# Conduction Due to the different thermal conductivity and heat capacity of paving stones and grass, the paving stone outline can be seen through this melting snow.

Heat conduction - also known as heat diffusion or conduction - is a mechanism for transporting thermal energy . According to the second law of thermodynamics , heat only flows in the direction of a lower temperature. No thermal energy is lost; the law of conservation of energy applies .

In physics , heat conduction is understood as the heat flow in or between a solid , a fluid or a gas as a result of a temperature difference . A measure of the thermal conduction in a certain substance is the thermal conductivity . The analogy to electrical current can often be used to calculate heat conduction , see thermal resistance . Then thermal conductivity and temperature calculations with the methods of electrical engineering are possible.

Further mechanisms for the transport of thermal energy are convection and thermal radiation .

## Heat flow, Fourier's law Wall piece of area and thickness . is the temperature of the warmer wall surface; is the temperature of the colder wall surface${\ displaystyle A}$ ${\ displaystyle d}$ ${\ displaystyle T_ {1}}$ ${\ displaystyle T_ {2}}$ The heat output transferred by conduction is described by Fourier's law (1822, after Jean Baptiste Joseph Fourier ), which for the simplified case of a solid body with two parallel wall surfaces reads: ${\ displaystyle {\ dot {Q}}}$ ${\ displaystyle {\ dot {Q}} = \ lambda \ cdot A \ cdot {\ frac {T_ {W_ {1}} - T_ {W_ {2}}} {d}}}$ Unit for is watt (W) ${\ displaystyle {\ dot {Q}}}$ The individual symbols stand for the following quantities :

• ${\ displaystyle T_ {W_ {1}}}$ : Temperature of the warmer wall surface
• ${\ displaystyle T_ {W_ {2}}}$ : Temperature of the colder wall surface
• ${\ displaystyle A}$ : Area through which the heat flows
• ${\ displaystyle \ lambda}$ : Thermal conductivity (temperature-dependent substance size)
• ${\ displaystyle d}$ : Thickness of the body measured from wall to wall

The transferred heat output is therefore

From today's perspective, the heat transport is described by the more precise term of heat flow density . The approaches to this go back to Fourier and Newton. The following relationship with the temperature gradient applies : ${\ displaystyle {\ vec {q}}}$ ${\ displaystyle {\ vec {\ dot {q}}} = - \ lambda \, \ operatorname {grad} \, T}$ ## Thermal equation

Mathematically, the phenomenon “heat conduction” is described by a partial differential equation. It has a parabolic characteristic. In its general form, this partial differential equation can be given in the following form.

${\ displaystyle {\ frac {\ partial u ({\ vec {r}}, t)} {\ partial t}} = a \, \ Delta u ({\ vec {r}}, t)}$ If you specialize this equation on the so-called heat conduction equation , it must be noted that this form of the heat conduction equation only applies to homogeneous, isotropic media. So only for media that have the same composition everywhere and that do not have a preferred orientation (preferred orientations occur, for example, through fibers in composite materials, but also through so-called grain stretching in rolled sheet metal, etc.). For these cases - and only for these - the material properties of the medium under consideration can be assumed to be variables that are exclusively dependent on the temperature. Strictly speaking, the equation formulated in this way only applies if no heat is introduced into or removed from the body under consideration through external effects. If this is the case, a so-called source term would have to be added. With these restrictions, the following form of the heat conduction equation applies:

${\ displaystyle {\ frac {\ partial T ({\ vec {r}}, t)} {\ partial t}} = a (T) \, \ times \ Delta T ({\ vec {r}}, t )}$ The differential equation generally describes transport processes (such as diffusion processes - by which one understands a material transport due to a concentration difference , or in the case of the heat conduction equation just a "migration" of the temperature distribution in a body due to a temperature gradient). The analytical solution of this equation is not possible in many cases. Today, technically relevant thermal conductivity tasks are calculated using the finite element method . As a result, one knows the temporal and spatial temperature distribution (temperature field). In this way, one can, for example, infer the spatial expansion behavior of the components, which in turn determines the local stress state. The temperature field calculation thus becomes an important basis for all technical design tasks in which the thermal component load cannot be neglected.

In inhomogeneous media with heat sources the heat conduction equation reads

${\ displaystyle \ rho ({\ vec {r}}) \ cdot c ({\ vec {r}}) \ cdot {\ frac {\ partial T ({\ vec {r}}, t)} {\ partial t}} \ = \ nabla \ cdot \ left [\ lambda ({\ vec {r}}) \ cdot \ nabla T ({\ vec {r}}, t) \ right] + q ({\ vec {r }})}$ where is the Nabla operator , the mass density , the specific heat capacity , the thermal conductivity and the heat flow introduced per volume by external or internal sources . ${\ displaystyle \ nabla}$ ${\ displaystyle \ rho}$ ${\ displaystyle c}$ ${\ displaystyle \ lambda}$ ${\ displaystyle q}$ ## Calculation method of stationary heat conduction processes using shape coefficients

In bodies on whose surfaces there are constant thermal conditions of the 1st type (surface temperature), 2nd type (heat flow density) or 3rd type (fluid temperature and heat transfer coefficient), very complex temperature fields are usually formed. In special cases, these can be calculated analytically by solving Laplace's differential equation. As a rule, however, numerically working simulation models are used. With knowledge of the temperature field, the heat flows can also be determined. In numerous cases, the user is only interested in the heat flows occurring on the body surfaces and / or the temperatures at certain locations within the solid. If such a body has been examined with the adjacent thermal conditions for states that do not represent any linear combinations among one another, a shape coefficient matrix can be determined from it. With this uniquely determined matrix of shape coefficients, the heat flows on the surfaces and selected local temperatures within the solid can then be determined, for example for variable surface or adjacent fluid temperatures or for impressed heat flow densities with simple calculation programs.

## Mechanisms

Thermal conductivity of selected materials
material Thermal conductivity at
20 ° C in W / (m K)
Silver (pure) 430
Copper (pure) 403
Iron (pure) 83.5
Glass 0.76
water 0.58
oil 0.145
air 0.0261
xenon 0.0051

### Dielectric solids

In dielectric solids ( insulators ), heat conduction occurs only through lattice vibrations, the phonons . The movement of the atoms is passed on from neighbor to neighbor. All electrons are bound to atoms and therefore cannot contribute to heat conduction. At very low temperatures, the thermal conductivity of non-metallic crystalline solids is limited by interface scattering.

### Electrically conductive solids

In electrically conductive bodies such as metals , the electrons can also transport heat and thus contribute to heat conduction. In metals, the heat conduction through the electrons even predominates. This connection leads to the Wiedemann-Franz law . Better electrical conductors such as copper therefore transfer heat better than poorer electrical conductors such as iron. In the superconducting state, the electrons no longer contribute to the conduction of heat - superconductors are therefore not good heat conductors.

### Liquids and gases

Light atoms or molecules conduct heat better than heavy ones, because they move faster with the same energy content . In contrast to convection , no vortices are formed with pure heat diffusion in liquids and gases.

In general, gases are considered to be poor conductors of heat. The thermal conductivity of liquids is generally about a power of ten higher than that of gases. As an example, the table shows the thermal conductivity of various substances at a temperature of 20 ° C in W / (m · K) (a detailed table can be found in the article on thermal conductivity ).

### Superfluids

In superfluids , heat is not transported by diffusion, as is usual, but by temperature pulses with a wave character . This effect is called the second sound .

## Examples

• In the case of a radiator , heat pipe or immersion heater , the heat energy from the hot interior reaches the outside by means of heat conduction through the housing.
• In the case of a soldering iron, a highly conductive metal such as copper must be installed between the heating element and the tip to transfer the thermal energy. Other metals like iron do not conduct heat well enough.
• With the Stirling engine or hot gas engine - in contrast to the Otto engine - the entire drive energy must be transferred from the external heat source to the working gas in the cylinder chamber by means of heat conduction. The thermal conductivity of the materials used limits the maximum achievable performance of the Stirling engine.
• Fridges are sheathed with glass wool or foamed polystyrene to keep the heat flow from the outside to the inside as low as possible.
• In a thermos or a vacuum tube collector for solar systems , u. a. Vacuum used to prevent convection and heat conduction.
• Multi-pane insulating glass with a very low heat transfer coefficient is used for windows in order to keep heating costs low with the loss of heat (see also Energy Saving Ordinance ). In this case, the distance is chosen so that the air / gas layer is sufficiently thick (gases are poor heat conductors), but thin enough that no significant convection takes place.

## literature

• Jochen Fricke, Walter L. Borst: Energy, A textbook of the physical basics Oldenbourg Verlag, Munich / Vienna 1984.
• Charles Kittel: Introduction to Solid State Physics . Different editions. Oldenbourg, Munich.