microbial concentration, X, in the reactor is assumed. This is believed to be valid as
long as (θ c /θ) > 5. Substituting the term X in Eq. (3.1) with the average microbial
concentration X , integrating the equation over the hydraulic retention time of the
wastewater in the aeration tank and simplifying, one obtains the following equation:
1
θ c
¼
Yk m S 0 À S 1
ð
Þ
S 0 À S 1
ð
ÞþeK s
À b
ð3:11aÞ
in which e ¼ (1 + R) ln[(RS 1 + S 0 )/(1 + R)S 1 ] and S 1 ¼ effluent substrate concentration of the plug flow process. When R, the recirculated flow ratio Q r /Q,
approaches zero, e ¼ ln(S 0 /S 1 ); therefore
1
θ c
¼
Yk m S 0 À S 1
ð
Þ
S 0 À S 1
ð
ÞþK s ln S 0 =S 1
ð
Þ
À b
ð3:11bÞ
The equation applies as long as the volumetric recycle ratio (ratio of return sludge
flow to influent wastewater flow) is less than unity. Equation (3.11b) shows that θ c is
a function of the influent as well as the effluent wastewater concentration, which is a
unique characteristic of a plug flow process.
With a given set of values for the coefficients K s , b, Y, and k m , one can calculate,
from the equations given above, the sludge retention time θ c required to produce a
predetermined effluent substrate concentration. For the treatment of wastewater to
obtain an effluent BOD of 20 mg/L or below, a shorter sludge retention time is
required for the plug flow process. In other words, for a given θ c value, a plug flow
process can obtain a lower effluent substrate concentration than that of a complete
mix process.
2.1.4 Sludge Growth
Previously, it has been shown that the amount of cell synthesis depends on the length
of time the cells are exposed to aeration. This phenomenon is expressed by Eq. (3.4)
for a no sludge recycle process or Eq. (3.8) for a sludge recycle process. Both
equations state that a shorter retention time results in a higher specific growth rate
and therefore more sludge growth. Since the effluent microbial concentration for a
no sludge recycle process is found to be X ¼ Y(S 0 À S)/(1 + bθ) in Eq. (3.6), the daily
microbial sludge production can be calculated as:
X θ ¼
VY S o À S
À
Á
θ 1 À bθ
ð
Þ
ð3:12Þ
in which X θ ¼ daily microbial sludge production with no sludge recycle, mass/d
when θ is expressed in d. A negligible amount of sludge loss with the secondary
clarifier effluent is assumed in the calculation.
3 Biological Processes
91
long as (θ c /θ) > 5. Substituting the term X in Eq. (3.1) with the average microbial
concentration X , integrating the equation over the hydraulic retention time of the
wastewater in the aeration tank and simplifying, one obtains the following equation:
1
θ c
¼
Yk m S 0 À S 1
ð
Þ
S 0 À S 1
ð
ÞþeK s
À b
ð3:11aÞ
in which e ¼ (1 + R) ln[(RS 1 + S 0 )/(1 + R)S 1 ] and S 1 ¼ effluent substrate concentration of the plug flow process. When R, the recirculated flow ratio Q r /Q,
approaches zero, e ¼ ln(S 0 /S 1 ); therefore
1
θ c
¼
Yk m S 0 À S 1
ð
Þ
S 0 À S 1
ð
ÞþK s ln S 0 =S 1
ð
Þ
À b
ð3:11bÞ
The equation applies as long as the volumetric recycle ratio (ratio of return sludge
flow to influent wastewater flow) is less than unity. Equation (3.11b) shows that θ c is
a function of the influent as well as the effluent wastewater concentration, which is a
unique characteristic of a plug flow process.
With a given set of values for the coefficients K s , b, Y, and k m , one can calculate,
from the equations given above, the sludge retention time θ c required to produce a
predetermined effluent substrate concentration. For the treatment of wastewater to
obtain an effluent BOD of 20 mg/L or below, a shorter sludge retention time is
required for the plug flow process. In other words, for a given θ c value, a plug flow
process can obtain a lower effluent substrate concentration than that of a complete
mix process.
2.1.4 Sludge Growth
Previously, it has been shown that the amount of cell synthesis depends on the length
of time the cells are exposed to aeration. This phenomenon is expressed by Eq. (3.4)
for a no sludge recycle process or Eq. (3.8) for a sludge recycle process. Both
equations state that a shorter retention time results in a higher specific growth rate
and therefore more sludge growth. Since the effluent microbial concentration for a
no sludge recycle process is found to be X ¼ Y(S 0 À S)/(1 + bθ) in Eq. (3.6), the daily
microbial sludge production can be calculated as:
X θ ¼
VY S o À S
À
Á
θ 1 À bθ
ð
Þ
ð3:12Þ
in which X θ ¼ daily microbial sludge production with no sludge recycle, mass/d
when θ is expressed in d. A negligible amount of sludge loss with the secondary
clarifier effluent is assumed in the calculation.
3 Biological Processes
91