Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter December 13, 2014

An Analysis of a Standard Reactor Cascade and a Step-Feed Reactor Cascade for Biological Processes Described by Monod Kinetics

Harvinder S. Sidhu, Mark Ian Nelson ORCID logo and Easwaran Balakrishnan

Abstract

We analyse the steady-state operation of two types of reactor cascade without recycle. The first is a standard reactor cascade in which the feed stream enters into the first reactor. The second is a step-feed reactor cascade in which an equal proportion of the feed stream enters each reactor in the cascade. The reaction is assumed to be a biological process governed by Monod growth kinetics with a decay coefficient for the microorganisms. The steady-states of both models are found for an arbitrary number of reactors and their stability determined as a function of the residence time. We show that in a step-feed reactor cascade the substrate and biomass concentrations leaving the reactor of the cascade are identical to those leaving the first reactor of the cascade. We further show that this result is true for a general specific growth rate of the form μ (S,X). Thus for such processes the non-standard cascade offers no advantage over that of a single reactor. This is surprising because the use of a non-standard cascade has been proposed as a mechanism to improve the biological treatment of wastewater.

Funding statement: Research funding: Sultan Qaboos University (Grant/Award Number: “IG/SCI/DOMS/08/04”, “IG/SCI/DOMS/14/04”).

Acknowledgments

During part of this work MIN was a Visiting Fellow in the School of Physical Environmental & Mathematical Sciences (PEMS), UNSW@ADFA. He thanks the members of PEMS for their collegiality.

Appendix A

A.1 Steady-state analysis

Here we show that the solution of eq. (16) in the second reactor of a cascade (i=2) is only physically meaningful (S2>0 and X2>0) when the negative square root sign is taken. In doing so the only property of S1 and X1 that we use is that they are positive.

The steady-state equations for the concentrations inside the third reactor of a cascade (i=3) are identical to those of the second reactor in the cascade except that the all indices have increased by one. It immediately follows that the physically meaningful steady-state solution in the third reactor, and hence any reactor in the cascade, is given by the negative square root in eq. (16).

In Section A.1.1 we establish the desired result for the non-degenerate case with ai0. In Section A.1.2 we show that the steady-state solution is physically meaningful in the degenerate case that ai=0.

A.1.1 Positivity of the steady-state solution (ai0)

Consider the function

(25)GS2=a2S22+b2S2+c2,a2=1kdτ21b2=a2S1τ2X11+kdτ2c2=1+kdτ2S1.

The coefficient c2 is strictly positive because kd0, S1>0 and τ2>0.

The roots of eq. (25) are given by

(26)S2=b2±b224a2c22a2.

By calculation we have

(27)GS2=0=c2>0,
(28)GS2=S1=τiS1X1<0,

as S1>0, X1>0 and τ1>0.

In the case when a2>0 the calculations (27) and (28) lead to the conclusion that eq. (25) always has two solutions: one in the region 0<S2<S1 and one in the region S1<S2. The solution with S1<S2 is not physically meaningful: eq. (14) shows that the corresponding concentration of microorganisms is negative. It follows that the solution of interest corresponds to the negative square root sign in eq. (26).

In the case when a2<0 the calculations (27) and (28) lead to the conclusion that eq. (25) always has two solutions: a positive solution in the region 0<S2<S1 and a negative solution S2<0. The latter solution is not physically meaningful. It follows that the solution of interest corresponds to the negative square root sign in eq. (26).

A.1.2 Positivity of the steady-state solution (ai=0)

In this section we show that the solution of eq. (16) is positive in the degenerate case when a2=0. The case a2=0 happens when τ2=1+kdτ2. Now a straightforward calculation shows that

S2=S11+X1<S1.

Appendix B

B.1 Symbols used

A subject j refers to a property of the jth reactor in a reactor cascade containing n reactors.

FFlowrate through the bioreactor.Lday1
KsMonod constant.mgCODL1
SjSubstrate concentration.mgCODL1
SjDimensionless substrate concentration. Sj=Sj/Ks(—)
SˆDimensionless substrate concentration along the no-washout solution branch.(–)
S0Substrate concentration in the feed.mgCODL1
S0Dimensionless substrate concentration in the feed. S0=S0/Ks(—)
SFRCStep-feed reactor cascade.
SRCStandard reactor cascade.
VjVolume of a bioreactor.L
XjConcentration of microorganisms.mgMLSSL1
XjDimensionless microorganism concentration. Xj=Xj/αKs(—)
X0Concentration of microorganisms in the feed.mgMLSSL1
X0Dimensionless microorganism concentration in the feed. X0=X0/αKs(—)
kdDeath coefficient.day1
kdDimensionless death coefficient. kd=kd/μm(—)
nThe number of reactors in a SRC or a SFRC.
tTime.day1
tDimensionless time. t=μmt(—)
αYield factor.mgMLSSmgCOD1
μSpecific growth rate model.day1
μmMaximum specific growth rate.day1
τResidence time.(day).
τavAverage residence time in a SFRC. τav=τnom.(day).
τnomNominal residence time through a SFRC with reactors of equal size. τnom=nτi(day).
τDimensionless residence time. τ=Vμm/F(—)
τtrThe value of the dimensionless residence time at the transcritical bifurcation.(—)

References

1. Grady JrC, DaiggerG, LimH. Biological wastewater treatment, 2nd ed., Chapter 7. Boca Raton, FL: CRC Press, 1999a:23194.Search in Google Scholar

2. HenzeM, Grady JrC, GujerW, MaraisG, MatsuoT. A general model for single-sludge wastewater treatment systems. Water Res1987;21:50515.10.1016/0043-1354(87)90058-3Search in Google Scholar

3. MonodJ. The growth of bacterial culture. Annu Rev Microbiol1949;3:37194.10.1146/annurev.mi.03.100149.002103Search in Google Scholar

4. NelsonM, KerrT, ChenX. A fundamental analysis of continuous flow bioreactor and membrane reactor models with death and maintenance included. Asia Pac J Chem Eng2008;3:7080. dx.doi.org/10.1002/apj.10610.1002/apj.106Search in Google Scholar

5. Grady JrC, Daigger GLimH. Biological wastewater treatment, Chapter 12. New York and Basel: Marcel Dekker, Inc., 1980a:365432.Search in Google Scholar

6. Grady JrC, DaiggerG, LimH. Biological wastewater treatment, 2nd ed. Boca Raton, FL: CRC Press, 1999b.Search in Google Scholar

7. Abu-ReeshI. Optimal design of multi-stage bioreactors for degradation of phenolic industrial wastewater: theoretical analysis. J Biochem Technol2010;2:17581.Search in Google Scholar

8. DraméA, HarmandJ, RapaportA, LobryC. Multiple steady state profiles in interconnected biological systems. Math Comput Model Interconnected Biol Syst2006;12:37993.10.1080/13873950600723277Search in Google Scholar

9. AlqahtaniR, NelsonM, WorthyA. A fundamental analysis of continuous flow bioreactor models with recycle around each reactor governed by Contois kinetics. III. Two and three reactor cascades. Chem Eng J2012;183:42232. dx.doi.org/10.1016/j.cej.2011.12.06110.1016/j.cej.2011.12.061Search in Google Scholar

10. AlqahtaniR, NelsonM, WorthyA. A fundamental analysis of continuous flow bioreactor models governed by Contois kinetics. IV. Recycle around the whole reactor cascade. Chem Eng J2013;218:99107.10.1016/j.cej.2012.12.022Search in Google Scholar

11. CarlsoonB, ZambranoJ. Analysis of simple bioreactor models—a comparison between Monod and Contois kinetics in IWA Special International Conference: Activated Sludge—100 years and Counting, 2014.Search in Google Scholar

12. YoonS-H. Important operational parameters of Membrane Bioreactor-Sludge Disintegration (MBR-SD) system for zero excess sludge production. Water Res2003;37:192131.10.1016/S0043-1354(02)00578-XSearch in Google Scholar

13. HerbertD. Multi-stage continuous culture. In: MalekI, editor. Continuous culture of microorganisms, proceedings of the 2nd international symposium on continuous culture. San Diego, CA: Academic Press, 1964:2344.Search in Google Scholar

14. Grady JrC, LimH. Biological wastewater treatment. New York and Basel: Marcel Dekker, Inc, 1980b.Search in Google Scholar

Published Online: 2014-12-13
Published in Print: 2015-3-1

©2015 by De Gruyter