Problems to hand in:

$5-113$ Consider two$-$dimensional transient heat transfer in an

L$-$shaped solid bar that is initially at a uniform temperature of

$140C$ and whose cross section is given in the figure. The ther$-$

mal conductivity and diffusivity of the body are $k=15\frac{W}{m}*K$

and $=3\mathrm{.}2{10}^{-6}\frac{m^2}{s},$ respectively, and heat is generated

in the body at a rate of $e^{}=2{10}^7\frac{W}{m^3}.$ The right surface of

the body is insulated, and the bottom surface is maintained at

a uniform temperature of $140C$ at all times. At time $t=0,$ the

entire top surface is subjected to convection with ambient air at

$T_{}=25C$ with a heat transfer coefficient of $h=80\frac{W}{m^2}*K,$

and the left surface is subjected to uniform heat flux at a rate of

$q_L^{}=8000\frac{W}{m^2}.$ The nodal network of the problem consists

of $13$ equally spaced nodes with $x=y=1\mathrm{.}5cm.$

Write down the formulas for the new temperature $T_m^{i+1}$ at the nodes $m=1,2,3,4,5,6,7,$ and $8,$

respectively. $($using the explicit method similar to example $5-7)$

Evaluate the maximum time step $t_{\text{m}\text{a}\text{x}}$ can be used without causing stability issues.

Write the code $($similar to the first $35-$line MATLAB code of Example $5-7)$ to draw the time$-$dependent

temperatures at nodes $1,3,$ and $8$ in the same figure. $($Use the time step $t=0\mathrm{.}01t_{\text{m}\text{a}\text{x}},$ the total time of $10$

minutes$)$ The examples of temperature at node $5$ and $10$ are shown on the next page. Please attach your

code at the end.

Redraw the time dependent$-$temperature at node $3$ using time step $t=0\mathrm{.}01t_{\text{m}\text{a}\text{x}},t=0\mathrm{.}1t_{\text{m}\text{a}\text{x}},t=$

$0\mathrm{.}5t_{\text{m}\text{a}\text{x}},$ and $t=1\mathrm{.}4t_{\text{m}\text{a}\text{x}}$ What do you find when $t>t_{\text{m}\text{a}\text{x}}?$