of multiple steady-states in continuous cultures has been discussed, a m o n g others, by Andrews (1968), Pawlowsky et al. (1973)--who also reported experimental results--and C h i e t al. (1974). DiBiasio et al. (1978, 1981) fully addressed the occurrence, stability and control of multiple steady-states in a continuous stirred tank bioreactor (CSTBR) without recycle, both theoretically and experimentally. A complete dynamic analysis and simulation of this same case is reported, for example, by Agrawal et al. (1982). The effects of substrate inhibited kinetics in an activated sludge reactor have been considered by Rozich and G a u d y (1984) and G a u d y and Rodney Sharp (1985). They showed the substrate conversion dependence of the dilution rate in a C S T B R with recycle of biomass, analyzed the influence o f R and XR as operating parameters and provided critical points curves for predicting how to avoid washout 169
170
ALBERTOBERTUCCOet al.
conditions. Rozich and Gaudy (1985) reported experimental results about the response to shock loadings of an activated sludge process using phenol as a substrate. Recently, Godrej and Sherrard (1988) addressed the problem of an activated sludge CSTBR for treatment of toxic organic wastewater by considering the sludge age as the reference operating variable. They also underlined the importance of quantifying the inhibitory effects of substrate in bioreactors. From all of these studies, it is apparent that substrate inhibition models are fundamental in predicting the results of wastewater systems containing toxic compounds. However, the theoretical models and experimental evidence proposed so far to demonstrate multilevel stability referred only to a one-CSTBR configuration and did not consider the effect of various types of mixing within the reactor. In two other papers (Klei et al., 1987; Bertucco et al., 1988), we have developed a general purpose computer program for the simulation of biochemical reactors which can be utilized to represent activated sludge configurations with recycle and substrate inhibition kinetics. By also using this program, we propose a different approach for characterizing the stability of steady-state conditions, for pointing out the importance of the sludge age as an independent variable, and for taking into account the effect of mixing within the reactor. These concepts and cases are discussed using dimensionless groups, according to the procedure presented by Sundstrom et al. (1973). The Haidane equation has been selected among the kinetic expressions available for describing substrate inhibition; according to Mulchandani and Luong (1989) it is accurate enough and requires only three adjustable parameters, therefore it is favourite with respect to more complicated equations. Although we do not present experimental results, we point out that, for all of the examples presented in our paper, values of the kinetic constants were investigated which are consistent with those calculated in kinetic studies for phenol degradation (Pawlowsky and Howell, 1973; D'Adamo et al., 1984; Rozich et al., 1985) and for nitrification process Oo SO' XO ~ PO
~ = - -s
y=
R . K"
~ = k2
KM
ko
i
Reactor
x
r,
rs = =
korsKM Oko
~ =
koKM y-------'~
vL Pe = - -
~c = Oc ko
Dz
(Castens and Rozich, 1986). Moreover, values of the operating variables were used, which are typical of current operations of activated sludge plants. MODEL BASED ON RECYCLE BIOMASS CONCENTRATION
The reference schematic of an activated sludge reactor with settler and recycle of biomass is reported in Fig. 1. General model equations for steady-state simulation and algorithms for their solutions have been developed in terms of dimensionless variables elsewhere (Bertucco et al., 1988). Table 1 summarizes the definitions used in the present paper. In developing our approach, the simplest model, i.e. a CSTBR with ideal settler is assumed as a first case. The material balances for substrate and biomass around the reactor are given by: S0 - S + Fs ~ = dS/dt"
+ (J:x- fl,~)~ = d,~'/df
(2)
with SY ex= - e , = 1 + g + S 2 / g
(3)
At steady-state, the right hand sides of equations (1) and (2) are zero. Under this hypothesis the combination of the two equations provides: So - S - ~f'(1 + R + f l O ) + :?0 + R-~'R = 0.
(2a)
and by introducing equations (2a) and (3) into equation (1), one can obtain a third-order equation A
Su, Xu ,Pu
O,
S e )Xe )
OR $R,×R,PR
(1)
,~'o + R-~R -- ~ ( 1 + R + fl~)
V, O R
St, Xz, Pz
I. Definition of dimensionless variables
Table
OW
Fig. I. Reference schematic of the activated sludge reactor with settler and recycle.
Stability of bioreactors
171
for E , at constant recycle biomass concentration )TR:
(4)
aE 3 + bE" + c,~ - 1 = 0 with
O.4
a
I/(S'oR)
=
0.2
[
b=
1/Eo-1//~
c = ~l/g° - 1
°
go(l+R+BO)
1
0
(l + R + ~ g ) + Eo(l + R + / t g ) J "
Equation (4) can be solved for g once g0, -~'0, R, )7,, and the kinetic constants have been assigned. Then the corresponding values for )7 can be calculated from equation (2a). Figure 2 shows E as a function of the space time for increasing dimensionless inhibition constant R. According to D ' A d a m o et al. (1984), the inhibition constant KH should always be greater than KM; we reported the curve at g = 1 as a limiting case of the actual situation. At very high Ks the expected Monod behavior is achieved, where for each ~ only one (g,)7) pair satisfies the material balance equations. On the other hand, at lower Ks a range of ~ arises where multiple solutions occur, namely a low and a high conversion state, which are each stable, and the intermediate unstable one. To better explain this matter, we consider a further representation of the solution, with )7 chosen as an independent variable and the conversion ~ as a dependent one. ~ is defined by:
E = (~o - g)lgo.
(5)
At a given ~ , the solutions for (E,)7") can be obtained also by solving the system: qx.p ----?x.n = E)7/(1 + E q- g : / R ) - / ~ ) 7
(6)
q,.,
(7)
=
[(R + 1))7 - R)Tr~- )7o1/~.
t6
12
g
~
30
40
~----~
1 + E0(l _ E) + E20(1 _ E)2/K + B/g0
2 = (2o + R)TR + Ego)/(R + 1 +/~).
I
3
I 4.
Fig. 2. CSTBR simulation with R--0.25, ~'R = 30t) and ~o = 20. S vs O profiles at different inhibition constants.
(9)
-]
g
+/~] - $(1 + E + E2/R)
D+R
2
(8)
In an ~ vs )7 diagram, equation (8) shows up as a sigmoid, while equation (9) appears as a straight line; both curves depend on ~ (see Fig. 3), but only the second one is a function of the recycle variables. It has to be pointed out that E can be rigorously used for expressing the production rate only when /~=0. The two curves in Fig. 3 intersect each other in three points (A, B, C), which are the solutions of the system of equations (8)--(9).Points A and C represent inherently stable operating situations. From a graphical point of view, itis apparent that any perturbation occurring on )7 inside the reactor will be annulled: in fact, on the right (or left) hand side of points A and C, the removal of biomass is higher (or lower) than its generation rate. Thus, the system is able to come back to its original steady-state. From an algebraic point of view, the conditions for stability can be applied to the system of equations (I)-(2), according to Liapunov's method (Morbidelli et al., 1987), in order to get:
L~
q
i
60
Equation (6) expresses the net production of biomass per unit reactor volume, and equation (7) the net removal of biomass from the reactor, in the same units. At steady-state, generation and withdrawal of biomass must equal each other, so that equations (6) and (7) are satisfied simultaneously. In terms of the selected variables, by recalling equations (2a) and (5), one can rearrange equations (6) and (7) to obtain:
33. (E2/R-I))7,>o
4
0
J
50
Fig. 3. Biomass production rate and biomass removal rate at different R and Ji'R= 300, O -- 1.05, ~o -- 20 and R = 3.
1 [I+R 8
10
2+R
(10)
E -¢ 1 + E + E:IR E a / R - 1))7
+(1 +E+E:/R) 2 /~>o.
(11)
172
ALBERTO BERTUCCO et al.
Condition (11) is satisfied for all three points while condition (10) is not for point B, which is then an unstable steady-state. The (E, ,~') plot is also helpful to inspect how the reactor conversion is affected by a change in the space time, due for example to a step change in the inlet flowrate, Q0. Starting from an initial steady-state [point A~ of Fig. 4(a)], an increase of Q0 makes the reactor move through states At, down to A2 where a limiting space time g * is reached. If ~ is further decreased, the solution jumps down to A~, the stable solution at low conversion, with a discontinuity for E. However, when operating at A3, an increase in does not shift the stable point back to high conversions until point A4 (at ~**) is attained. If ~ is further increased, the solution jumps to point A5 for the reasons previously discussed. This is the so-called hysteresis effect which is associated with substrate inhibition kinetics. For assigned R and avR, the values of ~*, g** can be calculated by equating equations (8) and (9) with E = Et and E = E2, where Et, E2are two solutions of the fourth-order equation obtained by imposing a tangency condition. The same phenomenon can be described by means of a ~ vs O plot, as is shown by the path evidenced in Fig. 4(b): here the shaded area represents the hysteresis loop, and the dotted part of the curve defines the range of instability, as it can be checked by applying equations (! 0)-(1 ! ) as well. Experimental evidence of hysteresis behavior has been provided by Pawiowsky et aL (1973) for a CSTBR without recycle. The effect of a step change in the recycle flowrate can be illustrated either by the ~ vs g plot of Fig. 5 or by the E vs ~' diagram of Fig. 3. It can be seen, for example, that starting from a high conversion steady-state situation (point A) at ~ = 1.05, R = 0.1, a decrease of R to a value of 0.05 at constant XR makes the system jump down to a low conversion steady-state (point D), which represents a totally
:oo,
_:_ _ x . . .
0.6 OA
.....
o.~ • / L /
2
0
3
4
5
6
7
Fig. 5. Effect of recycle ratio R in an ~ vs ~ plot. ~R = 200. S0=20and g = 3 . unacceptable performance. Very similar figures and conclusions could be derived if XR was varied and R kept constant. In Fig. 5 the curve at R = 0 is also reported; even though this case could be described by simplified relationships, it is easily simulated also as a limiting case of the general model developed above. It is remarkable to note that, without recycle and with -~'o= 0, washout conditions can be readily derived by differentiating [for equation (13)] or by setting S = So [for equation 04)], in the following: 1/O =
1 I/S + 1 + S / K
fl
which relates 0 and S at R = 0 and Xo = O. Washout occurs whenever 0 is less than its minimum O~n, given by the lower value between OM and OpR: 1 l/OM
=2/~-K_
i/erR =
1 /~ I
l/So + 1 + So/K
(13) -- #.
(b)
A ~
O.e
- .
1.14
A5
/~j' 0.6
0.4
0.4
0.2 ~ ' 1 /
~/
0.2 °A 3
O
20
40I
60I
(14)
When OpR > OM, OpR is an "activation" value of space time in the hysteresis loop: if OpR < O M no
leA2
0.6
(12)
0
0.5
1.0
1.5
Fig. 4. Hysteresis loop f o r a C S T B R according to the J/R-based model. ~' R = 200, R = 0.1, ~'0 = 20 and • = 3. (a) In an ~ vs ~' plot. (b) In an ~ vs O plot.
Stability of bioreactors hysteresis effect is possible because the inhibition is low enough to give results closer to Monod kinetics. It is also apparent from Fig. 5 that working with a recycle will lead to lower values of ~M, that is to less stringent operability limits for the reactor, and that complete washout is always avoided when ~R is kept constant.
173
Therefore, at assigned O and Q0, the sludge age depends upon Qw only, if R is kept constant by a recycle pump; in this way, Oc can be varied by simply opening or closing the biomass purging device. If equation (15), expressed in dimensionless form, is combined with equation (2), we also get: O c = ~x - fl;f' - £ 0
(18)
MODEL BASED ON SLUDGE AGE
Similar results for a CSTBR with substrate inhibition kinetics can be obtained on the basis of the variable called sludge age (Oc), defined as the ratio of biomass in the reactor to the net rate of biomass generation (Bertucco et al., 1988): VX VX OC= QoXo + rx.n = QwXR + Q~X .
(15)
Although Oc is not exactly the mean cell residence time, it is more commonly used because of the difficulty of determining the death constants of wastewater microbial populations; however, O c is usually an acceptable approximation of the biomass residence time. As it is known, Oc is related to the settleability of the sludge. In the above section we have seen that operating at constant recycle biomass concentration will never drive the system into total washout condition, so that a conversion of the substrate, even tiny, will always take place because the biomass concentration in the reactor inlet stream is never zero. Indeed, from a practical point of view, keeping XR constant means having a settling device capable of ensuring the desired recycle concentration at the bottom, no matter what is going on inside the reactor. This is a kind of attractive situation, but rather unrealistic when using a gravity settler, as usual in activated sludge plants. Therefore, it seems more appropriate in this case to adopt Oc as an independent variable. Moreover, Oc is related to parameters such as the flowrates, which can be predetermined and controlled easily by means of pumps and valves. To show the controllability of Oc, consider a (desirable) situation where no cells are wasted in the overflow stream Q,. Then, equation (I5) reduces to: Oc = ~
VX
On the other hand, by rearranging equation (15) and taking into account the material balance of biomass around the settler, one can get: O
R-4- I - R X R/X
(16)
that relates the concentration of biomass in the reactor to the recycle variables. A combination between equations (15a) and (16) finally provides: o~ = ~
1 llOc= ll~ + l +,~/R
ft.
(19)
When ~c is assigned, two solutions for ~ can be found, which are independent of ~; curves at constant ~ c are then given by straight lines on a ~ vs O plot, like the ones reported in Fig. 5. Equation (19) exhibits a minimum value OC.M and an "activation" value OC.PR: 1/~C,~a = -
-
1
2/x/~-
1
fl
(20)
1 I/~c'PR = 1/,~0 + 1 + ~'0/~ - fl"
(21)
Equations (19), (20) and (21) are the same as equations (12), (13) and (14) for R = 0, when ~ is replaced by ~c. Similar to previously considered values of ~, we now have to consider a washout value of ~ c given by ~c.~.., which is the minimum between ~C.M and ~C.PR. Due to the hysteresis effect, there are two washout conditions of ~ c to be determined: one is the ~c.mi. and the other an "activation" value, ~C.PR, if ~gC.PR > ~c.M.
The removal of biomass [equation (9)] is now expressed in terms of sludge age: = ~0~ + g0 .__~c
1 +fl~c
~
(9a)
whilst the biomass production is still given by equation (8). By looking at equation (9a) and recalling that for a CSTBR without recycle it holds:
(15a)
QwXR
Oc =
and, upon introducing the Haldane kinetics into equation (18), we have (for ~'0 = 0):
+ 1 .
(17)
1 + flO
(22)
lira Oc = ~
(23)
one can see that:
as would be expected from a correct definition of sludge age. As a consequence of the Oc-based model, we point out that the washout condition is always one of the stable solutions of the system given by equations (8) and (9a), as is evidenced in Fig. 6, in analogy with Fig. 3. Again, we have a sigmoid and a straight line, both depending on ~ (the straight line also depends
174
ALBERTOBERTUCCOet
on Oc); their intersections provide the possible steady-states A, B, C, but now the one marked by C is always at zero conversion. Point B represents an unstable steady-state. It can be seen that, if O is changed while Oc is kept constant, the system shifts to new possible operating points; increasing O at constant Oc reduces the cell concentration needed within the reactor, even though the substrate conversion remains unchanged. It is remarkable to note that complete washout occurs whenever Oc is lower than Oc.~,, since here a cell concentration in the recycle is not guaranteed, such as when operating at ,~'R= constant. Starting from Oc.min and increasing Oc with O, R is kept constant, a hysteresis behavior can be found, in analogy with the case at assigned ,I'R. Figure 6 shows details: the sigmoid not depending upon ~c, it is possible to follow the hysteresis loop by simply rotating the straight line. It is also apparent that above a value of dlc -OC.m~ (OC.~,~ = OC.PR if diC.PR> OC,M) only one solution is found, the high conversion one. It can be shown that for any value of O, the tangent point corresponding to OC,M provides a substrate conversion given by: Ev = I - x , ~ / ~ ' o
(24)
which is independent of O, being determined by the kinetic constants and the initial substrate concentration only. By imposing the two tangency conditions, equations (20) and (21) are obtained again, thus confirming that the Oc limiting values are not a function of O. The dependence of .~'t on Oc can be obtained analytically by combining equations (2a) and (16)-in dimensionless form--and (19). The solution is illustrated by means of Fig. 7: plots of Oc vs ,:I'R (bold lines for one CSTBR) exhibit minima and multiple solution ranges, which depend on the inhibition constant J~. It is remarkable to see that operating on substrate inhibited wastes requires values of ,f'R much higher than those used with Monod-like wastes: they can be achieved only by considerably raising the sludge age. All three curves are relative to cases with no cell death term in the kinetics; the introduction of a typical value for the factor ]~ results in a dramatic
o.
"21 0
20
40
60
80
I00
'120
'140
Fig. 6. Hysteresis loop for a CSTBR according to the Oc-based model. R = 0.25, ~ = 1.05. ~'0 -- 20 and /L'-- 3.
al.
2c ~
R, I
//
,
A.3 \ \\ '\ \\
~z
0
NON01~ 100
I 200
)¢Q-l_csrsR /fi~\scsrBR
I 300
t 40O
I
.
XR
Fig. 7. ~¢ vs "~'Rprofiles: for a CSTBR at different inhibition
constants and for different numbers of CSTBRs at /~ = 3 and ~"=0.1, R =0.25, (~= 1.26 and g0 = 20. raise of the sludge age required to ensure a desired .I'R or substrate conversion, as shown by the dashed curve for one CSTBR in Fig. 7. EFFECT OF MIXING
All considerations and remarks presented about the behavior of a CSTBR with substrate inhibited kinetics can be extended to cases when a different reactor configuration or a different hydraulic mixing model within the reactor are adopted for simulation. The definition of sludge age is the same, but the resulting expression in terms of recycle variables is slightly more complicated, since we now have: 0c =
;f'A~0 (R + l),t' - R~'R
(16a)
with
lff 3~ d V'.
~Av = ~
If a configuration with N CSTBR in series is considered, the material balances for ~, ~t' at the ith tank are now provided by:
o
S~_, - •, + ~s,, R + ! = 0
(25)
O .~_ ~ - ~ + r-~.,., R + I = 0
(26)
which have to be solved together with equation (16a). It is not possible to express these quantities by simple relationships as for a CSTBR and numerical solution is needed. The resulting Oc vs "]'R plot is somehow different, as indicated by the dotted lines in Fig. 7, which are calculated with the same/~ and operating conditions as for one CSTBR. Taking the number of CSTBR as a parameter (at constant total 0 ) and starting from a one tank situation, which is actually the lower limit, higher Ns lead to bigger values of Oc.mi, which are obtained at consistently higher ,I'a s. The upper limit is represented by the plugflow reactor (N --, oo). We note also the disappearance of intervals of multiple solutions, as N increases.
Stability of bioreactors The influence of the extent of mixing within a given reactor can be described by solving the pertinent material balance equations for S, .~ at different values of the Peclet number and constant space time and inlet concentrations ~0, -~'0:
175
O.8
0.6
1 d:~
dS
Pe dS:
d5
1 d2X
d.Y
Ped-'-
d5
,E -
-
R + 1
-
-
R + 1
~s = 0
(27)
e=OO
0.4
~x., = o
(28) 0.2 /
with the boundary conditions: d~ -~=Pe($'-.~0;
(a.~=0:
d..~
(~ 5=1:
dg
-
d J7 d.f
d.~ -~-=Pe(Y-J?l)
=0.
The system given by equations (27) and (28) can be solved by using an orthogonal collocation method, as outlined in a previous paper (Bertucco et al., 1988). An ~ vs z / L plot shows up, for example, as in Fig. 8, at constant XR = 300, and as in Fig. 9, at constant Oc = 9.1. We note a dramatic change in the profile when moving from a CSTBR situation (Pe = 0) to a plugflow one (Pe ~ oo). This also affects the outlet substrate concentration, especially at constant X R. An optimal mixing value can be found that maximizes the substrate conversion. In many cases, the pollutant removal gets worse as a plugflow situation is achieved, so that a certain extent of mixing is often ad~,isable when handling substrate inhibited wastes. It can be concluded that the degree of mixing could be used as an extra variable which affects the stability of the reactor. The influence of the degree of inhibition is finally illustrated by Fig. l0 at constant axial mixing (Pe = 3). Again, a critical range o f / ~ values can be identified, around which a sharp drop in the substrate conversion profiles takes place.
o. i •
0
/ /
~
I O.2
Pea oo
I 0.4
I 0.6
I 0.8
z/L
Fig. 8. Effect of mixing at -~'R= constant = 300, R = 0.l, (~ = 2.1, -~'o= 20 and g' = 3.
O
I
t
I
I
0.2
0.4
0.6
O.8
z/L Fig. 9. Effect of mixing at ~c = constant = 9.1, R = 0.1, = 2.1, ,go = 20 and E = 3.
0.8
0.6
Q4
0.2
I
I
I
I
0.2
0.4
0.6
0.8
z/L
Fig. 10, Effect of the degree of substrate inhibition in a reactor with constant axial dispersion: Pe = 3, R = 0.25, -~R = 200, ~ = 1.26, So = 20 and g = 3. CONCLUSIONS
The biological treatment of toxic wastewaters is of great interest for the future but also of great concern due to the problems arising from substrate inhibition effects. This is true either in designing and in operating activated sludge reactors dedicated to this scope. In the present paper the behavior of a bioreactor with a variable degree of internal mixing has been investigated taking into account a settler and a recycle stream in terms of dimensionless groups. Two different approaches have been considered, depending on which independent variable for the recycle is selected, namely the cell concentration ~R or the sludge age ~ c . From the reported results, four main conclusions can be derived. The first one is that special care should be taken about the possibility of kinetic effects due to substrate inhibition and to the death term, since the calculated steady-states of the system may deviate considerably from the common Monod-like ones. This is clearly shown, for example, by Figs 2 and 7, where the sensitivity of the system to relatively small changes in ~ or ~ c is large, if it is operated around critical conditions.
176
ALBERTOBERTUCCOet al.
As a second point, the occurrence of washout has to be considered as an actual danger. At A'R = constant, complete washout is never reached, but in the more common situation with ~ c assigned, any change in the space time or in the recycle ratio or in the sludge withdrawal flowrate will vary ~ c , thus requiring a careful check with respect to the desired performance of the plant. Also the degree of mixing can affect the critical point of operation. A third conclusion is related to the hysteresis effects: the dramatic changes of the overall substrate conversion occurring upon small perturbations of the system variables around their steady-state values are not reversible, as is represented by Figs 4 and 6. The "activation" value of ~ or ~ c , that is the threshold to overcome in startup or transient plant operation, is likely to be much higher than their minimum values at steady-state. Finally, we point out the remarkable effect of mixing, as is represented in Figs 8, 9 and 10: depending on it is the fact that the situation in any point within the reactor can be changed from a low to a high conversion one. All of these considerations are more relevant when the plant works at low values of recycle ratio. Therefore, both the design and the operation of activated sludge reactors for toxic wastes could take advantage of the capability of simulating their performance as a function of the inlet substrate and biomass concentration and the process variables, that is: space time, recycle ratio, biomass recycle concentration (or, preferably in our opinion, sludge age), kind of reactor configuration and degree of internal mixing. It is well known that for a given plant the flowrate and concentration of the feed tend to rise with time, so that the system could be led to operate in a "dangerous" region, and possibly fall into failure. On the other hand, even though the target values are not exceeded, the occurrence of random step changes in one of the above mentioned variables could be dangerous for the same reasons. All of these anomalous and unforeseen situations should be evaluated and their effects simulated, in order to design an effective control system and policy. We recognize that a lot of work has to be done by microbiologists so that proper microbial populations can be characterized and selected for processing toxic wastes; however, from the reaction engineering point of view, it also seems valuable to provide general equations and diagrams which allow one to predict and check the performance of a system where a substrate inhibited kinetic is dominant. Acknowledgements--The authors gratefully acknowledge Professor David DiBiasio from Worcester Polytechnic Institute, Worcester, Mass., for reviewing the manuscript and the Department of Higher Education (State of Connecticut) for financial assistance under High Technology Grant in developing computer models.
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