Cooling of a Composite Slab
Cooling of a Composite Slab
This Demonstration simulates the transient cooling of a composite slab that is suddenly immersed in a cooling bath.
Consider a composite slab of thickness , consisting of two parallel, perfectly fused layers of different thermal properties initially at uniform temperature , that is immersed at time in a wellstirred, insulated tank containing a fluid at constant temperature .
L=1
T
0
t=0
T
∞
The heat equation describing this system is:
∂T
∂t
∂T
∂
2
x
with
k+h(T)=0
∂T
∂x
T
∞
at both and , and
x=0
x=L
T(0,x)=
T
0
where is the space coordinate (), is the thermal diffusivity (/sec), is the thermal conductivity () and is the heat transfer coefficient between the slab and the surrounding fluid ().
x
cm
α
2
cm
k
W/cm·K
h
W·/K
2
cm
To solve this problem, it is convenient to introduce the following dimensionless variables:
Θ=
T
T
∞
T
0
T
∞
and
ξ=
x
L
∂Θ
∂t
∂Θ
∂
2
ξ
∂Θ
∂ξ
at both and , and
ξ=0
ξ=1
Θ(0,ξ)=1
with
α=

and
Bi=

where 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; is the dimensionless coordinate of separation between the two layers; and the subscripts and refer to thermal properties of the layers and , respectively.
Bi=
hL
k
λ
1
2
x≤λ
x>λ