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Analysis of a chemostat model with variable yield coefficient: Tessier kinetics

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Abstract

We investigate a chemostat model in which the growth rate is given by a Tessier expression with a variable yield coefficient. We combine analytical results with path-following methods. The washout conditions are found. When washout does not occur we establish the conditions under which the reactor performance and reactor productivity are maximised. We also determine the parameter region in which oscillations may be generated in the reactor. We briefly discuss the implications of our results for comparing the performance of a single bioreactor against a cascade of two bioreactors.

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Abbreviations

F :

Flowrate (l h−1)

K s :

Tessier constant (g l−1)

P :

Reactor productivity P  =  F· X s (g h−1)

P * :

Dimensionless reactor productivity \({P^* = \frac{P}{\alpha\mu_m K_sV}}\)(–)

S :

Substrate concentration (g l−1)

S * :

Dimensionless substrate concentration \({S_{i}^{*} = \frac{S_{i}}{K_{s}} }\)(–)

S 0 :

Substrate concentration in the feed (g l−1)

\({S_{0}^{*}}\) :

Dimensionless substrate concentration in the feed \({S_{0}^{*} = \frac{S_{0}}{K_{s}} }\)(–)

V :

Reactor volume (l)

X :

Cell mass concentration (g l−1)

X s :

Steady-state cell mass concentration (g l−1)

X * :

Dimensionless cell mass concentration \({X_{i}^{*} = \frac{X_{i}}{\alpha K_{s}} }\)(–)

\({X*_{\rm max}}\) :

The maximum (physically meaningful) value of the cell mass concentration on the no-washout solution branch (–)

X 0 :

Cell mass concentration in the feed (g l−1)

\({X_{0}^{*}}\) :

Dimensionless cell mass concentration in the feed \({X_{0}^{*} = \frac{X_{0}}{\alpha K_{s}} }\)(–)

\({X_2^*}\) :

Cell mass concentration in the second reactor of a cascade (–)

Y (S):

Cell mass yield coefficient (–)

t :

Time (h)

t * :

Dimensionless time t * = μ m t (–)

α :

Constant in yield coefficient (–)

β :

Constant in yield coefficient (l g−1)

β * :

Dimensionless yield coefficient \({\beta^{*} = \frac{\beta K_{s}}{\alpha} }\)(–)

μ (S):

Specific growth rate (h−1)

μ m :

Maximum specific growth rate (h−1)

τ :

Residence time \({\tau_{i} = \frac{V_{i}}{F} (h)}\)

τ * :

Dimensionless residence time \({\tau_{i}^{*} = \mu_{m}\cdot\frac{V_{i}}{F}}\)(–)

\({\tau^{*}_{\rm max}}\) :

The value of τ *, should it exist, at which the dimensionless cell mass concentration obtains its maximum value \({(X_{\rm max}^*) }\)(–)

\({\tau^*_i}\) :

The dimensionless residence time in the ith bioreactor in a bioreactor cascade.

\({\tau^*_{t}}\) :

The total dimensionless residence time in a cascade of two bioreactors. Parameter values K s = 1.75 g l−1, α = 0.01, β = 0.03 l g−1, μ m  = 0.3h−1. These give β * = 5.25

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Authors and Affiliations

  1. School of Mathematics and Applied Statistics, The University of Wollongong, Wollongong, NSW, 2522, Australia

    M. I. Nelson

  2. School of Physical, Environmental and Mathematical Science, UNSW at ADFA, Canberra, ACT, 2600, Australia

    H. S. Sidhu

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  1. M. I. Nelson
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  2. H. S. Sidhu
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Correspondence to M. I. Nelson.

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Nelson, M.I., Sidhu, H.S. Analysis of a chemostat model with variable yield coefficient: Tessier kinetics. J Math Chem 46, 303–321 (2009). https://doi.org/10.1007/s10910-008-9463-7

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