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