Thermal dilation

Dilation coefficients thermal $\ alpha$

General formulas: isotropic case

One can calculate for all isotropic materials the variation length and thus volume according to the temperature variation:

$\ Delta L = \ alpha \ cdot L_0 \ cdot \ T$
Delta

With:

$\ Delta L, \$ variation length in meter (m);
$\ alpha, \$ the linear dilation coefficient in 1 Kelvin ( $K^ {- 1}$ );
$L_0, \$ the initial length measures some (m);
$\ Delta T = T-T_0, \$ temperature variation in Kelvin (K) or Degree Celsius (°C).

Note: Since a variation is used, a difference in temperature, the difference in origin between Kelvin and degree Celsius is cancelled, the distinction is thus not necessary.

One can also directly calculate the length according to the temperature:

$L (T) = L+ \ Delta L = L (T_0) \ cdot (1+ \ alpha \ cdot (T-T_0))$

With:

$L, \$ the length measures some (m) according to the temperature;
$T, \$ the temperature considered in Kelvin (K);
$T_0, \$ the initial temperature in Kelvin (K).

Application

That is to say a steel rail of 30 m in winter with -20 °C ; in summer, the temperature is of 40°C .
The rail thus undergoes a temperature variation

\ Delta T =60 K, \

its variation length will be:

$\ Delta L = \ alpha_ {steel} \ cdot L_0 \ cdot \ Delta T = 12 \ cdot 10^ {- 6} \ times 30 \ times 60 = 2,16 \ cdot 10^ {- 2} m$

Thus the rail lengthens 21,6 mm , its length in summer is of 30,0216 m .

Thermal tensor of dilation

The materials Cristal cubic flaxes not present a thermal dilation anisotropic: one does not observe the same dilation coefficient

\ alpha \,

in all the directions. For this reason, one uses a symmetrical Tenseur of order 2 to describe dilation in anisotropic materials:

$\ begin {bmatrix} \ alpha_ {11} & \ alpha_ {12} & \ alpha_ {13} \ \ \ alpha_ {21} = \ alpha_ {12} & \ alpha_ {22} & \ alpha_ {23} \ \ \ alpha_ {31} = \ alpha_ {13} & \ alpha_ {32} = \ alpha_ {23} & \ alpha_ {33} \ end {bmatrix}$

Thus, in the general case of a Triclinic material , six dilation coefficients thermal are necessary. These coefficients referring to an orthogonal reference mark, the dilation coefficients inevitably do not have a direct relationship with the crystallographic axes of material. Indeed, the eigenvalues and clean vectors of a tensor of order 2 always form (if the eigenvalues are positive) an ellipsoid of revolution, whose axes are perpendicular the ones to the others: it is said that a tensor of order 2 always has at least the specific symmetry orthorhombic maximum $\ frac {2} {m} \ frac {2} {m} \ frac {2} {m}$ .

For an orthorhombic crystal for example, where $\ alpha_ {12} = \ alpha_ {13} = \ alpha_ {23} =0, \,$ the tensor of dilation is diagonal and $\ alpha_ {11}, \,$ $\ alpha_ {22} \,$ and $\ alpha_ {33} \,$ describes dilation along the three crystallographic directions $a, \,$ $b \,$ and $c \,$ of material. On the other hand, in the Monoclinical system , $\ alpha_ {13} \,$ is nonnull: whereas $\ alpha_ {22} \,$ represents thermal dilation along $b, \,$ the relation between $\ alpha_ {11}, \,$ $\ alpha_ {33}, \,$ $\ alpha_ {13} \,$ and the parameters of mesh $a, \,$ $c, \,$ $\ beta \,$ is not also commonplace. By convention, the orthogonal reference mark $(\ vec {E} _1, \ vec {E} _2, \ vec {E} _3)$ selected to describe thermal dilation in monoclinical materials is such as $\ vec {E} _2$ is parallel to $\ vec {B},$ axis of symmetry of the crystal, $\ vec {E} _3$ is parallel to $\ vec {C}$ and $\ vec {E} _1$ is parallel to the vector of the reciprocal Réseau $\ vec {has} ^ {\, *} = \ frac {\ vec {B} \ wedge \ vec {C}} {V}$ ( V being the volume of the mesh), which forms by definition a direct Trièdre with $\ vec {B}$ and $\ vec {C}$ : $\ alpha_ {33} \,$ represents thermal dilation along $\ vec {C},$ whereas $\ alpha_ {11} \,$ represents dilation along $\ vec {has} ^ {\, *} \ \ vec {has}$ .

Eigenvalues of the thermal tensor of dilation, or linear principal dilation coefficients $\ alpha_1 \,$ , $\ alpha_2 \,$ and $\ alpha_3 \,$ , also make it possible to obtain the voluminal dilation coefficient, trace of the tensor: $\ beta= \ alpha_1+ \ alpha_2+ \ alpha_3= \ alpha_ {11} + \ alpha_ {22} + \ alpha_ {33}, \,$ since the trace of a matrix square is invariant by change of bases.

Measure linear dilation coefficients

In the case of the crystalline materials, thermal dilation measures in a precise way by diffraction of x-rays. A method usually used consists in measuring the cell parameters of the crystal for various temperatures and to deduce the linear dilation coefficients from them. However, the intermediate calculation of the cell parameters introduces additional errors into the calculation of the coefficients and it is preferable to obtain them starting from the variation in temperature of the angle of diffraction

\ theta \,

. Several programs provide the components of the tensor of dilation starting from the variations of

\ theta \,

Linear dilation coefficients for principal materials

The coefficients given below are valid for temperatures ranging between 0°C and 100°C. Actually these coefficients depend on the temperature, the law of lengthening is thus not linear for very high differences in temperature.

Anomalies

water presents an anomaly; indeed while heating it contracts between 0 °C and + 4 °C.

Problems due to dilation

The solid dilation of the S is compensated on the Pont S by grooves: with the differences in exposures to the Sun and the heating of the atmosphere, a solid of several tens of meters can lengthen few centimetres. Without the space left by the grooves, the bridge would become deformed.

the dilation of a Liquide is often negligible compared to its boiling, but can explain certain phenomena, in particular with rigid containers.

It is not the cause of the overflow of the Lait only one heats too much, who is a phenomenon specific to the proteins pulps.

the breaking of abruptly heated glasses is explained by dilation.

Blocking of wheel. If a wheel is of a matter different from that of its axis, it will be able to be blocked at certain temperatures if the mechanical tolerances were badly calculated.

Applications of dilation

Thermometer thermal switch
Thermoscope
Dilatometer

Personalities having worked on dilation

Louis Joseph Gay-Lussac discoverer of the law of dilation of the Gases
Pierre Louis Dulong
Charles Edouard Guillaume, Nobel Prize of physics (1920), discovered alloys having low dilation coefficients.

See too

Vasodilatation
Dilation of the pupils
Dilation of time
Dilation of space in geometry

References

Random links:

Commune of Sigtuna | Bartolomeo Rastrelli | Morbio Superiore | Supreme Pity | Jean-Louis Laruette