WOLFRAM|DEMONSTRATIONS PROJECT

Cooling of a Composite Slab

​
t
2.
α
1
0.05
Bi
1
1.
λ
0.4
α
2
0.02
Bi
2
0.5
This Demonstration simulates the transient cooling of a composite slab that is suddenly immersed in a cooling bath.
Consider a composite slab of thickness
L=1
, consisting of two parallel, perfectly fused layers of different thermal properties initially at uniform temperature
T
0
,
that is immersed at time
t=0
in a well-stirred, insulated tank containing a fluid at constant temperature
T
∞
.
The heat equation describing this system is:
∂T
∂t
=α
∂T
∂
2
x
with
k
∂T
∂x
+h(T-
T
∞
)=0
at both
x=0
and
x=L
, and
T(0,x)=
T
0
,
where
x
is the space coordinate (
cm
),
α
is the thermal diffusivity (
2
cm
/sec
),
k
is the thermal conductivity (
W/cm·K
) and
h
is the heat transfer coefficient between the slab and the surrounding fluid (
W·
2
cm
/K
).
To solve this problem, it is convenient to introduce the following dimensionless variables:
Θ=
T-
T
∞
T
0
-
T
∞
and
ξ=
x
L
.Thus the equations become
∂Θ
∂t
=α
∂Θ
∂
2
ξ
,
∂Θ
∂ξ
+BiΘ=0
,
at both
ξ=0
and
ξ=1
, and
Θ(0,ξ)=1
with
α=
α
1
ξ≤λ
α
2
ξ>λ
and
Bi=
Bi
1
ξ≤λ
Bi
2
ξ>λ
,
where
Bi=
hL
k
is the Biot number, the ratio of internal resistance to conductive heat transfer in the slab to the external resistance of convective heat transfer from the slab to the surrounding fluid;
λ
is the dimensionless coordinate of separation between the two layers; and the subscripts
1
and
2
refer to thermal properties of the layers
x≤λ
and
x>λ
, respectively.