*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 it also can be represented as:
{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/N{~}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
_v_ is the terminal velocity \[L/T{^}2^\]
_L_ is the distance of the column above turbidimeter \[L\]

_t_ can be redefined as:

*Equation 2*
{latex}
$$
t = {d \over v}
$$
{latex}
substitution of Equation (2) in Equation (1) yields:

*Equation 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 both axes in Equation 3 gives:

*Equation 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 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*
{latex}
$$
{{\sum\limits_{j = 0}^{2i} {Y_j } } \over {2i + 1}}
$$
{latex}
*Equation 5b*
{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*
{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