PROBLEM THREE $[9$ marks$]$

A cylindrical nuclear fuel element is $10\mathrm{.}16cm$ long and $10\mathrm{.}77$ cm in diameter. The fuel generates

heat uniformly at a rate of $5170k\frac{W}{m^3}.$ The fuel is placed in an environment having a temperature

of $370$ K with a surface heat transfer coefficient of $4540\frac{W}{m^2}.K.$ The fuel material has a thermal

conductivity of $k=33\mathrm{.}9\frac{W}{m}.K.$ For the steady$-$state operation and with neglecting end effects,

determine:

a$)$ Express the differential equation that governs the variation of temperature in the fuel

element along with the boundary conditions. $[2$ marks$]$

b$)$ Solve the differential equation to obtain the temperature profile as a function of radial

position. $[3$marks$]$

c$)$ Determine the maximum temperature in the fuel element. $[2$ mark$]$

d$)$ Determine the surface temperature of the fuel element. $[2$ mark$]$PROBLEM THREE $[9$ marks$]$

A cylindrical nuclear fuel element is $10\mathrm{.}16cm$ long and $10\mathrm{.}77$ cm in diameter. The fuel generates

heat uniformly at a rate of $5170k\frac{W}{m^3}.$ The fuel is placed in an environment having a temperature

of $370$ K with a surface heat transfer coefficient of $4540\frac{W}{m^2}.K.$ The fuel material has a thermal

conductivity of $k=33\mathrm{.}9\frac{W}{m}.K.$ For the steady$-$state operation and with neglecting end effects,

determine:

a$)$ Express the differential equation that governs the variation of temperature in the fuel

element along with the boundary conditions. $[2$ marks$]$

b$)$ Solve the differential equation to obtain the temperature profile as a function of radial

position. $[3$marks$]$

c$)$ Determine the maximum temperature in the fuel element. $[2$ mark$]$

d$)$ Determine the surface temperature of the fuel element. $[2$ mark$]$