Microfiltration $($MF$)$ membranes are tested for water permeation at $293\mathrm{.}15K$ where the dynamic

viscosity of water is $=0\mathrm{.}001N*\frac{s}{m^2}.$ The membranes have uniformly sized cylindrical pores

with tortuosity of $=1\mathrm{.}5.$ Using spreadsheets to estimate the water flux $(J_w$;$\frac{m^3}{m^2*h})$ as a function of

a membrane structural parameter $($e$.$g$.,$ pore radius $r,$ membrane thickness $,$ and porosity $)$ and

as a function of an operation condition $($e$.$g$.,$ the applied pressure $P$ and operating temperature $T$

using the $-T$ dependence$)$ as specified below:

$(1)$ When $P=1$ bar, porosity $()=0\mathrm{.}5$ and membrane thickness $()=10m,$ calculate and

graph $J_w$ as a function of pore radius $(r)$ for $r$ varying from $0\mathrm{.}05$ to $5\mathrm{.}0m.$

$(2)$ When $P=1$ bar, $=0\mathrm{.}5$ and $r=0\mathrm{.}5m,$ calculate and plot $J_{w}vs$ for $$ varying from $2$ to

$50m.$

$(3)$ When $P=1$ bar, $=10m$ and $r=0\mathrm{.}5m,$ calculate and plot $J_{w}vs$ for $$ varying from $0\mathrm{.}1$

to $0\mathrm{.}6.$

$(4)$ When $=10m,r=0\mathrm{.}5m,$ and $=0\mathrm{.}5,$ plot $J_{w}vsP($i$.$e$.$ flux $J_w$ vs$.$ applied pressure $P)$

for $P$ from $0\mathrm{.}5$ to $\frac{10}{b}ar.$

$(5)$ When $P=1$ bar, $=10m,r=0\mathrm{.}5m,$ and $=0\mathrm{.}5,$ calculate and plot $J_{w}vsT$ for

operating temperature $T(K)$ varying from $283\mathrm{.}15K$ to $353\mathrm{.}15K.$ The temperature dependence

of dynamic viscosity is given by $=0\mathrm{.}12508-0\mathrm{.}0010922T+3\mathrm{.}2074{10}^{-6}{T}^2-3\mathrm{.}1566{10}^{-}$

${?}^9{T}^3,$ where unit of $$ is $\frac{N*s}{m^2}.($Assume constant water density over the temperature range.$)$

Viscous flow: $J_w=\frac{*r^2}{8}*\frac{P}{}$