Bulletin of Mathematical Biology

, Volume 74, Issue 4, pp 1001–1026 | Cite as

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

Original Article

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.

Keywords

Biofilm Mathematical model CSTR Wall attachment 

Nomenclature

A

area (m2)

au

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

aw

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

D

dilution rate (1/day)

Dc

diffusion coefficient (m2/day)

E

erosion parameter (1/m⋅day)

F

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

Kλ

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

Ku

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

kλ

death rate of biofilm (1/day)

ku

death rate of suspended bacteria (1/day)

kw

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

mu

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

mw

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

Q

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

S

substrate concentration (g/m3)

Sin

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)

wmax

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

W=w/wmax

wall occupancy fraction (–)

Greeks

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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Copyright information

© Society for Mathematical Biology 2011

Authors and Affiliations

  1. 1.School of TechnologyMalmö UniversityMalmöSweden
  2. 2.Centre for Mathematical SciencesLund UniversityLundSweden
  3. 3.Dept. Mathematics and StatisticsUniversity of GuelphGuelphCanada

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