Heat Transfer Lecture: Focus on Conduction

Introduction

• Overview of three modes of heat transfer:
  • Conduction: Heat transfer within a solid or stationary fluid.
  • Convection: Heat transfer from a surface to a moving fluid.
  • Radiation: Typically between surfaces.
• Rate equations:
  • Fourier’s Law for conduction.
  • Newton's Law of Cooling for convection.
  • Stefan-Boltzmann Law for radiation.

Conduction

• Heat transfer through a solid or stationary fluid caused by random motion of atoms, molecules, or electrons.
• Driving Force: Temperature difference.
• Heat Flux (q"): Denoted by q double prime.

Thought Experiment

• Metal Rod Scenario:
  • Uniform temperature initially.
  • Heat source (e.g., Bunsen burner) introduces energy at one end.
  • Atoms/molecules vibrate more, causing temperature rise and subsequent energy transfer along the rod.

Fourier’s Law

• Mathematical Representation:
  • Heat flux is proportional to the negative of the thermal conductivity (k) times the temperature gradient.
  • Formula: [ q'' = -k \frac{dT}{dx} ]
  • Units for temperature gradient: °C/m or K/m.
  • Indicates heat flows against the temperature gradient.
• Constant Slope:
  • In a one-dimensional system, if temperature gradient (dT/dx) is constant, Fourier's law can be expressed in terms of temperature difference over a distance.

Application of Fourier's Law

• Example Problem:
  • Given temperature profile: [ T(x) = 5x + 100 ]
  • Thermal conductivity: 10 W/m·K.
  • Calculate heat flux.
  • Key insight: Negative flux indicates heat flows against the coordinate system.
• Units and Practical Application:
  • Heat flux (units: W/m²).
  • Total rate of heat loss (units: Watts).
  • Application in homes to estimate energy loss due to conduction.

Important Concepts

• Temperature Profile: Shows how temperature changes with respect to spatial coordinates.
• Thermal Conductivity (k): Measure of a material's ability to conduct heat.
• Conceptual Understanding:
  • Essential to conceptualize direction and magnitude of heat flow.
  • Ensure understanding of coordinate systems and their impact on sign (positive/negative) of heat flux.
• Unit Check: Ensure equation units align to confirm correct calculations.

Conclusion

• Fourier’s Law is essential for quantifying conduction.
• Importance of solving temperature profiles to understand heat distribution.

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This lecture emphasized understanding and applying Fourier's Law to conduction problems and highlighted the practical significance of conduction in real-world scenarios.