Ch9.4 Heat Conduction and Fourier's Law

Ch9.2 Results

From Ch9.2, we learned that

\[ Q = cm\frac{dU}{dt} \]

Newton's Law of Cooling: \( Q \) is proportional to the temperature difference between the object and its surroundings:

\[ \frac{dU}{dt} = -\frac{hS}{cm} \left( U(t) - u_s \right) \]

Ch9.4 Heat Conduction & Fourier's Law

  • We examine the basic physics of heat conduction.
  • We define heat flux to describe the flow of heat.
  • Fourier's law relates heat flux to temperature.

Humor



What did the fish say when he hit the wall?

Dam.

Modes of Heat Transfer

  • Heat conduction involves transfer of heat energy by vibration of molecules.
  • Another heat transport mechanism is convection (current flow).
  • Radiation involves heat transfer by electromagnetic waves, and doesn't require a propagation medium.

Heat and Temperature

  • Temperature starts with initial state and then, as a body is cooled or heated, temperature changes.
  • If temperature is not same at every point, then heat conducts through body from regions of higher temperature to regions of lower temperature.
  • If body is liquid or gas, then this is in addition to heat transfer through mixing of convection currents.

Thermal Equilibrium

  • Temperature will tend to some equilibrium state where there may still be a flow of heat through body by conduction.
  • However, temperature at any point will not change with time.
  • When temperature is in equilibrium state, none of heat is used to increase temperature, nor is heat released to decrease temperature.

Thermal Equilibrium

In the compartment diagram below, equilibrium temperature corresponds to heat outflow and heat inflow being the same.

\[ \begin{Bmatrix} \mathrm{rate \, of \, change} \\ \mathrm{of \, heat ~ content } \end{Bmatrix} = 0 \]

Heat Conduction and Heat Flux

  • Observation demonstrates that the rate heat conducts through body is directly proportional to cross-sectional area through which heat flows.
  • Heat flux is the rate of heat flow per unit area through a cross-section (Watts/\( m^2 \), where Watts = Joules/sec).

\[ J = \begin{Bmatrix} \mathrm{rate \, of \, flow ~ of ~ heat} \\ \mathrm{per ~unit ~ time ~ per \, unit ~ area } \end{Bmatrix} \]

Fourier and Heat Flux

  • Some substances conduct heat better than others.
  • Heat flows more easily through certain metals than through substances such as brick and stone.
  • Fourier did some controlled experiments where he measured heat conducted through thin plates of different materials.

Fourier's Law

Heat flux is proportional to gradient of temperature as a function of distance \( x \).

  • \( J(x) \) denotes heat flux at \( x \)
  • \( U(x) \) is temperature at \( x \)

\[ J(x) = -k \frac{dU}{dx} \]

  • Conductivity \( k >0 \) depends on material
  • Units on \( k \) are \( [k] \) =W/(m K)
  • To compare, \( [c] \) = J/(kg K)

Fourier's Law

Need minus sign for positive rate of heat conduction:

  • Decreasing temperature: \( U' < 0 \) and heat flows in positive direction.
  • Increasing temperature: \( U' > 0 \) and heat flows in negative direction.

\[ J(x) = -k \frac{dU}{dx} \]

Conductivity Constant k

  • Over large temperature ranges, conductivity \( k \) of a material is not strictly constant.
  • Assuming constant conductivity where reasonable enables us to make sufficiently accurate predictions.

title