1. Introduction to Conduction

By applying an energy balance to a control volume, we acquire the governing equation for heat transfer due to conduction:

Where:

• • is the internal thermal energy of the material.

• • is the energy supplied due to an applied heat flux.

• • is the energy generated due to the rate of thermal energy generation.

Substitute in Fourier's Law

:

If incompressible, and therefore:

If one dimensional conduction, and assuming the specific heat

is constant:

If no heat generation,

:

If steady-state, then , giving the heat equation:

Boundary Conditions

The heat equation is second order, therefore two boundary conditions must be specified. There are various scenarios for this:

• • Constant temperature:

• • Applied flux:

• • Ideally insulated, with no applied flux:

• • If the surface is in contact with fluid, Newton's law can be applied: