Question

A circular (annular) fin of radius r2 is attached to a pipe of radius 11. The fin is exposed to an ambient fluid at temperature T. having heat transfer coefficient h and the fin base is maintained at temperature Tb. The fin thickness (t) is small compared to the fin length and the heat loss from the fin tip can be considered negligible compared with that from the top and bottom surfaces of the fin. Derive the differential equation and the boundary conditions for this fin by writing an energy balance around a differential volume element. Write your assumptions in detail. Do not solve the differential equation!

Answer 1

fp dat pipe H to ambient fluid Pemperature 2- radius of fin da radius of the pipe , Fox - Temperature of Ambient Fluid h- coefficient of heat transfer To - Fin base Temperature t- thickness of fin Before applying heat Balance, we need to assume a ring of thickness 48 Now Appeying heat Balance over this . ring' General heat Balance Heatin - Heatout & Heat generation - Heat consumption = Heat accumulation

Heat accumulation is zero (SteadyState) theat generation is zero Heat consumption is zero so final Heat Balance : Heatin - Heat out zol Applying this heat Balance to the ring of .24389%-21Ttrolar 2480rb (TT) = 0

21489/2-2 11 8996/8+48 2X2 Trith (FT) - peuiding the above expression by 21+ 48 (rof-891440) -24 48 9h (T-To) = 0 . KAKAN atta (o por for – 890|490– 23h (T-T) =0 Ar Applying Limit Arto

ryo AT dr . t Charly - par liter) - 20:{T-Ta) =0 - 26997) – 20h. (T-T6)=0 9r=-g ( from fourjes/as) (**) -235 (T-Ts= Kata ) - 20h (T-Tos) =0 peujding By K both sides. Create an (T-To)=0) This is the differential equation asked in question. Hence et is answer

We have already Discussed Answer But if you interested to solve this Differential equation, I can tell you, method to solve it. You need to make a dimension less differential equation a concert all variable to dimens/on- - less by putting &= r e da T-To toto you will get fedohn Gdet d& tk you can solueet By_mathematics - of Differential equations Thank you