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Licensed Unlicensed Requires Authentication Published by De Gruyter December 16, 2017

A Mathematical Model for End-Product Toxicity

Mark Ian Nelson

Abstract

Alcohol based biofuels, such as bio-butanol, have considerable potential to reduce the demand for petrochemical fuels. However, one of the main obstacles to the commercial development of biological based production processes of biofuels is end-product toxicity to the biocatalyst. We investigate the effect of end-product toxicity upon the steady-state production of a biofuel produced through the growth of microorganisms in a continuous flow bioreactor. The novelty of the model formulation is that the product is assumed to be toxic to the biomass. The increase in the per-capita decay rate due to the presence of the product is assumed to be proportional to the the concentration of the product. The steady-state solutions for the model are obtained, and their stability determined as a function of the residence time. These solutions are used to investigate how the maximum yield and the reactor productivity depend upon system parameters. Unlike systems which do not exhibit toxicity there is a value of the feed concentration which maximises the product yield. The maximum reactor productivity is shown to be a sharply decreasing function of both the feed concentration and the toxicity parameter. In conclusion, alternative reactor configurations are required to reduce the effects of highly toxic products.

Appendix

A Attracting Region

In this appendix we show that the region R is both positively invariant and attracting.

A.1 Solution components may not become negative

We first show that if the initial conditions are non-negative that the solution remains non-negative. In order for the solution to become negative it’s value must reduce in value to zero. We have

dSdtS=0=S0τ0,dXdtX=0=0,dPdtP=0=SX1+S0.

Note that second of these derivatives shows that the line X=0 is itself (positively) invariant.

A.2 The substrate component is bounded

We now show that the region 0SS0 is both (positively) invariant and exponentially attracting. We have

dSdt1τS0S,(as X0 and S0).

Let Z1 be the solution of the differential equation

dZ1dt=1τS0Z1

with initial condition Z10=S0. It follows from the classical scalar comparison theorem for ordinary differential equations that StZt. Hence

StS0S0S0exptτ.

This inequality shows that the region 0SS0 is invariant, because if the initial condition is within the invariant region, i.e. S0S0, then the solution remains within the invariant region, i.e. SS0. Furthermore, if the initial condition is outside the invariant region, S0>S0, then the solution must eventually enter the invariant region, i.e. SS0.

A.3 The biomass component is bounded

Let Z=S+X with initial condition Z0=S0+X0. Then adding eqs (5) we have and (6)

dZdtS0τβτZ,(as X0, P0, and 0β1).

Applying the scalar comparison theorem, as in A.2, we have

Z=Xt+StS0βS0βS(0X0)expβτt.

Combining this result with our earlier bound on the scaled substrate concentration it follows that the region

0SS0,0XS0βS

is bounded and exponentially attracting.

A.4 The product component is bounded

Let Z=S+P with initial condition Z0=S0+P0. Then adding eqs (5) we have and (7)

dZdt=S0τZτ,

Applying the scalar comparison theorem, as in A.2, we have

Z=Pt+St=S0S0S(0P0exptτ.

Combining this result with our earlier bound on the scaled substrate concentration it follows that the region

0SS0,0PS0S

is bounded and exponentially attracting.

Furthermore, taking the limit t we obtain

P=S0+P0S.

B The optimal value for the feed concentration to maximise the yield

After some algebra we find that

(37)ddS0Yτ==0,
(38)GS0=a1S02+b1S0+c1=0,

where the coefficients are given by

(39)a1=1kdkp2,
(40)b1=2kdkpkp1kd,
(41)c1=1+kp+kd24kdkd.

The discriminant of the quadratic eq. (38) is positive

(42)b124a1c1=4kdkp2kdkp12.

Consequently eq. (38) has two solutions. As the coefficient a1 is strictly positive we have

(43)S0,+>S0,

where

(44)S0,+=b1+b124a1c12a1,
(45)S0,=b1b124a1c12a1.

The condition for the no-washout branch to be physically meaningful, eq. (15), can be written in the equivalent form

(46)S0>S0,cr=kd1kd,0<kd<1.

We demonstrate that the solution S0, is not physically meaningful because

(47)S0,<S0,cr.

We have

(48)S0,=kdkd1kdkd+kdkp,
(49)<kd1kd,as 0<kd<1 and kp>0,
(50)=S0,cr.

It similarly follows that

(51)S0,+>S0,cr.

Thus the yield is maximised when the feed concentration is given by S0=S0,max=S0,+ where

(52)S0,+=kd+kd1kd+1kdkdkp.

C Symbols used

DSpecific decay rate.hr1
FFlowrate through the bioreactor.dm3hr1
GSingularity equation.(—)
JJacobian matrix.(—)
KsMonod constant.gdm3
PProduct concentration within the bioreactor.gdm3
PDimensionless product concentration.(—)
P=αpP/αsKs
P0The scaled product concentration inside the reactor at time t=0.(—)
PrDimensionless product productivity.(—)
Pr=Pτ
SSubstrate concentration within the bioreactor.gdm3
SDimensionless substrate concentration.(—)
S=S/Ks
SˆThe dimensionless substrate concentration along the no-washout solution branch.(—)
S0The scaled substrate concentration inside the reactor at time t=0.(—)
S0Substrate concentration in the feed (S0>0).gdm3
S0Dimensionless substrate concentration in the feed (S0>0).(—)
S0=S0/Ks
S0,crThe no-washout solution is only physically meaningful for S0>S0,cr.(—)
S0,cr=kd1kd
S0,maxThe value of the feed concentration which maximises the yield.(—)
Defined by eq. (35).
VVolume of the bioreactor.dm3
XConcentration of microorganisms within the bioreactor.gdm3
XDimensionless microorganism concentration.(—)
X=X/αsKs
YYield.(—)
Y=PS0
YDimensionless yield.(—)
Y=PS0
kdDecay coefficient, representing a combination of endogenous respiration,hr1
predation, and cell death followed by subsequent lysis kd>0.
kdDimensionless decay coefficient (kd>0).(—)
kd=kd/μm
kpProduct toxicity constant.hr1g1dm3
kpDimensionless product toxicity constant.(—)
kp=kpμmαsKsαp
tTime.hr1
tDimensionless time.(—)
t=μmt
αpProduct yield factor, the ratio of the weight of product produced to the(—)
weight of substrate consumed.
αsSubstrate yield factor, the ratio of the weight of product produced to the(–)
weight of substrate consumed.
βReactor parameter model.(—)
μSSpecific growth rate model.hr1
μmMaximum specific growth rate.hr1
τresidence time (τ>0).hr
τ=V/F
τDimensionless residence time (τ>0).(—)
τ=Vμm/F
τcr*The value of the dimensionless residence time at the transcritical bifurcation.(—)

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Received: 2017-09-12
Revised: 2017-11-07
Accepted: 2017-11-08
Published Online: 2017-12-16

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