Heat Conduction in Circular Electric Wire: Temperature & Heat Flux Distributions

Slide 1: Introduction

• Title: Heat Conduction with Electrical Heat Source
• Subtitle: Analysis of Temperature and Heat Flux Distributions in a Circular Electric Wire

Slide 2: Background

• Definition of Electrical Conductivity (ke): Measure of a material's ability to conduct electric current, measured in ohm-1cm-1.
• Introduction to Electric Current and Current Density (I): Flow of electric charge per unit time, and current per unit cross-sectional area, measured in amp/cm2.

Slide 3: Understanding the Meaning of Se (Heat Source)

• Se (Heat Source): Represents the electrical heat generated within the wire due to the passage of electric current.
• Heat Source Calculation: Se = ke * I^2, where Se is the heat source in watts/cm3, ke is the electrical conductivity, and I is the current density.

Slide 4: Derivation of Temperature Distribution

• Assumptions: Steady-state conditions, no heat generation, and radial symmetry.
• Governing Equation: Laplace's equation: ∇^2T = 0, where ∇^2 represents the Laplacian operator and T is the temperature.
• Boundary Condition: T(R) = T0, where R is the radius of the wire and T0 is the temperature at the wire's surface.
• Solution: T = T0 - (Se / (4πk)) * ln(r / R), where k is the thermal conductivity and r is the radial distance from the wire's center.

Slide 5: Heat Flux Distribution

• Definition of Heat Flux: Rate of heat transfer per unit area, measured in watts/cm2.
• Heat Flux Calculation: Heat Flux = -k * (∇T), where k is the thermal conductivity and ∇T is the temperature gradient.
• Heat Flux Distribution: Heat Flux = (Se / (2R)) * (1 - (R^2 / r^2)), where Se is the heat source and R is the radius of the wire.

Slide 6: Similarity with Viscous Flow in a Circular Tube

• Parallels between Heat Conduction and Viscous Flow:
  • Both involve circular cross-sections.
  • Both have governing equations based on gradients (heat conduction: ∇^2T = 0, viscous flow: ∇P = μ (∇^2v)).
  • Both exhibit radial symmetry in some cases.

Slide 7: Similarity with Viscous Flow in a Circular Tube (contd.)

• Relationship between Temperature and Velocity:
  • Temperature Distribution (T = T0 - (Se / (4πk)) * ln(r / R)).
  • Velocity Distribution (v = Vmax * (1 - (r^2 / R^2)), Hagen-Poiseuille flow in a circular tube).
  • Both follow similar mathematical patterns.

Slide 8: Conclusion

• Recap of the key points discussed:
  • Understanding the meaning of Se (heat source) in an electric wire.
  • Derivation of temperature and heat flux distributions in a circular electric wire.
  • Similarities between the heated wire problem and viscous flow in a circular tube.

Slide 9: Questions and Discussion

• Open the floor for questions and engage in a discussion with the audience.

Note: This PowerPoint presentation provides a simplified overview of the heat conduction with an electrical heat source in a circular electric wire. Please note that the derivations and equations presented are for illustrative purposes and may require further analysis and consideration of additional factors in practical applications.