Question:

Discuss about the Thermodynamics and give brief solution.

Answer:

Given data

Fin length L =110 mm

Fin thickness t = 1 mm

Thermal conductivity K =325 W/mk

Heat transfer coefficient h = 90 W/m^2k

1) Find position and length of minimum temperature .

Consider the equation

…………………………1)

ta = atmospheric temperature

C1 , C2 =constant

m =

P =Perimeter =2*2 =4mm^2

A =Area =1 mm^2

Put values

m =

m = 33.28 m^-1

Use boundary condition

At x = 0 t =100 deg C , At x = 0.11 m , t = 70 deg C

Put above condition in equation 1)

We get

90 = C1 + C2 ……..2)

30 =38.89 C1 +0.00257 C2 ………..3)

Solve the equation 2) , 3) We get ,

C1 =0.702 , C2 = 89.29

Put C1 , C2 values in equation 1)

……………………4)

For minimum value of temperature =0

Now differentiate equation 4 w.r.t x

Apply log on both sides

x = 0.0728 m

**x = 72.8 mm from the left side **

Apply value of x at equation 4)

Heat loss

** **

Fin efficiency

2) Calculate the minimum length

Consider the equation

…………………………5)

ta = atmospheric temperature

C1 , C2 =constant

m =

P =Perimeter =2*2 =4mm^2

A =Area =1 mm^2

Put values

m =

m = 33.28 m^-1

Use boundary condition

At x = 0 t =100 deg C , At x = 0.11 m , t = 70 deg C

Put above condition in equation 5)

We get

90 = C1 + C2 ……..2)

30 =38.89 C1 +0.00257 C2 ………..6)

Solve the equation 5) , 6) We get ,

C1 =0.702 , C2 = 89.29

Put C1 , C2 values in equation 5)

……………………7)

For minimum value of temperature =0

Now differentiate equation 7 w.r.t x

Apply log on both sides

x = 0.0728 m

**x = 72.8 mm from the left side **

Apply value of x at equation 7)

The temperature is at lowest value of 25.84 ,So it is considered that the length of fin should be of length = 72.8 , after that the temperature would rise due to effect of other plate .After this length , further increase in length is of no use, as it do not cause any reduce in temperature .