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Conduction

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Quick
Heat transfer that occurs across a stationary medium in which there exists a temperature gradient.


Equations

qx"  =  –k dT dx	Fourier's Law for one-dimensionl heat transfer
qx"  =  k T1 – T2 L   =  k ΔT L	heat flux, or rate of heat transfer per unit area under steady-state conditions and linear temperature distribution
qx = qx" A	rate of heat loss, or heat rate by conduction through a plane wall of area, A, the heat rate, qx, is expressed in terms of power

Nomenclature

symbol	description
k	thermal conductivity
T1	usually the warmer temperature
T2	usually the colder temperature
L	thickness of material
A	surface area of material
qx	rate of heat loss, or heat rate
qx"	heat flux, or rate of heat loss per unit area, or rate of heat transfer per unit area

Visualization

The stationary medium may be a fluid or solid T1 is temperature at left surface T2 is temperature at right surface q" is the heat transfer through the medium The difference in temperatures result in the temperature gradient.

Details

Conduction is a mode of heat transfer that results from processes that occur at atomic and molecular levels of activity. Conduction may be viewed as the transfer of energy from the more energetic to the less energetic particles of a substance through interactions between the particles.

The physical mechanism of conduction is most easily explained by considering a gas and using ideas familiar from the background of thermodynamics. Consider a gas in which there exists a temperature gradient and assume that there is no bulk motion. The gas may occupy the space between two surfaces that are maintained at different temperatures, as shown in the following figure. The temperature at any point within the energy field of gas molecules is associated to the proximity of that point. This energy is related to the random translational motion, as well as to the internal rotational and vibrational motions, of the molecules.

Higher temperatures are associated with higher molecular energies, and when neighboring molecules collide, as they are constantly doing, a transfer of energy from the more energetic to the less energetic molecules must occur. In the presence of a temperature gradient, energy transfer by conduction must then occur in the direction of decreasing temperature. This would even be true in the absence of collisions, as is evident from the following figure. The hypothetical plane at x_o is constantly being crossed by molecules from above and below due to their random motion. However, molecules from above are associated with a larger temperature than those from below, in which case there must be a net transfer of energy in the positive x direction. Collisions between molecules enhance this energy transfer. We may speak of the net transfer of energy by random molecular motion as a diffusion of energy.

The situation is much the same in liquids, although the molecules are more closely spaced and the molecular interactions are stronger and more frequent. Similarly, in a solid, conduction may be attributed to atomic activity in the form of lattice vibrations. The modern view is to ascribe the energy transfer to lattice waves induced by atomic motion. In an electrical nonconductor, the energy transfer is exclusively via these lattice waves; in a conductor it is also due to the translational motion of the free electrons.

It is possible to quantify heat transfer processes in terms of appropriate rate equations. These equations may be used to compute the amount of energy being transferred per unit time. For heat conduction, the rate is known as Fourier's law. For the one-dimensional plane wall shown in the following figure, having a temperature distribution T(x), the rate equation is expressed as

qx"  =  –k dT dx

Fourier's Law for one-dimensionl heat transfer

The heat flux q_x" is the heat transfer rate in the x direction per unit area perpendicular to the direction of transfer, and it is proportional to the temperature gradient, dT/dx, in this direction. The parameter k is the thermal conductivity. The minus sign is a consequence of the fact that heat is transferred in the direction of decreasing temperature. Under the steady-state conditions shown in the figure below, where the temperature distribution is linear, the temperature gradient may be expressed as:

dT dx

=

T2 – T1 L

and the heat flux is then:

qx"  =  –k

T2 – T1 L

or

qx"  =  k

T1 – T2 L

=  k

ΔT L

Note that this equation provides a heat flux, that is, the rate of heat transfer per unit area. The heat rate by conduction, or rate of heat loss, q_x, through a plane wall of area, A, is then the product of the flux and the area:

qx = qx" A

Molecules of matter have translational (kinetic), rotational, and vibrational energy. Energy in these modes can be transmitted to nearby molecules by interactions (collisions) or by exchange of molecules such that energy is given out by molecules that have more in the average (higher temperature) to those that have less in the average (lower temperature). This energy exchange between molecules is heat transfer by conduction, and it increases with the temperature difference and the ability of the substance to make the transfer. This is expressed in Fourier's law of conduction.


Examples

The exposed end of a metal spoon suddenly immersed in a cup of hot coffee will eventually be warmed due to the conduction of energy through the spoon. On a winter day there is significant energy loss from a heated room to the outside air. This loss is principally due to conduction heat transfer through the wall that separates the room air from the outside air.

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Related
▪ L - Fourier's Law, The Conduction Rate Equation