Lecture on Thermal Conduction and Heat Transfer

Introduction

• Sponsored by CuriosityStream.
• Example: Car disc brake rotor heats up due to friction.
• Heat transfer mechanisms:
  • Conduction: Transfer of heat through direct contact.
  • Convection: Transfer of heat by the movement of fluids.
  • Radiation: Transfer of heat through electromagnetic waves.
• Focus: Understanding thermal conduction, important for engineering applications.

Molecular Scale of Thermal Energy

• Thermal energy = Random motion of atoms and molecules.
• Increase in temperature = Increase in atomic/molecular motion.
• Near absolute zero = Minimal motion.
• Atoms connected by bonds in a lattice, causing vibrations (lattice structure) that redistribute thermal energy.

Mechanisms of Heat Conduction

1. Vibrations in the atomic lattice:
   • More regular lattice and stiffer bonds = Easier energy transfer.
2. Metallic materials:
   • Free electrons travel through lattice, colliding and redistributing energy.
   • Metals: Good thermal conductors due to free electrons and lattice vibrations.
3. Gases and liquids:
   • Conduction through collisions of atoms and molecules.

Heat Transfer Rate

• Interested in determining the heat transfer rate (q): Joules per second (Watts).

Fourier's Law

• Applies to heat transfer through solid walls:
  • Variables: Area (A), Thickness (L), Temperature difference (T1-T2).
  • Heat transfer direction: From higher to lower temperature.
  • Equation: q = -kA(dT/dx).
  • Thermal Conductivity (k): Material property, Watts/Meter-Kelvin.
  • Heat Flux: q per unit area (Watts/square meter).
• Positive heat transfer rate when temperature gradient is negative.
• Example calculation: Heat loss through a solid steel wall.

Properties of Materials

• Thermal conductivity varies by material:
  • Gases and non-metallic liquids: Low.
  • Non-metallic solids: Moderate.
  • Alloys and metals: High.
  • Diamond: High (due to crystalline lattice and strong bonds).
  • Aerogel: Low.
• Conductivity often assumed constant in thermal analyses.

Multi-Dimensional Heat Transfer

• One-dimensional heat transfer: Temperature function of x.
• Two and three-dimensional cases: Heat flow in multiple directions (x, y, z).
• Isotherms: Constant temperature lines/surfaces.
  • Heat flows perpendicular to isotherms in isotropic materials.
  • Reformulation using Del operator for more general Fourier’s law.

Heat Equation and Solving for Temperature Field

• Heat Equation: Describes heat flow, solving gives temperature field.
• Heat Equation: Partial differential equation (simple energy balance).
• Consider a small volume and energy transfer:
  • Left = Net thermal energy transfer.
  • Right = Rate of energy change stored.
• Thermal Diffusivity (α): Conductivity vs. heat storage ability.
• Equation involves density (ρ), specific heat capacity (Cp).
• Generalized form includes internal heat generation (e.g., power cables).

Solving Heat Transfer Problems

• Steady State: Temperature distribution doesn’t change with time (transient term = 0).
• Boundary Conditions: Used for calculating constants in integration.
• Software Tools: Numerical methods for complex cases.
• Thermal Resistance: Simplifies analysis of conduction through layers.
  • Related to critical insulation thickness problem.

Additional Resources

• Nebula: Streaming service for educational creators.
• Content includes videos on thermal resistance, dimensional analysis.
• Bundled offer with CuriosityStream for documentaries.

Conclusion

• Thermal conduction and heat equation overview.
• Sponsors: CuriosityStream and Nebula.