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