The thermal conduction is the mode of Transfer of heat caused by a difference in temperature between two areas of the same medium or between two mediums in contact without appreciable displacement of matter. It is in fact the thermal agitation which is transmitted gradually, a molecule or an atom yielding part of kinetic sound energy to its neighbor (the vibration of the atom slows down with the profit of the vibration of the neighbor).

General information

Fourier analysis

This spontaneous transfer of Chaleur of an area of high Température towards an area of lower temperature obeys the law known as of Fourier (established mathematically by Jean-Baptiste Biot in 1804 then in experiments by Fourier in 1822).

The density flux of heat is proportional to the Gradient of temperature.

$\backslash \; varphi=\; -\; \backslash \; lambda\; \backslash \; overrightarrow\; \{\backslash \; mathrm\; \{grad\}\}\; (T),$

The constant of proportionality λ is named thermal Conductivité material. It is always positive. With the units of the international system, thermal conductivity λ is expressed in J.m^-1.K^-1.s^-1 or, that is to say of W.m^-1.K^-1.

Equation of heat

An assessment of energy, and the expression of the Fourier analysis leads to the general equation of conduction of heat. It is given here in its unidimensional form:

$\backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; X\}\; \backslash \; left\; (\backslash \; lambda\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; X\}\; \backslash \; right)\; +\; P\; =\; \backslash \; rho\; \backslash ,\; C\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; T\}$,

where:

• P is the energy produced within material in W.m^-3. It is often null (case of the deposits of heat on the surface of walls, for example), but one can quote of many cases where it is not it; let us quote among others the study of the transfer of heat by conduction within nuclear fuel, or the absorption of the light or the microwaves within semi-transparent materials…,
• ρ is the density of material in kg.m^-3,
• and C is the specific heat of material in J.kg^-1.K^-1

(establishment of the equation of conduction of heat)

If P is null and where one makes moreover the approximation that thermal conductivity λ does not depend on the position:

$\backslash \; lambda\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial^2\; T\}\; \{\backslash \; partial\; x^2\}\; =\; \backslash \; rho\; \backslash ,\; C\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial\; T\}\; \{\backslash \; partial\; T\}$.

Maybe, in permanent mode (when the temperature does not evolve/move any more with time):

$\backslash \; lambda\; \backslash ,\; \backslash \; frac\; \{\backslash \; partial^2\; T\}\; \{\backslash \; partial\; x^2\}\; =\; 0$.

The solution of this equation is then:

$T\; =\; has\; X\; +\; B$,

where has and B are constants to be fixed according to the boundary conditions.

Conduction in stationary mode

Simple plane surface

The material is a conducting medium thermically limited by two parallel plans (case of a wall). Each plan at a temperature T homogeneous on all its surface. It is considered that the plans have infinite dimensions to be freed from the effects edges. Consequently entering flow is equal to outgoing flow, it does not have there losses of heat on the edges.

The heat flux is written:

$\backslash \; Phi\; =\; \backslash \; frac\; \{\backslash \; lambda\; S\}\; \{E\}\; (T\_1-T\_2)\; \backslash ,$

The density flux thermal surface is written:

$\backslash \; varphi=\; \backslash \; frac\; \{\backslash \; lambda\}\; \{E\}\; (T\_1-T\_2)$.

electric Analogy By analogy with the electricity (Law of Ohm) in the particular case where the surface of contact between each material is constant (surface flow $\backslash \; varphi$ constant) we let us can put in parallel the two expressions:

$(U\_1-U\_2)\; =\; IH\; \backslash ,$

$(T\_1-T\_2)\; =\; \backslash \; frac\; \{E\}\; \{\backslash \; lambda\; S\}\; \backslash \; Phi\; \backslash ,$

We can put in parallel the tension and the intensity

$(U\_1-U\_2)\; \backslash \; leftrightarrow\; (T\_1-T\_2),$

$I\; \backslash \; leftrightarrow\; \backslash \; Phi,$

With thermal resistance

$R\; \backslash \; leftrightarrow\; R\_\; \{thc\}\; =\; \backslash \; frac\; \{E\}\; \{\backslash \; lambda\; S\}\; \backslash ,$

where $S$ is the surface of materials and $E$ its thickness. Thermal resistance $R\_\; \{thc\}$ is homogeneous with of K.W^-1

Plane surfaces in series

One considers materials IS B and C respective thickness $e\_\; \{has\}$, $e\_\; \{B\}$ and $e\_\; \{C\}$ and of respective radiative conductivity $\backslash \; lambda\_\; \{has\}$, $\backslash \; lambda\_\; \{B\}$ and $\backslash \; lambda\_\; \{C\}$.

The assumptions are identical to those of a simple plane surface. It is considered that the contact between each layer is perfect what means that the temperature with the interfaces between 2 materials is identical in each materials (Not jump of temperature to the passage of an interface).

Finally the surface of contact between each material is constant what implies a surface flow $\backslash \; varphi$ constant.

thermal resistances are added:

$T\_1-\; T\_4=\; (\backslash \; frac\; \{e\_A\}\; \{\backslash \; lambda\_A\; S\}\; +\; \backslash \; frac\; \{e\_B\}\; \{\backslash \; lambda\_B\; S\}\; +\; \backslash \; frac\; \{e\_C\}\; \{\backslash \; lambda\_C\; S\})\; \backslash \; Phi\; \backslash \; =\; (R\_\; \{thA\}\; +\; R\_\; \{thB\}\; +\; R\_\; \{thC\})\; \backslash \; Phi$

profile of the temperatures
For each material the temperature variation follows a law of the type:

$T=\; T\_1-\; \backslash \; frac\; \{e\_X\}\; \{\backslash \; lambda\_X\; S\}\; \backslash \; Phi\; \backslash ,$

The temperature variation is thus linear in the thickness of material considered. The slope depends on λ (thermal Conductivité) characteristic of each material. More thermal conductivity will be the weak (thus more the material will be insulating) more slope will be strong.

electric Analogy
Same manner that electrical resistances in series are added, thermal resistances in series are added.

Plane surfaces in parallel

One considers juxtaposed plane materials side by side. Each material homogeneous and is limited by two parallel plans. It is for example the case of a wall with a window. The assumptions are identical to those of a simple plane surface. In supplement, one considers that the temperature is uniform on the surface of each element (T₁ and T₂). That is to say S_A, S_B and S_C respective surfaces of the elements has, B and C.

Thereafter, the assumption is made that flow is always perpendicular to the made up wall; this is not realistic since the temperature of surface of each element which compose it is different and which there exists consequently a side variation in temperature (at the origin of the cold bridges). Also, it is necessary to correct the heat flow calculated in the made up wall using linear coefficients of loss, specific to each junction of wall (and being able to be negligible, cf thermal regulation TH 2000)

The thermal conductances are added:

$C\_\; \{HT\}\; =\; \backslash \; frac\; \{1\}\; \{R\_\; \{HT\}\}\; =\; \backslash \; frac\; \{1\}\; \{\backslash \; frac\; \{e\_A\}\; \{\backslash \; lambda\_A\; S\_A\}\}\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; frac\; \{e\_B\}\; \{\backslash \; lambda\_B\; S\_B\}\}\; +\; \backslash \; frac\; \{1\}\; \{\backslash \; frac\; \{e\_C\}\; \{\backslash \; lambda\_C\; S\_C\}\}\; =\; \backslash \; frac\; \{1\}\; \{R\_\; \{thA\}\}\; +\; \backslash \; frac\; \{1\}\; \{R\_\; \{thB\}\}\; +\; \backslash \; frac\; \{1\}\; \{R\_\; \{thC\}\}$

electric Analogy
It is thus also possible to make an analogy between an electric assembly of resistances in parallel.

Simple cylindrical surface

The simple tube consists of only one homogeneous material. The temperature is homogeneous on each surface of the tube. One considers that the tube with an infinite length in order to free itself from the effects edge.

The temperature variation is written:

$\backslash \; T\_1-T\_2=\; \backslash \; frac\; \{\backslash \; Phi\}\; \{2\; \backslash \; pi\; \backslash \; lambda\; L\}\; \backslash \; ln\; \backslash \; frac\; \{R\_2\}\; \{R\_1\}$

Concentric cylindrical surfaces

The concentric tube consists of tubes laid out in concentric layers. It is considered that the contact is perfect between the tubes. The temperature is homogeneous on each surface of the tube. One considers that the tube with an infinite length L in order to free itself from the effects edge.

The total resistance of the tube is expressed according to a law of the type “series” like the made up wall series:

$\backslash \; R\_\; \{VHV\}\; =\; R\_\; \{thA\}\; +\; R\_\; \{thB\}$

Conduction in dynamic mode

August 1st

Random links:

Chatila | Hwange national park | Carol Owens | Stevan Knićanin | Servius Cornelius Merenda