Derive 3D general conduction equation for homogeneous material
To derive the 3D general conduction equation for a homogeneous material, we start with the heat conduction equation in one dimension and extend it to three dimensions.
The one-dimensional heat conduction equation is given by:
∂(T)/∂(t) = α ∂²(T)/∂(x²)
where ∂(T)/∂(t) represents the rate of change of temperature with respect to time, ∂²(T)/∂(x²) represents the second derivative of temperature with respect to the spatial coordinate x, and α is the thermal diffusivity of the material.
To extend this equation to three dimensions, we introduce the spatial coordinates y and z. We will assume that there are no heat sources or sinks within the material, and the material is homogeneous throughout.
The 3D heat conduction equation in Cartesian coordinates is then given by:
∂(T)/∂(t) = α (∂²(T)/∂(x²) + ∂²(T)/∂(y²) + ∂²(T)/∂(z²))
This equation describes the rate of change of temperature with time at any point within a homogeneous material, as determined by the thermal diffusivity α and the second derivatives of temperature with respect to each spatial coordinate.
By solving this equation, we can predict the distribution of temperature within a homogeneous material over time, depending on the initial conditions and boundary conditions.