Parameter Formulation – Characterize Collision Potential
Introduction
The formulation of parameters
\large$\theta \varepsilon ^
$
,
\large$K_
$
,
\large$\Pi _
$
and are described below, for characterizing flocculation potential using numerical solutions from CFD simulations. Note that this is a work in progress, so the notation of variables and interpretation of equations still need to be further clarified.
\large$\theta \varepsilon ^
$
:
Calculating a flow weighted average of
\large$\theta \varepsilon ^
$
:
\large${\theta _{baffle}}{\varepsilon ^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 3}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$3$}}}} =
\sum\limits_
\theta _{fe
\varepsilon _
^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 3}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$3$}}}{Q_
}}$
, where
\large$${\theta _{fe}} = {{{\forall _
}} \over Q_{fe
}} = \Delta x\Delta y} \over {\left| {{v_x}\Delta y} \right| + \left| {{v_y}\Delta x} \right|
$$
,
Thus
\large$${\theta _{baffle}}{\varepsilon ^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 3}}\right.\kern-\nulldelimiterspace}!\lower0.7ex\hbox{$3$}}}} =
\sum\limits_
\forall _{fe
\varepsilon _
^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 3}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$3$}}}} $$
\large
${K_{baffle}}$
:
\large
$${\varepsilon _{fe}} = g{h_l
\over \theta _{fe
}}$$
\large
$$
= {K_{baffle}}{V^2
\over {2g}}$$
\large
$$
= {\varepsilon _{fe
{\theta _
}} \over g}$$
\large
$${K_{baffle}} =
2g} \over {{V^2
}$$
where
\large
$${\theta _{fe}} = {{{\forall _
}} \over Q_{fe
}} = \Delta x\Delta y} \over {\left| {{v_x}\Delta y} \right| + \left| {{v_y}\Delta x} \right|
$$
\large
$${K_{baffle}} =
\sum\limits_
{{\varepsilon _{fe
{\theta _
}} \over g}2g} \over {{V^2
}{Q_
}} $$
\large
$${K_{baffle}} = {2 \over {Q
}}\sum\limits_
\varepsilon _{fe
{\forall _
}} $$
\large
$${K_{baffle}} = {2 \over {bw
}}\sum\limits_
\varepsilon _{fe
{\forall _
}} $$
where
\large
$$Q = Vbw$$
\large
${\Pi _{cell}}$
:
\large
$${\varepsilon {cell}} = {K
{baffle
{V^3}} \over {2{\Pi _{cell}}b}}$$
, plug in
\large
${K_{baffle}}$
and simplify:
\large
$${\Pi {cell}} = {1 \over b^2}w
{{{{\left( {\sum\limits
\forall _{fe
\varepsilon _
^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 3}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$3$}}}} } \right)}^{{\raise0.7ex\hbox{$3$} !\mathord{\left/
{\vphantom {3 2}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$2$}}}}} \over {{{\left( {\sum\limits_
\forall _{fe
{\varepsilon _
}} } \right)}^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 2}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$2$}}}}}}$$
\large
$G\theta $
:
\large
$$G{\theta _{baffle}} =
\sum\limits_
G_{fe
{\theta {fe}}{Q
}} $$
\large
$${\theta _{fe}} = {{{\forall _
}} \over Q_{fe
}} = \Delta x\Delta y} \over {\left| {{v_x}\Delta y} \right| + \left| {{v_y}\Delta x} \right|
$$
\large
$$G{\theta _{baffle}} =
\sum\limits_
G_{fe
{\forall _
}} $$
\large
$${G_{fe}} \propto \sqrt {{{{\varepsilon _
}} \over \nu }} $$
\large
$$G{\theta _{baffle}} =
\sum\limits_
{\sqrt {{{{\varepsilon _
}} \over \nu }} {\forall _
}} $$
\large
$$G{\theta {baffle}} = {1 \over {Q\sqrt \nu }}\sum\limits
{\varepsilon _
^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 2}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$2$}}}{\forall _
}} $$
All parameters are calculated from summing over all nodes (finite element)
\large
$$\sum \forall _{fe
\varepsilon _
^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 3}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$3$}}}} $$
,
\large
$$\sum \forall _{fe
\varepsilon _
^{{\raise0.7ex\hbox{$1$} !\mathord{\left/
{\vphantom {1 2}}\right.\kern-\nulldelimiterspace}
!\lower0.7ex\hbox{$2$}}}} $$
,
\large
$$\sum \forall _{fe
{\varepsilon _
}} $$
, which can be calculated using the following UDF script:performance.c.