Problem $3.$ Some independent conceptual questions.

$($a$)$ For $1-$D steady$-$state radial heat conduction in a cylindrical shell without heat generation,

how does the absolute value of heat flux, i$.$e$.,|q_r^{\text{'}}|,$ change as the radius $r$ increases?

$($increase$,$ decrease, or remain the same$).$

$($b$)$ To reduce the heat loss for a hot aluminum tube, whose outside diameter is $25$ mm

$(k_{Al}=200\frac{W}{m}*K),$ an engineer wrapped it with a $10-$mm$-$thick foam

$0\mathrm{.}12\frac{W}{m}*K.$ Assume that the heat transfer coefficient is $h=10\frac{W}{m^2}*K$ outside the

foam. Will the foam reduce or increase the heat loss from the pipe to the ambient air?

$($c$)$ The total thermal resistance of the composite wall shown in the figure below is $($in terms

of given symbols$)$

$R_{\text{t}\text{o}\text{t}}=$

Please show your derivations for each question.