The rate of heat loss and cost of the uninsulated pipe will be calculated first.
The thermal circuit for the uninsulated pipe is shown below
![image](https://sciemce.com/files/2/cc5f45c1a78b466bd1.png)
Evaluating the individual resistances
![image](https://sciemce.com/files/2/e3eef313e27522d2.png)
The rate of heat transfer for the uninsulated pipe is
![image](https://sciemce.com/files/2/3b1982b511bc43d3.png)
The cost to supply this heat loss is
cost = (3148 w/m) (J/W s) (3600 s/h) (24 h/day) (365 days/yr) ($5/109J) = $496/(yr m)
For the insulated pipe the thermal circuit is
![image](https://sciemce.com/files/2/d98c741a774d815d4.png)
The resistance of the insulation is given by:
![image](https://sciemce.com/files/2/2518c99604fd343d5.png)
(a) The rate of heat transfer for the pipe with 5 cm of insulation is
![image](https://sciemce.com/files/2/2f9cd19f656e3b96d6.png)
The cost of this heat loss is
cost = (166 w/m) (J/W s) (3600 s/h) (24 h/day) (365 days/yr) ($5/109J) = $26/yr m
Comparing this cost to that of the uninsulated pipe we can calculate the payback period
![image](https://sciemce.com/files/2/1ab18c11f15a4a2cd7.png)
Payback period = 0.43 yr = 5 months
(b) The rate of heat loss for the pipe with 10 cm of insulation is
![image](https://sciemce.com/files/2/0d12f10b3669ffd3d8.png)
The cost of this heat loss
cost = (95.7 w/m) (J/W s) (3600 s/h) (24 h/day) (365 days/yr) ($5/109J) = $15/yr m
Comparing this cost to that of the uninsulated pipe we can calculate the payback period
![image](https://sciemce.com/files/2/019a6d7f19a5c2d5d9.png)