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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


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”).


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


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

The roots of eq. (25) are given by


By calculation we have


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


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.
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.(—)


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

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