Problem $3\mathrm{.}75$

Consider the clad fuel element of Problem $3\mathrm{.}74.3\mathrm{.}74$ A nuclear fuel element of thickness $2L$ is covered with

a steel cladding of thickness $b.$ Heat generated within

the nuclear fuel at a rate $q^{}$ is removed by a fluid at $T_{},$

which adjoins one surface and is characterized by a

convection coefficient $h.$ The other surface is well insu$-$

lated, and the fuel and steel have thermal conductivities

of $k_f$ and $k_s,$ respectively.

$($a$)$ Obtain an equation for the temperature distribu$-$

tion $T(x)$ in the nuclear fuel. Express your results in

terms of $q^{},k_f,L,b,k_s,h,$ and $T_{}.$

$($b$)$ Sketch the temperature distribution $T(x)$ for the

entire system.

$($a$)$ Using appropriate relations from Tables C$.1$ and

C$\mathrm{.}2,$ obtain an expression for the temperature distri$-$

bution $T(x)$ in the fuel element. For $k_f=60\frac{W}{m}*K,$

$L=15mm,b=3mm,k_s=15\frac{W}{m}*K,h=10,000$

$\frac{W}{m^2}*K,$ and $T_{}=200C,$ what are the largest

and smallest temperatures in the fuel element if

heat is generated uniformly at a volumetric rate of

$q^{}=2{10}^7\frac{W}{m^3}?$ What are the corresponding

locations?

$($b$)$ If the insulation is removed and equivalent convec$-$

tion conditions are maintained at each surface, what

is the corresponding form of the temperature distri$-$

bution in the fuel element? For the conditions of part

$($a$),$ what are the largest and smallest temperatures in

the fuel? What are the corresponding locations?

$($c$)$ For the conditions of parts $($a$)$ and $($b$),$ plot the tem$-$

perature distributions in the fuel element.