# Thermal expansion

Under thermal expansion (also thermal expansion ) is the change in the geometric dimensions ( length , area , volume ) of a body caused by a change in its temperature . The reversal of this process through cooling is often referred to as heat shrinkage (also called thermal contraction ). The characteristic value is the coefficient of expansion .

## Impact and Applications

Measuring point for determining the thermal expansion at the Elbe bridge in Torgau
Bridge compensation element on the Danube bridge in Krems , Austria

A change in density is always associated with thermal expansion . In the case of fluid bodies, this can lead to changed pressure conditions. In particular, the resulting convection manifests itself in ocean currents and air currents and is part of the weather. This is used in thermal power plants and in gliding . Other examples of the use of thermal expansion are bimetal strips , many types of thermometers and temperature controllers , the Stirling engine , all internal combustion engines, and hot air balloons .

If there are different thermal expansions in a body or in mechanically connected bodies, mechanical stresses can arise, which in extreme cases can lead to damage or destruction of a component. This is impressively demonstrated in the so-called bolt explosion test . Certain dimensions change in the opposite direction to the change in length of the components. Thus, constructively provided distances between components can be reduced or closed as they expand. A difference in expansion can be caused by a temperature difference or a combination of materials with different thermal expansion behavior.

Architects , civil engineers and designers keep different thermal expansion low by using suitable materials. Additionally or alternatively, are expansion joints , sufficient clearance between components or compensation of size differences through expansion joints used. Position deviations caused by thermal expansion in electronically controlled machines, such as robots, can also be compensated for by control technology.

### Heat shrinkage

The term heat shrinkage is used for various processes.

• Because during casting, the warm material shrinks after it has solidified in the mold due to the release of heat, this is sometimes referred to as heat shrinkage. The cause of this is thermal expansion. The measure for heat shrinkage is the shrinkage , which is specified as a percentage of the finished size depending on the material or based on absolute dimensions in the respective length unit. The degree of shrinkage is largely determined by the material, since the second influencing variable, the temperature difference, is largely determined by the material via the suitable casting temperature.
• On the other hand, however, heat shrinkage is often also referred to as deformation due to heating, which is desirable with shrink tubing and, as with other permanent material shrinkage due to the action of heat, is not caused by thermal expansion.

## causes

In a solid, every single atom oscillates around a point of equilibrium . If it were harmonic oscillations , the distance between the atoms should on average remain the same as the equilibrium distance, because the atoms oscillate to the same extent in the direction of a neighboring atom and in the opposite direction. Therefore, the thermal expansion cannot be described with the approximation of the harmonic potential, but it must be taken into account that the potential energy increases more when two atoms approach each other than when they move away from each other. Due to the steeper potential curve, the deflection in the direction of a closer neighboring atom is smaller and at the same time the restoring force is greater than in the oscillation away from the neighboring atom (or in the direction of an atom further away); as a result, the atom spends less time in the vicinity of the neighboring atom, the distances between the atoms are on average larger than the equilibrium distance. If the vibrations take place with low energies, the potential is still relatively symmetrical; the higher the energies, the further the atoms vibrate in the asymmetrical area of ​​the potential and thus enlarge their vibration space. Higher energies are available at higher temperatures, which is why substances expand when heated. A quantitative description is given with the help of the Grüneisen parameter .

In the case of gases, the pressure increases at constant volume with increasing temperature, because the higher particle energy means more momentum per particle z. B. is delivered to a vessel wall, and the speed of the particles is higher, which leads to more impacting particles per unit of time. If the pressure is to remain constant, the volume must be increased so that the lower particle density compensates for the effects mentioned above. In the case of gases whose behavior deviates from that of the ideal gas , forces of attraction between the gas particles, which reduce thermal expansion, and the volume of an individual particle also play a role.

In liquids , the thermal expansion has the same causes as in gases, only it is greatly reduced by the forces of attraction between the particles .

## Equations of physics

Since the thermal expansion, especially in the case of solid bodies, is heavily dependent on the lattice structure or the bond conditions, the linear equations only represent approximations in the range of the standard conditions . Exact formulas and the derivation of the approximation can be found in the article " Expansion coefficient ".

In the case of transitions in the crystal structure, sudden changes can occur or non-linearities become apparent in the case of larger temperature differences, so that equations of the second or even higher order with two or more coefficients must be used.

### Solid

Linear expansion
length surface volume
${\ displaystyle {\ begin {matrix} \ Delta l & = & l_ {0} \ alpha \ Delta T \\ l_ {1} & = & l_ {0} (1+ \ alpha \ Delta T) \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} \ Delta A & \ approx & A_ {0} 2 \ alpha \ Delta T \\ A_ {1} & = & A_ {0} (1+ \ alpha \ Delta T) ^ {2} \ end {matrix}}}$ ${\ displaystyle {\ begin {matrix} \ Delta V & \ approx & V_ {0} 3 \ alpha \ Delta T \\ V_ {1} & = & V_ {0} (1+ \ alpha \ Delta T) ^ {3} \ end {matrix}}}$

### liquids

 ${\ displaystyle {\ begin {matrix} \ Delta V & = & V_ {0} \ gamma \ Delta T \\ V_ {1} & = & V_ {0} (1+ \ gamma \ Delta T) \ end {matrix}}}$

### Gases

Gases also tend to expand when the temperature rises. However, a cubic expansion coefficient corresponding to that of the formulas for liquids can only be defined for a certain starting temperature. For an ideal gas at an outlet temperature of 0 ° C it is ${\ displaystyle \ gamma}$

${\ displaystyle \ gamma}$= 1 / 273.15 K −1

In general, according to the equation of state for ideal gases under constant pressure , i. H. . This means that when the absolute temperature is doubled, the volume also doubles. ${\ displaystyle V \ propto T}$${\ displaystyle V_ {2} = V_ {1} \ cdot T_ {2} / T_ {1}}$

### Formula symbol

 ${\ displaystyle l}$, , ,${\ displaystyle l_ {0}}$${\ displaystyle l_ {1}}$${\ displaystyle \ Delta l}$ Length, start length, end length, length difference in m ${\ displaystyle A}$, , ,${\ displaystyle A_ {0}}$${\ displaystyle A_ {1}}$${\ displaystyle \ Delta A}$ Area, starting area, end area, area difference in m 2 ${\ displaystyle V}$, , ,${\ displaystyle V_ {0}}$${\ displaystyle V_ {1}}$${\ displaystyle \ Delta V}$ Volume, initial volume, final volume, volume difference in m 3 ${\ displaystyle \ Delta T}$ Temperature difference in K ${\ displaystyle \ alpha}$ Coefficient of linear expansion in K −1 ${\ displaystyle \ gamma}$ Room expansion coefficient in K −1

## particularities

Some materials such as zirconium tungstate or carbon fiber reinforced plastics ( CFRP ) can have a negative expansion coefficient ( density anomaly ). In the case of CFRP, this is anisotropic (direction-dependent).

In some temperature ranges, water also has a negative coefficient of expansion ( anomaly of water ).