analysis of transient heat conduction in different geometries - ethesis ...

∂θ ∂ ⎛ ∂θ
⎞
x dx = ⎜x⎟dx ∂τ∂x⎝ ∂x
⎠
1
m
1
m
0 0
∫ ∫
Simplifying the above equation we may write
Considering the average temperature we may write
∫
1
0
∂θ
dx = −Bθ
∂τ
Substituting the value of θ at equation (3.105) we have
Integrating the equation (3.106) we may write as
38
(3.119)
∂ θ
= −Bθ
∂ τ
(3.120)
∂ θ 3Bθ
=−
∂ τ B + 3
(3.121)
∂ θ 3B
= − ∂τ
θ B + 3
1 1
∫ ∫
0 0
Thus by simplifying the above equation we may write
θ = exp( −Pτ)
Or we may write
Where
(3.122)
⎛ 3B
⎞
θ = exp⎜−
τ ⎟
⎝ B + 3 ⎠ (3.123)
(3.124)
3B
P =
B + 3
(3.125)
Several profiles have been considered for the analysis. The corresponding modified Biot number,
P, has been deduced for the analysis and is shown in Table 4.2.