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Equation 1
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*Equation 1* {latex} $$ 1 - {N \over {N_0 }} = {{\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } t} \over {K_S + t}} $$ {latex} |
or
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it
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also
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can
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be
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represented
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as:
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{latex} $$ {N \over {N_0 }} = {{K_S + t - \left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } t} \over {K_S + t}} $$ {latex} |
where
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N
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is
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the
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effluent
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turbidity
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[-
...
]
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No
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is
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the
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maximum/initial
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effluent
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turbidity
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during
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settling
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state
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[-
...
]
...
(N/N0)max is the maximum value that the hyperbolic function will asymptomatically approach to.
t is the time [T]
KS is the rate of settling flocs [T].
Terminal velocity is the velocity of the flocs settling in the column.
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{~}0~)~max~ is the maximum value that the hyperbolic function will asymptomatically approach to. _t_ is the time \[T\] _K{_}{_}{~}S{~}_ is the rate of settling flocs \[T\]. Terminal velocity is the velocity of the flocs settling in the column. {latex} $$ v = {L \over t} $$ {latex} |
where
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v
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is
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the
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terminal
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velocity
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[L/T2]
L is the distance of the column above turbidimeter [L]
t can be redefined as:
Equation 2
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{^}2^\] _L_ is the distance of the column above turbidimeter \[L\] _t_ can be redefined as: *Equation 2* {latex} $$ t = {d \over v} $$ {latex} |
substitution
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of
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Equation
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(2)
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in
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Equation
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(1)
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yields:
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Equation 3
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3* {latex} $$ 1 - {N \over {N_0 }} = {{\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } L} \over {K_S v + L}} $$ {latex} |
Reciprocating
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both
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axes
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in
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Equation
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3
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gives:
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Equation 4
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4* {latex} $$ {1 \mathord{\left/ {\vphantom {1 {\left( {1 - {N \over {N_0 }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - {N \over {N_0 }}} \right)}} = {{K_S v} \over {\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } L}} + {1 \over {\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } }} $$ {latex} |
For
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the
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experimental
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data
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that
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is
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less
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than
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number
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of
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window
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specify
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(at
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lower
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end),
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Equation
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5a
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is
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used
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and
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Equation
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5b
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is
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used
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at
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the
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upper
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end.
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Equation 5a
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5a* {latex} $$ {{\sum\limits_{j = 0}^{2i} {Y_j } } \over {2i + 1}} $$ {latex} * |
Equation 5b
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5b* {latex} $$ {{\sum\limits_{j = 2i - n + 1}^{n - 1} {Y_j } } \over {2n - 2i - 1}} $$ {latex} |
For
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any
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other
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experimental
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data
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in
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between
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these
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two
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condition,
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Equation
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6
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is
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used.
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Equation 6
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6* {latex} $$ {{\sum\limits_{j = i - m}^{i + m} {Y_j } } \over w} $$ {latex} |
where
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w
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is
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the
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number
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of
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window
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(odd
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number)
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Y
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is
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the
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data
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n
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is
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the
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length
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of
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experimental
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data
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m
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is
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(w-1)/2