E V A L U A T I O N OF BIOMASS AGE IN A C T I V A T E D SLUDGE

P R O C E S S E S

M. C R I S T I N A A N N E S I N I

Cattedra di Principi di Ingegneria Chimica, Facolth di Ingegneria, Universitgl di Roma via Eudossiana 18, 00184 Roma, Italy

(Received October 27, 1981; Revised March 3, 1982)

Abstract. A criterion is proposed for evaluating biomass age in the activated sludge treatment process in which the biological reaction may be likened to completely mixed reactors operated in series. Expressions for a single-unit reactor have been deduced from the general expression, and these are shown to be equivalent to those reported in the art, and for plug flow reactor,

I. Introduction

Micro-organism activity in a biochemical process not only depends on environmental factors (pH, temperature, substrate features, etc.) but on micro-organism physiological and morphological states as well.

'Biomass age' has recently been introduced as a feature to describe the biomass state in the activated sludge process (Giona et al., 1979). It not only typifies activated sludge processes better than the 'sludge age' previously used (Jenkins and Garrison, 1968; Lawrence and Mc Carty, 1970; Burchett and Tcobanoglous, 1974), but it may also be used as a parameter to express the functional relationship between microbial population metabolism values and biomass 'state'.

An earlier paper (Giona et al., 1979) defined the expressions used for biomass age determination in a completely mixed reactor (CMR). From an analysis of experimental data, the kinetics of the activated sludge process was derived in terms of biomass age and temperature conditions.

The aim of this paper is to propose a general formulation for expressing biomass age that may be used regardless of aeration tank set up.

2. Biomass Age in a CMR with Recycle

The biomass age t s in an activated sludge CMR with recycle may be defined by:

t, = 1/#,, (1)

where #n is the biomass specific net growth rate inclusive of endogenous phenomena. Equation (1), which is clearly only valid when #n ~> 0 has been used by a number of

authors to calculate sludge age (Sherrard and Schroeder, 1972; Goodman and Englande, 1974). It may be noted that the expression for #, gives the same values for concentrations expressed in terms of VSS (volatile suspended solids) as well as for those that only refer

Water, Air, and Soil Pollution 19 (1983) 55-61. 0049-6979/83/0191-0055501.05. Copyright © 1983 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

56 M. CRISTINA ANNESINI

to the active part of sludge. Consequently, Equation (1) leads to biomass and sludge ages being expressed with the same numerical values.

Now, Equation (1) may be expressed in different, though mutually equivalent forms. In fact, if we consider the in-reactor biomass balance equation (Figure 1):

A o x o + ]AnX1V -.= A l x l , (2)

where A indicates the flow rate, x the biomass concentration and Vthe reactor volume,

A0'xD [ Alex1

l Ar,xu

Fig. 1.

Ae,xe

Au,Xu _

Schematic representation of the activated sludge process.

the subscripts 0 and 1 are pertaining respectively to the influent and effluent streams, we obtain

A ix1 - Aoxo

x1V

and, because

A I x I - Aoxo = AeX e + A , x u

if flow A contains no biomass, we obtain

XlV t~ - ( 3 )

Aexe + AuG

where the subscripts e and u refer to the clarified effluent stream and to the waste sludge stream respectively.

Equation (3) thereby shows that biomass age can be calculated as the ratio between in-reactor and out-flowing biomass (Jenkins and Garrison, 1968; Goodman and Englande, 1974).

From Equation (2), and because A o = A~, we have

q~ = Xo/X 1 = 1 - #nv (4)

where ~ is the volumetric residence time inside the reactor, defined by:

= V I A o. (5)

EVALUATION OF B1OMASS AGE IN ACTIVATED SLUDGE PROCESSES 57

Obtaining #n from Equation (4) and substituting it into Equation (1), we have

t, = z / (1 - (p) (6)

which enables us to define ts as the real mean in-reactor residence time for micro- organisms (Bisogni and Lawrence, 1971). It may be noted that, because cp can be written a s

q~ = A o x o / A l X l = A~x./Alxl, (7)

where the subscript r refers to the recycle stream this parameter expresses the probability of a biomass element being recycled through the reactor.

Fig. 2.

Ao,Xo _

ts

~f',x u , t S

~n,Xl ,V, O - - - - -

Ao,X 0

t s ÷ ~

Ao,X 1 , ts

Schematic representation for evaluating the biomass age in a single CMR with recycle.

Furthermore, Figure 2 enables us to determine biomass age as the weight average of the age t of the biomass formed inside the CMR, (equal to G, x1V) , and the age q + z of the recycling biomass, equal to A o x o.

Consequently, we have

A l X l t , = # , X l V T + Aoxo( t s + z) =

= A l X l ' C + Aoxot , (8)

i.e.,

t , = z + opt,

which leads us back to Equation (6).

It may be noted that this approach, though it only contemplates the overall effects of biomass formation phenomena, enables us to obtain a relationship for ts that coincides with Equations (1), (3), and (6) for a CMR with recycle and that it may be also extrapolated to calculate t s in different circumstances.

3. Series of CMRs with recycle

The above-mentioned criterion may also be extended to multi-stage oxidation treatment processes and in particular to the process shown in Figure 3,

58 M. CR1STINA ANNESIN1

Fig. 3. Schematic representation of the multi-stage process.

A , X h _ 1

ts,h--1

O ~ . . D m m ~ l ~ D m m m STAGE

h

p,n,h,X h,Vh-'C'h

- I A'Xh'tsh_, A,Xh_ 1

ts , h-1 + ~h

Fig. 4. Schematic representation for evaluation the biomass age in an h th stage.

Considering any h th stage as presented in Figure 4, Equation (8) can be written as:

Axhts, h = AXh- l ( t s , h-1 + ~h) + t~h, hXhVhZh,

where the subscript h refers to the h th stage and to the stream effluent from the h th stage,

and because

AXh_ 1 + lAn, hXhVh = A x h (9)

we have

ts, h

and for stage N

ts, h = zh + ( x ~ _ l / x h ) t s , ~ - i

Now, working back stage-by-stage to the first, we obtain

h

(Xi/Xh) z, + (Xo/Xh)ts, 0 i = 1

N

ls, N : 2 (Xi/XN)Zi + (Xo/XN)ts, 0 i = I

But due to recycle, we have

ts, o = t s , N

consequently, Equation (12) may be written as:

(10)

(11)

(12)

(13)

1 N t s 'N- - XN(1 - ~0) i=IZ Xi~i

(14)

EVALUATION OF BIOMASS AGE IN ACTIVATED SLUDGE PROCESSES

with

~o = Xo/XN.

Now, substituting Equation (14) back into Equation (11) we obtain

59

(15)

1 x iz i+~o ~ xiz i (16) ts'h-xh(1L o) ; = , =h+l

G, h can thereby be determined. Biomass age in any t th stage can also be expressed in terms of the biomass specific

net growth rates for each stage. If:

~h = (1 -- #n, h~h), (17)

Equation (10) may be written as

t,,h = Zh + ~ d , , h - l "

Now, working backwards, we have

h - - 1

G,h = 2 flh,,%-i + qoht,,o, (18) i = O

where

i

ilk,, = I-[ % - k + , (19) k - 1

with ilk, o = 1, and

h % = [-[ ak

k = l

Expressing Equation (18) for final stage N and taking into account Equation (14), we find

1 N - 1

G,N = G,o - Z flN, iZN-i, (20) 1-~p i=o

where

N

~0 = (PN = I ~ O(k" (21) k = l

Now, substituting Equation (20) into Equation (18), we have

h - 1 N - 1

t,. h = ~ fib. tZh-t + q ~ ~ fiN. i'~N--i (22) i=O 1 - q) i=o

60 M. CRISTINA ANNESINI

But Equation (22) is equivalent to Equation (16) and constitutes an useful expression for calculating b iomass age in multi-stage operations.

For a two-stage process (N = 2) Equation (22) enables us to state that

[s,X = [q21 -b (1 - #., , 'el)Z2]/(1 - q))

ts, 2 = [% + (1 - #., j 2 ) z , ] / ( 1 - cp)

with

~0 = (1 - #n, ~ ) ( 1 - #. ,2)-

For N = 1 (h = 1) we again find Equation (6).

The approach adopted requires that/~n, h > 0. If, on the other hand, #n, h < 0, the preceding equations are valid only w i t h / % h = 0.

4. Series of CMRs without Recycle

Evaluating biomass age in a multi-stage process without recycle where every sfage can be likened to a CMR, may be easily carried out with Equations (11) or (18), omitting

Equation (13). In this case tso o, which represents in-going first stage biomass age, may be nil if flow A o contains no biomasss.

In a single-stage process (N = 1), Equation (19) gives

ts, 1 = rl + (1- /~ , , l z~) t s , o. (23)

I f #,, 1 > 0, Equation (23) simplifies to

ts. 1 = ts, o + ~1. (24)

If/~n, 1 < 0, as is the case when the excess sludge is digested aerobically, the age of stabilized biomass is equal to the sum of incoming biomass and volumetric residence time.

5. Plug Flow Reactor With Recycle

As N approaches o% Equation (14), which gives biomass age for a series of N C M R s with recycle, may be extended to the case of a plug flow reactor with biomass recycle.

In fact, at distance z, we have

- x d z + q) x d z (25) t . , . L x . ( 1 - q~)

0 z

where ~ = V/Ao, L is the length of the reactor and xz is the biomass concentrat ion at distance z. This enables us to evaluate biomass age at distance z.

EVALUATION OF BIOMASS AGE IN ACTIVATED SLUDGE PROCESSES 61

For z = L, i.e. where the flow leaves the reactor, biomass age is given by

L V

i f - x d z - z d V . ( 2 6 ) t , ,L L X L ( 1 - q)) AoXL(1 - cp)

o o

Because the denominator of Equation (26) represents the out-going biomass flow rate and the integral equals the total quantity of in-reactor biomass, we obtain that Equation (26) leads to Equation (3) for CMRs.

6. Conclusions

The need to determine biomass age for treatment processes in which biochemical reactions occur in reactors which cannot be considered as if they were CMRs with recycle has led to a biomass age evaluation criterion that can easily be extended to multi-stage and plug-flow processes.

The expressions for t~ derived as per multi-stage reactors, each of which is likened to a separate CMR, hold true in general and enable the above-mentioned processes to be studied more thoroughly.

Kinetic relationships deduced in an earlier paper (Giona et al., 1979) express metabo- lic values in terms of t s, and this was taken as a parameter to characterize the bacterial population activity.

An examination of multi-staged processes without recycle shows that whenever the overall effect of phenomena connected to biomass growth is negative (i.e. when the endogenous phenomena prevail), biomass age can be deduced from the volumetric residence times of the single reactors.

R e f e r e n c e s

Bisogni, J. J. and Lawrence, A. W.: 1971, Water Res. 5, 753. Burchett, M. E. and Tchobanoglous, G.: 1974, J. Wat. Pollut. Control Fed. 46, 973. Giona, A. R., Annesini, M. C., Toro, L., and Gerardi, W.: 1979, J. Wat. Pollut. Control Fed. 51,999. Goodman, B. L. and Englande, A.J.: 1974, J. Wat. Pollut. Control Fed. 46, 312. Jenkins, D. and Garrison, W. E.: 1968, J. Wat. Pollut. Control Fed. 40, 1905. Lawrence, A. W. and Mc Carty, P. L.: 1970, J. Sanit. Engng. Div. Am. Soc. Civ. Engrs. 96, 757. Sherrard, J. F. and Schroeder, E. D.: 1972, Water Res. 6, 1039.