$(53$ of $95$

EXAMPLE $5\mathrm{.}3$: Flat Plate with Constant Heat Flux

A $1\mathrm{.}0-$kW heater is constructed of a glass plate with an electrically conducting film that produces a constant heat flux. The plate is $60$ cm by $60$ cm and placed in an airstream at $27C,1$atm with $u_{}=5\frac{m}{s}.$ Calculate the average temperature difference along the plate.

Solution

Properties should be evaluated at the film temperature, but we do not know the plate temperature. for an initial calculation, we take the properties at the free$-$stream conditions of

At $T_{}=27C$ we find the properties of the fluid

$v=15\mathrm{.}96**{10}^{-6}\frac{m^2}{s},Pr=0\mathrm{.}708,k=0\mathrm{.}02624\frac{W}{m}*C.$

$R{e}_L=\frac{u_{}L}{v}=\frac{5*0\mathrm{.}6}{15\mathrm{.}96**{10}^{-6}}=1\mathrm{.}88={10}^5$ therefore the flow $is$ laminar$\frac{}{b}$

$ar(T_w-T_{})=\frac{\frac{q_{w^2}}{L}}{0\mathrm{.}6795R{e}_L^{\frac{1}{2}p\frac{r^2}{3}}}\frac{?}{b}$

$ar(T_w-T_{})=\frac{(\frac{3000}{a{6}^2})(\frac{0\mathrm{.}6}{0\mathrm{.}02624})}{0\mathrm{.}6795(1\mathrm{.}88*{10}^5{)}^{\frac{1}{2}(0\mathrm{.}708{)}^{\frac{2}{3}}}}=241\mathrm{.}85C$

$T_w=241\mathrm{.}85+27=268\mathrm{.}85C$

Now we find the properties at the film temperature

$T_f=\frac{T_w+T_{}}{2}=\frac{268\mathrm{.}85+27}{2}=(147\mathrm{.}927+273)=421K$

$v=28\mathrm{.}22**{10}^{-6}\frac{m^2}{s},Pr=0\mathrm{.}687,k=0\mathrm{.}035\frac{W}{m}*C.$

$R{e}_L=\frac{5*0\mathrm{.}6}{28\mathrm{.}22*{10}^{-6}}=1\mathrm{.}06={10}^5\frac{?}{b}$

$ar(T_w-T_{})=\frac{(\frac{3000}{0\mathrm{.}2})(\frac{0\mathrm{.}6}{0\mathrm{.}035})}{0\mathrm{.}6795(1\mathrm{.}06*{10}^5{)}^{\frac{1}{2}(0\mathrm{.}687{)}^{\frac{2}{3}}}}=243\mathrm{.}6C$

${}_x=\frac{q_{w}x}{k(T_w-T_{})}$

$T_w-T_{}=\frac{q_{w}x}{k{}_x}$