Question $2$: You work in a steam generation power plant and your manager is worried about the high consumption of natural gas to keep the steam pressure and temperature condition required by your customers. After a quick inspection, you found out that that a $100-$m$-$long section of the steam pipeline was passing through an open space at $\backslash (20^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ with no insulation. The pipe has internal and external diameters of $6$ and $10$ cm $,$ respectively, and it transports steam at $\backslash (250^\{\backslash$circ$\}\backslash$mathrm$\{$C$\}\backslash )$ with an internal convection heat transfer coefficient of $\backslash (70\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash ).$ Assume that the convection heat transfer coefficient at the external environment is $\backslash (15\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}^\{2\}\backslash$cdot $\backslash$mathrm$\{$~K$\}\backslash )($natural convection$),$ and the thermal conductivity of the pipe material is $\backslash (50\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}\backslash ).$K$.$ The annual cost of energy lost through the pipeline is $\backslash (\backslash$$ $0\mathrm{.}55/\backslash )$ therm, in which $1$ therm is equal to $\backslash (105,500\backslash$mathrm$\{$~kJ$\}\backslash ).$

a$)$ Calculate rate of heat loss from the $100-$m section of the steam pipe using the thermal resistance approach and the total annual cost for this loss.

b$)$ In order to solve this problem, you proposed the utilization of fiberglass insulation $($thermal conductivity equal to $\backslash (0\mathrm{.}035\backslash$mathrm$\{$~W$\}/\backslash$mathrm$\{$m$\}.\backslash$mathrm$\{$K$\}\backslash ))$ through the pipeline. Determine the thickness of the insulation needed to eliminate $\backslash (90\backslash \%\backslash )$ of the heat loss through the pipeline.

c$)$ If the insulation will cost the company approximately $\backslash (\backslash$$ $0\mathrm{.}03\backslash )$ per $\backslash (\backslash$mathrm$\{$cm$\}^\{3\}\backslash )($including material and maintenance$)$ calculate the payback time of this investment based on the annual energy savings.