With the eigenvalues as
x
hL
αt
Bi cos λ n − λ n sin λ n = 0 for the slab, η = , Bi =
, Fo =
L
k
L 2
r
hr o
αt
λ n J 1 (λ n ) − BiJ o (λ n ) = 0 for the cylinder, η = , Bi =
, Fo =
r o
k
r 2
o
r
hr o
αt
λ n cos λ n + (Bi − 1) sin λ n = 0 for the sphere, η = , Bi =
, Fo =
k
r 2
r o
o
78
Analytical Heat Transfer
4.3 1-D Transient Heat Conduction in a Semiinfinite
Solid Material
4.3.1 Similarity Method for Semiinfinite Solid Material
The semiinfinite solid is characterized by a single identifiable surface. If a
sudden change in temperature occurs at this surface as shown in Figure 4.7,
then transient 1-D conduction will take place within the solid. The similarity
method can be employed to solve this kind of problem [3]. The 1-D transient
heat conduction equation in a semiinfinite solid without heat generation is
T i
x
t
Heating
q″
T s
T (x,t)
Cooling
t
q″
q″
T s
FIGURE 4.7
1-D transient heat conduction in a semiinfinite solid material.