Derivation of 3-dimensional Heat Conduction Equation

Derivation of the 3D Heat Conduction Equation in Cartesian Coordinates

Assumptions

• Material is homogeneous and isotropic
• Thermal properties \( k, \rho, c_p \) are constant
• No fluid motion (pure conduction)
• Heat generation is considered (optional)

1. Control Volume and Energy Balance

Consider a differential control volume of size \( dx \times dy \times dz \). The energy conservation principle gives:

Net rate of heat accumulation = Heat in - Heat out + Internal heat generation

2. Fourier’s Law of Heat Conduction

In the x-direction:

Heat conducted in: \( q_x = -k \left( \frac{\partial T}{\partial x} \right) \big|_x \)
Heat conducted out: \( q_{x+dx} = -k \left( \frac{\partial T}{\partial x} \right) \big|_{x+dx} \)

Net heat flux in x-direction:

\( \frac{\partial}{\partial x} \left( -k \frac{\partial T}{\partial x} \right) dx\,dy\,dz \)

Similarly for y and z directions.

3. Total Heat Conducted In

\( \nabla \cdot (-k \nabla T) \cdot dx\,dy\,dz \)

If \( k \) is constant:

\( = -k \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) dx\,dy\,dz \)

4. Heat Stored in the Element

\( \rho c_p \frac{\partial T}{\partial t} \cdot dx\,dy\,dz \)

5. Internal Heat Generation

\( \dot{q} \cdot dx\,dy\,dz \)

6. Energy Balance Equation

Combining all terms:

\( \rho c_p \frac{\partial T}{\partial t} = k \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + \dot{q} \)

7. Final Form using Thermal Diffusivity

Let \( \alpha = \frac{k}{\rho c_p} \), then:

\[ \boxed{ \frac{\partial T}{\partial t} = \alpha \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) + \frac{\dot{q}}{\rho c_p} } \]

Special Cases

• No internal heat generation: \( \dot{q} = 0 \)
• Steady-state condition: \( \frac{\partial T}{\partial t} = 0 \)