�
�
�
�
�
�
�
�
�
and
∂T �
∂θ �
q s = −k
= −k(T s − T i )
∂x
∂x
x=0
x=0
∂
2 η
2
= −k(T s − T i )
1 − √
e
−u du
∂x
π 0
x=0
�
�
(4.21)
2
1
−η 2
= −k(T s − T i ) − √ e
√
π
4αt η=0
k(T s − T i )
= √ παt
81
Transient Heat Conduction
1
4αt
T−T i
T s −T i
=
η =
θ
x
θ = erfc (η)
FIGURE 4.8
Solution of 1-D transient heat.
4.3.2 Laplace Transform Method for Semiinfinite Solid Material
The 1-D transient conduction problem in a semiinfinite solid material can be
solved by the Laplace transform method [1]. The heat diffusion equation for
1-D transient conduction is of the form
∂ 2 T
1 ∂T
=
∂x 2
α ∂t
Case 1: Constant surface temperature BC
T(0, t) = T s
T(∞, t) = T i
Initial condition:
T(x, 0) = T i .
�
�
�
�
�
�
�
�
and
∂T �
∂θ �
q s = −k
= −k(T s − T i )
∂x
∂x
x=0
x=0
∂
2 η
2
= −k(T s − T i )
1 − √
e
−u du
∂x
π 0
x=0
�
�
(4.21)
2
1
−η 2
= −k(T s − T i ) − √ e
√
π
4αt η=0
k(T s − T i )
= √ παt
81
Transient Heat Conduction
1
4αt
T−T i
T s −T i
=
η =
θ
x
θ = erfc (η)
FIGURE 4.8
Solution of 1-D transient heat.
4.3.2 Laplace Transform Method for Semiinfinite Solid Material
The 1-D transient conduction problem in a semiinfinite solid material can be
solved by the Laplace transform method [1]. The heat diffusion equation for
1-D transient conduction is of the form
∂ 2 T
1 ∂T
=
∂x 2
α ∂t
Case 1: Constant surface temperature BC
T(0, t) = T s
T(∞, t) = T i
Initial condition:
T(x, 0) = T i .