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

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

Consider a thin bar of length with initial temperature . The right end and the sides of the bar are insulated. For times , the left end is connected to a well-mixed insulated reservoir at an initial temperature . This Demonstration determines the transient temperature of the bar and the reservoir.

L

T

0

t≥0

T

r0

We use the following dimensionless variables:

Θ(ξ,τ)=-

T-

T

0

T

r0

T

0

ξ=

z

L

τ=

αt

2

L

Here are the dimensionless equations describing the system.

For the bar:

∂Θ

∂τ

2

∂

∂

2

ξ

with

Θ(ξ,0)=0

∂Θ(0,τ)

∂ξ

Θ

r

∂Θ(1,τ)

∂ξ

For the reservoir:

β=-(-Θ)

d

Θ

r

dτ

Θ

r

with

Θ

r

Here and are the temperatures of the bar and reservoir,

Θ

Θ

r

β=

m

r

C

p

r

m

b

C

p

b

m

C

p

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 and dimensionless time .

β

τ