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 1

μ = μ̂ SS/(KS + SS)

1/τ = μ̂ SS/(KS + SS) - b 2

rearranging EQUATION 2:

SS = KS(1 + bτ) / [μ̂τ - (1 + bτ)] 3

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)] 4

S Smin 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) 5

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) 6

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.