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Heat Transfer between a Bar and a Fluid Reservoir: A Coupled PDE-ODE Model

​
time
0.5
m
C
p
ratio
2.
bar
reservoir
bar
reservoir
Consider a thin bar of length
L
with initial temperature
T
0
. The right end and the sides of the bar are insulated. For times
t≥0
, the left end is connected to a well-mixed insulated reservoir at an initial temperature
T
r0
. This Demonstration determines the transient temperature of the bar and the reservoir.
We use the following dimensionless variables:
Θ(ξ,τ)=
T-
T
0
T
r0
-
T
0
is the dimensionless temperature,
ξ=
z
L
is the dimensionless space coordinate,
τ=
αt
2
L
is dimensionless time.
Here are the dimensionless equations describing the system.
For the bar:
∂Θ
∂τ
=
2
∂
Θ
∂
2
ξ
,
with
Θ(ξ,0)=0
,
∂Θ(0,τ)
∂ξ
=-(Θ-
Θ
r
)
and
∂Θ(1,τ)
∂ξ
=0
.
For the reservoir:
β
d
Θ
r
dτ
=-(
Θ
r
-Θ)
,
with
Θ
r
(0)=1
.
Here
Θ
and
Θ
r
are the temperatures of the bar and reservoir,
β=
m
r
C
p
r
m
b
C
p
b
is a mass-heat capacity ratio,
m
and
C
p
represent the mass and heat capacities of the reservoir and the bar respectively.
The coupled system of one partial and one ordinary differential equation is solved using the built-in Mathematica function NDSolve. The temperatures of the bar and the reservoir are shown for different values of the mass heat capacity ratio
β
and dimensionless time
τ
.
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