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FATE AND TRANSPORT
Microbial Kinetics
BOD Exertion
1 ( 1 e ) kt −
y = L −
t
where
k1 = deoxygenation rate constant (base e, days –1 ),
L = ultimate BOD (mg/L),
t = time (days), and
yt = the amount of BOD exerted at time t (mg/L).
Stream Modeling: Streeter Phelps
kL 1 o
D = exp ( −kt) −exp( − k t) + D exp −k
t
1 2 o
2
k − k
2
1
[ ] ( )
( k − k )
1 ⎡k⎛ 2
t = ln 1
c ⎢ ⎜ − Do
k − k k 2 1 ⎣ ⎝ 1
DO = DOsat – D, where
⎞⎤
2 1
⎟⎥
k L ⎠
1 o ⎦
D = dissolved oxygen deficit (mg/L),
k1 = deoxygenation rate constant, base e (days –1 ),
t = time (days),
k2 = reaeration rate, base e (days –1 ),
Lo = initial BOD ultimate in mixing zone (mg/L ),
Do = initial dissolved oxygen deficit in mixing zone
(mg/L),
tc = time which corresponds with minimum
dissolved oxygen (days),
DOsat = saturated dissolved oxygen concentration
DO =
(mg/L), and
dissolved oxygen concentration (mg/L).
Monod Kinetics—Substrate Limited Growth
Continuous flow systems where growth is limited by one
substrate (chemostat):
S
µ = µ max , where
K + S
µ = specific growth rate (time –1 ),
s
µmax = maximum specific growth rate (time –1 ),
S = concentration of substrate in solution (mass/unit
volume), and
Ks = half-velocity constant = half-saturation constant
(i.e., substrate concentration at which the specific
growth rate is one-half µmax) (mass/unit volume).
148
GROWTH RATE CONSTANT (µ, 1/t)
µ m
µ
m
2
K s
ENVIRONMENTAL ENGINEERING (continued)
♦ Monod growth rate constant as a function of
limiting food concentration.
LIMITING FOOD CONCENTRATION S (mg/L)
Multiple Limiting Substrates
µ
= ⎡
⎣µ 1( S1) ⎤⎡
⎦⎣µ 2( S2) ⎤
⎦⎣
⎡µ 3( S3) ⎤
⎦… ⎡
⎣µ n( Sn)
⎤
µ
⎦
m
Si
where µ i = for i = 1 to n
Ksi + Si
Non-steady State Continuous Flow
dx
= Dxo + ( µ−kd − D) x
dt
Steady State Continuous flow
µ = D with kd > Ss, and
Ks = saturation constant on half-velocity constant
[= concentration (mg/l) at µm/2].
f = flow rate (hr –1 ),
Vr = culture volume (l),
D = dilution rate (flow f / reactor volume Vr; hr –1 ),
µi = growth rate with one or multiple limiting substrates
(hr –1 ),
Si = substrate i concentration (mg/l),
So = initial substrate concentration (mg/l),
YP/S = product yield per unit of substrate (mg/mg)
p = product concentration (mg/l).
x = cell number (g/l),
xo = initial cell number (g/l),
t = time (hr)
kd = death rate (hr –1 )
♦ Davis, M.L. and S.J. Masten, Principles of Environmental Engineering and Science, McGraw-Hill,
2004. Used with permission of McGraw-Hill Companies.