Q3 When a fin is insulated at its tip, the temperature distribution in the fin is given by Eq. 3.1. coshm(L-x) coshmL Eq.3.1 T-T T₁-Tf = hP Where the symbols have their usual meaning, and m = kAc i. Derive the equation of temperature gradient along the fin. ii. [4 marks] Derive the equation of heat transfer rate at the root of the [4 marks] fin. iii. A cylindrical aluminium fin has the diameter of 0.002m and length of 0.02m. The fin is heated to 100°C at one end, and fully insulated at another end. It is surrounded by air at 20°C, the thermal conductivity of air is k=0.027W/mK, the obtained average Nusselt number around the fin is 0.5. The thermal conductivity of aluminium is k=230W/mK. [6 marks] Calculate the temperature at the midpoint of the fin. iv. Calculate the heat transfer rate of the above fin. V. Consider a case when the cylindrical aluminium fin is replaced by a fin of the rectangular cross section 0.0005mx0.00628m, while keeping the cross section area and the fin length the same, and assuming that the convective heat transfer coefficient is the same. Discuss the effect of the rectangular fin on the heat transfer rate comparing to the cylindrical fin. [2 marks] [4 marks]