Ch9.4 Heat Conduction and Fourier's Law

Ch9.1 & 9.2 Results

Temperature is a measure of average kinetic energy due to vibration of molecules in a substance.
At \( 0^{\circ} \) K, there is no vibration; above \( 0^{\circ} \) K there is.
Heat is the transfer of energy (Joules) from a higher temperature to a lower temperature.
Clearly there is a mathematical connection between rate of change \( Q \) of heat and rate of change \( U'(t) \) of temperature.
From Ch9.2, we have

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

Ch9.1 & Ch9.2 Results

So far we have \[ \frac{dU}{dt} = \frac{Q}{cm} \]
For an object, the greater the surface area, the greater the heat exchange \( Q \) with the surroundings.
Similarly, the greater the temperature difference between the object and the surroundings, the greater \( Q \) is.
This leads to Newton's Law of Cooling:

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

Ch9.2 & Ch9.3: Modeling Temperature

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Coffee cup:

\[ \frac{dU}{dt} = - \frac{h S}{cm} (U - u_s) \]

Hot Water Heater:

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

Ch9.4 Heat Conduction & Fourier's Law

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

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.

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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.

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Thermal Equilibrium

For thermal equilibrium, the word equation that characterizes the system in this state is shown below.

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

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Modes of Heat Transfer

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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 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).

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

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.

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Fourier's Law

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Heat flux is proportional to gradient of temperature expressed as a function of distance through the plate.
\( J(x) \) denotes heat flux at \( x \)
\( U(x) \) is temperature at \( x \)

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

Conductivity \( k >0 \) depends on material

Fourier's Law

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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.

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