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The fate of soluble, recalcitrant, and adsorbing compounds in activated sludge plants, II
ECOTOXICOLOGY
AND
ENVIRONMENTAL
SAFETY
10,290-294
( 1985)
The Fate of Soluble, Recalcitrant, and Adsorbing in Activated Sludge Plants, II’
Compounds
P. WIERICH Department
of Ecology,
Henkel
KGaA. Received
4000 Dtisseldorf October
I, Federal
Republic
of Germany
1984
An inhomogeneous first-order differential equation allows description of the time-dependent concentration change of an adsorbing compound in an activated sludge plant. Besides those parameters formerly considered, i.e., adsorption constant and surplus sludge wastage rate, the difference in dry matter content of the wasted sludge and that remaining in the activated sludge stage are now taken into account as well. This facilitates a more exact calculation of the elimination. The essential advantage of the new model lies in its simpler and faster applicability. The equations are steady and can be integrated facilitating, e.g., the calculation of the mean retention time of the adsorbed material in the activated sludge stage. Other mechanisms such as biodegradation, precipitation, flocculation, and stripping are, as in our older model, not taken into account. 0 1985 Academic
F’fes, Inc.
INTRODUCTION
Some time ago a mathematical model was presented which allows calculation of the distribution and elimination of soluble, recalcitrant, and adsorbing compounds in activated sludge plants. The present paper describes a simpler solution of the same problem. Elimination and adsorption in static systems. If a soluble, recalcitrant, but strongly adsorbing compound is added to activated sludge, an equilibrium is established between substance adsorbed onto the sludge and remaining in solution within a certain time. If more of the same substance is added, the absolute amount of adsorbed compound will increase; however, the relative proportion of adsorbed and dissolved material will remain constant until the linear range of the adsorption isotherm is exceeded. Further compound additions result in an increase of the absolute amount of adsorbed substance until the maximum adsorption capacity is reached. In static systems the elimination due to adsorption corresponds strictly to the ratio of adsorbed to the total amount of added substance. Elimination and adsorption in continuous jlow systems. Continuous flow systems such as activated sludge plants tend toward a steady-state between mass input and output. Thus, if a certain concentration of substance is added to the influent, the concentration of this substance will increase in the effluent as well as in the continuously wasted surplus sludge until a mass balance is established. It is obvious that the resulting substance elimination depends, on the one hand, on the distribution of the material between supernatant and activated sludge (i.e., on the ’
Dedicated
to Dr. Konrad Henkel on the occasion of his 70th birthday,
0147-6513185 $3.00 Cqyright @ 1985 by Academic Preq Inc. AU rights of reproduction in any form IcscTyed.
290
FATE OF COMPOUNDS
IN ACTIVATED
SLUDGE
PLANTS,
II
291
adsorption) and, on the other hand, on the fraction of wasted sludge. Thus, it corresponds numerically only in exceptional cases to that determined in static tests. THE
MATHEMATICAL
MODEL
According to the model an activated sludge plant corresponds essentially to a compartment with an influent and two output possibilities, i.e., via effluent and sludge wastage (for symbols see appendix).
The time-dependent concentration change- here with regard to the influent concentration-in the activated sludge plant results from the comparison of the momentary influent and effluent concentrations in analogy to a mass balance.
(1) The rate constants are as follows: k, = rij/V,
(2)
k,,pI$.
(3)
p’ ss
Here the density difference between activated sludge and surplus sludge is taken into account. k2 = kl - k3 = $ .
(4)
P
The effluent concentration
depends on the adsorption constant k’: (5)
The substance concentration ponents:
contained in the wasted surplus sludge has two com.h+(l ds
c, = k’/V,.c, -M
a
-k/VP)-c, (6)
b
a: fraction adsorbed on the sludge; b: fraction dissolved in the supematant. The general solution of Eq. (1) is p)=;+&e”. I
(7)
292
P. WIERICH
with
A =(kl -k,)(l +)+kJl+$($ and leads, on insertion of the starting condition $! (t) = :.(I I
l)]
(8)
cP/ci (t = 0) = 0, to the special solution - eA”).
(9)
Equation (9) allows the calculation of the substance concentration in the activated sludge stage at any moment. Contrary to the former model, this just requires a pocket calculator. dss/ds = 1 yields the same result as the old model. Substance Concentration
in the Dynamic Equilibrium
Of course, in an activated sludge plant conditions prevailing at dynamic equilibrium are of special interest. Equation (9) yields for t - co the total concentration-adsorbed and dissolvedunder steady-state conditions:
k, ?(f +CO)=-. A I Combining
Eqs. (5) and (10) yields the equilibrium -co)=(l
Steady-state Elimination
effluent concentration
-k.)+
Eficiency
The elimination efficiency of an activated sludge stage is correctly described by a mass balance in influent and effluent:
k?i- riz, 11=- rizi This yields with ti = p. c and under consideration
v,=l--.In practice, however, influent and effluent a high surplus sludge not differ significantly
of Eqs. (2) to (4) Eq. (13)
k, - k3 ce (t - al). k, G
the elimination efficiency is more often described by comparing concentrations. It can generally be assumed-except in case of wastage rate-that influent and effluent volume throughputs do (i.e., k3 = 0): 72
G = 1 --(t-+03). G
(14)
FATE OF COMPOUNDS
Mean Retention
IN ACTIVATED
SLUDGE PLANTS,
II
293
Time of the Substance in the Activated Sludge Stage
The determination of the mean retention time of a substance requires [at first] establishment of the function of the concentration change upon a one time addition: -f&k
c+k
dt
2’ e
(15)
c
3’ ‘*
Inserting Eqs. (5) and (6) results in: 2=-{k2(l
+)+k+
+g(2-
Separating the variables and integration
I)]}+
yields for to = 0:
C,(t) = c,(O) - e-A.f. This allows the calculation lication:
(16)
(17)
of the mean retention time in analogy to the former pub02 cp(t) . t - dt s t= Orm cp(t) - dt
J 0
Inserting Eq. ( 17) and subsequent integration
leads finally to: (19)
Since k2 is generally much larger than k,, an increase of the adsorption constant results usually in a prolongation of the mean retention time 7. CONCLUSION Comparison with the Former Method Compared with the old model, the new one offers the following advantages: (1) The calculations are simpler and much faster since the equilibrium concentrations may be calculated directly from the pertinent parameters and not, as in the old model, after thousands of iterative steps by computer. (2) The mean retention time of the substance in an activated sludge stage may be determined as a function of the adsorption constant, the effluent velocity, and the surplus sludge wastage rate. (3) Taking into account the fact that wasted surplus sludge usually has a higher dry matter content than that remaining in the activated sludge stage facilitates a more accurate prediction of the elimination efficiency. The old as well as the new model implies that the adsorption speed suffices for the establishment of a sorptive equilibrium within the given hydraulic mean retention time. This indicates that the predicted elimination efficiency often constitutes a maximum value.
294
P. WIERICH
APPENDIX: G Ci
CP G 4
d ss k, kz k3 k’ me mi
t t ti ti
.I
vss v, 171
v2
mg/liter mg/liter mg/liter mg/liter % org. dry matter % org. dry matter l/hr 1/hr l/hr liters mglhr mg/hr hr hr liters/hr liters/hr liters/hr liters
NOMENCLATURE
adsorbate concentration in the effluent adsorbate concentration in the influent adsorbate concentration in the plant adsorbate concentration in the surplus sludge dry matter content of activated sludge dry matter content of surplus sludge rate constant of the influent rate constant of the effluent rate constant of the surplus sludge wastage proportionality constant of adsorption effluent mass throughput of the adsorbate influent mass throughput of the adsorbate time mean residence time of the adsorbate volume flow rate of the effluent volume flow rate of the influent volume flow rate of the surplus sludge plant volume elimination efficiency/mass balance elimination efficiency/cone balance REFERENCE
P., AND GERM, P. (198 1). The fate of soluble, recalcitrant, and adsorbing compounds in activated sludge plants. Ecotoxicol. Environ. Sa&ty 5, 16 1- 170.
WIERICH,
AND
ENVIRONMENTAL
SAFETY
10,290-294
( 1985)
The Fate of Soluble, Recalcitrant, and Adsorbing in Activated Sludge Plants, II’
Compounds
P. WIERICH Department
of Ecology,
Henkel
KGaA. Received
4000 Dtisseldorf October
I, Federal
Republic
of Germany
1984
An inhomogeneous first-order differential equation allows description of the time-dependent concentration change of an adsorbing compound in an activated sludge plant. Besides those parameters formerly considered, i.e., adsorption constant and surplus sludge wastage rate, the difference in dry matter content of the wasted sludge and that remaining in the activated sludge stage are now taken into account as well. This facilitates a more exact calculation of the elimination. The essential advantage of the new model lies in its simpler and faster applicability. The equations are steady and can be integrated facilitating, e.g., the calculation of the mean retention time of the adsorbed material in the activated sludge stage. Other mechanisms such as biodegradation, precipitation, flocculation, and stripping are, as in our older model, not taken into account. 0 1985 Academic
F’fes, Inc.
INTRODUCTION
Some time ago a mathematical model was presented which allows calculation of the distribution and elimination of soluble, recalcitrant, and adsorbing compounds in activated sludge plants. The present paper describes a simpler solution of the same problem. Elimination and adsorption in static systems. If a soluble, recalcitrant, but strongly adsorbing compound is added to activated sludge, an equilibrium is established between substance adsorbed onto the sludge and remaining in solution within a certain time. If more of the same substance is added, the absolute amount of adsorbed compound will increase; however, the relative proportion of adsorbed and dissolved material will remain constant until the linear range of the adsorption isotherm is exceeded. Further compound additions result in an increase of the absolute amount of adsorbed substance until the maximum adsorption capacity is reached. In static systems the elimination due to adsorption corresponds strictly to the ratio of adsorbed to the total amount of added substance. Elimination and adsorption in continuous jlow systems. Continuous flow systems such as activated sludge plants tend toward a steady-state between mass input and output. Thus, if a certain concentration of substance is added to the influent, the concentration of this substance will increase in the effluent as well as in the continuously wasted surplus sludge until a mass balance is established. It is obvious that the resulting substance elimination depends, on the one hand, on the distribution of the material between supernatant and activated sludge (i.e., on the ’
Dedicated
to Dr. Konrad Henkel on the occasion of his 70th birthday,
0147-6513185 $3.00 Cqyright @ 1985 by Academic Preq Inc. AU rights of reproduction in any form IcscTyed.
290
FATE OF COMPOUNDS
IN ACTIVATED
SLUDGE
PLANTS,
II
291
adsorption) and, on the other hand, on the fraction of wasted sludge. Thus, it corresponds numerically only in exceptional cases to that determined in static tests. THE
MATHEMATICAL
MODEL
According to the model an activated sludge plant corresponds essentially to a compartment with an influent and two output possibilities, i.e., via effluent and sludge wastage (for symbols see appendix).
The time-dependent concentration change- here with regard to the influent concentration-in the activated sludge plant results from the comparison of the momentary influent and effluent concentrations in analogy to a mass balance.
(1) The rate constants are as follows: k, = rij/V,
(2)
k,,pI$.
(3)
p’ ss
Here the density difference between activated sludge and surplus sludge is taken into account. k2 = kl - k3 = $ .
(4)
P
The effluent concentration
depends on the adsorption constant k’: (5)
The substance concentration ponents:
contained in the wasted surplus sludge has two com.h+(l ds
c, = k’/V,.c, -M
a
-k/VP)-c, (6)
b
a: fraction adsorbed on the sludge; b: fraction dissolved in the supematant. The general solution of Eq. (1) is p)=;+&e”. I
(7)
292
P. WIERICH
with
A =(kl -k,)(l +)+kJl+$($ and leads, on insertion of the starting condition $! (t) = :.(I I
l)]
(8)
cP/ci (t = 0) = 0, to the special solution - eA”).
(9)
Equation (9) allows the calculation of the substance concentration in the activated sludge stage at any moment. Contrary to the former model, this just requires a pocket calculator. dss/ds = 1 yields the same result as the old model. Substance Concentration
in the Dynamic Equilibrium
Of course, in an activated sludge plant conditions prevailing at dynamic equilibrium are of special interest. Equation (9) yields for t - co the total concentration-adsorbed and dissolvedunder steady-state conditions:
k, ?(f +CO)=-. A I Combining
Eqs. (5) and (10) yields the equilibrium -co)=(l
Steady-state Elimination
effluent concentration
-k.)+
Eficiency
The elimination efficiency of an activated sludge stage is correctly described by a mass balance in influent and effluent:
k?i- riz, 11=- rizi This yields with ti = p. c and under consideration
v,=l--.In practice, however, influent and effluent a high surplus sludge not differ significantly
of Eqs. (2) to (4) Eq. (13)
k, - k3 ce (t - al). k, G
the elimination efficiency is more often described by comparing concentrations. It can generally be assumed-except in case of wastage rate-that influent and effluent volume throughputs do (i.e., k3 = 0): 72
G = 1 --(t-+03). G
(14)
FATE OF COMPOUNDS
Mean Retention
IN ACTIVATED
SLUDGE PLANTS,
II
293
Time of the Substance in the Activated Sludge Stage
The determination of the mean retention time of a substance requires [at first] establishment of the function of the concentration change upon a one time addition: -f&k
c+k
dt
2’ e
(15)
c
3’ ‘*
Inserting Eqs. (5) and (6) results in: 2=-{k2(l
+)+k+
+g(2-
Separating the variables and integration
I)]}+
yields for to = 0:
C,(t) = c,(O) - e-A.f. This allows the calculation lication:
(16)
(17)
of the mean retention time in analogy to the former pub02 cp(t) . t - dt s t= Orm cp(t) - dt
J 0
Inserting Eq. ( 17) and subsequent integration
leads finally to: (19)
Since k2 is generally much larger than k,, an increase of the adsorption constant results usually in a prolongation of the mean retention time 7. CONCLUSION Comparison with the Former Method Compared with the old model, the new one offers the following advantages: (1) The calculations are simpler and much faster since the equilibrium concentrations may be calculated directly from the pertinent parameters and not, as in the old model, after thousands of iterative steps by computer. (2) The mean retention time of the substance in an activated sludge stage may be determined as a function of the adsorption constant, the effluent velocity, and the surplus sludge wastage rate. (3) Taking into account the fact that wasted surplus sludge usually has a higher dry matter content than that remaining in the activated sludge stage facilitates a more accurate prediction of the elimination efficiency. The old as well as the new model implies that the adsorption speed suffices for the establishment of a sorptive equilibrium within the given hydraulic mean retention time. This indicates that the predicted elimination efficiency often constitutes a maximum value.
294
P. WIERICH
APPENDIX: G Ci
CP G 4
d ss k, kz k3 k’ me mi
t t ti ti
.I
vss v, 171
v2
mg/liter mg/liter mg/liter mg/liter % org. dry matter % org. dry matter l/hr 1/hr l/hr liters mglhr mg/hr hr hr liters/hr liters/hr liters/hr liters
NOMENCLATURE
adsorbate concentration in the effluent adsorbate concentration in the influent adsorbate concentration in the plant adsorbate concentration in the surplus sludge dry matter content of activated sludge dry matter content of surplus sludge rate constant of the influent rate constant of the effluent rate constant of the surplus sludge wastage proportionality constant of adsorption effluent mass throughput of the adsorbate influent mass throughput of the adsorbate time mean residence time of the adsorbate volume flow rate of the effluent volume flow rate of the influent volume flow rate of the surplus sludge plant volume elimination efficiency/mass balance elimination efficiency/cone balance REFERENCE
P., AND GERM, P. (198 1). The fate of soluble, recalcitrant, and adsorbing compounds in activated sludge plants. Ecotoxicol. Environ. Sa&ty 5, 16 1- 170.
WIERICH,