# Thermal diffusivity

The thermal conductivity or thermal diffusivity , and occasionally " thermal diffusivity " (from English thermal diffusivity ), is a material property that the spatial distribution of the description of the temporal change temperature by heat conduction serves as a consequence of a temperature gradient. ${\ displaystyle a}$

It is related to thermal conductivity , which is used to describe energy transport. ${\ displaystyle \ lambda}$

## Definition and unity

The thermal diffusivity is defined as:

${\ displaystyle a = {\ frac {\ lambda} {\ rho \ cdot c}}}$

With

${\ displaystyle \ lambda}$- thermal conductivity
${\ displaystyle \ rho}$- density
${\ displaystyle c}$- specific heat capacity .

The thermal diffusivity has the SI unit . In the US, the specification in is also common. ${\ displaystyle \ mathrm {m} ^ {2} / \ mathrm {s}}$${\ displaystyle \ mathrm {ft} ^ {2} / \ mathrm {h}}$

It is a temperature-dependent material property, since all underlying parameters are temperature-dependent.

## Thermal equation

The spatial and temporal distribution of the temperature in a body can be calculated using Fourier's law (according to JBJ Fourier ) and the heat conduction equation that follows. In the first considerations it goes back to Newton and expresses a simple fact: The change in the heat content of a spatial area flows as a heat flow through its envelope. ${\ displaystyle T ({\ vec {x}}, t)}$

For isotropic bodies with inhomogeneous thermal conductivity but constant thermal capacity per volume, the following applies:

${\ displaystyle {\ frac {\ partial T ({\ vec {x}}, t)} {\ partial t}} \ = \ nabla \ left [a ({\ vec {x}}, T) \, \ cdot \ nabla T ({\ vec {x}}, t) \ right]}$

In mathematical symbolism:

${\ displaystyle {\ vec {x}}}$: Position vector (symbolized by the vector arrow above the position variable )${\ displaystyle x}$
${\ displaystyle \ nabla}$ : Nabla operator : Differentiation rule with regard to the local derivatives, which can be applied in different ways to scalar quantities, vectors and operators.

For homogeneous, isotropic media, the thermal conduction equation, assuming a temperature-independent thermal diffusivity, is simplified to:

${\ displaystyle {\ frac {\ partial T ({\ vec {x}}, t)} {\ partial t}} \ = a \ cdot \ Delta T ({\ vec {x}}, t)}$.

In mathematical symbolism:

${\ displaystyle \ Delta}$: Laplace operator : Differentiation rule with regard to the local derivatives, which is applied here to the scalar variable temperature .

The differential equation is called the heat conduction equation and generally describes transport processes such as B. also the diffusion , or as here a migration of the temperature distribution in a body due to a temporary temperature gradient. From a mathematical point of view, the thermal diffusivity is therefore the “ transport coefficient of the heat conduction problem”. The two specified variants of the heat conduction equation only apply if no heat is generated or consumed in the body. If that were the case, a so-called source term would have to be added.

## Practical use

The analytical calculation of the unsteady temperature distribution is not possible in many cases. Thermal conduction problems are therefore often calculated numerically using the finite element method . The result is temporal and spatial temperature distributions (temperature fields). So you can z. B. infer the spatial expansion behavior of components or determine the local internal stress state . The temperature field calculation is therefore an important basis for technical design tasks in which temporary thermal residual stresses cannot be neglected.

Another example of the importance of thermal diffusivity is thermal insulation that is exposed to changing temperature gradients. These are, for example, fire doors or house insulation. The resistance of a fire door is expressed by the time it takes for the heat to penetrate the door. The door must not only insulate well from heat, but the insulating material should also have a low thermal conductivity. The situation is similar with a house insulation layer, for example in the roof area facing south: here, the low thermal conductivity of a less thick insulation can prevent the interior from heating up with temporary solar radiation.

Thermal diffusivity of selected metals at 20 ° C

Density ρ
(kg / dm 3 )
specific
heat capacity (kJ / (kg K)) ${\ displaystyle c}$
Thermal conductivity
λ
(W / (m · K))
Thermal
conductivity a
(mm 2 / s)
aluminum 2.7 0.888 237 98.8
lead 11.34 0.129 35 23.9
bronze 8.8 0.377 62 18.7
chrome 6.92 0.44 91 29.9
Cr-Ni steel
(X 12 CrNi 18.8 )
7.8 0.5 15th 3.8
iron 7.86 0.452 81 22.8
gold 19.26 0.129 316 127.2
cast iron 7.8 0.54 42 ... 50 10… 12
Steel (<0.4% C ) 7.85 0.465 45… 55 12… 15
copper 8.93 0.382 399 117
magnesium 1.74 1.02 156 87.9
manganese 7.42 0.473 21st 6th
molybdenum 10.2 0.251 138 53.9
sodium 0.97 1.22 133 112
nickel 8.85 0.448 91 23
platinum 21.37 0.133 71 25th
silver 10.5 0.235 427 173
titanium 4.5 0.522 22nd 9.4
tungsten 19th 0.134 173 67.9
zinc 7.1 0.387 121 44
Tin (white) 7.29 0.225 67 40.8
Silicon 2.33 0.700 148 87
Thermal diffusivity of selected non-metals at 20 ° C

Density ρ
(kg / dm 3 )
specific
heat capacity (kJ / (kg K)) ${\ displaystyle c}$
Thermal conductivity
λ
(W / (m · K))
Thermal
conductivity a
(mm 2 / s)
Acrylic glass (plexiglass) 1.18 1.44 0.184 0.108
asphalt 2.12 0.92 0.70 0.36
concrete 2.4 0.88 2.1 0.994
Ice (0 ° C) 0.917 2.04 2.25 1.203
Dirt (grobkiesig) 2.04 1.84 0.52 0.14
Sandy soil (dry) 1.65 0.80 0.27 0.20
Sandy soil (moist) 1.75 1.00 0.58 0.33
Clay soil 1.45 0.88 1.28 1.00
Window glass 2.48 0.70 0.87 0.50
Mirror glass 2.70 0.80 0.76 0.35
Quartz glass 2.21 0.73 1.40 0.87
Glass wool 0.12 0.66 0.046 0.58
plaster 2.2 to 2.4 1.09 0.51 0.203
granite 2.75 0.89 2.9 1.18
Carbon (graphite) 2.25 0.709 119 ... 165 74… 103
Cork boards 0.19 1.88 0.041 0.115
marble 2.6 0.80 2.8 1.35
mortar 1.9 0.80 0.93 0.61
paper 0.7 1.20 0.12 0.14
Polyethylene 0.92 2.30 0.35 0.17
Polytetrafluoroethylene 2.20 1.04 0.23 0.10
Polyvinyl chloride 1.38 0.96 0.15 0.11
Porcelain (95 ° C) 2.40 1.08 1.03 0.40
sulfur 1.96 0.71 0.269 0.193
Hard coal 1.35 1.26 0.26 0.15
Fir wood (radial) 0.415 2.72 0.14 0.12
Plastering 1.69 0.80 0.79 0.58
Brick 1.6 ... 1.8 0.84 0.38 ... 0.52 0.28 ... 0.34
air 0.0013 1.01 0.026 20th
water 1.0 4.18 0.6 0.14

## literature

• Ralf Bürgel: Handbook high temperature materials technology. 3. Edition. Friedrich Vieweg & Sohn Verlag, Wiesbaden 2006, ISBN 978-3-528-23107-1 .
• M. ten Bosch: The heat transfer. A textbook and reference book for practical use, third edition, Springer Verlag, Berlin / Heidelberg 1936.

## Individual evidence

1. The term number should be avoided because it is not a dimensionless ratio, but a quantity of the dimension .${\ displaystyle \ mathrm {m} ^ {2} / \ mathrm {s}}$
2. ^ John H. Lienhard IV and John H. Lienhard V: A Heat Transfer Textbook, 3rd edition, 2001, p. 55, Gl. 2.10.