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.

We use the following dimensionless variables:

is the dimensionless temperature,

is the dimensionless space coordinate,

is dimensionless time.

Here are the dimensionless equations describing the system.

For the bar:

,

with

,

and

.

For the reservoir:

,

with

.

Here and are the temperatures of the bar and reservoir,

is a mass-heat capacity ratio,

and 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 and dimensionless time .