Problems$\_$chapt...

Problem $2\mathrm{.}26$

One$-$dimensional, steady$-$state conduction with uniform internal energy generation occurs in a plane wall with a thickness of $50$ mm and a constant thermal conductivity of $5\frac{W}{m}*K.$ For these conditions, the temperature distribution has the form $T(x)=a+bx+c{x}^2.$ The surface at $x=0$ has a temperature of $T(0)=T_0=120C$ and experiences convection with a fluid for which $T_{}=20C$ and $h=500\frac{W}{m^2}*K.$ The surface at $x=L$ is well insulated.

$($a$)$ Applying an overall energy balance to the wall, calculate the internal energy generation rate, $q^{}$;

$($b$)$ Determine the coefficients $a,b,$ and c by applying the boundary conditions to the prescribed temperature distribution. Use the results to calculate and plot the temperature distribution;

$($c$)$ Consider conditions for which the convection coefficient is halved, but the internal energy generation rate remains unchanged. Determine the new values of $a,b,$ and $c,$ and use the results to plot the temperature distribution. Hint: recognize that $T(0)$ is no longer $120C.$

$($d$)$ Under conditions for which the internal energy generation rate is doubled, and the convection coefficient remains unchanged