Steady-State Conduction in Solids

Steady-State Conduction

Fourier's Law

• Fourier's Law states that the heat flux (q) is proportional to the negative gradient of temperature (dT/dx) in a given direction:
  • q = -k * (dT/dx)
  • k: thermal conductivity of the material
  • dT/dx: temperature gradient in the direction of heat flow
• The law is used to describe the conduction of heat in a solid

Conduction Equations

• The conduction equation is a partial differential equation that describes the flow of heat in a solid:
  • ∂u/∂t = α * ∇²u
  • u: temperature distribution
  • α: thermal diffusivity of the material
  • t: time
  • ∇²: Laplacian operator
• The equation is used to solve for the temperature distribution in a solid under steady-state conditions

Thermal Resistance

• Thermal resistance (R) is a measure of how much a material resists heat flow:
  • R = L / (k * A)
  • L: length of the material
  • k: thermal conductivity of the material
  • A: cross-sectional area of the material
• Thermal resistance is used to calculate the heat flux through a material

Temperature Distribution

• The temperature distribution in a solid under steady-state conditions can be described by the following equation:
  • T(x) = T0 + (qx / k) * (1 - x/L)
  • T(x): temperature at a given point x
  • T0: temperature at one end of the material
  • qx: heat flux
  • k: thermal conductivity of the material
  • L: length of the material
• The equation is used to calculate the temperature distribution in a solid under steady-state conditions