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The use of dimensionless groups in the design of activated sludge reactors
Water Research
Pergamon Press 1973. Vol, 7, pp, 1905-1913. Printed in Great Britain
THE USE OF DIMENSIONLESS GROUPS IN THE DESIGN OF ACTIVATED SLUDGE REACTORS DONALD W . SUNDSTROM, HERBERT E. K_LEI a n d ALLEN E. MOLVAR Department of Chemical Engineering, The University of Connecticut, Storrs, Connecticut 06268, U.S.A. (Received
21
November
1972)
Abstract--Dimensional analysis techniques were applied to the differential equations for a well mixed activated sludge reactor to obtain generalized information on the effect of process variables on conversion of substrate. The Monod model was used to represent the reaction kinetics since it also contains first and second order kinetics as special cases. The performance of a steady state reactor could be described in terms of five dimensionless groups involving the important combinations of process variables. A simple criterion was developed to predict washout conditions for reactors without recycle of biological solids. Under certain operating conditions, activated sludge reactors were shown to possess an inherent degree of self control to compensate for changes in feed concentration.
E k kd K Q S t U V X :f
Y fl 7
7, T 0
NOMENCLATURE fractional conversion of substrate maximum specific growth rate, time- t endogenous rate constant, time- ~ Michaelis--Menten constant, nag 1- t flow rate of fresh influent, vol time- t substrate concentration, mg 1- t = S / K = dimensionless substrate concentration ----time = recycle ratio, dimensionless = volume of reactor = biological solids concentration, nag 1- t = X / ( Y K ) ~ dimensionless solids concentration = cell yield constant, mg cells rag-t substrate = k d k = dimensionless endogenous rate = kO ----dimensionless growth rate = (1 + 1/.re)I(1 -- fill + 1/~¢o]) = critical value of ~, for washout without recycle = t/O = dimensionless time = V / Q = residence time based on fresh feed. = = = = = =
TIlE a c t i v a t e d sludge process is a n e c o n o m i c a l a n d widely a c c e p t e d m e t h o d f o r r e m o v ing o r g a n i c c o m p o n e n t s f r o m wastewater. T h e r e a c t i o n s are a u t o c a t a l y t i e in n a t u r e since t h e sludge f o r m e d b y t h e r e a c t i o n can react with the o r g a n i c substrate. Thus, t h e efficiency o f t h e process can be m o d i f i e d b y recycle o f m i c r o o r g a n i s m s . T h e p e r f o r m a n c e o f a c t i v a t e d sludge reactors is affected b y m a n y process a n d kinetic variables. RAMANATHAN a n d GAUDY (1971) h a v e illustrated t h e effect o f m a n y o f these i n d i v i d u a l variables on r e a c t o r b e h a v i o r . Because o f the n u m e r o u s variables involved, several process p a r a m e t e r s were h e l d c o n s t a n t o n each figure. A m o r e concise m e t h o d o f presenting this i n f o r m a t i o n is t h r o u g h the use o f dimensionless groups. By d i m e n s i o n a l analysis techniques, t h e process variables are collected into a smaller n u m b e r o f dimensionless groups. I n this p a p e r , d i m e n s i o n a l analysis is a p p l i e d t o t h e e q u a t i o n s describing a well m i x e d activated sludge reactor. A well m i x e d r e a c t o r is c o n s i d e r e d because o f its W.R. 7112--L
1905
1906
DONALD W. SUNDSTROM, HERBERT E. KLE! and ALLEN E. MOLVAR
ability to dampen shock loads and to equalize oxygen consumption rates. The biological reactions are described by the kinetic model of MONOD (1950), which implies that microbial growth is limited by the availability of one substance and that all other growth requirements are present in excess. In this case, the organic waste constituents are considered as the growth limiting substance for the heterotrophic organisms. GAUDY (1967) and RAMANATHAN(1969) have substantiated the use of the Monod model to describe the kinetic behavior of heterogeneous microbial populations in completely mixed reactors. D E V E L O P M E N T OF D I M E N S I O N L E S S G R O U P S A schematic diagram of an activated sludge process with recycle of biological solids is shown in FIG. 1. The waste stream arriving at the process has a flow rate Q
Q,So
-]
Reactor
X l ' SI
r...- V, X I S t
UQ,
Xn,, S E
FIG. l. and a substrate concentration So. The effluent from the well mixed reactor is settled and a portion of the concentrated sludge is returned to the reactor with recycle ratio U and concentration X,. The recycle stream is assumed to contain the same substrate concentration as the effluent from the reactor. Material balance equations for the organic substrate and microorganisms are as follows:
v d S t = aSo + OUS~ - Q(1 + U) st dt
kS1 X1 V Y(K + $1)
V dX, : Q u x , - Q(1 + u ) x , + kS, Xt V dt K + $1
ka X, V.
(1) (2)
The residence time based on the flow of fresh feed is defined as 0 -- V/Q.
dS, 1 dt = -0 (So -- St)
kSt 7(1 Y(K + S~)
dX1 1 kS1 X, dt = -0 [UX, -- (1 + U) Xt] + K + S-------~ ka X~.
(3)
(4)
The method described by HELLUMS and CHURCHILL(1964) is used to convert the equations to dimensionless form. dS1 = So -- St dr
dr
9,St X1 1 + St
(5)
1 + ~-------] ~ Y l
(6)
The Use of DimensionlessGroups in the Design of Activated SludgeReactors
1907
where .: = t/O S = S]K = X] Y K y = kO fl = ka/k
= = = = =
dimensionless time dimensionless substrate concentration dimensionless sludge concentration dimensionless growth rate group dimensionless endogenous rate group.
To illustrate the use of these equations, they will be applied to characterize the performance of steady state activated sludge reactors. For this particular case, the equations become:
~o -- Sl
ySI X1 1 + ~1 ~ 0
U~r -- (1 + U + fly) Xx + 1 + g------~~ 0.
(7) (8)
The solution of these equations for the effluent concentration groups Sx and X~ requires specification of five dimensionless groups (So, U, ?Tr,/3 and y). To solve the same equations in dimensional form, it would be necessary to specify nine parameters (V, Q, U, So, X,, k, ka, K and Y). Therefore, the use of dimensionless equations is more general and efficient since the important combinations of the variables are specified rather than the individual parameters. The effluent substrate concentration can be replaced in the equations by fractional conversion of input substrate: -
E =
$,
(9)
S--~
The fractional conversion or efficiency provides somewhat better insight into the performance of the reactor and will be plotted in the figures instead of S,. REACTOR WITHOUT RECYCLE Washout of the reactor may occur if no microorganisms enter the reactor with the fresh feed or recycle stream. Under washout conditions, the concentration of biological solids in the reactor drops to zero and there is no conversion of the substrate. By applying a stability analysis to the material balances, the necessary and sufficient conditions for washout can he developed. For a biological reactor following Monod kinetics, washout will take place when
1 Y--~ 1 - - f l ( 1 + ~ o )
~yw"
(10)
The washout criterion is illustrated by FIG. 2, which shows that calculated substrate conversion becomes zero at
Y----Yw----
1 + 1/So 1 --/3(1 + 1/~o)"
1908
DONALD W. SUNDSTROM,HERBERT E. KLEI and ALLEN E. MOLVAR
In this and all subsequent figures, the value of the dimensionless endogenous group (/3) was taken as 0.01. GAUDY (1967) studied the kinetic behavior of a well mixed biological reactor without recycle in the region of washout. Although complete washout was not observed, the measured concentration of biological solids decreased to a very low level for 7 < :¢w. FAN (1970) developed a similar washout criterion for well mixed biological reactors without recycle. When U = 0 and y > yw, the material balances can be solved to give S~ --
1
~-
+3r 1 -/3/
(11)
For stable reactor operation without recycle, then, the effiuent substrate concentration does not depend upon the magnitude of the entering substrate concentration. With a given value of residence time (constant 7 with no recycle), FIG. 2 shows that conU=O 1.0
ILl 0"8
E 0"6
o= "~ o
0.4
O 0-2
Dimensionless growi'h ra-i-e, 7 FIG. 2.
version of substrate increases with increasing influent concentration to maintain a constant output substrate concentration. This response represents an inherent "self control" by the reactor since changes in feed concentration do not affect the output substrate concentration. The mechanism for this seLfcontrol action is an increase in biological solids concentration that is sufficient to handle the higher loading of substrate. STORERand GAtrDV (1969) studied the transient response of a well mixed biological reactor without recycle to step changes in feed concentration. Glucose was the growth limiting nutrient and the microorganisms were of sewage origin. FIGURE3 shows the observed response of the effluent COD to a three fold step change in feed concentration. As predicted by theory, the effluent COD returns to about the same level that existed before the step change in feed concentration. REACTOR
WITH
RECYCLE
The return of biological solids from the elarifier to the reactor generaUy increases
The Use of Dimensionless Groups in the Design of Activated Sludge Reactors
1909
500
E
E
400
/ \
3oo
n
200
I O0
o
.J ;
",<._ I
2
I
4
I
6
~
!
,o
"l'Ime a f a r sl-ep Increase, hr F[o. 3.
the cell concentration, but also dilutes the substrate concentration of the feed and lowers the residence time. Thus, recycle of microorganisms may not always benefit the performance of the reactor. As long as some biological solids arc present in the recycle stream, however, complete washout of the reactor is not possible. The presence of cells in the entering stream ensures that cells will also exist in the reactor and that some finite conversion of substrate will occur. The model developed in this paper assumes that the thickener provides a recycle stream with relatively constant biological solids concentration. In some analyses of activated sludge reactors, the recycle solids are specified by assuming that the ratio of the output to input solids concentration for the thickener is a constant. If a constant ratio is used in the process model, washout is possible since the recycle solids concentration approaches zero as the reactor solids concentration approaches zero. The concentration of solids leaving a thickener changes with the amount of solids entering in the feed. If the thickener is designed to handle the maximum solids loading, the output solids concentration varies by a smaller percentage than the ratio of output to input solids concentration. RAMANATHANand GAUDY (1971) discuss these two methods for spec/fying recycle solids and conclude that use of a constant biological solids concentration in the recycle is more realistic. The effect of the growth rate group (7) and recycle ratio (U) on conversion of substrate is shown in FIG. 4. As discussed previously, there is washout with no recycle for all ~, ~ 7w. If biological solids are fed to the reactor through a recycle stream, the conversion increases monotonically with both increasing ), and recycle ratio. All curves with recycle terminate at the origin since washout is not possible for these cases.
The effect of recycle solids concentration on reactor performance differs for values of 7 above and below Yw. When 7 ~ 7w (FIG. 5), the reactor will washout when recycle ratio is zero. Conversion of substrate then increases with both increasing recycle ratio and recycle solids concentration. For each value of A~,,the conversion approaches an asymptote at high values of U where the biological solids concentration in the reactor becomes nearly equal to ~,. With a given recycle solids concentration, then,
1910
DONALD W. SUNDSTROM, HERBERT E. KLEI and ALLEN E. MOLVAR
~o = 3
~R = I0
ilO
0.6
~ 0.4.
U 0.2
Dimensionlessgrowthrate, 7" FIG. 4.
So=
3
Y= 0.75
"
~
0.8
Io
0.4
(") 0 0
2 0'5
~ 1"0
1'5
.2"0
2'5
Recycleratio, U FIG. 5.
increasing the recycle ratio beyond a certain point will give only a marginal improvement in conversion. When 7 > 7w, washout cannot occur and there is a finite conversion at U = 0. Since recycle dilutes the feed substrate concentration and reduces the residence time, conversion may decrease with increasing recycle ratio at low recycle solids concentration. By differentiating the material balance equations for 7 > 7w, the following characteristics result:
The Use of Dimensionless Groups in the design of Activated Sludge Reactors
1911
dE -- < 0 dU
for
,~,<
,-¢o 1 +37
1 7-- 1 --fly
(12a)
dE -- =0 dU
for
)7,=
go 1 --1 +fit y-- 1 --fit
(12b)
for
~o "Y' > 1 + 3 7
dE d--U > 0
1 7--1--t37"
(12c)
FIGURE 6 illustrates this behavior for fixed values of 7 = 2.0 and go = 3.0. In this case, conversion of substrate is independent of recycle ratio for ,Y, = 1.92, increases for .~', > 1.92 and decreases for .Y, < 1.92. Thus, when the recycle solids concentration is below a certain critical value, the performance of the reactor deteriorates with greater amounts of recycle.
7" = 2
~o'3
~# I00
I'0
t~ 0'8 h
o ,e., .~
~R- 2 0'6
(~ 0.4
"E %
r,.)
0.2
0
I
I
!
!
I
0"5
1.0
t 5
2'0
2"5
Recycle
mi-io,
U
FIG. 6.
Most activated sludge reactors operate with 1 < y < 3, 25 < ,Y, < 200, and 1 < ~o < 10 so that their response lies either on FIe. 5 or on the upper half of FIG. 6. Since their efficiency increases as the recycle ratio increases, the performance of the reactor can be controlled by adjusting the amount of recycled solids. The greatest benefits from sludge recycle are achieved at recycle ratios below 0.5; increases in recycle ratio above 1.0 generally give only a small improvement in performance. The conversion of substrate is also enhanced by a higher solids concentration in the recycle stream but the increase in conversion is usually marginal for ,~, values above 100. There is also a critical dimensionless parameter for the effect of input substrate
1912
DONALD W. SUNI~TROM, HI~.BERT E. KLEI and ALLEN E. MOLVAR
concentration on conversion. In the usual case where fl < 1, the material balances can be differentiated to show that dE -- < 0 dgo
for
I+U y < ~ 1 -- fl
(13a)
dE -- =0 d~o
for
I+U ~ , = ~ 1 -/3
(13b)
dE dg----~ > 0
for
l+ U ~' > 1---~--B"
(13c)
FIGUI~ 7 illustrates the effect of S0 on E for fixed values of U = 0.25 and ,~', = 10. This value of .~', is lower than normally used in activated sludge processes but was selected to permit a clearer presentation of the variables. u=o.25
I'0
X¢lo
w . 0.8
~'=5o
Y = 1'25,
2
0.6 "S = o
',~ 0'4 0.2 ~
'
~ t IO
210
3~)
Y,. 0.5 1 i 40 50
Dimen$10nless subsfmfe, So FIG. 7.
The behavior of the reactor for y > (1 + U)/(1 -- t9) represents another form of "self control" in that conversion increases with increasing input substrate concentration without any external control action. This is a desirable design condition for activated sludge reactors with recycle since the reaction generates a higher conversion when increases occur in the feed concentration. In summary, the application of dimensional analysis to biological reactors reduces the number of state variables needed to specify th e system. Washout of the reactor is shown to occur only if no biological solids enter the reactor and if y ~ ~w. For most activated sludge reactors in practice, the efficiency is improved by increasing the recycle ratio. When the recycle solids concentration is low, however, increasing the recycle ratio can actually lower the reactor efficiency. The conditions for "self control" of biological reactors in response to changes in feed concentration are readily expressed
The Use of Dimensionless Groups in the Design of Activated Sludge Reactors
1913
in terms o f dimensionless groups. A l t h o u g h m o s t o f these effects have been mentioned previously in the literature, dimensional analysis has allowed t h e m to be described m o r e concisely for use in design o f activated sludge reactors. Acknowledgeawnt~-The author wishes to acknowledge partial support of this work by the Water
Quality Office of the Environmental Protection Agency under Grant Number IT2-WP-244-OIAI, and the assistance of Mr. ROBBRTSCHIRRIPAin performing the computer calculations. REFERENCES FAN L. T. et aL (1970) Effect of mixing on the washout and steady state performance of continuous cultures. Biotech. Bioengng 12, 1019--1068.
GAUDYA. F., RAMANATHANM. and RAo B. S. (1967) Kinetic behavior of heterogeneous populations in completely mixed reactors. Biotech. Bioengng 9, 387--411. HELLUMSJ. D. and C-'m~CmLLS. W. (1964) Simplification of the mathematical description of boundary and initial value problems. AIChEJ. 10, 110-114. MOLVARA. E. (1971) Optimal control of biological reactors. Ph.D. thesis. Univ. of Conn. MONOD J. (1950) La technique de culture continue; Theorie et applications. Ann. I~t. Pasteur 79, 390-410. RAMANATmmM. and GAUDYA. F. (1969) Effect of high substrate concentration and cell feedback on kinetic behavior of heterogeneous populations in completely mixed systems. Biotech. Bloe~ng 11,207-237. RAMANATHANM. and GAUDYA. F. (1971) Steady state model for activated sludge with constant recycle sludge concentration. Blotech. Bioengng 13, 124-145. STOmSRF. F. and GAUDYA. F. (1969) Computational analysis of transient response to quantitative shock loadings of heterogeneous populations in continuous culture. Environ. Sci. Tectmol. 3, 143-149.
Pergamon Press 1973. Vol, 7, pp, 1905-1913. Printed in Great Britain
THE USE OF DIMENSIONLESS GROUPS IN THE DESIGN OF ACTIVATED SLUDGE REACTORS DONALD W . SUNDSTROM, HERBERT E. K_LEI a n d ALLEN E. MOLVAR Department of Chemical Engineering, The University of Connecticut, Storrs, Connecticut 06268, U.S.A. (Received
21
November
1972)
Abstract--Dimensional analysis techniques were applied to the differential equations for a well mixed activated sludge reactor to obtain generalized information on the effect of process variables on conversion of substrate. The Monod model was used to represent the reaction kinetics since it also contains first and second order kinetics as special cases. The performance of a steady state reactor could be described in terms of five dimensionless groups involving the important combinations of process variables. A simple criterion was developed to predict washout conditions for reactors without recycle of biological solids. Under certain operating conditions, activated sludge reactors were shown to possess an inherent degree of self control to compensate for changes in feed concentration.
E k kd K Q S t U V X :f
Y fl 7
7, T 0
NOMENCLATURE fractional conversion of substrate maximum specific growth rate, time- t endogenous rate constant, time- ~ Michaelis--Menten constant, nag 1- t flow rate of fresh influent, vol time- t substrate concentration, mg 1- t = S / K = dimensionless substrate concentration ----time = recycle ratio, dimensionless = volume of reactor = biological solids concentration, nag 1- t = X / ( Y K ) ~ dimensionless solids concentration = cell yield constant, mg cells rag-t substrate = k d k = dimensionless endogenous rate = kO ----dimensionless growth rate = (1 + 1/.re)I(1 -- fill + 1/~¢o]) = critical value of ~, for washout without recycle = t/O = dimensionless time = V / Q = residence time based on fresh feed. = = = = = =
TIlE a c t i v a t e d sludge process is a n e c o n o m i c a l a n d widely a c c e p t e d m e t h o d f o r r e m o v ing o r g a n i c c o m p o n e n t s f r o m wastewater. T h e r e a c t i o n s are a u t o c a t a l y t i e in n a t u r e since t h e sludge f o r m e d b y t h e r e a c t i o n can react with the o r g a n i c substrate. Thus, t h e efficiency o f t h e process can be m o d i f i e d b y recycle o f m i c r o o r g a n i s m s . T h e p e r f o r m a n c e o f a c t i v a t e d sludge reactors is affected b y m a n y process a n d kinetic variables. RAMANATHAN a n d GAUDY (1971) h a v e illustrated t h e effect o f m a n y o f these i n d i v i d u a l variables on r e a c t o r b e h a v i o r . Because o f the n u m e r o u s variables involved, several process p a r a m e t e r s were h e l d c o n s t a n t o n each figure. A m o r e concise m e t h o d o f presenting this i n f o r m a t i o n is t h r o u g h the use o f dimensionless groups. By d i m e n s i o n a l analysis techniques, t h e process variables are collected into a smaller n u m b e r o f dimensionless groups. I n this p a p e r , d i m e n s i o n a l analysis is a p p l i e d t o t h e e q u a t i o n s describing a well m i x e d activated sludge reactor. A well m i x e d r e a c t o r is c o n s i d e r e d because o f its W.R. 7112--L
1905
1906
DONALD W. SUNDSTROM, HERBERT E. KLE! and ALLEN E. MOLVAR
ability to dampen shock loads and to equalize oxygen consumption rates. The biological reactions are described by the kinetic model of MONOD (1950), which implies that microbial growth is limited by the availability of one substance and that all other growth requirements are present in excess. In this case, the organic waste constituents are considered as the growth limiting substance for the heterotrophic organisms. GAUDY (1967) and RAMANATHAN(1969) have substantiated the use of the Monod model to describe the kinetic behavior of heterogeneous microbial populations in completely mixed reactors. D E V E L O P M E N T OF D I M E N S I O N L E S S G R O U P S A schematic diagram of an activated sludge process with recycle of biological solids is shown in FIG. 1. The waste stream arriving at the process has a flow rate Q
Q,So
-]
Reactor
X l ' SI
r...- V, X I S t
UQ,
Xn,, S E
FIG. l. and a substrate concentration So. The effluent from the well mixed reactor is settled and a portion of the concentrated sludge is returned to the reactor with recycle ratio U and concentration X,. The recycle stream is assumed to contain the same substrate concentration as the effluent from the reactor. Material balance equations for the organic substrate and microorganisms are as follows:
v d S t = aSo + OUS~ - Q(1 + U) st dt
kS1 X1 V Y(K + $1)
V dX, : Q u x , - Q(1 + u ) x , + kS, Xt V dt K + $1
ka X, V.
(1) (2)
The residence time based on the flow of fresh feed is defined as 0 -- V/Q.
dS, 1 dt = -0 (So -- St)
kSt 7(1 Y(K + S~)
dX1 1 kS1 X, dt = -0 [UX, -- (1 + U) Xt] + K + S-------~ ka X~.
(3)
(4)
The method described by HELLUMS and CHURCHILL(1964) is used to convert the equations to dimensionless form. dS1 = So -- St dr
dr
9,St X1 1 + St
(5)
1 + ~-------] ~ Y l
(6)
The Use of DimensionlessGroups in the Design of Activated SludgeReactors
1907
where .: = t/O S = S]K = X] Y K y = kO fl = ka/k
= = = = =
dimensionless time dimensionless substrate concentration dimensionless sludge concentration dimensionless growth rate group dimensionless endogenous rate group.
To illustrate the use of these equations, they will be applied to characterize the performance of steady state activated sludge reactors. For this particular case, the equations become:
~o -- Sl
ySI X1 1 + ~1 ~ 0
U~r -- (1 + U + fly) Xx + 1 + g------~~ 0.
(7) (8)
The solution of these equations for the effluent concentration groups Sx and X~ requires specification of five dimensionless groups (So, U, ?Tr,/3 and y). To solve the same equations in dimensional form, it would be necessary to specify nine parameters (V, Q, U, So, X,, k, ka, K and Y). Therefore, the use of dimensionless equations is more general and efficient since the important combinations of the variables are specified rather than the individual parameters. The effluent substrate concentration can be replaced in the equations by fractional conversion of input substrate: -
E =
$,
(9)
S--~
The fractional conversion or efficiency provides somewhat better insight into the performance of the reactor and will be plotted in the figures instead of S,. REACTOR WITHOUT RECYCLE Washout of the reactor may occur if no microorganisms enter the reactor with the fresh feed or recycle stream. Under washout conditions, the concentration of biological solids in the reactor drops to zero and there is no conversion of the substrate. By applying a stability analysis to the material balances, the necessary and sufficient conditions for washout can he developed. For a biological reactor following Monod kinetics, washout will take place when
1 Y--~ 1 - - f l ( 1 + ~ o )
~yw"
(10)
The washout criterion is illustrated by FIG. 2, which shows that calculated substrate conversion becomes zero at
Y----Yw----
1 + 1/So 1 --/3(1 + 1/~o)"
1908
DONALD W. SUNDSTROM,HERBERT E. KLEI and ALLEN E. MOLVAR
In this and all subsequent figures, the value of the dimensionless endogenous group (/3) was taken as 0.01. GAUDY (1967) studied the kinetic behavior of a well mixed biological reactor without recycle in the region of washout. Although complete washout was not observed, the measured concentration of biological solids decreased to a very low level for 7 < :¢w. FAN (1970) developed a similar washout criterion for well mixed biological reactors without recycle. When U = 0 and y > yw, the material balances can be solved to give S~ --
1
~-
+3r 1 -/3/
(11)
For stable reactor operation without recycle, then, the effiuent substrate concentration does not depend upon the magnitude of the entering substrate concentration. With a given value of residence time (constant 7 with no recycle), FIG. 2 shows that conU=O 1.0
ILl 0"8
E 0"6
o= "~ o
0.4
O 0-2
Dimensionless growi'h ra-i-e, 7 FIG. 2.
version of substrate increases with increasing influent concentration to maintain a constant output substrate concentration. This response represents an inherent "self control" by the reactor since changes in feed concentration do not affect the output substrate concentration. The mechanism for this seLfcontrol action is an increase in biological solids concentration that is sufficient to handle the higher loading of substrate. STORERand GAtrDV (1969) studied the transient response of a well mixed biological reactor without recycle to step changes in feed concentration. Glucose was the growth limiting nutrient and the microorganisms were of sewage origin. FIGURE3 shows the observed response of the effluent COD to a three fold step change in feed concentration. As predicted by theory, the effluent COD returns to about the same level that existed before the step change in feed concentration. REACTOR
WITH
RECYCLE
The return of biological solids from the elarifier to the reactor generaUy increases
The Use of Dimensionless Groups in the Design of Activated Sludge Reactors
1909
500
E
E
400
/ \
3oo
n
200
I O0
o
.J ;
",<._ I
2
I
4
I
6
~
!
,o
"l'Ime a f a r sl-ep Increase, hr F[o. 3.
the cell concentration, but also dilutes the substrate concentration of the feed and lowers the residence time. Thus, recycle of microorganisms may not always benefit the performance of the reactor. As long as some biological solids arc present in the recycle stream, however, complete washout of the reactor is not possible. The presence of cells in the entering stream ensures that cells will also exist in the reactor and that some finite conversion of substrate will occur. The model developed in this paper assumes that the thickener provides a recycle stream with relatively constant biological solids concentration. In some analyses of activated sludge reactors, the recycle solids are specified by assuming that the ratio of the output to input solids concentration for the thickener is a constant. If a constant ratio is used in the process model, washout is possible since the recycle solids concentration approaches zero as the reactor solids concentration approaches zero. The concentration of solids leaving a thickener changes with the amount of solids entering in the feed. If the thickener is designed to handle the maximum solids loading, the output solids concentration varies by a smaller percentage than the ratio of output to input solids concentration. RAMANATHANand GAUDY (1971) discuss these two methods for spec/fying recycle solids and conclude that use of a constant biological solids concentration in the recycle is more realistic. The effect of the growth rate group (7) and recycle ratio (U) on conversion of substrate is shown in FIG. 4. As discussed previously, there is washout with no recycle for all ~, ~ 7w. If biological solids are fed to the reactor through a recycle stream, the conversion increases monotonically with both increasing ), and recycle ratio. All curves with recycle terminate at the origin since washout is not possible for these cases.
The effect of recycle solids concentration on reactor performance differs for values of 7 above and below Yw. When 7 ~ 7w (FIG. 5), the reactor will washout when recycle ratio is zero. Conversion of substrate then increases with both increasing recycle ratio and recycle solids concentration. For each value of A~,,the conversion approaches an asymptote at high values of U where the biological solids concentration in the reactor becomes nearly equal to ~,. With a given recycle solids concentration, then,
1910
DONALD W. SUNDSTROM, HERBERT E. KLEI and ALLEN E. MOLVAR
~o = 3
~R = I0
ilO
0.6
~ 0.4.
U 0.2
Dimensionlessgrowthrate, 7" FIG. 4.
So=
3
Y= 0.75
"
~
0.8
Io
0.4
(") 0 0
2 0'5
~ 1"0
1'5
.2"0
2'5
Recycleratio, U FIG. 5.
increasing the recycle ratio beyond a certain point will give only a marginal improvement in conversion. When 7 > 7w, washout cannot occur and there is a finite conversion at U = 0. Since recycle dilutes the feed substrate concentration and reduces the residence time, conversion may decrease with increasing recycle ratio at low recycle solids concentration. By differentiating the material balance equations for 7 > 7w, the following characteristics result:
The Use of Dimensionless Groups in the design of Activated Sludge Reactors
1911
dE -- < 0 dU
for
,~,<
,-¢o 1 +37
1 7-- 1 --fly
(12a)
dE -- =0 dU
for
)7,=
go 1 --1 +fit y-- 1 --fit
(12b)
for
~o "Y' > 1 + 3 7
dE d--U > 0
1 7--1--t37"
(12c)
FIGURE 6 illustrates this behavior for fixed values of 7 = 2.0 and go = 3.0. In this case, conversion of substrate is independent of recycle ratio for ,Y, = 1.92, increases for .~', > 1.92 and decreases for .Y, < 1.92. Thus, when the recycle solids concentration is below a certain critical value, the performance of the reactor deteriorates with greater amounts of recycle.
7" = 2
~o'3
~# I00
I'0
t~ 0'8 h
o ,e., .~
~R- 2 0'6
(~ 0.4
"E %
r,.)
0.2
0
I
I
!
!
I
0"5
1.0
t 5
2'0
2"5
Recycle
mi-io,
U
FIG. 6.
Most activated sludge reactors operate with 1 < y < 3, 25 < ,Y, < 200, and 1 < ~o < 10 so that their response lies either on FIe. 5 or on the upper half of FIG. 6. Since their efficiency increases as the recycle ratio increases, the performance of the reactor can be controlled by adjusting the amount of recycled solids. The greatest benefits from sludge recycle are achieved at recycle ratios below 0.5; increases in recycle ratio above 1.0 generally give only a small improvement in performance. The conversion of substrate is also enhanced by a higher solids concentration in the recycle stream but the increase in conversion is usually marginal for ,~, values above 100. There is also a critical dimensionless parameter for the effect of input substrate
1912
DONALD W. SUNI~TROM, HI~.BERT E. KLEI and ALLEN E. MOLVAR
concentration on conversion. In the usual case where fl < 1, the material balances can be differentiated to show that dE -- < 0 dgo
for
I+U y < ~ 1 -- fl
(13a)
dE -- =0 d~o
for
I+U ~ , = ~ 1 -/3
(13b)
dE dg----~ > 0
for
l+ U ~' > 1---~--B"
(13c)
FIGUI~ 7 illustrates the effect of S0 on E for fixed values of U = 0.25 and ,~', = 10. This value of .~', is lower than normally used in activated sludge processes but was selected to permit a clearer presentation of the variables. u=o.25
I'0
X¢lo
w . 0.8
~'=5o
Y = 1'25,
2
0.6 "S = o
',~ 0'4 0.2 ~
'
~ t IO
210
3~)
Y,. 0.5 1 i 40 50
Dimen$10nless subsfmfe, So FIG. 7.
The behavior of the reactor for y > (1 + U)/(1 -- t9) represents another form of "self control" in that conversion increases with increasing input substrate concentration without any external control action. This is a desirable design condition for activated sludge reactors with recycle since the reaction generates a higher conversion when increases occur in the feed concentration. In summary, the application of dimensional analysis to biological reactors reduces the number of state variables needed to specify th e system. Washout of the reactor is shown to occur only if no biological solids enter the reactor and if y ~ ~w. For most activated sludge reactors in practice, the efficiency is improved by increasing the recycle ratio. When the recycle solids concentration is low, however, increasing the recycle ratio can actually lower the reactor efficiency. The conditions for "self control" of biological reactors in response to changes in feed concentration are readily expressed
The Use of Dimensionless Groups in the Design of Activated Sludge Reactors
1913
in terms o f dimensionless groups. A l t h o u g h m o s t o f these effects have been mentioned previously in the literature, dimensional analysis has allowed t h e m to be described m o r e concisely for use in design o f activated sludge reactors. Acknowledgeawnt~-The author wishes to acknowledge partial support of this work by the Water
Quality Office of the Environmental Protection Agency under Grant Number IT2-WP-244-OIAI, and the assistance of Mr. ROBBRTSCHIRRIPAin performing the computer calculations. REFERENCES FAN L. T. et aL (1970) Effect of mixing on the washout and steady state performance of continuous cultures. Biotech. Bioengng 12, 1019--1068.
GAUDYA. F., RAMANATHANM. and RAo B. S. (1967) Kinetic behavior of heterogeneous populations in completely mixed reactors. Biotech. Bioengng 9, 387--411. HELLUMSJ. D. and C-'m~CmLLS. W. (1964) Simplification of the mathematical description of boundary and initial value problems. AIChEJ. 10, 110-114. MOLVARA. E. (1971) Optimal control of biological reactors. Ph.D. thesis. Univ. of Conn. MONOD J. (1950) La technique de culture continue; Theorie et applications. Ann. I~t. Pasteur 79, 390-410. RAMANATmmM. and GAUDYA. F. (1969) Effect of high substrate concentration and cell feedback on kinetic behavior of heterogeneous populations in completely mixed systems. Biotech. Bloe~ng 11,207-237. RAMANATHANM. and GAUDYA. F. (1971) Steady state model for activated sludge with constant recycle sludge concentration. Blotech. Bioengng 13, 124-145. STOmSRF. F. and GAUDYA. F. (1969) Computational analysis of transient response to quantitative shock loadings of heterogeneous populations in continuous culture. Environ. Sci. Tectmol. 3, 143-149.