A refrigerant having a temperature T$($inf$,1)$ and a heat transfer coefficient h$1$ flows inside the copper pipe. An insulation material was installed outside the copper pipe to cool this refrigerant. A fluid having a temperature T$($inf$,4)$ and a heat transfer coefficient h$4$ flows outside the insulation material.

Inside radius of copper tube r$1,$ temperature T$1$

Copper Tube Thickness delta

Outside radius r$2,$ temperature T$2$ of copper tube

Radius of insulation r$3,$ temperature T$3$

Copper thermal conductivity kc

Thermal conductivity of insulation materials ki

$($a$)$ Draw a thermal circuit of this system, specify the thermal resistance and temperature.

$($b$)$ Draw a thermal circuit by simplifying $($a$)$ above. Write down the reason.