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Equation 1
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{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 it also can be represented 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
N is the effluent turbidity [-]
No is the maximum/initial effluent turbidity during settling state [-]
(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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{latex} $$ v = {L \over t} $$ {latex} |
where
v is the terminal velocity [L/T2]
L is the distance of the column above turbidimeter [L]
t can be redefined as:
Equation 2
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{latex} $$ t = {d \over v} $$ {latex} |
substitution of Equation (2) in Equation (1) yields:
Equation 3
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{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 both axes in Equation 3 gives:
Equation 4
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{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 the experimental data that is less than number of window specify (at lower end), Equation 5a is used and Equation 5b is used at the upper end.
Equation 5a
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{latex} $$ {{\sum\limits_{j = 0}^{2i} {Y_j } } \over {2i + 1}} $$ {latex} |
Equation 5b
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{latex} $$ {{\sum\limits_{j = 2i - n + 1}^{n - 1} {Y_j } } \over {2n - 2i - 1}} $$ {latex} |
For any other experimental data in between these two condition, Equation 6 is used.
Equation 6
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{latex} $$ {{\sum\limits_{j = i - m}^{i + m} {Y_j } } \over w} $$ {latex} |
where
w is the number of window (odd number)
Y is the data
n is the length of experimental data
m is (w-1)/2