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The description of the experiments can be viewed in the previous reports. All of experimental data (i.e time, pressure, turbidity, flow rates, etc) was collected using Process Controller and saved in Excel files. Data of interest (i.e. effluent turbidity) was stored corresponding to the date of experiment run and the status file indicated the state of treatment process (i.e. flocculation state, settling state, etc).

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Assuming the dataset follows a hyperbolic curve function, therefore it can be represented as:

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/N₀)_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].

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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/T²]
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}

Equations can also be viewed here.

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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

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