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Equation 1

Unknown macro: {latex}

$$
1 - {N \over {N_0 }} = {{\left( {{N \mathord{\left/
{\vphantom {N

Unknown macro: {N_0 } }} \right.
\kern-\nulldelimiterspace}

}} \right)_

Unknown macro: {max }

t} \over {K_S + t}}
$$

or it also can be represented as:

Unknown macro: {latex}

$$
{N \over {N_0 }} = {{K_S + t - \left( {{N \mathord{\left/
{\vphantom {N

Unknown macro: {N_0 } }} \right.
\kern-\nulldelimiterspace}

}} \right)_

Unknown macro: {max }

t} \over {K_S + t}}
$$

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.

Unknown macro: {latex}

$$
v =

Unknown macro: {L over t}

$$

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

Unknown macro: {latex}

$$
t =

Unknown macro: {d over v}

$$

substitution of Equation (2) in Equation (1) yields:

Equation 3

Unknown macro: {latex}

$$
1 - {N \over {N_0 }} = {{\left( {{N \mathord{\left/
{\vphantom {N

Unknown macro: {N_0 } }} \right.
\kern-\nulldelimiterspace}

}} \right)_

Unknown macro: {max }

L} \over {K_S v + L}}
$$

Reciprocating both axes in Equation 3 gives:

Equation 4

Unknown macro: {latex}

$$
{1 \mathord{\left/
{\vphantom {1 {\left( {1 - {N \over

Unknown macro: {N_0 } }} \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {1 - {N \over

}} \right)}} = {{K_S v} \over {\left( {{N \mathord{\left/
{\vphantom {N

Unknown macro: {N_0 } }} \right.
\kern-\nulldelimiterspace}

}} \right)_

Unknown macro: {max }

L}} + {1 \over {\left( {{N \mathord{\left/
{\vphantom {N

Unknown macro: {N_0 } }} \right.
\kern-\nulldelimiterspace}

}} \right)_

}}
$$

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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$$
{{\sum\limits_

Unknown macro: {j = 0}

^

Unknown macro: {2i}

Unknown macro: {Y_j }

} \over {2i + 1}}
$$

Equation 5b

Unknown macro: {latex}

$$
{{\sum\limits_

Unknown macro: {j = 2i - n + 1}

^

Unknown macro: {n - 1}

Unknown macro: {Y_j }

} \over {2n - 2i - 1}}
$$

For any other experimental data in between these two condition, Equation 6 is used.
Equation 6

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$$
{{\sum\limits_

Unknown macro: {j = i - m}

^

Unknown macro: {i + m}

Unknown macro: {Y_j }

} \over w}
$$

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

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