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Lecture 5
CE260/Spring 2000
 Bacterial Growth – If binary fusion then doubling of the # of cells at each division
N t  N o e kt


Where:

Nt = number of microorganisms at time t

No = number of microorganisms at t = 0

k = growth constant
Rate of increase in microbes is:
dN
 kN
dt

Since bacteria need food, substrate S
dS dN

dt
dt


dS
1 dN

dt
Y dt
Y = yield factor (mass of micros formed/mass of substrate removed)
 Limiting factors to growth

If food is the limiting factor to growth then:
dN
 k ' SN
dt

k’ = rate constant

If batch system w/ pure culture of bacteria (Figure 6.6)

If batch system w/ mixed culture (Figure 6.7)

Monod (1949) equation:
dS
kXS

dt
K S


K = constant

S = substrate conc.

X = concentration of micros (mass/volume)

k = maximum specific rate of substrate removal
If by convention we let k = u (specific growth rate) then the substrate limited
growth is:
u
u max S
Ks  S
and
dX u max SX

dt
Ks  S
umax
u
½ umax
Ks
S mg/L

Since:
Y

dX
X dX
dt


dS
S dS
dt
Then:
dS 1  dX 
 
 and
dt Y  dt 




In the book the Monod equation is written as (v used in place of u):
v

dS 1  u max SX
 
dt Y  K s  S
vmax [ S ]
K  [S ]
Since S and u can be measured as the Monod equation can be linearized by
inverting and factoring out umax.
Ks
Ks
1 Ks  S
S
1
1






u u max S u max S u max S
u u max S u max
y  mx  B 

1 Ks

u u max
1
1
 
 S  u max
The above equation yields the Lineweaver-Burke plot:
1/u
Ks/umax
1/umax

1/S
Eadie-Hofstee plot (Figure 4.13)
 Summary

Lineweaver-Burke – weights the low concentration data

Edie-Hofstee – magnifies departures from linearity that may not be apparent
in the Lineweaver-Burke plot.