Question $7(14$ marks$)$

Figure $3$: The cross$-$section of the pipe in question $6$

Consider a circular pipe of length $L$ with thermal conductivity $,$ and internal and external radii $r_1$ and $r_2$

respectively as shown in Figure $3.$ There is some fluid passing through the cavity inside the pipe.

The temperature of the fluid inside the cavity of the pipe is $T_i$ and the ambient temperature $($outside the pipe$)$

is $T_o$ with $T_i>T_o.$ The convective heat transfer coefficients of the inside the cavity and outside the pipe are $h_i$

and $h_o$ respectively. The temperatures on the inner and outer surfaces of the pipe are $T_1$ and $T_2$ respectively.

Suppose that the heat transfer is in a steady state.

$($a$)$ By using the Fourier law for conduction, show that the rate of heat transfer $q$ from the inner surface to

the outer surface of the pipe satisfies

$T_1-T_2=\frac{q}{2L}ln(\frac{r_2}{r_1}).$

$(3$ marks$)$

$($b$)$ By using the steady$-$state heat transfer by convection, show that the rate of heat transfer $q$ from the outer

surface of the pipe to the air in the surroundings due to convection in the steady state is

$T_2-T_o=\frac{q}{2r_{2}L{h}_o}.$

$(2$ marks$)$

$($c$)$ By using the steady$-$state heat transfer by convection again, show that the rate of heat transfer $q$ from

the cavity to the inner surface of the pipe due to convection in the steady state is

$T_i-T_1=\frac{q}{2r_{1}L{h}_i}.$

$(2$ marks$)$

$($d$)$ Using the previous results, or otherwise, show that

$\frac{q}{L}=2(T_i-T_o)\{\frac{1}{r_1{h}_i}+\frac{1}{}ln(\frac{r_2}{r_1})+\frac{1}{r_2{h}_o}{\}}^{-1}.$

$(2$ marks$)$

$($e$)$ Given the data shown in Table $2,$ find the rate of heat loss per unit length of the pipe $\frac{q}{L},$ and the

temperatures at the inner and outer surfaces $T_1$ and $T_2.(5$ marks$)$