A mathematical model for the activated sludge process with a sludge disintegration unit

Salman S. Alsaeed; Mark I. Nelson; Maureen Edwards; Ahmed Msmali

doi:10.1515/cppm-2021-0064

Published by De Gruyter May 3, 2022

A mathematical model for the activated sludge process with a sludge disintegration unit

Salman S. Alsaeed , Mark I. Nelson , Maureen Edwards and Ahmed Msmali

From the journal Chemical Product and Process Modeling

https://doi.org/10.1515/cppm-2021-0064

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Abstract

We develop and investigate a model for sludge production in the activated sludge process when a biological reactor is coupled to a sludge disintegration unit (SDU). The model for the biological reactor is a slimmed down version of the activated sludge model 1 in which only processes related to carbon are retained. Consequently, the death-regeneration concept is included in our model which is an improvement on almost all previous models. This provides an improved representation of the total suspended solids in the biological reactor, which is the key parameter of interest. We investigate the steady-state behaviour of this system as a function of the residence time within the biological reactor and as a function of parameters associated with the operation of the SDU. A key parameter is the sludge disintegration factor. As this parameter is increased the concentration of total suspended solids within the biological reactor decreases at the expense increasing the chemical oxygen demand in the effluent stream. The existence of a maximum acceptable chemical oxygen demand in the effluent stream therefore imposes a maximum achievable reduction in the total suspended solids. This paper improves our theoretical understanding of the utility of sludge disintegration as a means to reduce excess sludge formation. As an aside to the main thrust of our paper we investigate the common assumption that the sludge disintegration processes occur on a much shorter timescale than the biological processes. We show that the disintegration processes must be exceptional slow before the inclusion of the biological processes becomes important.

Keywords: activated sludge model no. 1 (ASM1); cell lysing; excess activated sludge; ozonation; sludge disintegration; sludge reduction

Corresponding author: Mark I. Nelson, School of Mathematics and Applied Statistics, University of Wollongong, Wollongong, NSW 2522, Australia, E-mail: mnelson@uow.edu.au

Acknowledgments

Salman Alsaeed is a PhD student at the University of Wollongong. He gratefully acknowledges the award of a PhD scholarship by Jouf University (Saudi Arabia). The authors thank the reviewers for their detailed and considered comments on our manuscript. These have led to significant improvements in the paper.

Author contributions: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A: Parameter values

The default parameter values in the model are shown in Table 3.

Table 3:

The model parameters and their values.

Symbols	Explanation	Value	Unit
C	The recycle concentration factor	–	(−)
COD	Chemical oxygen demand in the reactor	–	mg COD L − 1
D	Sludge disintegration factor	0.2	(−)
F	Flow rate through the bioreactor	–	dm 3 h − 1
K L , A	Oxygen transfer coefficient	96 [26]	day − 1
K O , H	Oxygen half-saturation coefficient	0.2 [26]	mg O 2 L − 1
K s	Monod constant for biomass	20 [26]	g COD L − 1
K s	Contois coefficient for hydrolysis of particulate biodegradable substrate	0.03 [26]	g COD
M 2	Monod kinetics for readily biodegradable soluble substrate	–	(−)
M 8 h	Monod kinetics for the component S 0 with respect to biomass	–	(−)
MLVSS	Mixed liquor suspended solids	–	mg COD L − 1
R	Recycle ratio	0.4 [26]	(−)
R *	Effective recycle parameter	–	(−)
S O	Concentration of soluble oxygen	–	mg O 2 L − 1
S O , in	Soluble oxygen concentration in the feed	2.0 [26]	mg O 2 L − 1
S O , max	Maximum concentration of soluble oxygen	10.0 [26]	mg O 2 L − 1
S s	Soluble substrate concentration	–	mg COD L − 1
S s , in	Concentration soluble substrate in the feed	200 [26]	mg COD L − 1
TSS	Total suspended solids	–	g SS L − 1
V	Bioreactor volume	2.0 [39]	L
V s	SDU volume	0.8 [39]	L
X B , H	Concentration of heterotrophic biomass	–	mg COD L − 1
X P	Concentration of particulate products arising from biomass decay	–	mg COD L − 1
X s	Concentration of slowly biodegradable particulates	–	mg COD L − 1
X s , in	Concentration of slowly biodegradable particulates in the feed	100 [26]	mg COD L − 1
Y H	Heterotrophic yield coefficient	0.67 [26]	(−)
b H	Heterotrophic decay coefficient	0.22 [26]	day − 1
c 1	Conversion factor from COD to TSS for solution X s	0.75 [26]	g SS (g COD − 1 )
c 2	Conversion factor from COD to TSS for solution X B , H	0.90 [26]	g SS (g COD − 1 )
f p	The fraction of dead biomass	0.08 [26]	(−)
k h	Maximum hydrolysis rate	3.0 [26]	day − 1
k s	Disintegration rate of slowly biodegradable particulates within the SDU	1	day − 1
k sat	Saturation kinetics for hydrolysis	–	(−)
k p	Disintegration rate of non-biodegradable particulates within the SDU	1	day − 1
k x	Disintegration rate of biomass within the SDU	1	day − 1
t	Time	–	day − 1
μ max , H	Maximum specific growth rate for biomass	6.0 [26]	day − 1
τ	Residence time	–	day
α	Sludge solubilization efficiency [ 0 ≤ α ≤ 1 ]	0.5	(−)
β	Sludge solubilization efficiency [ 0 ≤ β ≤ 1 ]	0.5	(−)

Appendix B: The ideal settling unit model

In this section we describe the ideal settling unit model. Let X be the concentration of particulates in the fluid stream leaving the bioreactor, X e the concentration in the effluent stream, and X s the concentration in the recycle stream. Then a mass balance around the settling unit gives

F ( 1 + R ) X = F ( 1 − w ) X e + R F X s .

(The reader not familiar with the ideal settling unit model may benefit from referring back to Figure 3). The ideal settling unit model uses the assumption that

X s = C ⋅ X ,

where the parameter C is known as the concentration factor (If C < 1 the concentration of particulates in the recycle stream is lower than that of the stream leaving the biological reactor. This represents failure of the settling unit). After cancellation we obtain

(37) ( 1 + R ) X = ( 1 − w ) X e + R C X .

If we make the standard assumption that the settling unit captures all particulates then X e = 0 . Rearranging the previous equation we obtain the maximum possible value for the concentration factor

(38) C max = ( 1 + 1 R )

If the settling unit does not capture all the particulates then we have

C = α ⋅ C max ,

where 0 ≤ α < 1 . Substituting this expression into Eq. (37) and rearranging we find that the concentration of particulates in the effluent stream is

(39) X e = ( 1 + R ) ⋅ 1 − α 1 − w ⋅ X .

This equation provides a way to investigate the effectiveness of sludge disintegration techniques when the settling unit in not operating perfectly, at the expense of introducing an additional parameter into the model.

Appendix C: Stability

From Eq. (35) the coefficients b i are polynomials in the residence time (τ). These are

b 2 = b 2,4 τ 4 + b 2,3 τ 3 + b 2,2 τ 2 , b 2,4 = − α K L , A ( ( ( − b H K O , H + S O , max ( μ max , H − b H ) ) S s , i n − b H ( K s + X s , in ) K O , H + S O , max ( − b H K s + X s , i n ( μ max , H − b H ) ) ) D − ( R ∗ − 1 ) ( ( − b H K O , H + S O , max ( μ max , H − b H ) ) S s , in − b H K s ( K O , H + S O , max ) ) ) , b 2,3 = − ( − α K L , A ( K O , H + S O , max ) ( K s + X s , in + S s , in ) D 2 + α ( ( K O , H + S O , max ) ( R ∗ − 1 ) ( X s , in + 2 K s + 2 S s , in ) K L , A + ( − b H K O , H + S O , i n ( μ max , H − b H ) ) S s , in − b H ( K s + X s , in ) K O , H + ( − K s b H + X s , in ( μ max , H − b H ) ) S O , in ) D − α ( ( ( K s + X s , in ) ( K O , H + S O , max ) ( R ∗ − 1 ) K L , A + ( − b H K O , H + S O , in ( μ max , H − b H ) ) S s , in − b H K s ( K O , H + S O , in ) ) ( R ∗ − 1 ) ) , b 2,2 = ( α ( K O , H + S O , in ) ∗ ( X s , in + K s + S s , in ) D 2 − ( R ∗ − 1 ) α ( K O , H + S O , in ) ( X s , in + 2 K s + 2 S s , in ) D + α ( R ∗ − 1 ) 2 ( K O , H + S O , in ) ( K s + S s , in ) ) .

The coefficient b 1 is given by

b 1 = b 1,3 τ 3 + b 1,2 τ 2 + b 1,1 τ , b 1,3 = − α K L , A V ∗ ( ( − b H K O , H + S O , max ( μ max , H − b H ) ) S s , in − b H ( K s + S s , in ) K O , H + S O , max ( − b H K s + X s , in ( μ max , H − b H ) ) ) D 2 − K L , A ( α + 1 ) ( R ∗ − 1 ) V ∗ ( ( − b H K O , H + S O , max ( μ max , H − b H ) ) S s , in − b H ( K s ( K O , H + S O , max ) ) D ) , b 1,2 = ( − ( ( ( R ∗ − 1 ) ( K O , H + S O , max ) ( X s , in + K s + S s , in ) K L , A + ( − b H K O , H + S O , i n ( μ max , H − b H ) ) S s , in − b H ( X s , in + K s ) K O , H + ( − b H ( X s , in + K s ( μ max , H − b H ) ) S O , in ) α + ( K s + S s , in ) ( K O , H + S O , max ) ( R ∗ − 1 ) K L , A ) V ∗ D 2 + ( R ∗ − 1 ) ( α + 1 ) ( ( K s + S s , in ) ( K O , H + S O , max ) ( R ∗ − 1 ) K L , A + ( − b H K O , H + S O , in ( μ max , H − b H ) ) S s , in − b H K s ( K O , H + S O , in ) ) V ∗ D ) , b 1,1 = − ( R ∗ − 1 ) D V ∗ ( ( K O , H + S O , in ) ( ( X s , in + K s + S s , in ) α + K s + S s , in ) D − ( K O , H + S O , in ) ( K s + S s , in ) ( α + 1 ) ( R ∗ − 1 ) ) .

The coefficient b 0 is given by

b 0 = b 0,2 τ 2 + b 0,1 τ + b 0,0 , b 0,2 = ( R ∗ − 1 ) ( ( ( − b H K O , H + S O , max ( μ max , H − b H ) ) S s , in − b H K s ( S O , max + K O , H ) ) K L , A , b 0,1 = ( ( ( R ∗ − 1 ) ( K O , H + S O , max ) K L , A − b H K O , H + S O , in ( μ max , H − b H ) ) S s , in + ( ( R ∗ − 1 ) ( K O , H + S O , max ) K L , A − b H ( K O , H + S O , in ) ) K s ) , b 0,0 = ( R ∗ − 1 ) ( K O , H + S O , in ) ( K s + S s , in ) ) D 2 V ∗ 2 .

Appendix D: What happens when the biological processes in the sludge disintegration are included in the model?

In this section we explore the behaviour of the model when the SDU model contains both the biological processes and the sludge disintegration reactions. In Section D.1 we show how the original model, i.e. differential Eqs. (4)–(8) and (14)–(17), is modified to include the biological processes in the SDU. In Section D.2 we investigate whether the inclusion of the biological processes has any significant effect upon either the concentration or soluble substrate or the total suspended solids within the biological reactor. Having concluded that it is acceptable to ignore the biological processes in the SDU in Section D.3 we cast some light on why this is so.

D.1 Model equations

The model equations in the biological reactor are essentially the same as given before. However, there is a slight change to the differential equation for the concentration of soluble oxygen.

(40) V d S O d t = F ( S O , in − S O ) + V K L , A ( S O , max − S O ) − ( 1 − Y H ) Y H ⋅ μ max , H ⋅ M 2 ⋅ M 8 h ⋅ X B , H ⋅ V − D F ( S O − S O , SDU ) .

The new term in the model is given in red. This represents the exchange of dissolved oxygen between the biological reactor and the SDU. It was not required previously because the concentration of dissolved oxygen in the biological reactor was equal to that in the SDU as a consequence of the assumptions regarding the biological processes.

The model equations in the SDU are given below. The new terms are denoted in red. Note, as explained in the previous paragraph, there was no need for a differential equation for the soluble substrate concentration in the original model.

The rate of change of soluble substrate

V s d S s , SDU d t = D F ( S s − S s , SDU ) + α k s X s , SDU V s + β k p X P , SDU V s + α k x X B , H , SDU V s − μ max , H Y H ⋅ M 2 , SDU ⋅ M 8 h , SDU ⋅ X B , H , SDU ⋅ V s + k h ⋅ k sat , SDU ⋅ M 8 h , SDU ⋅ X B , H , SDU ⋅ V s .

The rate of change of heterotrophic biomass

V s d X B , H , SDU d t = D F ( X B , H − X B , H , SDU ) − k x X B , H , SDU V s − b H X B , H , SDU ⋅ V s + μ max , H ⋅ M 2 , SDU ⋅ M 8 h , SDU ⋅ X B , H , SDU ⋅ V s .

The rate of change of slowly biodegradable particulate substrate

V s d X s , SDU d t = D F ( X s − X s , SDU ) − α k s X s , SDU V s + ( 1 − α − f p ) k x X B , H , SDU V s + ( 1 − f p ) b H X B , H , SDU ⋅ V s − k h ⋅ k sat , SDU ⋅ M 8 h , SDU ⋅ X B , H , SDU ⋅ V s .

The rate of chance of soluble oxygen

V s d S O , SDU d t = D F ( S O − S O , SDU ) + V s K L , A , 2 ( S O , max − S O , SDU ) − ( 1 − Y H ) Y H ⋅ μ max , H ⋅ M 2 , SDU ⋅ M 8 h , SDU ⋅ X B , H , SDU ⋅ V s .

The rate of change of non-biodegradable particulate products

V s d X P , SDU d t = D F ( X P − X P , SDU ) − β k p X P , SDU V s + f p k x X B , H , SDU V s + f p b H X B , H , SDU V s .

Note that to avoid additional costs it is unlikely for the SDU to be aerated. Accordingly, we assume that K L , A , 2 = 0 .

The reaction rates in the SDU are

M 2 , SDU = S s , SDU K s + S s , SDU ,

M 8 h , SDU = S O , SDU K O , H + S O , SDU ,

k sat , SDU = X s , SDU K X X B , H , SDU + X s , SDU .

D.2 Numerical investigation: are biological processes in the SDU important?

In this section we investigate how the concentration of soluble substrate and the total suspended solids within the biological reactor vary as a function of the residence time when the biological processes are included in the SDU. We show results for four values of the sludge disintegration rate. In Figures 14 and 15 the solid line indicates the default model in which the biological processes in the SDU are not included in the model whereas the dashed line indicates the revised model in which they are included.

Figure 14 demonstrate that when k ≥ 100 h − 1 that the default and revised models give equivalent results. Even when the disintegration rate is reduced to k = 1 day − 1 there is little difference between the model predictions and what difference there is occurs only for values of the residence time near to the washout value.

Figure 14:

The dependence of the soluble substrate concentration in the biological reactor upon the residence and the value of the sludge disintegration rate. The solid line represents the case when the biological processes only occur in the bioreactor. The dash line represents the case when the processes occur in both the biological reactor and the SDU.

Figure 15 shows that the prediction of the two models for the total suspended solids are in complete agreement when k ≥ 1000 h − 1 . In the region when 100 ≤ k ≤ 150 h − 1 the extremely slight difference between the models along the no-washout branch is of no practical significance. A similar observation applies to the predictions along the no-washout branch when k = 1 h − 1 . There is a larger difference in the predictions in the vicinity of the washout point.

Figure 15:

The dependence of the soluble substrate concentration in the biological reactor upon the residence and the value of the sludge disintegration rate. The solid line represents the case when the biological processes only occur in the biological reactor. The dash line represents the case when the processes occur in both the biological reactor and the SDU.

In the approach to modelling sludge disintegration processes developed by Yoon [4] it is assumed these are infinitely fast so that biological processes within the SDU can be ignored. Although the sludge disintegration processes may operate on a very small time-scale they can not be infinitely fast. The question then arises as to how ‘fast’ these processes must be in order to justify the assumption that they operate on a shorter time-scale than the biological processes. We have shown that even if the rate of the disintegration processes is as low as 1 h − 1 that for practical operating conditions, i.e. away from the washout point, that the contribution of biological processes within the SDU to either the concentration of soluble substrate or that of the total suspended solids within the biological reactor is negligible, i.e. this assumption of the Yoon’s approach is valid. We cast some informal light on this observation in the next section.

D.3 Why are the biological processes not important?

In this section we offer some informal reasoning as to why the biological processes are unimportant when the sludge disintegration reactions are very quick. As way of explaining this we consider the rate at which biomass grow through consumption of the substrate ( μ growth ) inside the SDU. We have

μ growth = μ max , H · M 2 , SDU · M 8 h , SDU · X B , H · V s , < μ max , H · M 8 h , SDU · X B , H · V s , as M 2 , SDU = S S , SDU K S + S S , SDU < 1 , < μ max , H · X B , H · V s , as M 8 h , SDU = S O , SDU K O , H + S O , SDU < 1 .

The rate of removal of biomass through disintegration is k x X B , H , SDU V s . Comparison of the two expressions shows that when

k x ≫ μ max , H = 6.0 day − 1

growth through consumption of soluble substrate is insignificant. Note that in practice the effective value of k x at which growth becomes insignificant is lower than that predicted here. We obtained our approximation using crude bounds on the functions M 2 , SDU and M 8 h , SDU . These assume that the concentrations of soluble substrate and oxygen are both infinitely large.

Similar observations can be made for the other biological terms.

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Received: 2021-10-12

Accepted: 2022-04-08

Published Online: 2022-05-03

A mathematical model for the activated sludge process with a sludge disintegration unit

Abstract

Acknowledgments

Appendix A: Parameter values

Appendix B: The ideal settling unit model

Appendix C: Stability

Appendix D: What happens when the biological processes in the sludge disintegration are included in the model?

D.1 Model equations

D.2 Numerical investigation: are biological processes in the SDU important?

D.3 Why are the biological processes not important?

References

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