Kinetics of Microbial Growth

Yield Coefficient, Y

• Part of substrate goes to cell material production. Part goes to energy production.

• Substrate consumption often measured as BOD

• Can be computed from stoichiometrics or energetics for single compounds

• Measured in laboratory for complex mixtures• Aerobic Y typically 0.3 to 0.5.• “Synthesis” or “true” vs. “Observed” yield

consumedsubstrateofMass

producedmaterialcellofMassY =

Substrate Consumption

• Object of biological wastewater treatment

• Rate of consumption, dS/dt = rsu

• Michaelis-Mentenexpression

• Consequence of enzyme-substrate reaction kinetics

( )127.EqSK

kXS-r

ssu −

+=

Substrate conc. decreasing

PESESE +→+↔+

“Half Velocity” or M-M constant

0

100

200

300

400

500

600

700

0 5 10 15 20 25 30 35 40 45 50

Substrate Concentration

Su

bst

rate

Uti

liza

tio

n R

ate,

-r

su

Asmyptote

S >> Ks-rsu ≈ kX= 660 mg/L-d

Ks >> S

-rsu ≈ (S)kX/Ks

≈ k’S

Ks= 5

( )127.EqSK

kXS-r

ssu −

+=

After M&E Figure 7-11:k = 6.6 mg COD/mg VSSX = 100 mg VSS/LKs= 5 mg COD/L

660

330

k’

1

Bacterial Growth

• Consequence of biological wastewater treatment

• Rate of growth, dX/dt = rg

• Max specific growth rate = µm (=rgm/X)• Related to substrate consumption by Y• Substitute 7-14 into 7-12: ( )

( )147.EqY

k

137.EqkY

m

m

−µ=

−=µ

( ) ( )157.EqSKY

XS-r

s

msu −

+µ=

Microbial Growth

• rg also related to rsuby Y→rg = -Yrsu

• Endogenous decay→Cell material

consumed for maintenance energy

→Cell death and predation

→rd = kdX

• Therefore: ( )217.EqXkYrr dsug −−−=

Microbial Growth

( )217.EqXkYrr dsug −−−=

( )227.EqXkSK

kXSYr d

sg −−

+=

( )127.EqSK

kXS-r

ssu −

+=Substitute

Substitute

( )"a"227.EqXkSK

XSr d

s

mg −−

+µ=

( )147.EqY

k m −µ=

Specific Growth Rate, µgr

Xµ =

( )227.EqXkSK

kXSYr d

sg −−

+=

( )237.EqkSK

kSY d

s

−−+

=µ

Substitute into

Or substitute into ( )"a"227.EqXkSK

XSr d

s

mg −−

+µ=

( )a237.EqkSK

Sd

s

m −−+

µ=µ