QUESTION $2$

$2\mathrm{.}1$ Write the expression for the thermal conduction resistance of a long cylinder of internal radius $r_1,$ outside radius $r_2,$ thermal conductivity k and length $L.$

$2\mathrm{.}2$ Superheated steam at $150C$ flows inside a cylindrical $15$ cm schedule $40$ steel pipe $($inner diameter $15\mathrm{.}16$ cm $,$ outside diameter $16\mathrm{.}56$ cm and thermal conductivity $43\frac{W}{m}K)$ covered with a $7\mathrm{.}5$ cm thick layer of $85\%$ magnesia $($thermal conductivity $0\mathrm{.}06\frac{W}{m}K).$ The pipe is in a space at $16C$ and the heat transfer coefficient of the steam and outer space is $170\frac{W}{m^2}K$ and $30\frac{W}{m^2}K,$ respectively.

$($a$)$ draw the thermal circuit $($thermal resistance$)$ diagram for the insulated pipe

$($b$)$ calculate:

$($i$)$ the individual resistances and the total resistance, and

$($ii$)$ the rate of heat loss per metre.

$(3)$

$[12]$