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4.2. Solve transient temperature profiles of a convectively cooled sphere, as shown
in Figure 4.1c, by separation of variables.
SOLUTION
1 ∂ 2
1 ∂θ
(r θ) =
r ∂r 2
α ∂t
Boundary conditions:
i. r = r o − k ∂θ = hθ r o
∂r
ii. t = 0 θ = 1
Let U(r , t ) = r θ(r , t ),
∂ 2 U
1 ∂U
=
∂r 2
α ∂t
⎧
⎨U = 0 at r = 0
BCs ∂U
h
1
⎩
+
−
U = 0 at r = r o
∂r
k
r o
U = r for t = 0
U = R(r )τ(t )
∂τ
2
−αλ 2 t
+ αλ τ = 0; ⇒ τ(t ) = C 3 e
∂t
∂ 2 R + λ 2 R(r ) = 0
∂r 2
R(r ) = C 1 sin(λr ) + C 2 cos(λr )
at r = 0, U = 0, ⇒ C 2 = 0
h
1
C 1 λ cos(λr o ) +
−
C 1 sin(λr o ) = 0
k
r o
λ n = λr o
λ n cos(λ n ) + (B i − 1)sin(λ n ) = 0
where B i = (hr o /k ),
∞
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r
−λ 2 F 0
n
U =
C n sin λ n
e
r o
n=1
where F 0 = (αt /r 2 ).
o
95
Transient Heat Conduction
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4.2. Solve transient temperature profiles of a convectively cooled sphere, as shown
in Figure 4.1c, by separation of variables.
SOLUTION
1 ∂ 2
1 ∂θ
(r θ) =
r ∂r 2
α ∂t
Boundary conditions:
i. r = r o − k ∂θ = hθ r o
∂r
ii. t = 0 θ = 1
Let U(r , t ) = r θ(r , t ),
∂ 2 U
1 ∂U
=
∂r 2
α ∂t
⎧
⎨U = 0 at r = 0
BCs ∂U
h
1
⎩
+
−
U = 0 at r = r o
∂r
k
r o
U = r for t = 0
U = R(r )τ(t )
∂τ
2
−αλ 2 t
+ αλ τ = 0; ⇒ τ(t ) = C 3 e
∂t
∂ 2 R + λ 2 R(r ) = 0
∂r 2
R(r ) = C 1 sin(λr ) + C 2 cos(λr )
at r = 0, U = 0, ⇒ C 2 = 0
h
1
C 1 λ cos(λr o ) +
−
C 1 sin(λr o ) = 0
k
r o
λ n = λr o
λ n cos(λ n ) + (B i − 1)sin(λ n ) = 0
where B i = (hr o /k ),
∞
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r
−λ 2 F 0
n
U =
C n sin λ n
e
r o
n=1
where F 0 = (αt /r 2 ).
o
95
Transient Heat Conduction