Heat Transfer

Heat energy in transit due to a temperature difference

Conduction - A stationary medium that has a temperature gradient
Convection - heat transfer between a surface and a moving fluid when they are at different temperatures
Thermal radiation - energy emission in the form of electromagnetic waves
- radiation between two surfaces at different temperatures
- all surfaces emit thermal radiation

Conduction

At the atomic and molecular level, energy is transferred from more energetic particles to less energetic ones. This energy transfer always happens in the direction of decreasing temperature. The net transfer of energy by random molecular motion is called diffusion of energy.

In solids, conduction may be due to atomic activity due to vibrations. Materials that are particularly good at heat transfer are considered conductors. These conductors can conduct heat tranfer with lattice wave vibrations and by the translational motion of free elections. Non-conductors can only conduct heat with lattice wave vibrations.

Since the molecules in liquids are closer together, interactions are stronger and more frequent.

Fourier's Law for Heat Conduction

q_x'' = -k{dT \over dx}

Where:

q_x'' is the heat flux, the rate of heat transfer per unit area

k is the thermal conductivity

Convection

Two mechanisms:

Diffusion - Random molecular motion
Bulk transfer - Energy transferred by the macroscopic motion of the fluid

Convection between a fluid in motion and a bounding surface. At the surface, the velocity of the particles is zero. This is called the hydrodynamic velocity boundry.

If the temperature of the surface is greater than the temperature of the outer flow, convection heat transfer occurs from the surface to the outer flow.

The thermal boundary layer is the region of a fluid through which the temperature varies from the surface temperature to the outer flow temperature. At the surface, the fluid velocity is zero, so heat transfer can only occur by diffusion.

Radiation

Thermal energy radiation is emitted by matter that is at a finite temperature.

Can occur from liquids and gases
The emission may be attributed to changes in electron configurations of the molecules
Electromagnetic waves transport the energy
Doesn't require the persence of another material.
- Most efficient in a vacuum

Stefan-Boltzmann Law

Gives the maximum flux at which radiation may be emitted

q'' = \sigma T_s^4

Where:

T_s is the absolute temperature

\sigma is the Stefan-Boltzmann constant

For a real surface:

q'' = \varepsilon \sigma T_s^4

Where \varepsilon is the emissivity (0 \le \varepsilon \le 1).

If radiation is incident upon a surface, a portion will be absorbed. The rate of absorbtion per unit surface area:

q''_\text{abs} = \alpha q''_\text{inc}

Where (0 \le \alpha \le 1).

Radiation emission decreases the thermal energy of the matter, and radiation absorbtion increases the thermal energy of the matter.

*Special case where we have the net exchange between a small surface and a much larger surface completely surounding it. Assume \alpha = \varepsilon.

q'' = {q\over A} = \varepsilon \sigma (T_s^4 - T_\text{sur}^4)

The net rate of radiation per unit area.

Example

q_\text{rad} = h_r A(T_s - T_\text{sur})

q_\text{conv} = hA(T_s - T_\infty)

q_\text{total} = q_\text{conv} + q_\text{rad}

q_\text{total} = hA(T_s - T_\infty) + \varepsilon A \sigma(T_s - T_\text{sur})

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