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Persistence in a Single Species CSTR Model with Suspended Flocs and Wall Attached Biofilms

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Abstract

We consider a mathematical model for a bacterial population in a continuously stirred tank reactor (CSTR) with wall attachment. This is a modification of the Freter model, in which we model the sessile bacteria as a microbial biofilm. Our analysis indicates that the results of the algebraically simpler original Freter model largely carry over. In a computational simulation study, we find that the vast majority of bacteria in the reactor will eventually be sessile. However, we also find that suspended biomass is relatively more efficient in removing substrate from the reactor than biofilm bacteria.

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Abbreviations

A :

area (m2)

a u :

half-saturation Monod const. of suspended bacteria (Freter) (g/m3)

a w :

half-saturation Monod const. of wall-attached bacteria (Freter) (g/m3)

D :

dilution rate (1/day)

D c :

diffusion coefficient (m2/day)

E :

erosion parameter (1/m⋅day)

F :

flow (velocity) through the reactor (Freter) (m3/day)

K λ :

half-saturation Monod constant of biofilm (g/m3)

K u :

half-saturation Monod constant of suspended bacteria (g/m3)

k λ :

death rate of biofilm (1/day)

k u :

death rate of suspended bacteria (1/day)

k w :

death rate of wall-attached bacteria in Freter model (1/day)

m u :

maximum growth rate of suspended bacteria (Freter) (1/day)

m w :

maximum growth rate of wall-attached bacteria (Freter) (1/day)

Q :

flow (velocity) through the reactor (m3/day)

S :

substrate concentration (g/m3)

S in :

substrate concentration at inlet (g/m3)

u :

suspended bacteria (g)

u :

concentration of suspended bacteria (Freter) (g/m3)

V :

volume of the reactor (m3)

w :

areal biomass density of wall-attached bacteria (g/m2)

w max :

maximum areal biomass density of wall-attached bacteria (g/m2)

W=w/w max :

wall occupancy fraction (–)

α :

attachment rate (1/day)

β :

detachment rate (Freter) (1/day)

λ :

biofilm thickness (m)

\(\mu_{\lambda}^{\max}\) :

maximum growth rate of biofilm (1/day)

\(\mu_{u}^{\max}\) :

maximum growth rate of suspended bacteria (1/day)

γ :

yield (–)

ρ :

biofilm biomass density (g/m3)

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Authors and Affiliations

  1. School of Technology, Malmö University, 20506, Malmö, Sweden

    Alma Mašić

  2. Centre for Mathematical Sciences, Lund University, Box 118, 22100, Lund, Sweden

    Alma Mašić

  3. Dept. Mathematics and Statistics, University of Guelph, Guelph, ON, Canada

    Hermann J. Eberl

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  1. Alma Mašić
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  2. Hermann J. Eberl
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Correspondence to Alma Mašić.

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Mašić, A., Eberl, H.J. Persistence in a Single Species CSTR Model with Suspended Flocs and Wall Attached Biofilms. Bull Math Biol 74, 1001–1026 (2012). https://doi.org/10.1007/s11538-011-9707-8

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  • DOI: https://doi.org/10.1007/s11538-011-9707-8

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