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

.