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