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h2. Data Analysis

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h3. {toggle-cloak:id=Methods} Methods

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h4. Data source

Data from [previous experiments| Overall Tube Floc Research and Past Report] were collected using Process Controller and saved in Excel files. Data of interest (i.e. effluent turbidity) was stored in a file corresponding to the date of experiment run and the status file indicated the state of treatment process (i.e. flocculation state, settling state, etc). Then the dataset was extracted using [Meta Data|Meta Data] algorithm and analyzed in the steps provided below.

h4. Data repeatability

The details of the experiment conducted to test data repeatability can be viewed [here|Data Fluctuation].

h4. Model fitting

In the settling state (state 4 or 5, depending on the experimental and software setup), the analysis of settling dataset starts from its maximum turbidity reading. This will eliminate the data fluctuation due to a sudden stop of flow.  Data is normalized to the maximum reading of dataset and converted to a positive hyperbolic curve for data interpretation. The dataset will now read as the cumulative amount of settling in the tube. A hyperbolic curve can be linearized using double reciprocal method (or _Lineweaver-Burke_ plot), where the Y-axis is the reciprocal of the cumulative amount of settling and X-axis is the terminal velocity. The length of tube shone by the light is assumed to be 5 cm.

h4. Equations

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

*Equation 1*
{latex}
$$
\displaylines{
1 - {{NTU} \over {NTU_{\max } }} = {t \over {K + t}} \cr
{{NTU} \over {NTU_{\max } }} = {K \over {K + t}} \cr}
$$
{latex}
where
_NTU_ is the effluent turbidity \[-\]
_t_ is the time \[T\]
_K_ is the rate of settling flocs \[T\].

This simplification will enable the team to extract an important parameter of the curve (i.e. _K_)

Terminal velocity is the velocity of the flocs settling in the column.
{latex}
$$
v = {d \over t}
$$
{latex}
where
v is the terminal velocity

Assuming _d_ as the length of tube shone by the light, _t_ can be redefined as:
*Equation 2*
{latex}
$$
t = {d \over v}
$$
{latex}
substitution of Equation (2) in Equation (1) yields:

*Equation 3*
{latex}
$$
\displaylines{
1 - {{NTU} \over {NTU_{\max } }} = {d \over {Kv + d}} \cr
{{NTU} \over {NTU_{\max } }} = {{Kv} \over {Kv + d}} \cr}
$$
{latex}
Double reciprocal equation
*Equation 4*
{latex}
$$
{1 \mathord{\left/
{\vphantom {1 {\left( {1 - {{NTU} \over {NTU_{\max } }}} \right)}}} \right.
\kern-\nulldelimiterspace} {\left( {1 - {{NTU} \over {NTU_{\max } }}} \right)}} = {{Kv} \over d} + 1
$$
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Equations can also be viewed [here|TubeFloc - Equations].

h4. Window average

Algorithm and equations of window average is attached in Algorithm MathCad file.

h4. Mathcad files

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h3. {toggle-cloak:id=Results and Discussions} Results and Discussions

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h4. Data repeatability

The results and discussion of data fluctuation can be viewed [here|Data Fluctuation].

h4. Model fitting

By extracting the data using Meta Data files, we were able to choose any set of settling data of interest, plot and analyze them individually or collectively to see any if there is any [trend|First Look into the Data] to the data. We proposed polynomial fit as an option to evaluate the data and [error analysis algorithm|Error Analysis] was utilized to see which equations fit each datasets the best.

[Another datasets example|Another example...] is also analyzed to determine which polynomial fit suits the best. From sum squared error analysis, the third and fourth degree of polynomial were found to be the best.[Another raw sedimentation data|Raw Data of Settling Column] was evaluated to get a better sense of parameters that actually effect the curve.

The limitation of using polynomial fit in the data analysis is hard to determine the any parameter that is responsible to describe the dataset. Each time a dataset was analyzed, the polynomial coefficients were changing and we couldn't find any trend that best described the settling data.

A hyperbolic fit was proposed and a sample of experimental dataset was [normalized|Data Normalization] to its maximum turbidity reading to determine the percentage of flocs that has been settled over a period of time. _Lineweaver-Burke_ data analysis was applied.

Examples of [data linearization|Data Linearization] using _double reciprocal_ analysis.

Table 1 shows the _K_ values of 10 repeatable experimental runs to determine the efficiency of the system.




From table 2, the R{^}2^ values are improving when the data is window averaged. 

*Table 2* _K_ values and its corresponding R{^}2^ values for different flow rates of raw data and window averaged data (w=33).
| | Raw | | Smooth | |
| Q | K | R{^}2^ | K | R{^}2^ |
| 1.6 | 0.003799 | 0.514 | 0.003312 | 0.935 |
| 1.75 | 0.005004 | 0.522 | 0.004601 | 0.937 |
| 1.9 | 0.002506 | 0.329 | 0.00228 | 0.818 |
| 2.05 | 0.002778 | 0.749 | 0.002737 | 0.954 |
| 2.2 | 0.004961 | 0.632 | 0.00443 | 0.972 |
| 2.35 | 0.01 | 0.046 | 0.007944 | 0.839 |
| 2.5 | 0.008992 | 0.414 | 0.007599 | 0.638 |
| 2.65 | 0.009679 | 0.477 | 0.008392 | 0.909 |
| 2.95 | 0.002599 | 0.468 | 0.002769 | 0.807 |
| 3.1 | 0.003805 | 0.715 | 0.004098 | 0.891 |
| 3.25 | 0.018 | 0.331 | 0.009059 | 0.895 |
| 3.4 | 0.005315 | 0.741 | 0.005201 | 0.957 |

From the [data linearization|Data Linearization] results, we can that even though the linearization of raw data yields good range of R{^}2^ values, it gives different output when substituting into the original. This could be due to the effect of dataset reciprocal and recirculation problem that is occurring in the settling tube in settling state.

*Table 3* From new setup

|K	|R{^}2^    |
|4.04E-04|0.931|
|2.61E-03|0.567|
|1.23E-03|0.85|
|1.45E-03|0.856|
|1.27E-03|0.87|
|1.47E-03|0.871|
|1.01E-03|0.878|
|2.10E-03|0.325|
|1.74E-03|0.87|
|1.45E-03|0.777|

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h2. {toggle-cloak:id=Progress} Progress

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Apart from analyzing past datasets, the team also interested to analyze datasets from new setup, which addresses some of the problems encountered in the previous setup. The new setup can be viewed here and the analysis of experimental runs can be viewed [here|Data Analysis on New Setup].
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