CSTR with COD consumption, cell growth and decay

Q = 10,000 m³/d

SSO = 400 g/m³ COD

XBO = 0

Y = 0.6 g-cell COD/g-COD

μ̂ = 1 d⁻¹

b = 0.1 d⁻¹

Kₛ = 50 g/m³ COD

Analysis

Steady state mass balance on cells

VrXB = QXB

rXB = (μ − b)XB

(μ − b)XB = XB/τ

1/τ = μ − b

μ = μ̂ SS/(Kₛ + SS)

1/τ = μ̂ SS/(Kₛ + SS) − b

rearranging EQUATION 2:

SS = Kₛ(1 + bτ) / (μ̂τ − (1 + bτ))

note that mass balance on cells produces relation for substrate

COD in terms of kinetic and reactor parameters

also, SS is not dependent on SSO

other relations:

τmin is detention time associated with cell washout, at washout, no

cells and no COD consumption, SS → SSO

SSO = Kₛ(1 + bτmin) / (μ̂τmin − (1 + bτmin))

rearranging

τmin = (Kₛ + SSO) / (μ̂SSO − b(Kₛ + SSO))

SSmin indicates the minimum substrate concentration which will

sustain cell growth greater than loss from decay, corresponds to

maximum substrate removal, the maximum detention time, and the

minimum growth rate

1/τmax = μmin − b → 0

and

μmin = b

and

μ̂(SSmin)/(Kₛ + SSmin) = b

rearranging

SSmin = Kₛb / (μ̂ − b)

steady state mass balance on substrate

QSSO − QSS + VrS = 0

SSO − SS + τrS = 0

−rS = (SSO − SS)/τ

substituting: rS = −μXB/Y into mass balance

XB = Y(SSO − SS)/(μτ)

from equation 1: 1/τ = μ − b and μ = 1/τ + b

XB = Y(SSO − SS) / τ(1/τ + b) = Y(SSO − SS)/(1 + bτ)

for this system, mass balance on substrate produces relation for

cell concentration

where

SS = Kₛ(1 + bτ) / (μ̂τ − (1 + bτ))

Cell growth and substrate consumption are coupled.