# 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 and Easwaran Balakrishnan

# Abstract

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

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 (ai≠0)

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.

 F Flowrate through the bioreactor. Lday−1 Ks Monod constant. mgCODL−1 Sj Substrate concentration. mgCODL−1 Sj∗ Dimensionless substrate concentration. Sj∗=Sj/Ks (—) Sˆ∗ Dimensionless substrate concentration along the no-washout solution branch. (–) S0 Substrate concentration in the feed. mgCODL−1 S0∗ Dimensionless substrate concentration in the feed. S0∗=S0/Ks (—) SFRC Step-feed reactor cascade. SRC Standard reactor cascade. Vj Volume of a bioreactor. L Xj Concentration of microorganisms. mgMLSSL−1 Xj∗ Dimensionless microorganism concentration. Xj∗=Xj/αKs (—) X0 Concentration of microorganisms in the feed. mgMLSSL−1 X0∗ Dimensionless microorganism concentration in the feed. X0∗=X0/αKs (—) kd Death coefficient. day−1 kd∗ Dimensionless death coefficient. kd∗=kd/μm (—) n The number of reactors in a SRC or a SFRC. t Time. day−1 t∗ Dimensionless time. t∗=μmt (—) α Yield factor. mgMLSSmgCOD−1 μ Specific growth rate model. day−1 μm Maximum specific growth rate. day−1 τ Residence time. (day). τav Average residence time in a SFRC. τav=τnom. (day). τnom Nominal residence time through a SFRC with reactors of equal size. τnom=nτi (day). τ∗ Dimensionless residence time. τ∗=Vμm/F (—) τtr∗ The 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