$1.($a$)$ A $0\mathrm{.}02$ m thick steel pipe $($thermal conductivity $=43$ W m $$C$-1)$ with a $0\mathrm{.}06$ m inside diameter is being used to convey steam from a boiler to process equipment over a distance of $40$ m$.$ The inside pipe surface temperature is $115$C$,$ and the outside pipe surface temperature is $90$C$.$ Calculate the total heat loss to the surroundings under steady$-$state conditions.

$($b$)$ The temperature of a gas stream is to be measured by a thermocouple whose junction can be approximated as a $1$ mm diameter sphere as shown below. The properties of the junction are k $=35$ W m$-1$C$-1,$ density $=8,500$ kg m$-3$ and specific heat $=320$ J kg $1$C$-1,$ and the convection heat transfer coefficient between the junction and the gas is $210$ W m$-2$C $-1.$ Determine how long it will take for the thermocouple to read $99\%$ of the initial temperature difference.