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h1. Parameter Formulation -- Characterize Collision Potential (under construction...)h3. h3. Introduction The formulation of parameters {latex} \large $\theta \varepsilon ^{1/3} $ {latex} , {latex} \large $K_{baffle} $ {latex} , {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. {latex} \large $\theta \varepsilon ^{1/3} $ {latex} : Calculating a flow weighted average of {latex} \large $\theta \varepsilon ^{1/3} $ {latex} : {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 {latex} \large $\theta \varepsilon ^{1/3} $$${\theta _{fe}} = {{{\forall _{fe}}} \over {{Q_{fe}}}} = {{\Delta x\Delta y} \over {\left| {{v_x}\Delta y} \right| + \left| {{v_y}\Delta x} \right|}}$$ {latex} , Thus {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} {latex} \large $\theta \varepsilon ^{1/3} {K_{baffle}}$ {latex} : {latex} \large $\theta \varepsilon ^{1/3} $$${\varepsilon _{fe}} = {{g{h_l}} \over {{\theta _{fe}}}}$$ {latex} \\ {latex} \large $\theta \varepsilon ^{1/3} $$${h_e} = {K_{baffle}}{{{V^2}} \over {2g}}$$ {latex}\\ \\ {latex} \large $\theta \varepsilon ^{1/3} $$${h_l} = {{{\varepsilon _{fe}}{\theta _{fe}}} \over g}$$ {latex} \\ {latex} \large $\theta \varepsilon ^{1/3} $$${K_{baffle}} = {h_e}{{2g} \over {{V^2}}}$$ {latex}\\ where \\ {latex} \large $$${\theta \varepsilon ^{1/3} $_{fe}} = {{{\forall _{fe}}} \over {{Q_{fe}}}} = {{\Delta x\Delta y} \over {\left| {{v_x}\Delta y} \right| + \left| {{v_y}\Delta x} \right|}}$$ {latex} \\ {latex} \large $\theta \varepsilon ^{1/3} $$${K_{baffle}} = {1 \over Q}\sum\limits_{fe} {{{{\varepsilon _{fe}}{\theta _{fe}}} \over g}{{2g} \over {{V^2}}}{Q_{fe}}} $$ {latex} \\ {latex} \large $\theta \varepsilon ^{1/3} $$${K_{baffle}} = {2 \over {Q{V^2}}}\sum\limits_{fe} {{\varepsilon _{fe}}{\forall _{fe}}} $$ {latex} \\ {latex} \large $\theta \varepsilon ^{1/3} $$${K_{baffle}} = {2 \over {bw{V^3}}}\sum\limits_{fe} {{\varepsilon _{fe}}{\forall _{fe}}} $$ {latex} \\ \\ where {latex} \large $$Q = Vbw$$ {latex} {latex} \large ${\theta \varepsilon ^{1/3} Pi _{cell}}$ {latex} : {latex} \large $\theta \varepsilon ^{1/3} $$${\varepsilon _{cell}} = {{{K_{baffle}}{V^3}} \over {2{\Pi _{cell}}b}}$$ {latex} , plug in {latex} \large $\theta \varepsilon ^{1/3} {K_{baffle}}$ {latex} and simplify: {latex} \large $\theta \varepsilon ^{1/3} $ {latex} {latex} \large $\theta \varepsilon ^{1/3}$${\Pi _{cell}} = {1 \over {{b^2}w}}{{{{\left( {\sum\limits_{fe} {{\forall _{fe}}\varepsilon _{fe}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} } \right)}^{{\raise0.7ex\hbox{$3$} \!\mathord{\left/ {\vphantom {3 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}} \over {{{\left( {\sum\limits_{fe} {{\forall _{fe}}{\varepsilon _{fe}}} } \right)}^{{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}}}}$$ {latex} {latex} \large $G\theta $ {latex} : {latex} \large $\theta \varepsilon ^{1/3} $$$G{\theta _{baffle}} = {1 \over Q}\sum\limits_{fe} {{G_{fe}}{\theta _{fe}}{Q_{fe}}} $$ {latex} \\ {latex} \large $$${\theta \varepsilon ^{1/3} $_{fe}} = {{{\forall _{fe}}} \over {{Q_{fe}}}} = {{\Delta x\Delta y} \over {\left| {{v_x}\Delta y} \right| + \left| {{v_y}\Delta x} \right|}}$$ {latex} \\ {latex} \large $$$G{\theta \varepsilon ^{1/3} $_{baffle}} = {1 \over Q}\sum\limits_{fe} {{G_{fe}}{\forall _{fe}}} $$ {latex} \\ {latex} \large $\theta $${G_{fe}} \propto \sqrt {{{{\varepsilon ^{1/3} $_{fe}}} \over \nu }} $$ {latex} \\ {latex} \large $$$G{\theta _{baffle}} = {1 \over Q}\varepsilon ^{1/3} $sum\limits_{fe} {\sqrt {{{{\varepsilon _{fe}}} \over \nu }} {\forall _{fe}}} $$ {latex} \\ {latex} \large $$$G{\theta _{baffle}} = {1 \over {Q\sqrt \nu }}\sum\limits_{fe} {\varepsilon _{fe}^{1/3} ${{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}{\forall _{fe}}} $$ {latex} All parameters are calculated from summing over all nodes (finite element) {latex} \large $\theta $$\sum {{\forall _{fe}}\varepsilon _{fe}^{{\raise0.7ex\hbox{$1$} \!\mathord{1/3} $\left/ {\vphantom {1 3}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$3$}}}} $$ {latex} , {latex} \large $$$\thetasum {{\forall _{fe}}\varepsilon _{fe}^{{1/3} $\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-\nulldelimiterspace} \!\lower0.7ex\hbox{$2$}}}} $$ {latex} , {latex} \large $$$\thetasum {{\forall _{fe}}{\varepsilon ^{1/3_{fe}}} $$$ {latex} , which can be calculated using the following UDF script:[parameters.c|^parameters.c] |
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