# LaTeX2HTML 2002-2-1 (1.70)
# Associate images original text with physical files.


$key = q/displaystylefrac{9}{8}left(frac{u_{i,j}-u_{i-1,j}}{Deltax}right)-frac{1}{24}left(frac{u_{i+1,j}-u_{i-2,j}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="327" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img443.png"
 ALT="$\displaystyle \frac{9}{8}\left(\frac{u_{i,j} - u _{i-1,j}}{\Delta x}\right)
- \frac{1}{24}\left(\frac{u_{i+1,j} - u _{i-2,j}}{\Delta x}\right)$">|; 

$key = q/displaystyle+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{tkm-1(w)}left(DP{pi^{tau+Deltatau}}{z}right)_{i,km-1(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="481" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img375.png"
 ALT="$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
..._{km-1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km-1(w)}
\right\}$">|; 

$key = q/{displaymath}mbox{dcstaff@gfd-dennou.org}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="190" HEIGHT="35" BORDER="0"
 SRC="|."$dir".q|img495.png"
 ALT="\begin{displaymath}
\mbox{dcstaff@gfd-dennou.org}
\end{displaymath}">|; 

$key = q/displaystylegamma_{v}=alpha_{v}left(1-cos{frac{pi(z-z_{max}-d_{v})}{d_{v}}}right)^{3};;(z>z_{max}-d_{v}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img351.png"
 ALT="$\displaystyle \gamma_{v} = \alpha_{v} \left
( 1 - \cos{\frac{\pi (z - z_{max} - d_{v})}{d_{v}}}
\right)^{3}
\;\; (z &gt; z_{max} - d_{v}),$">|; 

$key = q/C_{varepsilon}=C_{m}=0.2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img318.png"
 ALT="$C_{\varepsilon} = C_{m} = 0.2$">|; 

$key = q/displaystyleleft[q_{{rm{NH_4SH}right]^{t+Deltat}=[q_{{rm{NH_3}]^{*}+[q_{{rm{H_2S}]^{*}-[q_{{rm{NH_3}]^{t+Deltat}-[q_{{rm{H_2S}]^{t+Deltat},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="461" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img292.png"
 ALT="$\displaystyle \left[ q_{\rm NH_4SH} \right]^{t + \Delta t}
= [q_{\rm NH_3}]^{*}...
...rm H_2S}]^{*}
- [q_{\rm NH_3}]^{t + \Delta t} - [q_{\rm H_2S}]^{t + \Delta t} ,$">|; 

$key = q/displaystylepi^{tau}_{i,1}-(1-beta)left(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rtheta}_{v}}right)_{1}left(DP{u^{tau+Deltatau}}{x}right)_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="528" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img132.png"
 ALT="$\displaystyle \pi^{\tau}_{i,1}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau...
...ar{\theta}_{v}}
\right)_{1}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,1}$">|; 

$key = q/1leqi(u)leqim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="119" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img15.png"
 ALT="$1 \leq i(u) \leq im$">|; 

$key = q/displaystyle-betaleft(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{thetabar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{km(w)}E_{i,km(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img376.png"
 ALT="$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}$">|; 

$key = q/displaystyle-DP{mu_{{rm{NH_4SH}}{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img463.png"
 ALT="$\displaystyle - \DP{\mu_{\rm NH_4SH}}{T}$">|; 

$key = q/displaystyleleft[{{rm{Diff}.uright]_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img187.png"
 ALT="$\displaystyle \left[ {\rm Diff}.u \right]_{i(u),k}^{t - \Delta t}$">|; 

$key = q/bar{pi};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img341.png"
 ALT="$\bar{\pi}$">|; 

$key = q/displaystyle-betaleft(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rho}bar{theta}_{v}^lphaDiv)^{tau}}{z}right}+F_{w}^{t}Deltatauright}right]_{i,km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="657" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img142.png"
 ALT="$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rh...
...\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau
\right\}
\right]_{i,km}$">|; 

$key = q/displaystyleA_{km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img151.png"
 ALT="$\displaystyle A_{km}$">|; 

$key = q/1leqkleqkm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="102" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img18.png"
 ALT="$1 \leq k \leq km$">|; 

$key = q/displaystyleleft[DP{u}{x}right]_{i,k}Deltax+frac{1}{3}left[DP[3]{u}{x}right]_{i,60}left[DP[5]{u}{x}right]_{i,k}left(frac{Deltax}{2}right)^{5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="448" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img421.png"
 ALT="$\displaystyle \left[\DP{u}{x} \right]_{i,k}\Delta x
+ \frac{1}{3}\left[\DP[3]{u...
...frac{1}{60}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{5}$">|; 

$key = q/0leqxleqx_{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img344.png"
 ALT="$0 \leq x \leq
x_{max}$">|; 

$key = q/displaystyle-frac{2}{3C_{m}^{2}l^{2}}left(DP{K_{m}^{2}}{x}right)_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img182.png"
 ALT="$\displaystyle - \frac{2}{3 C_{m}^{2} l^{2}}
\left( \DP{ K_{m}^{2} }{x} \right)_{i(u),k}^{t - \Delta t}$">|; 

$key = q/displaystylegamma_{h}=0hspace{10em}(d_{h}leqxleqx_{max}-d_{h}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="424" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img346.png"
 ALT="$\displaystyle \gamma_{h} = 0 \hspace{10em} ( d_{h} \leq x \leq x_{max} - d_{h}),$">|; 

$key = q/N^{2}=frac{g}{overline{theta}_{v}}DP{overline{theta}_{v}}{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="48" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img395.png"
 ALT="$N^{2} = \frac{g}{\overline{\theta}_{v}}\DP{
\overline{\theta}_{v}}{z}$">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{t+Deltatau}_{i,2}-pi^{tau+Deltatau}_{i,1}}{Deltaz}right)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="431" HEIGHT="70" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img368.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...\Delta \tau}_{i,2}
- \pi^{\tau + \Delta \tau}_{i,1}}{\Delta z}
\right)
\right\}$">|; 

$key = q/displaystyleleft[DP{phi}{x}right]_{i,k(w)}equivfrac{9}{8}left(frac{phi_{i(u),k(w}left(frac{phi_{i+1(u),k(w)}-phi_{i-2(u),k(w)}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="595" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img46.png"
 ALT="$\displaystyle \left[\DP{\phi}{x} \right]_{i,k(w)}
\equiv \frac{9}{8}
\left(\fra...
...{1}{24}
\left(\frac{\phi_{i+1(u), k(w)} - \phi_{i-2(u), k(w)}}{\Delta x}\right)$">|; 

$key = q/CN_{vc},EV_{cv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="96" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img219.png"
 ALT="$CN_{vc}, EV_{cv}$">|; 

$key = q/displaystyleu_{im+2(u),k}=u_{2(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img80.png"
 ALT="$\displaystyle u_{im+2(u), k} = u_{2(u), k}$">|; 

$key = q/[q_c]^{t+Deltat}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img269.png"
 ALT="$[q_c]^{t + \Delta t} &gt; 0$">|; 

$key = q/[q_{c}]^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img257.png"
 ALT="$[q_{c}]^{*}$">|; 

$key = q/displaystyleDD{p_{v}}{T}=frac{p_{v}L_{v}}{RT^{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img456.png"
 ALT="$\displaystyle \DD{p_{v}}{T} = \frac{p_{v} L_{v}}{R T^{2}}$">|; 

$key = q/(u_{i,j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img410.png"
 ALT="$(u_{i,j})$">|; 

$key = q/omega;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img399.png"
 ALT="$\omega$">|; 

$key = q/displaystyle[{{rm{Adv}.K_m]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img305.png"
 ALT="$\displaystyle [{\rm Adv}.K_m]_{i,k}^{t}$">|; 

$key = q/displaystyleX=Dinv{2}left{(p_{{rm{NH_{3}}+p_{{rm{H_{2}S})pmsqrt{(p_{{rm{NH_{3}}+}S})^{2}-4(p_{{rm{NH_{3}}cdotp_{{rm{H_{2}S}-e^{K_{p}})}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="536" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img281.png"
 ALT="$\displaystyle X = \Dinv{2}
\left\{
(p_{\rm NH_{3}} + p_{\rm H_{2}S})
\pm \sqrt{...
...m H_{2}S})^{2}
- 4 (p_{\rm NH_{3}} \cdot p_{\rm H_{2}S} - e^{K_{p}}) }
\right\}$">|; 

$key = q/displaystyle+gfrac{theta_{i,k(w)}}{overline{theta}_{i,k(w)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="79" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img58.png"
 ALT="$\displaystyle + g \frac{\theta_{i,k(w)}}{\overline{\theta}_{i,k(w)}}$">|; 

$key = q/[q_{r}]^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img258.png"
 ALT="$[q_{r}]^{*}$">|; 

$key = q/displaystyleDeltas=RTDP{K_{p}}{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img472.png"
 ALT="$\displaystyle \Delta s
= RT \DP{K_{p}}{T}$">|; 

$key = q/pi_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img21.png"
 ALT="$\pi_{i, k}$">|; 

$key = q/textstyle=;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img52.png"
 ALT="$\textstyle =$">|; 

$key = q/displaystylepi_{i(u),k}equivfrac{pi_{i+1,k}+pi_{i,k}}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img22.png"
 ALT="$\displaystyle \pi_{i(u),k} \equiv \frac{\pi_{i+1, k} + \pi_{i, k}}{2}$">|; 

$key = q/(u_{i-1(u),j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="81" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img415.png"
 ALT="$(u_{i-1(u),j})$">|; 

$key = q/_4;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img461.png"
 ALT="$_4$">|; 

$key = q/displaystyle-[{{rm{Adv}.K_m]_{i,k}^{t}+[{{rm{Buoy}.K_m]_{i,k}^{t-Deltat}+[{{rm{Shear}.K_m]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img301.png"
 ALT="$\displaystyle - [{\rm Adv}.K_m]_{i,k}^{t}
+ [{\rm Buoy}.K_m]_{i,k}^{t - \Delta t}
+ [{\rm Shear}.K_m]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyleDP{w_{i,k(w)}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img55.png"
 ALT="$\displaystyle \DP{w_{i,k(w)}}{t}$">|; 

$key = q/displaystyle(1-2gamma)u^{t}_{i(u),k}+gamma(u^{*}_{i(u),k}+u^{t-Deltat}_{i(u),k});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img332.png"
 ALT="$\displaystyle (1-2 \gamma)u^{t}_{i(u),k} +
\gamma (u^{*}_{i(u),k} + u^{t -\Delta t}_{i(u),k})$">|; 

$key = q/displaystyleleft{-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}}bar{theta}_{v}^{2}right)_{1(w)}right}pi^{tau+Deltatau}_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="707" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img131.png"
 ALT="$\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b...
...ho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\} \pi^{\tau + \Delta \tau}_{i,1}$">|; 

$key = q/displaystylepi^{tau}-(1-beta)frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rho}bar{thet{2}Deltatau}{bar{c_{p}}bar{theta}_{v}}DP{u^{tau+Deltatau}}{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img116.png"
 ALT="$\displaystyle \pi^{\tau}
-(1 - \beta)
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}}...
...}^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}}
\DP{u^{\tau + \Delta \tau}}{x}$">|; 

$key = q/PR_{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img247.png"
 ALT="$PR_{r}$">|; 

$key = q/i(u),k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img10.png"
 ALT="$i(u),k$">|; 

$key = q/dS=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img251.png"
 ALT="$dS=0$">|; 

$key = q/displaystylepi^{tau+Deltatau}_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img365.png"
 ALT="$\displaystyle \pi^{\tau + \Delta \tau}_{i,1}$">|; 

$key = q/displaystylenu_{v}=frac{alpha_{v}Deltaz^{2}}{Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img194.png"
 ALT="$\displaystyle \nu_{v} = \frac{\alpha_{v} \Delta z^{2}}{\Delta t}$">|; 

$key = q/overline{rho}_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img49.png"
 ALT="$\overline{\rho}_{i,k}$">|; 

$key = q/[theta]^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="32" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img255.png"
 ALT="$[\theta]^{*}$">|; 

$key = q/displaystyleleft[q_{v}right]^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img264.png"
 ALT="$\displaystyle \left[ q_{v} \right]^{t + \Delta t}$">|; 

$key = q/displaystyle+[{{rm{Src}.q_{v}]_{i,k}+[{{rm{Turb}.{overline{q_{v}}}]_{i,k}+[{{rm{Turb}.{q_{v}}]_{i,k},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="321" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img69.png"
 ALT="$\displaystyle + [{\rm Src}.q_{v}]_{i,k}
+ [{\rm Turb}.{\overline{q_{v}}}]_{i,k} + [{\rm Turb}.{q_{v}}]_{i,k},$">|; 

$key = q/displaystyleleft[DP{pi}{z}right]_{i,k(w)}equivfrac{pi_{i,k+1}-pi_{i,k}}{Deltaz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="211" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img33.png"
 ALT="$\displaystyle \left[\DP{\pi}{z} \right]_{i,k(w)}
\equiv \frac{\pi_{i, k+1} - \pi_{i, k}}{\Delta z}$">|; 

$key = q/displaystylepi^{tau+Deltatau}_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img356.png"
 ALT="$\displaystyle \pi^{\tau + \Delta \tau}_{i,k}$">|; 

$key = q/displaystyleL_{{rm{NH_4SH}=TDeltas=RT^{2}DP{K_{p}}{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img473.png"
 ALT="$\displaystyle L_{\rm NH_4SH} = T \Delta s = RT^{2} \DP{K_{p}}{T}$">|; 

$key = q/d_{h},d_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img354.png"
 ALT="$d_{h}, d_{v}$">|; 

$key = q/{displaymath}left[DP{overline{pi}}{z}right]_{i,k}=-frac{g}{{c_{p}}_{d}[overline{theta_{v}}]_{i,k}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="166" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img48.png"
 ALT="\begin{displaymath}
\left[\DP{\overline{\pi}}{z}\right]_{i,k} =
- \frac{g}{{c_{p}}_{d} [\overline{\theta_{v}}]_{i,k}}
\end{displaymath}">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{t{2}}left(bar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{k-1(w)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img154.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...lta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)},$">|; 

$key = q/displaystyle+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{tght)_{0(w)}left(DP{pi^{tau+Deltatau}}{z}right)_{i,0(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img367.png"
 ALT="$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{0(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}
\right\}$">|; 

$key = q/displaystyleleft(C_{m}^{2}l^{2}right)_{i,k}left[left(DP{u}{x}right)^{2}right]_{ight)_{i,k}left[left(DP{w}{z}right)^{2}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img310.png"
 ALT="$\displaystyle \left( C_{m}^{2} l^{2} \right)_{i,k}
\left[
\left( \DP{u}{x} \rig...
... \right)_{i,k}
\left[
\left( \DP{w}{z} \right)^{2}
\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/u=hat{u}e^{i(kx+lz-omegat)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="20" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img393.png"
 ALT="$u=\hat{u}e^{i(kx+lz -\omega t)}$">|; 

$key = q/k(w)=0(w);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img126.png"
 ALT="$k(w) = 0(w)$">|; 

$key = q/displaystyleA_{1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img149.png"
 ALT="$\displaystyle A_{1}$">|; 

$key = q/displaystyleleft[F_{q_{c}}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img213.png"
 ALT="$\displaystyle \left[F_{q_{c}}\right]_{i,k}^{t}$">|; 

$key = q/displaystyleleft[q_{c}right]_{i,k}^{t-Deltat}+2Deltat[F_{q_{c}}]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img204.png"
 ALT="$\displaystyle \left[ q_{c} \right]_{i,k}^{t-\Delta t}
+ 2 \Delta t [F_{q_{c}}]_{i,k}^{t}$">|; 

$key = q/displaystyleDP{u}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img379.png"
 ALT="$\displaystyle \DP{u}{t}$">|; 

$key = q/w^{tau+Deltatau};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img112.png"
 ALT="$w^{\tau +
\Delta \tau}$">|; 

$key = q/displaystylelnp^{*}=(A-Bslash(C+T-T_{0}))*log(10.0)+ln(133.322);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img448.png"
 ALT="$\displaystyle \ln p^{*}= (A - B / (C + T - T_{0})) * \log(10.0) + \ln(133.322)$">|; 

$key = q/textstyle-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img114.png"
 ALT="$\textstyle -$">|; 

$key = q/displaystyleu^{*}_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img328.png"
 ALT="$\displaystyle u^{*}_{i(u),k}$">|; 

$key = q/_3;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img297.png"
 ALT="$_3$">|; 

$key = q/mbox{Max}(epsilon)leq1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="98" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img403.png"
 ALT="$\mbox{Max}(\epsilon) \leq 1$">|; 

$key = q/displaystyleu_{im+1(u),k}=-u_{im-1(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img85.png"
 ALT="$\displaystyle u_{im+1(u),k} = - u_{im-1(u),k}$">|; 

$key = q/[Q_{rad}]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img240.png"
 ALT="$[Q_{rad}]_{i,k}$">|; 

$key = q/Deltaq_{{rm{NH_3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img295.png"
 ALT="$\Delta q_{\rm NH_3}$">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{tt)_{km(w)}left(DP{pi^{tau+Deltatau}}{z}right)_{i,km(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="447" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img374.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...ght)_{km(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km(w)}
\right\}$">|; 

$key = q/displaystyle-{c_{p}}_{d}[overline{theta_{v}}]_{i,k(w)}left[DP{pi}{z}right]_{i,k(w)}+left[{{rm{Turb}.{w}right]_{i,k(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="322" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img57.png"
 ALT="$\displaystyle - {c_{p}}_{d} [\overline{\theta_{v}}]_{i,k(w)}\left[\DP{\pi}{z}\right]_{i,k(w)}
+ \left[{\rm Turb}.{w}\right]_{i,k(w)}$">|; 

$key = q/w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img98.png"
 ALT="$w$">|; 

$key = q/{displaymath}overline{c}_{i,k}^{2}=frac{{c_{p}}_{d}R_{d}}{c_{v}}overline{pi}_{i,k}[overline{theta_{v}}]_{i,k}.{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="174" HEIGHT="47" BORDER="0"
 SRC="|."$dir".q|img63.png"
 ALT="\begin{displaymath}
\overline{c}_{i,k}^{2} = \frac{{c_{p}}_{d} R_{d}}{c_{v}}
\overline{\pi}_{i,k} [\overline{\theta_{v}}]_{i,k}.
\end{displaymath}">|; 

$key = q/displaystyleleft[{{rm{Adv}.{bar{phi}}right]_{i,k}^{t}=left[w_{i,k(w)}left[DP{overline{phi}}{z}right]_{i,k(w)}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img226.png"
 ALT="$\displaystyle \left[{\rm Adv}.{\bar{\phi}}\right]_{i,k}^{t} =
\left[
w_{i,k(w)} \left[ \DP{\overline{\phi}}{z} \right]_{i,k(w)}
\right]_{i,k}^{t}$">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{tght)_{1(w)}left(DP{pi^{tau+Deltatau}}{z}right)_{i,1(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img366.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,1(w)}
\right\}$">|; 

$key = q/{displaymath}omega^{4}+ialpha(k^{2}+l^{2})omega^{3}-[overline{c}^{2}(k^{2}+l^{2}lphak^{2}N^{2}omega+overline{c}^{2}k^{2}N^{2}=0,{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="552" HEIGHT="36" BORDER="0"
 SRC="|."$dir".q|img394.png"
 ALT="\begin{displaymath}
\omega ^{4} + i\alpha (k^{2}+l^{2})\omega ^{3} -
[\overli...
...}
- i\alpha k^{2}N^{2}\omega + \overline{c}^{2}k^{2}N^{2}=0,
\end{displaymath}">|; 

$key = q/R_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img459.png"
 ALT="$R_{v}$">|; 

$key = q/displaystyle+left[CN_{cr}+CL_{cr}-EV_{rv}right]_{i,k}^{t}+left[PR_{r}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img218.png"
 ALT="$\displaystyle + \left[CN_{cr} + CL_{cr} - EV_{rv} \right]_{i,k}^{t}
+ \left[ PR_{r} \right]_{i,k}^{t}$">|; 

$key = q/displaystyleleft[DP{w}{z}right]_{i,k}equivfrac{w_{i,k(w)}-w_{i,k-1(w)}}{Deltaz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="240" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img37.png"
 ALT="$\displaystyle \left[\DP{w}{z} \right]_{i,k}
\equiv \frac{w_{i, k(w)} - w_{i, k-1(w)}}{\Delta z}$">|; 

$key = q/t;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="11" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img107.png"
 ALT="$t$">|; 

$key = q/displaystyle-gfrac{sum[q_{v}]_{i,k(w)}^{t}+sum[q_{c}]_{i,k(w)}^{t}+sum[q_{r}]_{i,k(w)}^{t}}{1+sum[bar{q_{v}}]_{i,k(w)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img178.png"
 ALT="$\displaystyle - g \frac{\sum [q_{v}]_{i,k(w)}^{t}
+ \sum [q_{c}]_{i,k(w)}^{t} + \sum [q_{r}]_{i,k(w)}^{t}}
{1 + \sum [\bar{q_{v}}]_{i,k(w)}}$">|; 

$key = q/displaystyle(k=2,3,cdotskm),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="148" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img155.png"
 ALT="$\displaystyle (k = 2, 3, \cdots km),$">|; 

$key = q/displaystyleX={{rm{min}(P_{{rm{NH_3},P_{{rm{H_{2}S});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="178" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img284.png"
 ALT="$\displaystyle X = {\rm min}(P_{\rm NH_3}, P_{\rm H_{2}S} )$">|; 

$key = q/P_{{rm{NH_3}>P_{{rm{H_{2}S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img285.png"
 ALT="$P_{\rm NH_3} &gt; P_{\rm H_{2}S}$">|; 

$key = q/displaystyleleft[{{rm{Disp}.K_mright]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="118" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img316.png"
 ALT="$\displaystyle \left[{\rm Disp}.K_m\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystylefrac{q_{x}}{M_{x}(1slashM_{d}+sum_{v}q_{v}slashM_{v})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img494.png"
 ALT="$\displaystyle \frac{q_{x}}{M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})}$">|; 

$key = q/z;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img11.png"
 ALT="$z$">|; 

$key = q/displaystyleA_{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img146.png"
 ALT="$\displaystyle A_{k}$">|; 

$key = q/displaystylefrac{p_{x}M_{x}(1slashM_{d}+sum_{vneqx}q_{v}slashM_{v})}{p-p_{x}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img486.png"
 ALT="$\displaystyle \frac{ p_{x} M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v}) }{p - p_{x}}$">|; 

$key = q/displaystyleu_{i-2(u),j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img434.png"
 ALT="$\displaystyle u _{i-2(u),j}$">|; 

$key = q/nu_{h},nu_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="48" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img192.png"
 ALT="$\nu_{h}, \nu_{v}$">|; 

$key = q/mu_{vapar}=mu_{condensedphase};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="184" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img252.png"
 ALT="$\mu_{vapar} = \mu_{condensed phase}$">|; 

$key = q/k(w)=km(w);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img135.png"
 ALT="$k(w) = km(w)$">|; 

$key = q/displaystyleleft[PR_{r}right]_{i,k}^{t}=Dinv{[bar{rho}]_{i,k}}DP{}{z}([bar{rho}]_{i,k}[U_{r}]_{i,k}^{t}[q_{r}]_{i,k}^{t}).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="304" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img248.png"
 ALT="$\displaystyle \left[
PR_{r}
\right]_{i,k}^{t}
= \Dinv{[\bar{\rho}]_{i,k}} \DP{}{z}([\bar{\rho}]_{i,k}
[U_{r}]_{i,k}^{t}
[q_{r}]_{i,k}^{t}).$">|; 

$key = q/displaystyleleft{-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}ar{theta}_{v}^{2}right)_{k(w)}right}pi^{tau+Deltatau}_{i,k+1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="338" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img118.png"
 ALT="$\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b...
...} \bar{\theta}_{v}^{2}
\right)_{k(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k+1}$">|; 

$key = q/displaystylew_{i(u),k(w)}equivfrac{w_{i+1,k(w)}+w_{i,k(w)}}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="244" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img30.png"
 ALT="$\displaystyle w_{i(u),k(w)} \equiv \frac{w_{i+1, k(w)} + w_{i, k(w)}}{2}$">|; 

$key = q/q_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img221.png"
 ALT="$q_{v}$">|; 

$key = q/displaystyleE_{i,km(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img138.png"
 ALT="$\displaystyle E_{i,km(w)}$">|; 

$key = q/displaystyle-left[CN_{cr}+CL_{cr}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="156" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img215.png"
 ALT="$\displaystyle - \left[ CN_{cr} + CL_{cr} \right]_{i,k}^{t}$">|; 

$key = q/displaystyleu^{tau+(N_{tau}-1)Deltatau}_{i(u),k}-left[bar{c_{p}}bar{theta}_{v}De(alphaDiv)^{tau+(N_{tau}-1)Deltatau}}{x}right}right]_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img329.png"
 ALT="$\displaystyle u^{\tau + (N_{\tau}-1)\Delta \tau}_{i(u), k}
- \left[
\bar{c_{p}}...
...\DP{(\alpha Div)^{\tau + (N_{\tau}-1)\Delta \tau}}{x}
\right\}
\right]_{i(u),k}$">|; 

$key = q/displaystyleD;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img387.png"
 ALT="$\displaystyle D$">|; 

$key = q/displaystyleleft[DP{u}{x}right]_{i,k}24Deltax-frac{216}{60}left[DP[5]{u}{x}right]_{i,k}left(frac{Deltax}{2}right)^{5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="305" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img441.png"
 ALT="$\displaystyle \left[\DP{u}{x} \right]_{i,k}24\Delta x
- \frac{216}{60}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{5}$">|; 

$key = q/displaystyleleft[F_{q_{r}}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img216.png"
 ALT="$\displaystyle \left[F_{q_{r}}\right]_{i,k}^{t}$">|; 

$key = q/exp{(K_{p})}approx0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="109" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img283.png"
 ALT="$\exp{(K_{p})} \approx 0$">|; 

$key = q/displaystyleDeqref{uwpi:sabun}mbox{leftside};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img355.png"
 ALT="$\displaystyle \Deqref{uwpi:sabun}\mbox{left side}$">|; 

$key = q/displaystyleu_{i(u),j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img411.png"
 ALT="$\displaystyle u _{i(u),j}$">|; 

$key = q/displaystylepi_{im+2,k}=pi_{im-1,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="143" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img94.png"
 ALT="$\displaystyle \pi_{im+2,k} = \pi_{im-1,k}$">|; 

$key = q/q_{c}^{*}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img261.png"
 ALT="$q_{c}^{*} &gt; 0$">|; 

$key = q/{displaymath}left|frac{3}{640}left[DP[5]{u}{x}right]_{i(u),k}left(Deltaxright)^{4}right|{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="179" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img447.png"
 ALT="\begin{displaymath}
\left\vert\frac{3}{640}\left[\DP[5]{u}{x} \right]_{i(u),k}
\left(\Delta x\right)^{4}\right\vert
\end{displaymath}">|; 

$key = q/Q_{cnd};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="41" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img235.png"
 ALT="$Q_{cnd}$">|; 

$key = q/displaystyle-{c_{p}}_{d}[overline{theta_{v}}]_{i(u),k}left[DP{pi}{x}right]_{i(u),k}+left[{{rm{Turb}.{u}right]_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="312" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img54.png"
 ALT="$\displaystyle - {c_{p}}_{d} [\overline{\theta_{v}}]_{i(u),k}
\left[\DP{\pi}{x}\right]_{i(u),k}
+ \left[{\rm Turb}.{u}\right]_{i(u),k}$">|; 

$key = q/displaystyle+left[{{rm{Diff}.q_{v}right]_{i,k}^{t-Deltat}+left[EV_{rv}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img212.png"
 ALT="$\displaystyle + \left[ {\rm Diff}.q_{v} \right]_{i,k}^{t - \Delta t}
+ \left[ EV_{rv} \right]_{i,k}^{t}$">|; 

$key = q/T;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="18" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img276.png"
 ALT="$T$">|; 

$key = q/_2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img298.png"
 ALT="$_2$">|; 

$key = q/displaystyleleft{-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}}bar{theta}_{v}^{2}right)_{1(w)}right}pi^{tau+Deltatau}_{i,2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="337" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img369.png"
 ALT="$\displaystyle \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\b...
...ho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,2}$">|; 

$key = q/Q_{dis};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img237.png"
 ALT="$Q_{dis}$">|; 

$key = q/{displaymath}[Q_{dis}]_{i,k}^{t-Deltat}=frac{1}{{c_{p}}_{d}bar{pi}}frac{C_{varepac{(K_{m,i,k}^{t-Deltat})^{3}}{{C_{m}}^{2}l^{4}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="376" HEIGHT="57" BORDER="0"
 SRC="|."$dir".q|img238.png"
 ALT="\begin{displaymath}[Q_{dis}]_{i,k}^{t-\Delta t}
= \frac{1}{{c_{p}}_{d} \bar{\p...
...{\pi}}
\frac{(K_{m,i,k}^{t-\Delta t})^{3}}{{C_{m}}^{2} l^{4}}
\end{displaymath}">|; 

$key = q/u^{t+Deltat}_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img327.png"
 ALT="$ u^{t + \Delta t}_{i(u),k}$">|; 

$key = q/displaystyleleft[q_{{rm{NH_3}right]^{t+Deltat}=[q_{{rm{NH_3}]^{*}+Deltaq_{{rm{NH_3},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="242" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img290.png"
 ALT="$\displaystyle \left[ q_{\rm NH_3} \right]^{t + \Delta t}
= [q_{\rm NH_3}]^{*} + \Delta q_{\rm NH_3},$">|; 

$key = q/F_{u},F_{w};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img104.png"
 ALT="$F_{u}, F_{w}$">|; 

$key = q/displaystylepi^{tau+Deltatau};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img113.png"
 ALT="$\displaystyle \pi^{\tau + \Delta \tau}$">|; 

$key = q/u^{tau+Deltatau};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="19" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img109.png"
 ALT="$u^{\tau +\Delta \tau}$">|; 

$key = q/displaystyleleft[DP{u}{z}right]_{i(u),k(w)}equivfrac{9}{8}left(frac{u_{i(u),k+1}frac{1}{24}left(frac{u_{i(u),k+2}-u_{i(u),k-1}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="531" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img43.png"
 ALT="$\displaystyle \left[\DP{u}{z} \right]_{i(u),k(w)}
\equiv \frac{9}{8}\left(\frac...
...ight) -
\frac{1}{24}\left(\frac{u_{i(u), k+2} - u_{i(u), k-1}}{\Delta x}\right)$">|; 

$key = q/(Deltax)^{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img439.png"
 ALT="$(\Delta x)^{3}$">|; 

$key = q/1leqk(w)leqkm;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="130" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img17.png"
 ALT="$1 \leq k(w) \leq km$">|; 

$key = q/displaystylegamma_{v}=0hspace{12em}(zleqz_{max}-d_{v}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img350.png"
 ALT="$\displaystyle \gamma_{v} = 0 \hspace{12em} ( z \leq z_{max} - d_{v}),$">|; 

$key = q/displaystyles_{{rm{NH_4SH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="59" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img462.png"
 ALT="$\displaystyle s_{\rm NH_4SH}$">|; 

$key = q/displaystylefrac{p_{x}M_{x}(1slashM_{d}+sum_{vneqx}q_{v}slashM_{v})}{p_{0}pi^{{c_{p}}_{d}slashR_{d}}-p_{x}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="230" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img487.png"
 ALT="$\displaystyle \frac{ p_{x} M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v}) }
{p_{0} \pi^{{c_{p}}_{d}/R_{d}} - p_{x}}$">|; 

$key = q/displaystyleleft{1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho{theta}_{v}^{2}right)_{km-1(w)}right}pi^{tau+Deltatau}_{i,km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="397" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img139.png"
 ALT="$\displaystyle \left\{
1 +
\beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{...
...\bar{\theta}_{v}^{2}
\right)_{km-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,km}$">|; 

$key = q/displaystyleu_{i-1(u),j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img417.png"
 ALT="$\displaystyle u_{i-1(u),j}$">|; 

$key = q/p_{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img478.png"
 ALT="$p_{x}$">|; 

$key = q/displaystyle-u_{i,k}left[DP{theta}{x}right]_{i,k}-w_{i,k}left[DP{theta}{z}right]ta}right]_{i,k}+left[{{rm{Turb}.{overline{theta}}right]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="558" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img65.png"
 ALT="$\displaystyle - u_{i,k}\left[\DP{\theta}{x}\right]_{i,k}
- w_{i,k}\left[\DP{\th...
...Turb}.{\theta}\right]_{i,k}
+ \left[{\rm Turb}.{\overline{\theta}}\right]_{i,k}$">|; 

$key = q/theta;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img220.png"
 ALT="$\theta$">|; 

$key = q/displaystyleC_{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="26" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img156.png"
 ALT="$\displaystyle C_{k}$">|; 

$key = q/-Deltaxslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img409.png"
 ALT="$-\Delta x/2$">|; 

$key = q/K_{h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img234.png"
 ALT="$K_{h}$">|; 

$key = q/displaystyle+betaleft(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{thetat(bar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{0(w)}E_{i,0(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img371.png"
 ALT="$\displaystyle + \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...z}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{0(w)}
E_{i,0(w)}$">|; 

$key = q/displaystyle-u_{i,k}left[DP{q_{v}}{x}right]_{i,k}-w_{i,k}left[DP{q_{v}}{x}right]_{i,k}-w_{i,k}left[DP{bar{q_{v}}}{x}right]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="363" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img68.png"
 ALT="$\displaystyle - u_{i,k} \left[\DP{q_{v}}{x} \right]_{i,k}
- w_{i,k} \left[\DP{q_{v}}{x} \right]_{i,k}
- w_{i,k} \left[\DP{\bar{q_{v}}}{x} \right]_{i,k}$">|; 

$key = q/displaystyleleft[DP{pi}{x}right]_{i(u),k}equivfrac{9}{8}left(frac{pi_{i+1,k}-pi_ht)-frac{1}{24}left(frac{pi_{i+2,k}-pi_{i-1,k}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="437" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img40.png"
 ALT="$\displaystyle \left[\DP{\pi}{x} \right]_{i(u),k}
\equiv \frac{9}{8}\left(\frac{...
...\right) -
\frac{1}{24}\left(\frac{\pi_{i+2, k} - \pi_{i-1, k}}{\Delta x}\right)$">|; 

$key = q/T_{0}=273.15;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img450.png"
 ALT="$T_{0} = 273.15$">|; 

$key = q/L_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img457.png"
 ALT="$L_{v}$">|; 

$key = q/displaystylepi_{im+1,k}=pi_{im,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img93.png"
 ALT="$\displaystyle \pi_{im+1,k} = \pi_{im,k}$">|; 

$key = q/displaystyleF_{u,i(u),k}^{t}=-left[{{rm{Adv}.uright]_{i(u),k}^{t}+left[{{rm{Turbu),k}^{t-Deltat}+left[{{rm{Diff}.uright]_{i(u),k}^{t-Deltat},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="450" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img169.png"
 ALT="$\displaystyle F_{u,i(u),k}^{t} =
- \left[ {\rm Adv}.u \right]_{i(u),k}^{t}
+ \l...
...ight]_{i(u),k}^{t-\Delta t}
+ \left[{\rm Diff}.u \right]_{i(u),k}^{t-\Delta t},$">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{tght)_{k(w)}left(DP{pi^{tau+Deltatau}}{z}right)_{i,k(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="411" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img357.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...right)_{k(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,k(w)}
\right\}$">|; 

$key = q/displaystyleDP{u}{x}+DP{w}{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img388.png"
 ALT="$\displaystyle \DP{u}{x}+\DP{w}{z}$">|; 

$key = q/displaystyleoverline{theta_{v}}+theta_{v}^{'};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="61" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img324.png"
 ALT="$\displaystyle \overline{\theta_{v}} + \theta_{v}^{'}$">|; 

$key = q/{figure}center{{Depsf[120mm]{psslashnum.eps}{center{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="540" HEIGHT="170" BORDER="0"
 SRC="|."$dir".q|img9.png"
 ALT="\begin{figure}\begin{center}
\Depsf[120mm]{ps/num.eps}
\end{center} \end{figure}">|; 

$key = q/displaystyleleft[DP{pi}{x}right]_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="85" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img423.png"
 ALT="$\displaystyle \left[\DP{\pi}{x} \right]_{i(u),k}$">|; 

$key = q/displaystyles_{{rm{NH_4SH}+s_{{rm{NH_4SH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="136" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img467.png"
 ALT="$\displaystyle s_{\rm NH_4SH} + s_{\rm NH_4SH}$">|; 

$key = q/displaystyle(k=2,3,cdotskm-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img148.png"
 ALT="$\displaystyle (k = 2, 3, \cdots km-1),$">|; 

$key = q/displaystyleq_{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img479.png"
 ALT="$\displaystyle q_{x}$">|; 

$key = q/displaystyleDP{u_{i(u),k}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="62" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img51.png"
 ALT="$\displaystyle \DP{u_{i(u),k}}{t}$">|; 

$key = q/displaystyleleft[{{rm{Turb}.{phi}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img227.png"
 ALT="$\displaystyle \left[{\rm Turb}.{\phi} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyle[w]^{t+Deltat}=2Deltatleft{[{{rm{Adv}.w]^{t}+cdots+left{gamma_{h}(x)+gamma_{v}(z)right}[w]^{t-Deltat}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="468" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img339.png"
 ALT="$\displaystyle [w]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.w]^{t}
+ \cdots
+ \left\{
\gamma_{h}(x) + \gamma_{v}(z)
\right\} [w]^{t - \Delta t}
\right\}$">|; 

$key = q/displaystyleleft[DP{phi}{z}right]_{i(u),k}equivfrac{phi_{i(u),k(w)}-phi_{i(u),k-1(w)}}{Deltaz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img39.png"
 ALT="$\displaystyle \left[\DP{\phi}{z} \right]_{i(u),k}
\equiv \frac{\phi_{i(u), k(w)} - \phi_{i(u), k-1(w)}}{\Delta z}$">|; 

$key = q/displaystyleleft[DP{pi}{x}right]_{i(u),k}equivfrac{pi_{i+1,k}-pi_{i,k}}{Deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="209" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img32.png"
 ALT="$\displaystyle \left[\DP{\pi}{x} \right]_{i(u),k}
\equiv \frac{\pi_{i+1, k} - \pi_{i, k}}{\Delta x}$">|; 

$key = q/displaystyle+left[left{left(DP{K_{m}}{x}right)^{2}right}_{i(u),k}right]_{i,k}^{t(DP{K_{m}}{z}right)^{2}right}_{i,k(w)}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="435" HEIGHT="93" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img315.png"
 ALT="$\displaystyle +
\left[
\left\{
\left( \DP{K_{m}}{x}\right)^{2}
\right\}_{i(u),k...
...{
\left(\DP{K_{m}}{z}\right)^{2}
\right\}_{i,k(w)}
\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyleDP{u_{i(u),k}}{t}=-left[bar{c_{p}}bar{theta}_{v}DP{(pi-alphaDiv)}{x}right]_{i(u),k}+[F_{u}]_{i(u),k}^{t},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="398" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img100.png"
 ALT="$\displaystyle \DP{u_{i(u),k}}{t} = - \left[\bar{c_{p}} \bar{\theta}_{v}
\DP{(\pi - \alpha Div )}{x}\right]_{i(u),k}
+ [F_{u}]_{i(u),k}^{t},$">|; 

$key = q/epsilon;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img400.png"
 ALT="$\epsilon$">|; 

$key = q/i,k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img13.png"
 ALT="$i,k$">|; 

$key = q/displaystyleDinv{2}left[left{DP{}{x}left(DP{K_{m}^{2}}{x}right)_{i(u),k}right}_{(DP{K_{m}^{2}}{z}right)_{i,k(w)}right}_{i,k}^{t-Deltat}right];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="455" HEIGHT="85" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img314.png"
 ALT="$\displaystyle \Dinv{2}
\left[
\left\{
\DP{}{x}
\left(
\DP{K_{m}^{2}}{x}
\right)...
...}
\left(
\DP{K_{m}^{2}}{z}
\right)_{i,k(w)}
\right\}_{i,k}^{t-\Delta t}
\right]$">|; 

$key = q/left(Deltaxright)^{4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img445.png"
 ALT="$\left(\Delta x\right)^{4}$">|; 

$key = q/Deltatau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="30" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img1.png"
 ALT="$\Delta \tau$">|; 

$key = q/displaystyleu_{i,j}+left[DP{u}{x}right]_{i,k}frac{3Deltax}{2}+frac{1}{2!}left[DP!}left[DP[3]{u}{x}right]_{i,k}left(frac{3Deltax}{2}right)^{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="530" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img432.png"
 ALT="$\displaystyle u_{i,j}
+ \left[\DP{u}{x} \right]_{i,k}\frac{3\Delta x}{2}
+ \fra...
...rac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{3}$">|; 

$key = q/displaystyleq_{x}left(1-frac{p_{x}}{p}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="104" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img483.png"
 ALT="$\displaystyle q_{x} \left(1 - \frac{p_{x}}{p} \right)$">|; 

$key = q/displaystyletheta_{i,k}^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="46" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img199.png"
 ALT="$\displaystyle \theta_{i,k}^{t + \Delta t}$">|; 

$key = q/gamma;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img333.png"
 ALT="$\gamma$">|; 

$key = q/{displaymath}frac{alphaDeltatau}{mbox{Min}(Deltax^{2},Deltaz^{2})}leqfrac{1}{2}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="162" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img401.png"
 ALT="\begin{displaymath}
\frac{\alpha \Delta \tau}{\mbox{Min}(\Delta x^{2}, \Delta z^{2})} \leq \frac{1}{2}
\end{displaymath}">|; 

$key = q/displaystyleleft[F_{w}right]_{i,k(w)}^{t}=-left[{{rm{Adv}.{w}right]_{i,k(w)}^{t}(w)}^{t-Deltat}.+left[{{rm{Diff}.wright]_{i,k(w)}^{t-Deltat}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="597" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img106.png"
 ALT="$\displaystyle \left[F_{w}\right]_{i,k(w)}^{t} =
- \left[{\rm Adv}.{w}\right]_{i...
..._{i,k(w)}^{t - \Delta t}.
+ \left[{\rm Diff}.w \right]_{i,k(w)}^{t - \Delta t}.$">|; 

$key = q/u_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img19.png"
 ALT="$u_{i(u), k}$">|; 

$key = q/l=(DeltaxDeltaz)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img239.png"
 ALT="$l=(\Delta x\Delta z)^{1/2}$">|; 

$key = q/N_{tau}equiv2DeltatslashDeltatau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="120" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img326.png"
 ALT="$N_{\tau}\equiv 2\Delta t/\Delta
\tau$">|; 

$key = q/(i,j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="43" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img405.png"
 ALT="$(i,j)$">|; 

$key = q/t-Deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img303.png"
 ALT="$t - \Delta t$">|; 

$key = q/displaystyleDeltaq_{c}=MAX{0,[q_{v}]^{*}-q_{vsw}([theta]^{*})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img259.png"
 ALT="$\displaystyle \Delta q_{c} = MAX\{0, [q_{v}]^{*} - q_{vsw}([\theta]^{*})\}$">|; 

$key = q/u,w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img6.png"
 ALT="$u, w$">|; 

$key = q/displaystyle+[{{rm{Turb}.K_m]_{i,k}^{t-Deltat}+[{{rm{Disp}.K_m]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img302.png"
 ALT="$\displaystyle + [{\rm Turb}.K_m]_{i,k}^{t - \Delta t}
+ [{\rm Disp}.K_m]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyle+F_{u,i(u),k}^{t}Deltatau,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img330.png"
 ALT="$\displaystyle +
F_{u,i(u),k}^{t} \Delta \tau,$">|; 

$key = q/phi_{i(u),k(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img31.png"
 ALT="$\phi_{i(u), k(w)}$">|; 

$key = q/{displaymath}omega^{2}=frac{N^{2}k^{2}}{k^{2}+l^{2}},{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="107" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img396.png"
 ALT="\begin{displaymath}
\omega ^{2} = \frac{N^{2}k^{2}}{k^{2}+l^{2}},
\end{displaymath}">|; 

$key = q/displaystyles_{{rm{NH_3}+s_{{rm{H_2S}-RK_{p}-RTln{p_{0}}^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="257" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img471.png"
 ALT="$\displaystyle s_{\rm NH_3} + s_{\rm H_2S} - R K_{p}
- RT \ln {p_{0}}^{2}$">|; 

$key = q/displaystyleu_{-1(u),k}=u_{im-1(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="163" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img78.png"
 ALT="$\displaystyle u_{-1(u), k} = u_{im-1(u), k}$">|; 

$key = q/displaystyleleft[q_{r}right]_{i,k}^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img205.png"
 ALT="$\displaystyle \left[ q_{r} \right]_{i,k}^{t+\Delta t}$">|; 

$key = q/displaystyleleft[{{rm{Buoy}.K_mright]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="124" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img307.png"
 ALT="$\displaystyle \left[{\rm Buoy}.K_m\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyle2left[DP{}{z}left{left(K_{m}right)_{i,k}left(DP{w}{z}right)_{i,k}right}right]_{i,k(w)}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img184.png"
 ALT="$\displaystyle 2 \left[
\DP{}{z}\left\{
\left( K_{m} \right)_{i,k} \left( \DP{w}{z} \right)_{i,k}
\right\}
\right]_{i,k(w)}^{t - \Delta t}$">|; 

$key = q/displaystylepi^{tau}_{i,k}-(1-beta)left(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rv}}right)_{k}left(DP{u^{tau+Deltatau}}{x}right)_{i,k}+F_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="580" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img160.png"
 ALT="$\displaystyle \pi^{\tau}_{i,k}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau...
..._{v}}
\right)_{k}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,k}
+ F_{i,k}$">|; 

$key = q/displaystyle1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{eft(bar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{k-1(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img147.png"
 ALT="$\displaystyle 1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{...
...
+
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}$">|; 

$key = q/displaystylebetaleft(DP{pi^{tau+Deltatau}}{z}right)_{i,0(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="145" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img127.png"
 ALT="$\displaystyle \beta \left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,0(w)}$">|; 

$key = q/displaystylehspace{10mm}left.left.-left(bar{c_{p}}bar{theta}_{v}right)_{i,k(w)}D(w)}+left(F_{w}^{t}right)_{i,k(w)}Deltatauright}right]_{i,k}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="633" HEIGHT="75" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img123.png"
 ALT="$\displaystyle \hspace{10mm}
\left. \left.
- \left( \bar{c_{p}} \bar{\theta}_{v}...
...i,k(w)}
+ \left( F_{w}^{t} \right)_{i,k(w)} \Delta \tau
\right\}
\right]_{i,k}.$">|; 

$key = q/displaystyleu_{i,j}-left[DP{u}{x}right]_{i,k}frac{3Deltax}{2}+frac{1}{2!}left[DP!}left[DP[3]{u}{x}right]_{i,k}left(frac{3Deltax}{2}right)^{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="530" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img435.png"
 ALT="$\displaystyle u_{i,j}
- \left[\DP{u}{x} \right]_{i,k}\frac{3\Delta x}{2}
+ \fra...
...rac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{3}$">|; 

$key = q/displaystyle+betaleft(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{thetaar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{km(w)}E_{i,km(w)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img166.png"
 ALT="$\displaystyle + \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}.$">|; 

$key = q/alpha_h,alpha_v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img352.png"
 ALT="$\alpha_h, \alpha_v$">|; 

$key = q/displaystyleu_{i,k(w)}equivfrac{u_{i(u),k+1}+u_{i-1(u),k+1}+u_{i(u),k}+u_{i-1(u),k}}{4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img26.png"
 ALT="$\displaystyle u_{i,k(w)} \equiv \frac{u_{i(u), k+1} + u_{i-1(u), k+1}
+ u_{i(u), k} + u_{i-1(u), k}}{4}$">|; 

$key = q/displaystyle+frac{1}{4!}left[DP[4]{u}{x}right]_{i,k}left(frac{3Deltax}{2}right)^DP[5]{u}{x}right]_{i,k}left(frac{3Deltax}{2}right)^{5}+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img433.png"
 ALT="$\displaystyle + \frac{1}{4!}\left[\DP[4]{u}{x} \right]_{i,k}
\left(\frac{3\Delt...
...left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{5}
+ \cdots .$">|; 

$key = q/displaystyleDP{[q_{c}]_{i,k}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img70.png"
 ALT="$\displaystyle \DP{[q_{c}]_{i,k}}{t}$">|; 

$key = q/textstyleequiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img129.png"
 ALT="$\textstyle \equiv$">|; 

$key = q/displaystyle[F_{u}]_{i(u),k}^{t}=-left[{{rm{Adv}.{u}right]_{i(u),k}^{t}+left[{{r(u),k}^{t-Deltat}+left[{{rm{Diff}.uright]_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img105.png"
 ALT="$\displaystyle [F_{u}]_{i(u),k}^{t} =
- \left[{\rm Adv}.{u}\right]_{i(u),k}^{t}
...
...ht]_{i(u),k}^{t - \Delta t}
+ \left[{\rm Diff}.u\right]_{i(u),k}^{t - \Delta t}$">|; 

$key = q/displaystyle(k=1,2,cdotskm-1),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img158.png"
 ALT="$\displaystyle (k = 1, 2, \cdots km-1),$">|; 

$key = q/CL_{cr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img242.png"
 ALT="$CL_{cr}$">|; 

$key = q/displaystyle(p_{{rm{NH_{3}}-X)(p_{{rm{H_{2}S}-X)=e^{k_{p}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="235" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img279.png"
 ALT="$\displaystyle (p_{\rm NH_{3}} - X) ( p_{\rm H_{2}S} - X )
= e^{k_{p}}$">|; 

$key = q/displaystyleDP{phi}{t}=-{{rm{Adv}.phi+cdots+gamma_{h}(x)(phi-phi_{e})+gamma_{v}(z)(phi-phi_{e});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img334.png"
 ALT="$\displaystyle \DP{\phi}{t} = -{\rm Adv}.\phi + \cdots + \gamma_{h}(x) (\phi - \phi_{e})
+ \gamma_{v}(z) (\phi - \phi_{e})$">|; 

$key = q/displaystylefrac{rho_{x}}{rho_{d}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img480.png"
 ALT="$\displaystyle \frac{\rho_{x}}{\rho_{d}}$">|; 

$key = q/displaystyle-betaleft(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{thetaar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{i,0(w)}E_{i,0(w)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img163.png"
 ALT="$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{i,0(w)}
E_{i,0(w)},$">|; 

$key = q/_{4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img277.png"
 ALT="$_{4}$">|; 

$key = q/displaystyleleft[DP{u}{x}right]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img442.png"
 ALT="$\displaystyle \left[\DP{u}{x}\right]_{i,k}$">|; 

$key = q/gamma_{h},gamma_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img342.png"
 ALT="$\gamma_{h}, \gamma_{v}$">|; 

$key = q/displaystyleoverline{theta_{v}}+theta_{v}^{'};;;(for;;q_{c}=0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="180" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img322.png"
 ALT="$\displaystyle \overline{ \theta_{v}} + \theta_{v}^{'} \;\;\; (for \;\; q_{c} = 0)$">|; 

$key = q/displaystyleu_{i,k}equivfrac{u_{i(u),k}+u_{i-1(u),k}}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="192" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img25.png"
 ALT="$\displaystyle u_{i,k} \equiv \frac{u_{i(u), k} + u_{i-1(u), k}}{2}$">|; 

$key = q/displaystyleD_{i,km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img164.png"
 ALT="$\displaystyle D_{i,km}$">|; 

$key = q/(u_{i+1(u),j},u_{i-2(u),j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img430.png"
 ALT="$(u _{i+1(u),j}, u
_{i-2(u),j})$">|; 

$key = q/displaystyleu_{i(u),k(w)}equivfrac{u_{i(u),k+1}+u_{i(u),k}}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="231" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img27.png"
 ALT="$\displaystyle u_{i(u),k(w)} \equiv \frac{u_{i(u), k+1} + u_{i(u), k}}{2}$">|; 

$key = q/displaystyleleft[{{rm{Turb}.{u}right]_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img179.png"
 ALT="$\displaystyle \left[ {\rm Turb}.{u} \right]_{i(u),k}^{t - \Delta t}$">|; 

$key = q/displaystyleleft(DP{(alphaDiv)^{tau}}{z}right)_{i,km(w)}-(1-beta)left(DP{pi^{tau+left(Dinv{bar{c_{p}}bar{theta}_{v}}F_{w}^{t}right)_{i,km(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="512" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img137.png"
 ALT="$\displaystyle \left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,km(w)}
- (1 - \beta...
...i,km(w)}
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,km(w)}$">|; 

$key = q/displaystyleleft[q_{r}right]_{i,k}^{t-Deltat}+2Deltat[F_{q_{r}}]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img206.png"
 ALT="$\displaystyle \left[ q_{r} \right]_{i,k}^{t-\Delta t}
+ 2 \Delta t [F_{q_{r}}]_{i,k}^{t}$">|; 

$key = q/X=P_{{rm{H_{2}S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="84" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img286.png"
 ALT="$X = P_{\rm H_{2}S}$">|; 

$key = q/displaystyleu_{i,k(w)}left[DP{w}{x}right]_{i,k(w)}^{t}+w_{i,k(w)}left[DP{w}{z}right]_{i,k(w)}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img174.png"
 ALT="$\displaystyle u_{i,k(w)}\left[\DP{w}{x}\right]_{i,k(w)}^{t}
+ w_{i,k(w)}\left[\DP{w}{z}\right]_{i,k(w)}^{t}$">|; 

$key = q/displaystyleleft{u_{i(u),k}left(DP{K_{m}}{x}right)_{i(u),k}right}_{i,k}^{t}+left{w_{i,k(w)}left(DP{K_{m}}{z}right)_{i,k(w)}right}_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="430" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img306.png"
 ALT="$\displaystyle \left\{
u_{i(u),k} \left( \DP{K_{m}}{x} \right)_{i(u), k}
\right\...
...
+
\left\{
w_{i,k(w)} \left( \DP{K_{m}}{z} \right)_{i, k(w)}
\right\}_{i,k}^{t}$">|; 

$key = q/displaystylebetaleft(DP{pi^{tau+Deltatau}}{z}right)_{i,km(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="158" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img136.png"
 ALT="$\displaystyle \beta \left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,km(w)}$">|; 

$key = q/displaystyle+[{{rm{Fall}.q_{r}]_{i,k}+[{{rm{Turb}.{q_{r}}]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="208" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img74.png"
 ALT="$\displaystyle + [{\rm Fall}.q_{r}]_{i,k}
+ [{\rm Turb}.{q_{r}}]_{i,k}$">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{tz^{2}}left(bar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{k(w)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="286" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img157.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{k(w)},$">|; 

$key = q/displaystyleleft[{{rm{Turb}.{w}right]_{i,k(w)}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img183.png"
 ALT="$\displaystyle \left[ {\rm Turb}.{w} \right]_{i,k(w)}^{t - \Delta t}$">|; 

$key = q/displaystyleL_{v}=left{frac{Bln(10.0)}{(C+T-T_{0})^{2}}right}R_{v}T^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="241" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img460.png"
 ALT="$\displaystyle L_{v} = \left\{
\frac{B \ln(10.0)}{ (C + T - T_{0})^{2} }
\right\} R_{v} T^{2}$">|; 

$key = q/mu;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img253.png"
 ALT="$\mu$">|; 

$key = q/displaystyleleft[thetaright]^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img262.png"
 ALT="$\displaystyle \left[ \theta \right]^{t + \Delta t}$">|; 

$key = q/p^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img449.png"
 ALT="$p^{*}$">|; 

$key = q/displaystyle-left[{{rm{Adv}.{theta}right]_{i,k}^{t}-left[{{rm{Adv}.{bar{theta}}r,k}^{t-Deltat}+left[{{rm{Diff}.{theta}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="557" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img208.png"
 ALT="$\displaystyle - \left[{\rm Adv}.{\theta}\right]_{i,k}^{t}
- \left[{\rm Adv}.{\b...
...]_{i,k}^{t - \Delta t}
+ \left[{\rm Diff}.{\theta} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyle+left[1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rhta}_{v}^{2}right)_{k-1(w)}right}right]pi^{tau+Deltatau}_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="521" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img361.png"
 ALT="$\displaystyle + \left[
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}...
...theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\right]
\pi^{\tau + \Delta \tau}_{i,k}$">|; 

$key = q/displaystylepi_{0,k}=-pi_{1,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img87.png"
 ALT="$\displaystyle \pi_{0,k} = - \pi_{1,k}$">|; 

$key = q/displaystyleleft[EV_{rv}right]_{i,k}^{t}=4.85times10^{-2}([q_{vsw}]_{i,k}^{t}-[q_{v}]_{i,k}^{t})([bar{rho}]_{i,k}[q_{r}]_{i,k}^{t})^{0.65};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img246.png"
 ALT="$\displaystyle \left[ EV_{rv} \right]_{i,k}^{t} =
4.85 \times 10^{-2} ([q_{vsw}]_{i,k}^{t} - [q_{v}]_{i,k}^{t})
([\bar{\rho}]_{i,k} [q_{r}]_{i,k}^{t})^{0.65}$">|; 

$key = q/displaystyle[{{rm{Buoy}]^{t}_{i,k(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="95" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img175.png"
 ALT="$\displaystyle [{\rm Buoy}]^{t}_{i,k(w)}$">|; 

$key = q/displaystyle+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{t_{k-1(w)}left(DP{pi^{tau+Deltatau}}{z}right)_{i,k-1(w)}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="446" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img358.png"
 ALT="$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...t)_{k-1(w)}
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)_{i,k-1(w)}
\right\}$">|; 

$key = q/displaystyle-left{frac{3gC_{m}^{2}l^{2}}{2overline{theta}}left(DP{theta_{el}}{z}right)_{i,k(w)}right}_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="247" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img308.png"
 ALT="$\displaystyle - \left\{
\frac{3 g C_{m}^{2} l^{2}}{ 2 \overline{\theta}}
\left(\DP{\theta_{el}}{z} \right)_{i,k(w)}
\right\}_{i,k}^{t-\Delta t}$">|; 

$key = q/displaystyle{{rm{NH_{3}}+{{rm{H_{2}S}rightarrow{{rm{NH_{4}SH};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img274.png"
 ALT="$\displaystyle {\rm NH_{3}} + {\rm H_{2}S} \rightarrow {\rm NH_{4}SH}$">|; 

$key = q/{displaymath}overline{rho}_{i,k}=frac{p_{0}}{R_{d}}frac{[overline{pi}^{c_{v}slashR_{d}}]_{i,k}}{[overline{theta_{v}}]_{i,k}}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="155" HEIGHT="55" BORDER="0"
 SRC="|."$dir".q|img50.png"
 ALT="\begin{displaymath}
\overline{\rho}_{i,k} = \frac{p_{0}}{R_{d}}
\frac{[\overline{\pi}^{c_{v}/R_{d}}]_{i,k}}
{[\overline{\theta_{v}}]_{i,k}}
\end{displaymath}">|; 

$key = q/displaystyleleft[q_{c}right]^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img266.png"
 ALT="$\displaystyle \left[ q_{c} \right]^{t + \Delta t}$">|; 

$key = q/displaystyleleft[q_{v}right]^{t+Deltat},=[q_{v}]^{*}+[q_{c}]^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="185" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img272.png"
 ALT="$\displaystyle \left[ q_{v} \right]^{t + \Delta t},
= [q_{v}]^{*} + [q_{c}]^{*}$">|; 

$key = q/displaystyle-DP{mu_{{rm{NH_3}+mu_{{rm{H_2S}}{T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="133" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img468.png"
 ALT="$\displaystyle - \DP{\mu_{\rm NH_3} + \mu_{\rm H_2S}}{T}$">|; 

$key = q/displaystyleleft(array{{cccc}A_{1}<col_mark>B_{2}<col_mark><col_mark>0<row_mark>dots<col_mark>pi_{im,km}<row_mark>array{right)^{tau+Deltatau};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="500" HEIGHT="142" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img144.png"
 ALT="$\displaystyle \left(\begin{array}{cccc}
A_{1} &amp; B_{2} &amp; &amp; 0 \\\\
C_{1} &amp; \ddots ...
...m} &amp; \cdots &amp; \cdots &amp; \pi_{im, km} \\\\
\end{array}\right)^{\tau + \Delta \tau}$">|; 

$key = q/displaystyleDP{[q_{v}]_{i,k}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img67.png"
 ALT="$\displaystyle \DP{[q_{v}]_{i,k}}{t}$">|; 

$key = q/gamma=L_{v}slash(c_{p}Pi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="115" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img268.png"
 ALT="$\gamma = L_{v}/(c_{p} \Pi)$">|; 

$key = q/displaystyleu_{i+1(u),j}-u_{i-2(u),j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img436.png"
 ALT="$\displaystyle u _{i+1(u),j} - u_{i-2(u),j}$">|; 

$key = q/displaystyle-gfrac{sum[q_{v}]_{i,k(w)}+sum[q_{c}]_{i,k(w)}+sum[q_{r}]_{i,k(w)}}{1+sum[bar{q_{v}}]_{i,k(w)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="319" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img60.png"
 ALT="$\displaystyle - g \frac{\sum [q_{v}]_{i,k(w)} + \sum [q_{c}]_{i,k(w)} + \sum [q_{r}]_{i,k(w)}}
{1 + \sum [\bar{q_{v}}]_{i,k(w)}}$">|; 

$key = q/displaystylepi_{im+1,k}=-pi_{im,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="140" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img89.png"
 ALT="$\displaystyle \pi_{im+1,k} = - \pi_{im,k}$">|; 

$key = q/displaystyleleft[DP{}{x}left{left(K_{h}right)_{i(u),k}left(DP{phi}{x}right)_{i(u),k}right}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="280" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img228.png"
 ALT="$\displaystyle \left[ \DP{}{x}
\left\{
\left( K_{h} \right)_{i(u),k}
\left( \DP{\phi}{x} \right)_{i(u),k}
\right\}
\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyle+frac{3}{640}left[DP[5]{u}{x}right]_{i,k}left(Deltaxright)^{4}+O[(Deltax)^{6}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="270" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img444.png"
 ALT="$\displaystyle + \frac{3}{640}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\Delta x\right)^{4}
+ O[(\Delta x)^{6}]$">|; 

$key = q/displaystyle-frac{1}{1920}left[DP[5]{pi}{x}right]_{i(u),k}left(Deltaxright)^{4}+O[(Deltax)^{6}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="298" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img425.png"
 ALT="$\displaystyle - \frac{1}{1920}\left[\DP[5]{\pi}{x} \right]_{i(u),k}
\left(\Delta x\right)^{4}
+ O[(\Delta x)^{6}]$">|; 

$key = q/F_{K_m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="40" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img304.png"
 ALT="$F_{K_m}$">|; 

$key = q/displaystyleu_{i(u),j}-u_{i-1(u),j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="134" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img420.png"
 ALT="$\displaystyle u _{i(u),j} - u_{i-1(u),j}$">|; 

$key = q/displaystyleleft[{{rm{Diff}.wright]_{i,k(w)}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="103" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img189.png"
 ALT="$\displaystyle \left[ {\rm Diff}.w \right]_{i,k(w)}^{t - \Delta t}$">|; 

$key = q/X=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img289.png"
 ALT="$X = 0$">|; 

$key = q/phi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img5.png"
 ALT="$\phi$">|; 

$key = q/displaystyle[U_{r}]_{i,k}^{t}=12.2([q_{r}]_{i,k}^{t})^{0.125};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="199" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img249.png"
 ALT="$\displaystyle [U_{r}]_{i,k}^{t} = 12.2 ([q_{r}]_{i,k}^{t})^{0.125}$">|; 

$key = q/-1leqkleqkm+2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="149" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img82.png"
 ALT="$-1 \leq k \leq km + 2$">|; 

$key = q/displaystyleleft[DP{phi}{x}right]_{i,k(w)}equivfrac{phi_{i(u),k(w)}-phi_{i-1(u),k(w)}}{Deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img38.png"
 ALT="$\displaystyle \left[\DP{\phi}{x} \right]_{i,k(w)}
\equiv \frac{\phi_{i(u), k(w)} - \phi_{i-1(u), k(w)}}{\Delta x}$">|; 

$key = q/Deltaq_{c}>0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="70" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img260.png"
 ALT="$\Delta q_{c} &gt; 0$">|; 

$key = q/displaystyle-DP{}{T}left(mu_{{rm{NH_3}+mu_{{rm{H_2S}+RTK_{p}-RTln{p_{0}}^{2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img464.png"
 ALT="$\displaystyle - \DP{}{T}
\left(
\mu_{\rm NH_3} + \mu_{\rm H_2S} + RT K_{p}
- RT \ln {p_{0}}^{2}
\right)$">|; 

$key = q/displaystylefrac{q_{x}p_{0}pi^{{c_{p}}_{d}slashR_{d}}}{M_{x}(1slashM_{d}+sum_{v}q_{v}slashM_{v})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img492.png"
 ALT="$\displaystyle \frac{q_{x}p_{0} \pi^{{c_{p}}_{d}/R_{d}} }{M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})}$">|; 

$key = q/displaystylefrac{pi_{i+1,j}-pi_{i,j}}{Deltax}-frac{1}{24}left[DP[3]{pi}{x}right]_{i(u),k}left(Deltaxright)^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="288" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img424.png"
 ALT="$\displaystyle \frac{\pi _{i+1,j} - \pi _{i,j}}{\Delta x}
- \frac{1}{24}\left[\DP[3]{\pi}{x} \right]_{i(u),k}
\left(\Delta x\right)^{2}$">|; 

$key = q/displaystyle+left{-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rhotheta}_{v}^{2}right)_{k-1(w)}right}pi^{tau+Deltatau}_{i,k-1}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="379" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img362.png"
 ALT="$\displaystyle + \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{...
...bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k-1}.$">|; 

$key = q/_{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img453.png"
 ALT="$_{2}$">|; 

$key = q/displaystyle-overline{c}_{p}overline{theta}_{v}Dlaplapi+gDP{}{z}left(frac{theta}{overline{theta}_{v}}right)+alphaDlaplaD,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="274" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img390.png"
 ALT="$\displaystyle - \overline{c}_{p}\overline{\theta}_{v}\Dlapla \pi
+ g\DP{}{z}\left(\frac{\theta}{\overline{\theta}_{v}}\right)
+ \alpha \Dlapla D,$">|; 

$key = q/displaystyle[q_{v}]^{*}+frac{[theta]^{*}-[theta]^{t+Deltat}}{gamma},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="173" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img265.png"
 ALT="$\displaystyle [q_{v}]^{*} + \frac{[\theta]^{*} - [\theta]^{t + \Delta t}}{\gamma},$">|; 

$key = q/displaystyle-overline{c}_{p}overline{theta}_{v}DP{pi}{x}+alphaDP{D}{x},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img380.png"
 ALT="$\displaystyle - \overline{c}_{p}\overline{\theta}_{v}
\DP{\pi}{x} + \alpha\DP{D}{x} ,$">|; 

$key = q/(u_{i(u),j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img408.png"
 ALT="$(u_{i(u),j})$">|; 

$key = q/displaystylepi_{i,k(w)}equivfrac{pi_{i,k+1}+pi_{i,k}}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="176" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img23.png"
 ALT="$\displaystyle \pi_{i,k(w)} \equiv \frac{\pi_{i, k+1} + \pi_{i, k}}{2}$">|; 

$key = q/displaystyleDP{pi}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img383.png"
 ALT="$\displaystyle \DP{\pi}{t}$">|; 

$key = q/d_{h};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img343.png"
 ALT="$d_{h}$">|; 

$key = q/displaystyleleft[DP{u}{x}right]_{i,k}3Deltax+frac{1}{3}left[DP[3]{u}{x}right]_{i0}left[DP[5]{u}{x}right]_{i,k}left(frac{3Deltax}{2}right)^{5};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="476" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img437.png"
 ALT="$\displaystyle \left[\DP{u}{x} \right]_{i,k}3\Delta x
+ \frac{1}{3}\left[\DP[3]{...
...rac{1}{60}\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{3\Delta x}{2}\right)^{5}$">|; 

$key = q/displaystyle[K_{m}]_{i,k}^{t+Deltat}=[K_{m}]_{i,k}^{t-Deltat}+2Deltat[F_{K_m}]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="290" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img299.png"
 ALT="$\displaystyle [K_{m}]_{i,k}^{t + \Delta t} = [K_{m}]_{i,k}^{t - \Delta t}
+ 2 \Delta t [F_{K_m}]_{i,k}^{t}$">|; 

$key = q/EV_{rv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img245.png"
 ALT="$EV_{rv}$">|; 

$key = q/1(u)simim(u);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="113" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img75.png"
 ALT="$1(u)\sim im(u)$">|; 

$key = q/displaystyleoverline{theta_{v}}+theta_{v}^{'}+frac{sumLq_{v}}{{c_p}_{d}bar{pi}};;;(for;;q_{c}>0);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="256" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img323.png"
 ALT="$\displaystyle \overline{\theta_{v}} + \theta_{v}^{'} + \frac{ \sum L
q_{v}}{{c_p}_{d} \bar{\pi}}
\;\;\; (for \;\; q_{c} &gt; 0)$">|; 

$key = q/K_{p};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img466.png"
 ALT="$K_{p}$">|; 

$key = q/displaystyle+left[DP{}{x}left{left(K_{m}right)_{i(u),k(w)}left(DP{w}{x}right)_{ift(DP{u}{z}right)_{i(u),k(w)}right}right]_{i,k(w)}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="578" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img185.png"
 ALT="$\displaystyle +\left[ \DP{}{x}\left\{
\left( K_{m} \right)_{i(u),k(w)}
\left( \...
...}
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\} \right]_{i,k(w)}^{t - \Delta t}$">|; 

$key = q/displaystyleK_{p}=ln(p_{{rm{NH_{3}}cdotp_{{rm{H_{2}S})=61.781-frac{10834}{T}-ln{10^{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img455.png"
 ALT="$\displaystyle K_{p}
= \ln(p_{\rm NH_{3}} \cdot p_{\rm H_{2}S})
= 61.781 - \frac{10834}{T} - \ln{10^{2}}$">|; 

$key = q/displaystylepi^{tau}_{i,km}-(1-beta)left(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{eta}_{v}}right)_{km}left(DP{u^{tau+Deltatau}}{x}right)_{i,km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="588" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img141.png"
 ALT="$\displaystyle \pi^{\tau}_{i,km}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \ta...
...{\theta}_{v}}
\right)_{km}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,km}$">|; 

$key = q/displaystyleDlapla;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="28" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img391.png"
 ALT="$\displaystyle \Dlapla$">|; 

$key = q/{displaymath}left[DP{u}{x}right]_{i,k}=frac{9}{8}left(frac{u_{i,j}-u_{i-1,j}}{De{24}left(frac{u_{i+1,j}-u_{i-2,j}}{Deltax}right){displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="402" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img446.png"
 ALT="\begin{displaymath}
\left[\DP{u}{x}\right]_{i,k} =
\frac{9}{8}\left(\frac{u_{i...
...rac{1}{24}\left(\frac{u_{i+1,j} - u _{i-2,j}}{\Delta x}\right)
\end{displaymath}">|; 

$key = q/displaystyleleft[F_{q_{v}}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img210.png"
 ALT="$\displaystyle \left[F_{q_{v}}\right]_{i,k}^{t}$">|; 

$key = q/A,B,C;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="65" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img451.png"
 ALT="$A, B, C$">|; 

$key = q/displaystyletheta^{*}+frac{gamma([q_{v}]^{*}-q_{vsw}([theta]^{*}))}{1+gammaDP{q_{vsw}([theta]^{*})}{theta}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img263.png"
 ALT="$\displaystyle \theta^{*} +
\frac
{ \gamma ( [q_{v}]^{*} - q_{vsw}([\theta]^{*})) }
{ 1 + \gamma \DP{q_{vsw}([\theta]^{*})}{\theta} }$">|; 

$key = q/displaystyle[u]^{t+Deltat}=2Deltatleft{[{{rm{Adv}.u]^{t}+cdots+left{gamma_{h}(x)+gamma_{v}(z)right}[u]^{t-Deltat}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="458" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img338.png"
 ALT="$\displaystyle [u]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.u]^{t}
+ \cdots
+ \left\{
\gamma_{h}(x) + \gamma_{v}(z)
\right\} [u]^{t - \Delta t}
\right\}$">|; 

$key = q/displaystyle-left[{{rm{Adv}.q_{r}right]_{i,k}^{t}+left[{{rm{Turb}.q_{r}right]_{i,k}^{t-Deltat}+left[{{rm{Diff}.q_{c}right]^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="349" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img217.png"
 ALT="$\displaystyle - \left[ {\rm Adv}.q_{r} \right]_{i,k}^{t}
+ \left[ {\rm Turb}.q_{r} \right]_{i,k}^{t-\Delta t}
+ \left[ {\rm Diff}.q_{c} \right]^{t-\Delta t}$">|; 

$key = q/w,pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="38" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img3.png"
 ALT="$w, \pi$">|; 

$key = q/displaystylew(i,km(w))=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img125.png"
 ALT="$\displaystyle w(i,km(w)) = 0$">|; 

$key = q/displaystyle[theta]^{t+Deltat}=2Deltatleft{[{{rm{Adv}.theta]^{t}+cdots+left{gamma_{h}(x)+gamma_{v}(z)right}[theta]^{t-Deltat}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="454" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img340.png"
 ALT="$\displaystyle [\theta]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.\theta]^{...
...\left\{
\gamma_{h}(x) + \gamma_{v}(z)
\right\} [\theta]^{t - \Delta t}
\right\}$">|; 

$key = q/0leqzleqz_{max};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="111" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img349.png"
 ALT="$0 \leq z \leq z_{max}$">|; 

$key = q/displaystyle+frac{beta}{Deltaz}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}barar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{km(w)}E_{i,km(w)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="326" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img143.png"
 ALT="$\displaystyle + \frac{\beta}{\Delta z}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2...
...\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km(w)}
E_{i,km(w)}.$">|; 

$key = q/displaystyle-betaleft(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rho}bar{theta}_{v}^alphaDiv)^{tau}}{z}right}+F_{w}^{t}Deltatauright}right]_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="631" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img133.png"
 ALT="$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rh...
...(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau
\right\}
\right]_{i,1}$">|; 

$key = q/(i(u),j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="68" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img407.png"
 ALT="$(i(u),j)$">|; 

$key = q/displaystyleL_{{rm{NH_4SH}=frac{10834}{T^2}{RT^{2}}=10834R;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="260" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img474.png"
 ALT="$\displaystyle L_{\rm NH_4SH} = \frac{10834}{T^2} {R T^{2}} = 10834 R$">|; 

$key = q/displaystyleE_{i,k(w)}equivleft(DP{(alphaDiv)^{tau}}{z}right)_{i,k(w)}-(1-beta)l}+left(Dinv{bar{c_{p}}bar{theta}_{v}}F_{w}^{t}right)_{i,k(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="553" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img167.png"
 ALT="$\displaystyle E_{i,k(w)} \equiv
\left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,k...
..._{i,k(w)}
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,k(w)}$">|; 

$key = q/q_{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img222.png"
 ALT="$q_{c}$">|; 

$key = q/displaystyleDP{w_{i,k(w)}}{t}=-left[bar{c_{p}}bar{theta}_{v}DP{(pi-alphaDiv)}{z}right]_{i,k(w)}+[F_{w}]_{i,k(w)}^{t},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="409" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img101.png"
 ALT="$\displaystyle \DP{w_{i,k(w)}}{t} = - \left[\bar{c_{p}} \bar{\theta}_{v}
\DP{(\pi - \alpha Div )}{z}\right]_{i,k(w)}
+ [F_{w}]_{i,k(w)}^{t},$">|; 

$key = q/i(u),k(w);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="83" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img14.png"
 ALT="$i(u), k(w)$">|; 

$key = q/displaystyle[F_{K_m}]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="67" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img300.png"
 ALT="$\displaystyle [F_{K_m}]_{i,k}^{t}$">|; 

$key = q/displaystylegamma_{h}=alpha_{h}left(1-frac{x}{d_{h}}right)^{3}hspace{5em}(x<d_{h}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img345.png"
 ALT="$\displaystyle \gamma_{h} = \alpha_{h} \left( 1 - \frac{x}{d_{h}}\right)^{3}
\hspace{5em} (x &lt; d_{h}),$">|; 

$key = q/displaystyle+gfrac{sum[q_{v}]_{i,k(w)}^{t}slashM_{v}}{1slashM_{d}+sum[bar{q_{v}}]_{i,k(w)}slashM_{v}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img177.png"
 ALT="$\displaystyle + g \frac{\sum [q_{v}]_{i,k(w)}^{t}/M_{v}}{1/M_{d}
+ \sum [\bar{q_{v}}]_{i,k(w)}/M_{v}}$">|; 

$key = q/displaystyletheta_{i,k}^{t-Deltat}+2Deltat[F_{theta}]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img200.png"
 ALT="$\displaystyle \theta_{i,k}^{t - \Delta t} + 2 \Delta t [F_{\theta}]_{i,k}^{t}$">|; 

$key = q/displaystyleleft[DP{w}{x}right]_{i(u),k(w)}equivfrac{9}{8}left(frac{w_{i+1,k(w)}frac{1}{24}left(frac{w_{i+2,k(w)}-w_{i-1,k(w)}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="554" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img44.png"
 ALT="$\displaystyle \left[\DP{w}{x} \right]_{i(u),k(w)}
\equiv \frac{9}{8}\left(\frac...
...ight) -
\frac{1}{24}\left(\frac{w_{i+2, k(w)} - w_{i-1, k(w)}}{\Delta x}\right)$">|; 

$key = q/displaystylegfrac{theta_{i,k(w)}^{t}}{overline{theta}_{i,k(w)}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="69" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img176.png"
 ALT="$\displaystyle g \frac{\theta_{i,k(w)}^{t}}{\overline{\theta}_{i,k(w)}}$">|; 

$key = q/1slashalpha_{h},1slashalpha_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="91" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img353.png"
 ALT="$1/\alpha_{h}, 1/\alpha_{v}$">|; 

$key = q/displaystyle[F_{theta}]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="51" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img207.png"
 ALT="$\displaystyle [F_{\theta}]_{i,k}$">|; 

$key = q/displaystyle-overline{c}_{p}overline{theta}_{v}DP{pi}{z}+alphaDP{D}{z}+gfrac{theta}{overline{theta}_{v}},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="204" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img382.png"
 ALT="$\displaystyle - \overline{c}_{p}\overline{\theta}_{v}
\DP{\pi}{z} + \alpha\DP{D}{z}
+ g\frac{\theta}{\overline{\theta}_{v}},$">|; 

$key = q/alpha;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="17" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img397.png"
 ALT="$\alpha$">|; 

$key = q/{displaymath}left[DP{u}{x}right]_{i(u),k}=frac{u_{i+1,j}-u_{i,j}}{Deltax}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="196" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img427.png"
 ALT="\begin{displaymath}
\left[\DP{u}{x} \right]_{i(u),k} =
\frac{u_{i+1,j} - u_{i,j}}{\Delta x}
\end{displaymath}">|; 

$key = q/displaystyleleft[DP{}{z}left{left(K_{h}right)_{i,k(w)}left(DP{overline{phi}}{z}right)_{i,k(w)}right}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="283" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img231.png"
 ALT="$\displaystyle \left[ \DP{}{z}\left\{
\left( K_{h} \right)_{i,k(w)}
\left( \DP{\overline{\phi}}{z} \right)_{i,k(w)}
\right\} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyleu_{i(u),k}^{t}left[DP{u}{x}right]_{i(u),k}^{t}+w_{i(u),k}^{t}left[DP{u}{z}right]_{i(u),k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="287" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img172.png"
 ALT="$\displaystyle u_{i(u),k}^{t} \left[\DP{u}{x}\right]_{i(u),k}^{t}
+ w_{i(u),k}^{t} \left[\DP{u}{z}\right]_{i(u),k}^{t}$">|; 

$key = q/displaystyle+left{1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rh}bar{theta}_{v}^{2}right)_{1(w)}right}pi^{tau+Deltatau}_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="372" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img370.png"
 ALT="$\displaystyle +
\left\{
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}...
...ho} \bar{\theta}_{v}^{2}
\right)_{1(w)}
\right\} \pi^{\tau + \Delta \tau}_{i,1}$">|; 

$key = q/displaystyleleft[u_{i(u),k}left[DP{phi}{x}right]_{i(u),k}right]_{i,k}^{t}+left[w_{i,k(w)}left[DP{phi}{z}right]_{i,k(w)}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="81" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img225.png"
 ALT="$\displaystyle \left[
u_{i(u),k} \left[ \DP{\phi}{x} \right]_{i(u),k}
\right]_{i,k}^{t}
+
\left[
w_{i,k(w)} \left[ \DP{\phi}{z} \right]_{i,k(w)}
\right]_{i,k}^{t}$">|; 

$key = q/displaystyleu_{i,j}+left[DP{u}{x}right]_{i,k}frac{Deltax}{2}+frac{1}{2!}left[DP[3!}left[DP[3]{u}{x}right]_{i,k}left(frac{Deltax}{2}right)^{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img412.png"
 ALT="$\displaystyle u_{i,j}
+ \left[\DP{u}{x} \right]_{i,k}\frac{\Delta x}{2}
+ \frac...
...frac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{3}$">|; 

$key = q/displaystyle+O[(Deltax)^{7}];MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="93" HEIGHT="41" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img422.png"
 ALT="$\displaystyle + O[(\Delta x)^{7}]$">|; 

$key = q/displaystylew_{i(u),k}equivfrac{w_{i+1,k(w)}+w_{i,k(w)}+w_{i+1,k-1(w)}+w_{i,k-1(w)}}{4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="422" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img29.png"
 ALT="$\displaystyle w_{i(u),k} \equiv \frac{w_{i+1, k(w)} + w_{i, k(w)}
+ w_{i+1, k-1(w)} + w_{i, k-1(w)}}{4}$">|; 

$key = q/displaystyle-betafrac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rho}bar{theta}_{v}^{2}}l-DP{(alphaDiv)^{tau}}{z}right}+F_{w}^{t}Deltatauright}right].;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="588" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img117.png"
 ALT="$\displaystyle - \beta \frac{\bar{c}^{2}\Delta \tau}{\bar{c_{p}} \bar{\rho} \bar...
... \DP{(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau
\right\}
\right].$">|; 

$key = q/displaystyleleft[{{rm{Turb}.K_mright]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="121" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img313.png"
 ALT="$\displaystyle \left[{\rm Turb}.K_m\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyleleft[DP{w}{x}right]_{i(u),k(w)}equivfrac{w_{i+1,k(w)}-w_{i,k(w)}}{Deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="279" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img36.png"
 ALT="$\displaystyle \left[\DP{w}{x} \right]_{i(u),k(w)}
\equiv \frac{w_{i+1, k(w)} - w_{i, k(w)}}{\Delta x}$">|; 

$key = q/displaystyleleft[q_{c}right]^{t+Deltat}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="97" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img273.png"
 ALT="$\displaystyle \left[ q_{c} \right]^{t + \Delta t}
= 0$">|; 

$key = q/displaystyle=pi^{tau}_{i,k}-(1-beta)left(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{theta}_{v}}right)_{k}left(DP{u^{tau+Deltatau}}{x}right)_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="551" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img121.png"
 ALT="$\displaystyle =
\pi^{\tau}_{i,k}
- (1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \...
...ar{\theta}_{v}}
\right)_{k}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,k}$">|; 

$key = q/-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img419.png"
 ALT="$-$">|; 

$key = q/[q_{v}]^{*};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="39" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img256.png"
 ALT="$[q_{v}]^{*}$">|; 

$key = q/displaystylenu_{h}left{DP{}{x}left(DP{u}{x}right)_{i,k}right}_{i(u),k}^{t-Deltat}{z}left(DP{u}{z}right)_{i(u),k(w)}right}_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="428" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img188.png"
 ALT="$\displaystyle \nu_{h} \left\{ \DP{}{x} \left(\DP{u}{x}\right)_{i,k} \right\}_{i...
...\{ \DP{}{z} \left(\DP{u}{z}\right)_{i(u),k(w)} \right\}_{i(u),k}^{t - \Delta t}$">|; 

$key = q/displaystyle+gfrac{sum[q_{v}]_{i,k(w)}slashM_{v}}{1slashM_{d}+sum[bar{q_{v}}]_{i,k(w)}slashM_{v}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="214" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img59.png"
 ALT="$\displaystyle + g \frac{\sum [q_{v}]_{i,k(w)}/M_{v}}{1/M_{d}
+ \sum [\bar{q_{v}}]_{i,k(w)}/M_{v}}$">|; 

$key = q/phi_{e};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="23" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img335.png"
 ALT="$\phi_{e}$">|; 

$key = q/displaystyleleft[CL_{cr}right]_{i,k}^{t}=2.2[q_{c}]_{i,k}^{t}left([bar{rho}]_{i,k}left[q_{r}right]_{i,k}^{t}right)^{0.875};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="299" HEIGHT="53" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img244.png"
 ALT="$\displaystyle \left[ CL_{cr} \right]_{i,k}^{t}
= 2.2 [q_{c}]_{i,k}^{t}
\left(
[\bar{\rho}]_{i,k}
\left[ q_{r} \right]_{i,k}^{t}
\right)^{0.875}$">|; 

$key = q/displaystyleleft[left(DP{(alphaDiv)^{tau}}{z}right)-(1-beta)left(DP{pi^{tau}}{z}(Dinv{bar{c_{p}}bar{theta}_{v}}F_{w}^{t}right)right]_{i,0(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="420" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img363.png"
 ALT="$\displaystyle \left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
- (1 - \beta) \...
...)
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,0(w)}$">|; 

$key = q/displaystylepi_{im+2,k}=-pi_{im-1,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="157" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img90.png"
 ALT="$\displaystyle \pi_{im+2,k} = - \pi_{im-1,k}$">|; 

$key = q/displaystyle+left{-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rhoheta}_{v}^{2}right)_{km-1(w)}right}pi^{tau+Deltatau}_{i,km-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="405" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img140.png"
 ALT="$\displaystyle + \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{...
...ar{\theta}_{v}^{2}
\right)_{km-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,km-1}$">|; 

$key = q/*;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img254.png"
 ALT="$*$">|; 

$key = q/displaystylehspace{10mm}+left{-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c{theta}_{v}^{2}right)_{k-1(w)}right}pi^{tau+Deltatau}_{i,k-1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="419" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img120.png"
 ALT="$\displaystyle \hspace{10mm}+ \left\{
- \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delt...
...\bar{\theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\pi^{\tau + \Delta \tau}_{i,k-1}$">|; 

$key = q/F_u,F_w;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="57" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img99.png"
 ALT="$F_u, F_w$">|; 

$key = q/(i-1(u),j);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="101" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img414.png"
 ALT="$(i-1(u),j)$">|; 

$key = q/displaystyle-left(frac{K_{m}}{3}right)_{i,k}^{t-Deltat}left{left(DP{u}{x}right)_{i,k}^{t-Deltat}+left(DP{w}{z}right)_{i,k}^{t-Deltat}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="336" HEIGHT="76" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img312.png"
 ALT="$\displaystyle - \left( \frac{K_{m}}{3} \right)_{i,k}^{t - \Delta t}
\left\{
\le...
...ight)_{i,k}^{t-\Delta t}
+
\left( \DP{w}{z} \right)_{i,k}^{t-\Delta t}
\right\}$">|; 

$key = q/displaystylehspace{10mm}+left[1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{ta}_{v}^{2}right)_{k-1(w)}right}right]pi^{tau+Deltatau}_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="563" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img119.png"
 ALT="$\displaystyle \hspace{10mm}+ \left[
1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Del...
...theta}_{v}^{2}
\right)_{k-1(w)}
\right\}
\right]
\pi^{\tau + \Delta \tau}_{i,k}$">|; 

$key = q/k=1;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img364.png"
 ALT="$k = 1$">|; 

$key = q/displaystyleF_{w,i,k(w)}=-left[{{rm{Adv}.wright]_{i(u),k}^{t}+left[{{rm{Buoy}rigk(w)}^{t-Deltat}+left[{{rm{Diff}.wright]_{i,k(w)}^{t-Deltat}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="579" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img170.png"
 ALT="$\displaystyle F_{w,i,k(w)} =
- \left[ {\rm Adv}.w \right]_{i(u),k}^{t}
+ \left[...
..._{i,k(w)}^{t - \Delta t}
+ \left[ {\rm Diff}.w \right]_{i,k(w)}^{t - \Delta t}.$">|; 

$key = q/displaystyle+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{teltatau}_{i,k}-pi^{tau+Deltatau}_{i,k-1}}{Deltaz}right)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img360.png"
 ALT="$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...elta \tau}_{i,k}
- \pi^{\tau + \Delta \tau}_{i,k-1}}{\Delta z}
\right)
\right\}$">|; 

$key = q/displaystyle-u_{i,k(w)}left[DP{u}{x}right]_{i,k(w)}-w_{i,k(w)}left[DP{u}{z}right]_{i,k(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img56.png"
 ALT="$\displaystyle - u_{i,k(w)}\left[\DP{u}{x}\right]_{i,k(w)}
- w_{i,k(w)}\left[\DP{u}{z}\right]_{i,k(w)}$">|; 

$key = q/displaystyle[CN_{cr}]_{i,k}^{t}=(q_{c,i,k}^{t}-q_{c0})slashtau_{ac};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="225" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img243.png"
 ALT="$\displaystyle [CN_{cr}]_{i,k}^{t} = (q_{c,i,k}^{t} - q_{c0})/\tau _{ac}$">|; 

$key = q/displaystylew(i,0(w))=0,;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img124.png"
 ALT="$\displaystyle w(i,0(w)) = 0,$">|; 

$key = q/{figure}center{{Depsf[120mm]{psslashkoushi.eps}{center{{{{figure};LFS=12;FSF=1.6;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="539" HEIGHT="535" BORDER="0"
 SRC="|."$dir".q|img8.png"
 ALT="\begin{figure}\begin{center}
\Depsf[120mm]{ps/koushi.eps}
\end{center} \end{figure}">|; 

$key = q/theta_{el};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img320.png"
 ALT="$\theta_{el}$">|; 

$key = q/displaystylepi^{tau}_{i,1}-(1-beta)left(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rv}}right)_{1}left(DP{u^{tau+Deltatau}}{x}right)_{i,1}+F_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="576" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img162.png"
 ALT="$\displaystyle \pi^{\tau}_{i,1}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \tau...
..._{v}}
\right)_{1}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,1}
+ F_{i,1}$">|; 

$key = q/displaystyleDP{D}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img389.png"
 ALT="$\displaystyle \DP{D}{t}$">|; 

$key = q/overline{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="13" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img62.png"
 ALT="$\overline{c}$">|; 

$key = q/displaystyleX^{2}-(p_{{rm{NH_{3}}+p_{{rm{H_{2}S})X+p_{{rm{NH_{3}}cdotp_{{rm{H_{2}S}-e^{k_{p}}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="368" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img280.png"
 ALT="$\displaystyle X^{2} - (p_{\rm NH_{3}} + p_{\rm H_{2}S}) X
+ p_{\rm NH_{3}} \cdot p_{\rm H_{2}S}
- e^{k_{p}} = 0$">|; 

$key = q/x_{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img488.png"
 ALT="$x_{x}$">|; 

$key = q/displaystylefrac{p_{x}}{p}M_{x}(1slashM_{d}+sum_{vneqx}q_{v}slashM_{v});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img484.png"
 ALT="$\displaystyle \frac{p_{x}}{p} M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v})$">|; 

$key = q/displaystyleX=Dinv{2}left{(p_{{rm{NH_{3}}+p_{{rm{H_{2}S})-sqrt{(p_{{rm{NH_{3}}-p_{{rm{H_{2}S})^{2}+4e^{K_{p}}}.right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="418" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img287.png"
 ALT="$\displaystyle X = \Dinv{2}
\left\{
(p_{\rm NH_{3}} + p_{\rm H_{2}S})
- \sqrt{ (p_{\rm NH_{3}} - p_{\rm H_{2}S})^{2}
+ 4 e^{K_{p}} }.
\right\}$">|; 

$key = q/displaystyleleft[DP{u}{x}right]_{i,k}equivfrac{9}{8}left(frac{u_{i(u),k}-u_{i-1(frac{1}{24}left(frac{u_{i(u)+1,k}-u_{i-2(u),k}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="492" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img42.png"
 ALT="$\displaystyle \left[\DP{u}{x} \right]_{i,k}
\equiv \frac{9}{8}\left(\frac{u_{i(...
...ight) -
\frac{1}{24}\left(\frac{u_{i(u)+1, k} - u_{i-2(u), k}}{\Delta x}\right)$">|; 

$key = q/displaystyle[pi]^{t+Deltat}=2Deltatleft{[{{rm{Adv}.pi]^{t}+cdots+left{gamma_{h}(x)+gamma_{v}(z)right}(pi-bar{pi})^{t-Deltat}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="499" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img337.png"
 ALT="$\displaystyle [\pi]^{t + \Delta t} = 2 \Delta t
\left\{
[{\rm Adv}.\pi]^{t}
+ \...
...gamma_{h}(x) + \gamma_{v}(z)
\right\} (\pi - \bar{\pi})^{t - \Delta t}
\right\}$">|; 

$key = q/times27-;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img438.png"
 ALT="$\times 27 -$">|; 

$key = q/displaystylew_{i,k}equivfrac{w_{i,k(w)}+w_{i,k-1(w)}}{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="205" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img28.png"
 ALT="$\displaystyle w_{i,k} \equiv \frac{w_{i, k(w)} + w_{i, k-1(w)}}{2}$">|; 

$key = q/displaystylepi_{i(u),k(w)}equivfrac{pi_{i,k}+pi_{i+1,k}+pi_{i,k+1}+pi_{i+1,k+1}}{4};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="57" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img24.png"
 ALT="$\displaystyle \pi_{i(u),k(w)} \equiv
\frac{\pi_{i, k} + \pi_{i+1, k}+ \pi_{i, k+1}+ \pi_{i+1, k+1}}{4}$">|; 

$key = q/displaystyle[q_{v}]^{*}+[q_{c}]^{*}-[q_{v}]^{t+Deltat}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img267.png"
 ALT="$\displaystyle [q_{v}]^{*} + [q_{c}]^{*} - [q_{v}]^{t + \Delta t} .$">|; 

$key = q/displaystylebeta^{2}frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{theta}_theta}_{v}^{2}right)left(DP{pi^{tau+Deltatau}}{z}right)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img115.png"
 ALT="$\displaystyle \beta^{2}
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \bar{\r...
...\theta}_{v}^{2}\right)
\left(
\DP{\pi^{\tau + \Delta \tau}}{z}
\right)
\right\}$">|; 

$key = q/2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img76.png"
 ALT="$2$">|; 

$key = q/{displaymath}[Q_{cnd}]_{i,k}^{t}=-left[frac{L}{{c_{p}bar{pi}}_{d}bar{pi}}EV_{rv}}^{t})(bar{rho}_{i,k}[q_{r}]_{i,k})^{0.65}right}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="673" HEIGHT="61" BORDER="0"
 SRC="|."$dir".q|img236.png"
 ALT="\begin{displaymath}[Q_{cnd}]_{i,k}^{t}
= - \left[ \frac{L}{{c_{p} \bar{\pi}}_{...
...{i,k}^{t})
(\bar{\rho}_{i,k} [q_{r}]_{i,k})^{0.65}
\right\}
\end{displaymath}">|; 

$key = q/displaystyleu^{t+Deltat}_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="50" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img331.png"
 ALT="$\displaystyle u^{t+\Delta t}_{i(u),k}$">|; 

$key = q/{displaymath}left[DP{u}{x}right]_{i,k}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="56" HEIGHT="58" BORDER="0"
 SRC="|."$dir".q|img406.png"
 ALT="\begin{displaymath}
\left[\DP{u}{x}\right]_{i,k}
\end{displaymath}">|; 

$key = q/displaystyleu_{i+1(u),j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="66" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img431.png"
 ALT="$\displaystyle u _{i+1(u),j}$">|; 

$key = q/i,k(w);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="58" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img12.png"
 ALT="$i,k(w)$">|; 

$key = q/v;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img476.png"
 ALT="$v$">|; 

$key = q/2DeltatslashDeltatau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="72" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img378.png"
 ALT="$2\Delta t/\Delta \tau$">|; 

$key = q/displaystyle1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{z^{2}}left(bar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{1(w)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="285" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img150.png"
 ALT="$\displaystyle 1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{...
...Delta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{1(w)},$">|; 

$key = q/displaystyle-Dinv{2l^{2}}left(K_{m}^{2}right)_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="129" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img317.png"
 ALT="$\displaystyle - \Dinv{2 l^{2}} \left( K_{m}^{2} \right)_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystylehspace{2mm}-betaleft(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rho}bar{ar{theta}_{v}right)_{i,k(w)}left{w^{tau}_{i,k(w)}right.right.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="314" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img122.png"
 ALT="$\displaystyle \hspace{2mm} - \beta
\left(
\frac{\bar{c}^{2}\Delta \tau}{\bar{c_...
...ho} \bar{\theta}_{v} \right)_{i,k(w)}
\left\{
w^{\tau}_{i,k(w)}
\right. \right.$">|; 

$key = q/displaystyleleft[thetaright]^{t+Deltat}=[theta]^{*}-gamma[q_{c}]^{*},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="181" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img271.png"
 ALT="$\displaystyle \left[ \theta \right]^{t + \Delta t} =
[\theta]^{*} - \gamma [q_{c}]^{*},$">|; 

$key = q/Deltaq_{{rm{H_2S};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="54" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img296.png"
 ALT="$\Delta q_{\rm H_2S}$">|; 

$key = q/displaystyleDP{w}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="34" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img381.png"
 ALT="$\displaystyle \DP{w}{t}$">|; 

$key = q/Deltax,Deltaz;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img195.png"
 ALT="$\Delta x, \Delta z$">|; 

$key = q/displaystyleleft[q_{{rm{H_2S}right]^{t+Deltat}=[q_{{rm{H_2S}]^{*}+Deltaq_{{rm{H_2S},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img291.png"
 ALT="$\displaystyle \left[ q_{\rm H_2S} \right]^{t + \Delta t}
= [q_{\rm H_2S}]^{*} + \Delta q_{\rm H_2S},$">|; 

$key = q/displaystyle27(u_{i(u),j}-u_{i-1(u),j})-(u_{i+1(u),j}-u_{i-2(u),j});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="352" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img440.png"
 ALT="$\displaystyle 27(u _{i(u),j} - u_{i-1(u),j}) - (u _{i+1(u),j} - u_{i-2(u),j})$">|; 

$key = q/left(Deltaxright)^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="44" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img426.png"
 ALT="$\left(\Delta x\right)^{2}$">|; 

$key = q/displaystylefrac{q_{x}p}{M_{x}(1slashM_{d}+sum_{v}q_{v}slashM_{v})};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img491.png"
 ALT="$\displaystyle \frac{q_{x} p}{M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})}$">|; 

$key = q/displaystyle-frac{overline{c}^{2}}{overline{c}_{p}overline{rho}overline{theta}_{eta}_{v}u}{x}+DP{overline{rho}overline{theta}_{v}w}{z}right],;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="228" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img384.png"
 ALT="$\displaystyle - \frac{\overline{c}^{2}}{\overline{c}_{p}
\overline{\rho}\overli...
...rline{\theta}_{v}u}{x} +
\DP{\overline{\rho}\overline{\theta}_{v}w}{z}
\right],$">|; 

$key = q/pmDelta3xslash2;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="74" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img429.png"
 ALT="$\pm
\Delta 3x/2$">|; 

$key = q/displaystylep_{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img490.png"
 ALT="$\displaystyle p_{x}$">|; 

$key = q/displaystyleleft[{{rm{Adv}.{w}right]_{i,k(w)}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img173.png"
 ALT="$\displaystyle \left[ {\rm Adv}.{w} \right]_{i,k(w)}^{t}$">|; 

$key = q/displaystyleleft[{{rm{Adv}.{phi}right]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="82" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img224.png"
 ALT="$\displaystyle \left[{\rm Adv}.{\phi}\right]_{i,k}^{t}$">|; 

$key = q/u;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img2.png"
 ALT="$u$">|; 

$key = q/displaystyle-beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{teltatau}_{i,k+1}-pi^{tau+Deltatau}_{i,k}}{Deltaz}right)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="432" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img359.png"
 ALT="$\displaystyle - \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}...
...elta \tau}_{i,k+1}
- \pi^{\tau + \Delta \tau}_{i,k}}{\Delta z}
\right)
\right\}$">|; 

$key = q/displaystyleDP{theta}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="29" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img385.png"
 ALT="$\displaystyle \DP{\theta}{t}$">|; 

$key = q/displaystylex_{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="24" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img493.png"
 ALT="$\displaystyle x_{x}$">|; 

$key = q/u,pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="35" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img97.png"
 ALT="$u, \pi$">|; 

$key = q/displaystyle+frac{1}{4!}left[DP[4]{u}{x}right]_{i,k}left(frac{Deltax}{2}right)^{[DP[5]{u}{x}right]_{i,k}left(frac{Deltax}{2}right)^{5}+cdots.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="403" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img413.png"
 ALT="$\displaystyle + \frac{1}{4!}\left[\DP[4]{u}{x} \right]_{i,k}
\left(\frac{\Delta...
...\left[\DP[5]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{5}
+ \cdots .$">|; 

$key = q/displaystylenu_{h}left{DP{}{x}left(DP{w}{x}right)_{i(u),k(w)}right}_{i,k(w)}^{t-ft{DP{}{z}left(DP{w}{z}right)_{i,k}right}_{i,k(w)}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img190.png"
 ALT="$\displaystyle \nu_{h} \left\{ \DP{}{x} \left(\DP{w}{x}\right)_{i(u),k(w)} \righ...
... \left\{ \DP{}{z} \left(\DP{w}{z}\right)_{i,k} \right\}_{i,k(w)}^{t - \Delta t}$">|; 

$key = q/displaystyleu_{0(u),k}=u_{im(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="135" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img77.png"
 ALT="$\displaystyle u_{0(u), k} = u_{im(u), k}$">|; 

$key = q/displaystyle0leqXleq{{rm{min}(P_{{rm{NH_{3}},P_{{rm{H_{2}S});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="213" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img288.png"
 ALT="$\displaystyle 0 \leq X \leq {\rm min}(P_{\rm NH_{3}}, P_{\rm H_{2}S})$">|; 

$key = q/displaystyleleft[left(DP{(alphaDiv)^{tau}}{z}right)-(1-beta)left(DP{pi^{tau}}{z}Dinv{bar{c_{p}}bar{theta}_{v}}F_{w}^{t}right)right]_{i,km(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="433" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img372.png"
 ALT="$\displaystyle \left[
\left( \DP{(\alpha Div)^{\tau}}{z} \right)
- (1 - \beta) \...
...
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)
\right]_{i,km(w)}$">|; 

$key = q/{displaymath}left|frac{1}{24}left[DP[3]{u}{x}right]_{i(u),k}left(Deltaxright)^{2}right|{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="170" HEIGHT="64" BORDER="0"
 SRC="|."$dir".q|img428.png"
 ALT="\begin{displaymath}
\left\vert\frac{1}{24}\left[\DP[3]{u}{x} \right]_{i(u),k}
\left(\Delta x\right)^{2}\right\vert
\end{displaymath}">|; 

$key = q/displaystyleleft[thetaright]^{t+Deltat}=theta^{*}+gammaleft([q_{{rm{NH_3}]^{*}+[*}-[q_{{rm{NH_3}]^{t+Deltat}-[q_{{rm{H_2S}]^{t+Deltat}right).;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="493" HEIGHT="47" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img293.png"
 ALT="$\displaystyle \left[ \theta \right]^{t + \Delta t}
=
\theta^{*} + \gamma \left(...
...}]^{*}
- [q_{\rm NH_3}]^{t + \Delta t} - [q_{\rm H_2S}]^{t + \Delta t}
\right).$">|; 

$key = q/displaystyle-u_{i,k}left[DP{q_{c}}{x}right]_{i,k}-w_{i,k}left[DP{q_{c}}{x}right]_{i,k}+[{{rm{Src}.q_{c}]_{i,k}+[{{rm{Turb}.{q_{c}}]_{i,k},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="449" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img71.png"
 ALT="$\displaystyle - u_{i,k} \left[ \DP{q_{c}}{x} \right]_{i,k}
- w_{i,k} \left[ \DP{q_{c}}{x} \right]_{i,k}
+ [{\rm Src}.q_{c}]_{i,k} + [{\rm Turb}.{q_{c}}]_{i,k},$">|; 

$key = q/displaystyle2left[DP{}{x}left{left(K_{m}right)_{i,k}left(DP{u}{x}right)_{i,k}right}right]_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="262" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img180.png"
 ALT="$\displaystyle 2 \left[
\DP{}{x}\left\{
\left( K_{m} \right)_{i,k} \left( \DP{u}{x} \right)_{i,k}
\right\}
\right]_{i(u),k}^{t - \Delta t}$">|; 

$key = q/displaystylebar{theta_{v}}left{1+frac{theta}{bar{theta}}+frac{sumq_{v}slashM_{v}_{v}}-frac{sumq_{v}+sumq_{c}+sumq_{r}}{1+sumbar{q_{v}}}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="439" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img325.png"
 ALT="$\displaystyle \bar{\theta_{v}}
\left\{
1 + \frac{\theta}{\bar{\theta}}
+ \frac{...
...}
- \frac{\sum q_{v} + \sum q_{c} + \sum q_{r}}
{1 + \sum \bar{q_{v}}}
\right\}$">|; 

$key = q/K_{m};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img191.png"
 ALT="$K_{m}$">|; 

$key = q/k;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img81.png"
 ALT="$k$">|; 

$key = q/pi;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="16" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img96.png"
 ALT="$\pi$">|; 

$key = q/displaystyleleft(DP{(alphaDiv)^{tau}}{z}right)_{i,0(w)}-(1-beta)left(DP{pi^{tau}}+left(Dinv{bar{c_{p}}bar{theta}_{v}}F_{w}^{t}right)_{i,0(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="475" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img128.png"
 ALT="$\displaystyle \left( \DP{(\alpha Div)^{\tau}}{z} \right)_{i,0(w)}
- (1 - \beta)...
..._{i,0(w)}
+ \left(\Dinv{\bar{c_{p}} \bar{\theta}_{v}} F_{w}^{t}\right)_{i,0(w)}$">|; 

$key = q/displaystyleL_{v}=DD{lnp_{v}}{T}{R_{v}T^{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="147" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img458.png"
 ALT="$\displaystyle L_{v} = \DD{\ln p_{v}}{T} {R_{v} T^{2}}$">|; 

$key = q/displaystyle-u_{i,k}left[DP{q_{r}}{x}right]_{i,k}-w_{i,k}left[DP{q_{r}}{x}right]_{i,k}+[{{rm{Src}.q_{r}]_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="335" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img73.png"
 ALT="$\displaystyle - u_{i,k} \left[ \DP{q_{r}}{x} \right]_{i,k}
- w_{i,k} \left[ \DP{q_{r}}{x} \right]_{i,k}
+ [{\rm Src}.q_{r}]_{i,k}$">|; 

$key = q/displaystyleleft[q_{v}right]_{i,k}^{t-Deltat}+2Deltat[F_{q_{v}}]_{i,k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="171" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img202.png"
 ALT="$\displaystyle \left[ q_{v} \right]_{i,k}^{t-\Delta t}
+ 2 \Delta t [F_{q_{v}}]_{i,k}^{t}$">|; 

$key = q/displaystylepi^{tau+Deltatau}_{i,km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img373.png"
 ALT="$\displaystyle \pi^{\tau + \Delta \tau}_{i,km}$">|; 

$key = q/displaystyleE_{i,0(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img130.png"
 ALT="$\displaystyle E_{i,0(w)}$">|; 

$key = q/displaystyles_{{rm{NH_3}+s_{{rm{H_2S}-RTDP{K_{p}}{T}-RK_{p}-RTln{p_{0}}^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="347" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img465.png"
 ALT="$\displaystyle s_{\rm NH_3} + s_{\rm H_2S} - RT \DP{K_{p}}{T} - R K_{p}
- RT \ln {p_{0}}^{2}$">|; 

$key = q/2Deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img336.png"
 ALT="$2 \Delta t$">|; 

$key = q/displaystyleu_{i,j}-left[DP{u}{x}right]_{i,k}frac{Deltax}{2}+frac{1}{2!}left[DP[3!}left[DP[3]{u}{x}right]_{i,k}left(frac{Deltax}{2}right)^{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="502" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img418.png"
 ALT="$\displaystyle u_{i,j}
- \left[\DP{u}{x} \right]_{i,k}\frac{\Delta x}{2}
+ \frac...
...frac{1}{3!}\left[\DP[3]{u}{x} \right]_{i,k}
\left(\frac{\Delta x}{2}\right)^{3}$">|; 

$key = q/displaystyleleft[DP{phi}{z}right]_{i(u),k}equivfrac{9}{8}left(frac{phi_{i(u),k(w}left(frac{phi_{i(u),k+1(w)}-phi_{i(u),k-2(w)}}{Deltaz}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="593" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img47.png"
 ALT="$\displaystyle \left[\DP{\phi}{z} \right]_{i(u),k}
\equiv \frac{9}{8}\left(
\fra...
...{1}{24}\left(
\frac{\phi_{i(u), k+1(w)} - \phi_{i(u), k-2(w)}}{\Delta z}\right)$">|; 

$key = q/d_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img348.png"
 ALT="$d_{v}$">|; 

$key = q/displaystyle+left[DP{}{z}left{left(K_{h}right)_{i,k(w)}left(DP{phi}{z}right)_{i,k(w)}right}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="301" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img229.png"
 ALT="$\displaystyle + \left[ \DP{}{z}\left\{
\left( K_{h} \right)_{i,k(w)}
\left( \DP{\phi }{z} \right)_{i,k(w)}
\right\} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/F;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="19" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img198.png"
 ALT="$F$">|; 

$key = q/displaystylenu_{h}=frac{alpha_{h}Deltax^{2}}{Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="106" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img193.png"
 ALT="$\displaystyle \nu_{h} = \frac{\alpha_{h} \Delta x^{2}}{\Delta t}$">|; 

$key = q/displaystyleK_{p}equivln(p_{{rm{NH_{3}}cdotp_{{rm{H_{2}S})=61.781-frac{10834}{T}-ln{10^{2}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="388" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img275.png"
 ALT="$\displaystyle K_{p}
\equiv \ln(p_{\rm NH_{3}} \cdot p_{\rm H_{2}S})
= 61.781 - \frac{10834}{T} - \ln{10^{2}}$">|; 

$key = q/displaystylenu_{h}left{DP{}{x}left(DP{phi}{x}right)_{i(u),k}right}_{i,k}^{t-Delt{DP{}{z}left(DP{phi}{z}right)_{i,k(w)}right}_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img233.png"
 ALT="$\displaystyle \nu_{h} \left\{ \DP{}{x} \left(\DP{\phi}{x}\right)_{i(u),k} \righ...
...eft\{ \DP{}{z} \left(\DP{\phi}{z}\right)_{i,k(w)} \right\}_{i,k}^{t - \Delta t}$">|; 

$key = q/alpha_{h},alpha_{v};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="53" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img196.png"
 ALT="$\alpha_{h}, \alpha_{v}$">|; 

$key = q/displaystyleDP{[q_{r}]_{i,k}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="63" HEIGHT="63" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img72.png"
 ALT="$\displaystyle \DP{[q_{r}]_{i,k}}{t}$">|; 

$key = q/epsilonequivalphasqrt{k^{2}+l^{2}}slashoverline{c};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="137" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img398.png"
 ALT="$\epsilon \equiv \alpha \sqrt{k^{2}+l^{2}}/\overline{c}$">|; 

$key = q/displaystyletheta_{el};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="25" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img321.png"
 ALT="$\displaystyle \theta_{el}$">|; 

$key = q/-CN_{vc}+EV_{cv};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="125" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img250.png"
 ALT="$-CN_{vc} + EV_{cv}$">|; 

$key = q/displaystyleleft[{{rm{Shear}.K_mright]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="127" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img309.png"
 ALT="$\displaystyle \left[{\rm Shear}.K_m\right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyleX=Dinv{2}left{(p_{{rm{NH_{3}}+p_{{rm{H_{2}S})pmsqrt{(p_{{rm{NH_{3}}-p_{{rm{H_{2}S})^{2}+4e^{K_{p}}}right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="413" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img282.png"
 ALT="$\displaystyle X = \Dinv{2}
\left\{
(p_{\rm NH_{3}} + p_{\rm H_{2}S})
\pm \sqrt{ (p_{\rm NH_{3}} - p_{\rm H_{2}S})^{2}
+ 4 e^{K_{p}} }
\right\}$">|; 

$key = q/displaystyleu_{0(u),k}=u_{im(u),k}=0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="169" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img83.png"
 ALT="$\displaystyle u_{0(u),k} = u_{im(u),k} = 0$">|; 

$key = q/displaystyle+left[DP{}{z}left{left(K_{m}right)_{i(u),k(w)}left(DP{w}{x}right)_{ift(DP{u}{z}right)_{i(u),k(w)}right}right]_{i(u),k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="576" HEIGHT="83" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img181.png"
 ALT="$\displaystyle +\left[ \DP{}{z}\left\{
\left( K_{m} \right)_{i(u),k(w)}
\left( \...
...}
\left( \DP{u}{z} \right)_{i(u),k(w)}
\right\} \right]_{i(u),k}^{t - \Delta t}$">|; 

$key = q/displaystylepi^{tau+Deltatau}+betafrac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{rho}bar2}Deltatau}{bar{c_{p}}bar{theta}_{v}}DP{u^{tau+Deltatau}}{x}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="628" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img111.png"
 ALT="$\displaystyle \pi^{\tau + \Delta \tau}
+ \beta \frac{\bar{c}^{2}\Delta \tau}{\b...
...^{2} \Delta \tau}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u^{\tau + \Delta \tau}}{x}.$">|; 

$key = q/q_{x};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="21" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img477.png"
 ALT="$q_{x}$">|; 

$key = q/displaystyleleft[{{rm{Adv}.{u}right]_{i(u),k}^{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="100" HEIGHT="43" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img171.png"
 ALT="$\displaystyle \left[ {\rm Adv}.{u} \right]_{i(u),k}^{t}$">|; 

$key = q/displaystyleu_{im+1(u),k}=u_{1(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="152" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img79.png"
 ALT="$\displaystyle u_{im+1(u), k} = u_{1(u), k}$">|; 

$key = q/d;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="14" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img475.png"
 ALT="$d$">|; 

$key = q/displaystyleB_{k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img153.png"
 ALT="$\displaystyle B_{k}$">|; 

$key = q/displaystyle-left[{{rm{Adv}.q_{c}right]_{i,k}^{t}+left[{{rm{Turb}.q_{c}right]^{t-Deltat}+left[{{rm{Diff}.q_{c}right]^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="348" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img214.png"
 ALT="$\displaystyle - \left[ {\rm Adv}.q_{c} \right]_{i,k}^{t}
+ \left[ {\rm Turb}.q_{c} \right]^{t-\Delta t}
+ \left[ {\rm Diff}.q_{c} \right]^{t-\Delta t}$">|; 

$key = q/displaystyleDP{pi_{i,k}}{t}+frac{overline{c}_{i,k}^{2}}{{c_{p}}_{d}overline{rho}u}{x}+DP{overline{rho}overline{theta_{v}}u}{z}right]_{i,k}=0.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="381" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img61.png"
 ALT="$\displaystyle \DP{\pi_{i,k}}{t}
+ \frac{\overline{c}_{i,k}^{2}}{{c_{p}}_{d}
\ov...
...{v}} u}{x} +
\DP{\overline{\rho} \overline{\theta_{v}} u}{z}
\right]_{i,k}
= 0.$">|; 

$key = q/displaystyle-wDP{overline{theta}_{v}}{z},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="71" HEIGHT="68" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img386.png"
 ALT="$\displaystyle -w\DP{\overline{\theta}_{v}}{z},$">|; 

$key = q/displaystyle+[Q_{cnd}]_{i,k}^{t}+[Q_{rad}]_{i,k}^{t-Deltat}+[Q_{dis}]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="284" HEIGHT="42" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img209.png"
 ALT="$\displaystyle + [Q_{cnd}]_{i,k}^{t}
+ [Q_{rad}]_{i,k}^{t-\Delta t}
+ [Q_{dis}]_{i,k}^{t-\Delta t}$">|; 

$key = q/displaystyle-frac{2}{3C_{m}^{2}l^{2}}left(DP{K_{m}^{2}}{z}right)_{i,k(w)}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="182" HEIGHT="73" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img186.png"
 ALT="$\displaystyle - \frac{2}{3 C_{m}^{2} l^{2}}
\left( \DP{ K_{m}^{2} }{z} \right)_{i,k(w)}^{t - \Delta t}$">|; 

$key = q/displaystylepi^{tau}_{i,km}-(1-beta)left(frac{bar{c}^{2}Deltatau}{bar{c_{p}}bar{right)_{km}left(DP{u^{tau+Deltatau}}{x}right)_{i,km}+F_{i,km};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="652" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img165.png"
 ALT="$\displaystyle \pi^{\tau}_{i,km}
-(1 - \beta)
\left(
\frac{\bar{c}^{2}\Delta \ta...
...}}
\right)_{km}
\left(
\DP{u^{\tau + \Delta \tau}}{x}
\right)_{i,km}
+ F_{i,km}$">|; 

$key = q/displaystyle-betaleft(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{thetaar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{i,0(w)}E_{i,0(w)}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="311" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img134.png"
 ALT="$\displaystyle - \beta
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{p}} \b...
...\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{i,0(w)}
E_{i,0(w)}.$">|; 

$key = q/displaystyleleft[DP{u}{z}right]_{i(u),k(w)}equivfrac{u_{i(u),k+1}-u_{i(u),k}}{Deltaz};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="266" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img35.png"
 ALT="$\displaystyle \left[\DP{u}{z} \right]_{i(u),k(w)}
\equiv \frac{u_{i(u), k+1} - u_{i(u), k}}{\Delta z}$">|; 

$key = q/displaystylepi_{-1,k}=-pi_{1,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="114" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img88.png"
 ALT="$\displaystyle \pi_{-1,k} = - \pi_{1,k}$">|; 

$key = q/displaystyleq_{x}=frac{x_{x}M_{x}(1slashM_{d}+sum_{vneqx}q_{v}slashM_{v})}{1-x_{x}};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="273" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img489.png"
 ALT="$\displaystyle q_{x} =
\frac{ x_{x} M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v})}{ 1 - x_{x} }$">|; 

$key = q/displaystyleu^{tau+Deltatau}_{i(u),k}=u^{tau}_{i(u),k}-left[bar{c_{p}}bar{theta}aDiv)^{tau}}{x}right}right]_{i(u),k}+F_{u,i(u),k}^{t}Deltatau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="536" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img108.png"
 ALT="$\displaystyle u^{\tau + \Delta \tau}_{i(u),k}
= u^{\tau}_{i(u), k}
- \left[
\ba...
...\alpha Div)^{\tau}}{x}
\right\}
\right]_{i(u),k}
+
F_{u,i(u),k}^{t} \Delta \tau$">|; 

$key = q/displaystylew^{tau+Deltatau}=w^{tau}-bar{c_{p}}bar{theta}_{v}Deltatauleft{betaDPpi^{tau}}{z}-DP{(alphaDiv)^{tau}}{z}right}+F_{w}^{t}Deltatau.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="586" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img110.png"
 ALT="$\displaystyle w^{\tau + \Delta \tau}
= w^{\tau}
- \bar{c_{p}} \bar{\theta}_{v} ...
...{\pi^{\tau}}{z}
- \DP{(\alpha Div)^{\tau}}{z}
\right\}
+ F_{w}^{t} \Delta \tau.$">|; 

$key = q/displaystyleleft[{{rm{Diff}_{phi}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img232.png"
 ALT="$\displaystyle \left[ {\rm Diff}_{\phi} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystyleF_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="33" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img168.png"
 ALT="$\displaystyle F_{i,k}$">|; 

$key = q/displaystylefrac{p_{x}}{p}frac{M_{x}(1slashM_{d}+sum_{vneqx}q_{v}slashM_{v})}{left(1-frac{p_{x}}{p}right)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="234" HEIGHT="64" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img485.png"
 ALT="$\displaystyle \frac{p_{x}}{p}\frac{ M_{x} (1/M_{d} + \sum_{v \neq x} q_{v}/M_{v}) }
{\left(1 - \frac{p_{x}}{p} \right) }$">|; 

$key = q/displaystylepi_{-1,k}=pi_{1,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="99" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img92.png"
 ALT="$\displaystyle \pi_{-1,k} = \pi_{1,k}$">|; 

$key = q/displaystyle-left[{{rm{Adv}.q_{v}right]_{i,k}^{t}-left[{{rm{Adv}.bar{q_{v}}right^{t-Deltat}+left[{{rm{Turb}.bar{q_{v}}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="466" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img211.png"
 ALT="$\displaystyle - \left[ {\rm Adv}.q_{v} \right]_{i,k}^{t}
- \left[ {\rm Adv}.\ba...
...,k}^{t - \Delta t}
+ \left[ {\rm Turb}.\bar{q_{v}} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/displaystylepi_{0,k}=pi_{1,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="88" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img91.png"
 ALT="$\displaystyle \pi_{0,k} = \pi_{1,k}$">|; 

$key = q/displaystyleleft(frac{R_{d}T}{p}frac{1slashM_{d}+sum_{v}q_{v}slashM_{v}}{1slashM_{d}}right)frac{p_{x}}{R_{x}T};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="261" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img481.png"
 ALT="$\displaystyle \left(
\frac{R_{d} T}{p} \frac{1/M_{d} + \sum_{v} q_{v}/M_{v}}{1/M_{d}}
\right)
\frac{p_{x}}{R_{x}T}$">|; 

$key = q/Deltat;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="27" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img4.png"
 ALT="$\Delta t$">|; 

$key = q/displaystyle+Dinv{overline{pi}_{i,k}}left([Q_{cnd}]_{i,k}+[Q_{rad}]_{i,k}+[Q_{dis}]_{i,k}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img66.png"
 ALT="$\displaystyle + \Dinv{\overline{\pi}_{i,k}}
\left([Q_{cnd}]_{i,k} + [Q_{rad}]_{i,k} + [Q_{dis}]_{i,k}\right)$">|; 

$key = q/u_{i,j};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="31" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img416.png"
 ALT="$u_{i,j}$">|; 

$key = q/displaystylefrac{p_{x}}{p}M_{x}(1slashM_{d}+sum_{v}q_{v}slashM_{v});MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="210" HEIGHT="51" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img482.png"
 ALT="$\displaystyle \frac{p_{x}}{p} M_{x} (1/M_{d} + \sum_{v} q_{v}/M_{v})$">|; 

$key = q/displaystyleDP{pi_{i,k}}{t}+left[frac{bar{c}^{2}}{bar{c_{p}}bar{rho}bar{theta}_{ac{bar{c}^{2}}{bar{c_{p}}bar{theta}_{v}}DP{u}{x}right]_{i,k}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="370" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img102.png"
 ALT="$\displaystyle \DP{\pi_{i,k}}{t}
+ \left[\frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\rh...
... \left[\frac{\bar{c}^{2}}{\bar{c_{p}} \bar{\theta}_{v}} \DP{u}{x}\right]_{i,k}.$">|; 

$key = q/displaystyleleft[q_{c}right]_{i,k}^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="62" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img203.png"
 ALT="$\displaystyle \left[ q_{c} \right]_{i,k}^{t+\Delta t}$">|; 

$key = q/alphaDiv;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="49" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img103.png"
 ALT="$\alpha Div$">|; 

$key = q/gamma=L_{{rm{NH_4SH}slash({c_{p}}_{d}Pi);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="160" HEIGHT="37" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img294.png"
 ALT="$\gamma = L_{\rm NH_4SH}/({c_{p}}_{d} \Pi)$">|; 

$key = q/displaystyle-u_{i(u),k}left[DP{u}{x}right]_{i(u),k}-w_{i(u),k}left[DP{u}{z}right]_{i(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="302" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img53.png"
 ALT="$\displaystyle - u_{i(u),k}\left[\DP{u}{x}\right]_{i(u),k}
- w_{i(u),k}\left[\DP{u}{z}\right]_{i(u),k}$">|; 

$key = q/displaystyle-DP{}{T}left(mu^{circ}_{{rm{NH_3}+mu^{circ}_{{rm{H_2S}+RTK_{p}-RTln{p_{0}}^{2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="341" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img470.png"
 ALT="$\displaystyle - \DP{}{T}
\left(
\mu^{\circ}_{\rm NH_3} + \mu^{\circ}_{\rm H_2S}
+ RT K_{p} - RT \ln {p_{0}}^{2}
\right)$">|; 

$key = q/displaystyle+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{ttatau}_{i,km}-pi^{tau+Deltatau}_{i,km-1}}{Deltaz}right)right};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="480" HEIGHT="71" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img377.png"
 ALT="$\displaystyle + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}
{\bar{c_{p...
...ta \tau}_{i,km}
- \pi^{\tau + \Delta \tau}_{i,km-1}}{\Delta z}
\right)
\right\}$">|; 

$key = q/tau+Deltatau;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img95.png"
 ALT="$\tau +\Delta \tau$">|; 

$key = q/displaystyleD_{i,k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="37" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img159.png"
 ALT="$\displaystyle D_{i,k}$">|; 

$key = q/x;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="15" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img7.png"
 ALT="$x$">|; 

$key = q/displaystyle=left(array{{cccc}D_{1,1}<col_mark>D_{2,1}<col_mark>cdots<col_mark>Dol_mark>cdots<col_mark>D_{im,km}<row_mark>array{right)^{tau}.;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="291" HEIGHT="136" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img145.png"
 ALT="$\displaystyle =
\left(\begin{array}{cccc}
D_{1,1} &amp; D_{2,1} &amp; \cdots &amp; D_{im,1}...
...dots \\\\
D_{1,km} &amp; \cdots &amp; \cdots &amp; D_{im,km} \\\\
\end{array}\right)^{\tau}
.$">|; 

$key = q/displaystyleleft[{{rm{Turb}.{bar{phi}}right]_{i,k}^{t-Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="108" HEIGHT="54" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img230.png"
 ALT="$\displaystyle \left[{\rm Turb}.{\bar{\phi}} \right]_{i,k}^{t - \Delta t}$">|; 

$key = q/{displaymath}frac{alphaDeltatau}{mbox{Min}(Deltax^{2},Deltaz^{2})}leqfrac{1}{2}fc{overline{c}Deltatau}{mbox{Min}(Deltax,Deltaz)}{displaymath};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="272" HEIGHT="49" BORDER="0"
 SRC="|."$dir".q|img404.png"
 ALT="\begin{displaymath}
\frac{\alpha \Delta \tau}{\mbox{Min}(\Delta x^{2}, \Delta z...
...\frac{\overline{c}\Delta \tau}{\mbox{Min}(\Delta x, \Delta z)}
\end{displaymath}">|; 

$key = q/displaystyleleft[DP{pi}{z}right]_{i,k(w)}equivfrac{9}{8}left(frac{pi_{i,k+1}-pi_ht)-frac{1}{24}left(frac{pi_{i,k+2}-pi_{i,k-1}}{Deltax}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="440" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img41.png"
 ALT="$\displaystyle \left[\DP{\pi}{z} \right]_{i,k(w)}
\equiv \frac{9}{8}\left(\frac{...
...\right) -
\frac{1}{24}\left(\frac{\pi_{i, k+2} - \pi_{i, k-1}}{\Delta x}\right)$">|; 

$key = q/displaystyle+left(frac{C_{m}^{2}l^{2}}{2}right)_{i,k}left[left{left(DP{u}{z}right(DP{w}{x}right)_{i(u),k(w)}right}_{i,k}^{t-Deltat}right]^{2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="501" HEIGHT="92" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img311.png"
 ALT="$\displaystyle + \left( \frac{ C_{m}^{2} l^{2} }{2} \right)_{i,k}
\left[
\left\{...
...
\left( \DP{w}{x} \right)_{i(u),k(w)}
\right\}_{i,k}^{t - \Delta t}
\right]^{2}$">|; 

$key = q/displaystyle-DP{}{T}left(mu^{circ}_{{rm{NH_3}+mu^{circ}_{{rm{H_2S}+RTln(p_{{rm{NH_3}p_{{rm{H_2S})-RTln{p_{0}}^{2}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="423" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img469.png"
 ALT="$\displaystyle - \DP{}{T}
\left(
\mu^{\circ}_{\rm NH_3} + \mu^{\circ}_{\rm H_2S}
+ RT \ln( p_{\rm NH_3} p_{\rm H_2S}) - RT \ln {p_{0}}^{2}
\right)$">|; 

$key = q/displaystyleDP{theta_{i,k}}{t};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="45" HEIGHT="61" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img64.png"
 ALT="$\displaystyle \DP{\theta_{i,k}}{t}$">|; 

$key = q/displaystyleleft[DP{u}{x}right]_{i,k}equivfrac{u_{i(u),k}-u_{i-1(u),k}}{Deltax};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="227" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img34.png"
 ALT="$\displaystyle \left[\DP{u}{x} \right]_{i,k}
\equiv \frac{u_{i(u), k} - u_{i-1(u), k}}{\Delta x}$">|; 

$key = q/displaystyleleft[q_{v}right]_{i,k}^{t+Deltat};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="64" HEIGHT="45" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img201.png"
 ALT="$\displaystyle \left[ q_{v} \right]_{i,k}^{t+\Delta t}$">|; 

$key = q/displaystyleleft[DP{w}{z}right]_{i,k}equivfrac{9}{8}left(frac{w_{i,k(w)}-w_{i,k-frac{1}{24}left(frac{w_{i,k+1(w)}-w_{i,k-2(w)}}{Deltaz}right);MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="515" HEIGHT="66" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img45.png"
 ALT="$\displaystyle \left[\DP{w}{z} \right]_{i,k}
\equiv \frac{9}{8}\left(\frac{w_{i,...
...ight) -
\frac{1}{24}\left(\frac{w_{i, k+1(w)} - w_{i, k-2(w)}}{\Delta z}\right)$">|; 

$key = q/w_{i,k(w)};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="55" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img20.png"
 ALT="$w_{i, k(w)}$">|; 

$key = q/_{3};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img454.png"
 ALT="$_{3}$">|; 

$key = q/displaystyleDP[2]{}{x}+DP[2]{}{z};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="92" HEIGHT="65" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img392.png"
 ALT="$\displaystyle \DP[2]{}{x} + \DP[2]{}{z}$">|; 

$key = q/^{circ};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="12" HEIGHT="18" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img452.png"
 ALT="$^{\circ}$">|; 

$key = q/q_{c}^{t+Deltat}<0;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="80" HEIGHT="40" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img270.png"
 ALT="$q_{c}^{t + \Delta t} &lt; 0$">|; 

$key = q/X;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="22" HEIGHT="16" ALIGN="BOTTOM" BORDER="0"
 SRC="|."$dir".q|img278.png"
 ALT="$X$">|; 

$key = q/displaystyleu_{-1(u),k}=-u_{1(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="151" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img84.png"
 ALT="$\displaystyle u_{-1(u),k} = - u_{1(u),k}$">|; 

$key = q/displaystyleu_{im+2(u),k}=-u_{im-2(u),k};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="194" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img86.png"
 ALT="$\displaystyle u_{im+2(u),k} = - u_{im-2(u),k}$">|; 

$key = q/CN_{cr};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="47" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img241.png"
 ALT="$CN_{cr}$">|; 

$key = q/displaystyleD_{i,1};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="36" HEIGHT="35" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img161.png"
 ALT="$\displaystyle D_{i,1}$">|; 

$key = q/{eqnarraystar}mbox{Max}(epsilon)&=&frac{2alpha}{mbox{Min}(Deltax,Deltaz)overlineeltatau}{mbox{Min}(Deltax,Deltaz)}right.{eqnarraystar};MSF=1.6;LFS=12;TAGS=R;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="364" HEIGHT="103" BORDER="0"
 SRC="|."$dir".q|img402.png"
 ALT="\begin{eqnarray*}
\mbox{Max}(\epsilon)
&amp;=&amp; \frac{2\alpha}{\mbox{Min}(\Delta x...
...overline{c}\Delta \tau}{\mbox{Min}(\Delta x, \Delta z)}
\right.
\end{eqnarray*}">|; 

$key = q/1leqileqim;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="94" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img16.png"
 ALT="$1 \leq i \leq im$">|; 

$key = q/l=left(DeltaxDeltazright)^{1slash2};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="123" HEIGHT="46" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img319.png"
 ALT="$l = \left(\Delta x \Delta z \right)^{1/2}$">|; 

$key = q/displaystylealpha_{h}leDinv{8},hspace{3em}alpha_{v}leDinv{8};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="187" HEIGHT="59" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img197.png"
 ALT="$\displaystyle \alpha_{h} \le \Dinv{8}, \hspace{3em}
\alpha_{v} \le \Dinv{8}$">|; 

$key = q/displaystyle1+beta^{2}left(frac{bar{c}^{2}{Deltatau}^{2}}{bar{c_{p}}bar{rho}bar{2}}left(bar{c_{p}}bar{rho}bar{theta}_{v}^{2}right)_{km-1(w)},;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="328" HEIGHT="67" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img152.png"
 ALT="$\displaystyle 1 + \beta^{2}
\left(
\frac{\bar{c}^{2}{\Delta \tau}^{2}}{\bar{c_{...
...ta z^{2}}
\left(
\bar{c_{p}} \bar{\rho} \bar{\theta}_{v}^{2}
\right)_{km-1(w)},$">|; 

$key = q/q_{r};MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="20" HEIGHT="33" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img223.png"
 ALT="$q_{r}$">|; 

$key = q/displaystylegamma_{h}=alpha_{h}left(1-frac{(x_{max}-x)}{d_{h}}right)^{3};;(x>x_{max}-d_{h}),;MSF=1.6;LFS=12;AAT/;
$cached_env_img{$key} = q|<IMG
 WIDTH="377" HEIGHT="72" ALIGN="MIDDLE" BORDER="0"
 SRC="|."$dir".q|img347.png"
 ALT="$\displaystyle \gamma_{h} = \alpha_{h} \left( 1 - \frac{(x_{max} - x)}{d_{h}}\right)^{3}
\;\; (x &gt; x_{max} - d_{h}),$">|; 

1;