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 .
