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Data Analysis
Objective
To develop an analysis tool that can be used to characterize experimental datasets from tube flocculator experiments
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Analysis h3. Objective To develop an analysis tool that can be used to characterize experimental datasets from tube flocculator experiments {composition-setup} cloak.toggle.exclusive=false {composition-setup} h3. |
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Methods {cloak:id=Methods} h4. |
Data
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source
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The
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description
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of
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the
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experiments
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can
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be
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viewed
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in
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the
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.
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All
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of
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experimental
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data
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(i.e
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time,
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pressure,
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turbidity,
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flow
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rates,
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etc)
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was
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collected
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using
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Controller and saved in Excel files. Data of interest (i.e.
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effluent
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turbidity)
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was
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stored
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corresponding
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to
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the
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date
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of
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experiment
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run
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and
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the
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status
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file
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indicated
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the
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state
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of
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treatment
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process
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(i.e.
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flocculation
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state,
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settling
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state,
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etc).
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The
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dataset
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was
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extracted
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using
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algorithm
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and
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analyzed
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in
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the
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steps
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provided
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below.
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Model
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fitting
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In
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the
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settling
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state
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(state
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4
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or
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5,
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depending
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on
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the
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experimental
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and
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software
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setup),
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the
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analysis
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of
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settling
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dataset
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starts
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from
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its
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maximum
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turbidity
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reading.
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This
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will
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eliminate
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the
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data
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fluctuation
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due
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to
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a
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sudden
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stop
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of
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flow.
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Data
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is
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normalized
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to
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the
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maximum
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reading
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of
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dataset
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and
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converted
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to
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a
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positive
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hyperbolic
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curve
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for
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data
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interpretation.
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The
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dataset
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will
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now
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read
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as
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the
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amount
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of
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settling
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in
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the
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tube.
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A
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hyperbolic
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curve
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can
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be
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linearized
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using
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double
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reciprocal
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method
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(or
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Lineweaver-Burke
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plot),
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where
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the
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Y-axis
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is
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the
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reciprocal
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of
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the
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cumulative
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amount
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of
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settling
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and
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X-axis
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is
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the
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terminal
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velocity.
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The
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distance
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of
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column
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above
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turbiditimeter
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is
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assumed
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to
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be
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5
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cm.
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Data
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repeatability
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The
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details
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of
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the
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experiment
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conducted
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to
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test
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data
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repeatability
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can
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be
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viewed
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here.
Equations
Assuming the dataset follows a hyperbolic curve function, therefore it can be represented as:
Equation 1
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|Data Fluctuation]. h4. Equations 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} |
or it also can be represented as:
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$$
{N \over {N_0 }} = {{K_S + t - \left( {{N \mathord{\left/
{\vphantom {N {N_0 }}} \right.
\kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } t} \over {K_S + t}}
$$
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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].
This simplification will enable the team to extract important parameters of the curve (i.e. KS and ) and to use these parameters in comparing with other curves.
In order to linearize the hyperbolic function, double reciprocal on both axes can be applied. By using terminal velocity as the reciprocal of time, we can quantify the data using a meaningful parameter instead of using frequency.
Terminal velocity is the velocity of the flocs settling in the column.
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{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 \[T\]. This simplification will enable the team to extract important parameters of the curve (i.e. _K{_}{_}{~}S{~}_ and ) and to use these parameters in comparing with other curves. In order to linearize the hyperbolic function, double reciprocal on both axes can be applied. By using terminal velocity as the reciprocal of time, we can quantify the data using a meaningful parameter instead of using frequency. Terminal velocity is the velocity of the flocs settling in the column. {latex} $$ v = {L \over t} $$ {latex} where _v_ is the terminal velocity \ |
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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{^}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
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of
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Equation
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(2)
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in
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Equation
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(1)
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yields:
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Equation
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3
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} $$ 1 - {N \over {N_0 }} = {{\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } L} \over {K_S v + L}} $$ {latex} |
Reciprocating
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both
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axes
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in
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Equation
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3
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gives:
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Equation
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4
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} $$ {1 \mathord{\left/ {\vphantom {1 {\left( {1 - {N \over {N_0 }}} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {1 - {N \over {N_0 }}} \right)}} = {{K_S v} \over {\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } L}} + {1 \over {\left( {{N \mathord{\left/ {\vphantom {N {N_0 }}} \right. \kern-\nulldelimiterspace} {N_0 }}} \right)_{\max } }} $$ {latex} |
Equations
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can
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also
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be
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viewed here.
Window average
In statistics, a moving average is used to analyze time series data.
Algorithm and equations of window average is attached in Algorithm MathCad file. Here, we specify the specific window for the program to take an average from.
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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[here|TubeFloc - Equations]. h4. Window average In statistics, a moving average is used to analyze time series data. Algorithm and equations of window average is attached in Algorithm MathCad file. Here, we specify the specific window for the program to take an average from. 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
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5b
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} $$ {{\sum\limits_{j = 2i - n + 1}^{n - 1} {Y_j } } \over {2n - 2i - 1}} $$ {latex} |
For
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any
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other
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experimental
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data
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in
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between
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these
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two
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condition,
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Equation
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6
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is
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used.
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Equation
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6
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} $$ {{\sum\limits_{j = i - m}^{i + m} {Y_j } } \over w} $$ {latex} where _w_ is the number of window (odd number) _Y_ is the data _n_ is the length of experimental data _m_ is |
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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Mathcad
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files
Three Mathcad files:
- Data Processor Function: This file extracts and sorts raw data into column of states.
• Algorithm: This file stores information on algorithm, such as double reciprocal method, window average and minor calculations.
• Settling Analysis : This is the main file and it contains sections for metafilter, plots, etc.
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...
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and Discussions
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Discussions {cloak:id=Results and Discussions} h4. |
Model
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fitting
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By
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extracting
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the
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data
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using
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Meta
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Data
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files,
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we
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were
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able
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to
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choose
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any
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set
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of
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settling
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data
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of
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interest,
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plot
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and
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analyze
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them
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individually
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or
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collectively
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to
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see
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any
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if
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there
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is
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any
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to
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the
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data.
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At
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the
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beginning,
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we
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proposed
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polynomial
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fit
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as
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an
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option
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to
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evaluate
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the
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data
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and
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...
...
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was
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utilized
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to
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see
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which
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equations
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fit
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each
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datasets
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the
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best.
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From
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sum
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squared
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error
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analysis,
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the
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third
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and
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fourth
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degree
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of
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polynomial
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were
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found
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to
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be
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the
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best.
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The
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limitation
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of
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using
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polynomial
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fit
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in
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the
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data
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analysis
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is
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to
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specify
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any
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parameter
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that
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would
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be
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responsible
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for
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every
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dataset.
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Each
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time
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a
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dataset
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was
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modeled,
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the
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polynomial
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coefficients
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were
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changing
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and
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we
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couldn't
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find
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any
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trend
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that
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best
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described
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the
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settling
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data.
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It
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is
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also
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hard
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to
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fix
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one
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n
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th
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degree
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of
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polynomial
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for
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the
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entire
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set
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because
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each
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individual
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dataset
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gives
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different
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error
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analysis.
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A
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hyperbolic
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fit
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was
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proposed
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and
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a
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sample
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of
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experimental
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dataset
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was
...
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to
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its
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maximum
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turbidity
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reading
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to
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determine
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the
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percentage
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of
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flocs
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that
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has
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been
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settled
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over
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a
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period
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of
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time.
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Lineweaver-Burke
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or
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double
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reciprocal
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method
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was
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applied.
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We
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analyzed
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one
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set
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of
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data
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from
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iterating
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flow
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rate
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experiment
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(50
...
ft
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tube
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flocculator,
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50NTU
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initial
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turbidity
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and
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25
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mg/L
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alum),
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which
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is
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the
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result
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from
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flow
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rate
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of
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2.95
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mL/s
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for
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600s.
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Figure
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1
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shows
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the
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raw
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data
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of
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settling
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state
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normally
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that
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is
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normally
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obtained
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in
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the
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experiment.
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The
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initial
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drop
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in
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every
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curve
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represents
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the
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bigger
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flocs
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that
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drop
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faster
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than
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the
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smaller
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flocs.
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On
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the
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other
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hand,
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the
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tailing
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of
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the
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curve
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represents
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the
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settling
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of
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the
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smaller
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particles
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in
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the
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settling
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column.
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Ideally,
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the
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drop
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should
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be
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very
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steep
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and
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the
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tailing
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should
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approach
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0.
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The
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raw
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data
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is
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normalized
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in
...
order
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to
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compare
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the
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percentage
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of
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turbidity
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drop
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with
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other
...
sets
...
of
...
data
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in
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the
...
experiments,
...
since
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the
...
initial
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turbidity
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at
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the
...
inlet
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could
...
fluctuate.
...
The
...
turbidity
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drop
...
of
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5NTU
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in
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one
...
run
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would
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not
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mean
...
the
...
same
...
as
...
the
...
drop
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in
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the
...
other
...
run.
...
The
...
normalization
...
would
...
also
...
enable
...
the
...
team
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to
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look
...
at
...
the
...
efficiency
...
of
...
the
...
tube
...
flocculator
...
.
Figure 1. Typical raw data in the settling column
As previously mentioned, the experimental data was flipped on horizontal axis to indicate a "positive growth". Note that the data now is translated as the amount of particles that settle.
In order to quantify which parameter that best describes the data, we linearized the data by reciprocate both of the axes. The time axis (x-axis) was transformed into velocity by dividing the length of the tube above turbidimeter to the time. The result of the linearization is illustrated in Figure 2 and Figure 3.
Figure 2. Best fit linearization using double reciprocal method.
Figure 3. The parameters from linearization was substituted into Equation 1.
The R2 value obtained from the linearization was low (0.622) for this set of data considering the fluctuation in the raw data and the effects of reciprocation, which amplify the lower end and compress the upper end of the original data (x-axis). This also will result in amplification of error of the fluctuated data.
In this example, we did data smoothing with frame window of 33 (w=33). Moving average managed to increase the R2 value to 0.870. However, we can see that the Ks and N/N0 didn't have any improvement.
Figure 4. Raw data and its moving average of 33 (red line). The best fit from linearization is shown here (yellow line).
Table 1. The parameters obtained from linearization of iterating flow rates experimental data.
Flow rates | Ks | (N/N0)max | R2 | Ks 1 | (N/N0)max 1 | R2 |
1.3 | 0.001563 | 0.952 | 0.456 | 0.001417 | 0.934 | 0.709 |
1.45 | 0.0004205 | 0.743 | 0.456 | 0.0003706 | 0.74 | 0.798 |
1.6 | 0.0005225 | 0.828 | 0.647 | 0.000584 | 0.848 | 0.904 |
1.75 | 0.0001045 | 0.666 | 0.233 | 0.0001108 | 0.687 | 0.601 |
1.9 | 0.0005999 | 0.904 | 0.775 | 0.0006628 | 0.923 | 0.952 |
2.05 | 0.0006199 | 0.87 | 0.739 | 0.0007027 | 0.896 | 0.868 |
2.2 | 0.000774 | 0.91 | 0.545 | 0.0008465 | 0.932 | 0.638 |
2.5 | 0.001861 | 1.321 | 0.309 | 0.0008398 | 0.965 | 0.585 |
2.65 | 0.001861 | 1.321 | 0.309 | 0.0008398 | 0.965 | 0.585 |
2.8 | 0.0007028 | 0.854 | 0.654 | 0.0007923 | 0.878 | 0.792 |
2.95 | 0.0002372 | 0.823 | 0.668 | 0.0002776 | 0.839 | 0.893 |
3.1 | 0.0007587 | 0.798 | 0.529 | 0.0006515 | 0.793 | 0.929 |
1 Window average of 33 was applied in the determination of these parameters.
Data repeatability
The results and discussion of data fluctuation for previous experimental setup can be viewed here.
In the previous setup, the average Ks value derived from 10 repeat runs was 0.00391 and the standard deviation was 0.00247. After moving average was applied, the average was 0.00353 and the standard deviation was 0.00241. Both results yielded high standard deviation value. The (N/N0)max values for without and with moving average were 0.6121 and 0.6123, respectively and their standard deviations were 0.293 and 0.294, respectively.
Table 2 The average values of KS and (N/N0)max and their corresponding standard deviation for old and new experimental setups.
Setup | KS | Std dev. | KS 1 | Std dev. | (N/N0)max | Std dev. | (N/N0)max 1 | Std dev. |
Old | 0.00391 | 0.00247 | 0.00352 | 0.00241 | 0.6121 | 0.293 | 0.6123 | 0.294 |
New | 0.00173 | 0.000975 | 0.00199 | 0.00131 | 1.1243 | 0.193 | 1.2135 | 0.294 |
1 moving average of 33 was applied.
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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.
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{cloak} |
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