Thermal conduction

See also: electrical conduction

In heat transfer, conduction (or heat conduction) is the transfer of thermal energy between neighboring molecules in a substance due to a temperature gradient. It always takes place from a region of higher temperature to a region of lower temperature, and acts to equalize temperature differences. Conduction takes place in all forms of matter, viz. solids, liquids, gases and plasmas, but does not require any bulk motion of matter. In solids, it is due to the combination of vibrations of the molecules in a lattice and the energy transport by free electrons. In gases and liquids, conduction is due to the collisions and diffusion of the molecules during their random motion.

Heat can also be transferred by radiation and/or convection, and often more than one of these processes occur in a given situation.

Fourier's law

The law of Heat Conduction, also known as Fourier's law, states that the time rate of heat transfer through a material is proportional to the negative gradient in the temperature and to the area at right angles, to that gradient, through which the heat is flowing. We can state this law in two equivalent forms: the integral form, in which we look at the amount of energy flowing into or out of a body as a whole, and the differential form, in which we look at the flows or fluxes of energy locally.

Differential form

The differential form of Fourier's Law of thermal conduction shows that the local heat flux, ${\overrightarrow {q}}$, is proportional to the thermal conductivity, $k$, times the negative local temperature gradient, $-\nabla T$. The heat flux is the amount of energy that flows through a particular surface per unit area per unit time.

${\overrightarrow {q}}=-k{\nabla }T$

where (including the SI units)

${\overrightarrow {q}}$ is the local heat flux, [W·m⁻²]

${\big .}k{\big .}$ is the material's conductivity, [W·m⁻¹·K⁻¹],

${\big .}\nabla T{\big .}$ is the temperature gradient, [K·m⁻¹].

The thermal conductivity, $k$, is often treated as a constant, though this is not always true. While the thermal conductivity of a material generally varies with temperature, the variation can be small over a significant range of temperatures for some common materials. In anisotropic materials, the thermal conductivity typically varies with orientation; in this case $k$ is represented by a second-order tensor. In nonuniform materials, $k$ varies with spatial location

For many simple applications, Fourier's law is used in its one-dimensional form. In the x-direction,

$q_{x}=-k{\frac {dT}{dx}}$

Integral form

By integrating the differential form over the material's total surface $S$, we arrive at the integral form of Fourier's law:

${\frac {\partial Q}{\partial t}}=-k\oint _{S}{{\overrightarrow {\nabla }}T\cdot \,{\overrightarrow {dS}}}$

where (including the SI units)

${\big .}{\frac {\partial Q}{\partial t}}{\big .}$ is the amount of heat transferred per unit time [W] and

${\overrightarrow {dS}}$ is an oriented surface area element [m²]

The above differential equation, when integrated for a homogeneous material of 1-D geometry between two endpoints at constant temperature, gives the heat flow rate as:

${\big .}{\frac {\Delta Q}{\Delta t}}=-kA{\frac {\Delta T}{\Delta x}}$

where

A is the cross-sectional surface area,

$\Delta T$ is the temperature difference between the ends,

$\Delta x$ is the distance between the ends.

This law forms the basis for the derivation of the heat equation. R-value is the unit for heat resistance, the reciprocal of the conductance. Ohm's law is the electrical analogue of Fourier's law.

Conductance

Writing

${\big .}U={\frac {k}{\Delta x}},\quad$

where U is the conductance, in W/(m² K).

Fourier's law can also be stated as:

${\big .}{\frac {\Delta Q}{\Delta t}}=UA\,(-\Delta T).$

The reciprocal of conductance is resistance, R, given by:

${\big .}R={\frac {1}{U}}={\frac {\Delta x}{k}}={\frac {A\,(-\Delta T)}{\frac {\Delta Q}{\Delta t}}},\quad$

and it is resistance which is additive when several conducting layers lie between the hot and cool regions, because A and Q are the same for all layers. In a multilayer partition, the total conductance is related to the conductance of its layers by:

${\big .}{\frac {1}{U}}={\frac {1}{U_{1}}}+{\frac {1}{U_{2}}}+{\frac {1}{U_{3}}}+\cdots$

So, when dealing with a multilayer partition, the following formula is usually used:

${\big .}{\frac {\Delta Q}{\Delta t}}={\frac {A\,(-\Delta T)}{{\frac {\Delta x_{1}}{k_{1}}}+{\frac {\Delta x_{2}}{k_{2}}}+{\frac {\Delta x_{3}}{k_{3}}}+\cdots }}.$

When heat is being conducted from one fluid to another through a barrier, it is sometimes important to consider the conductance of the thin film of fluid which remains stationary next to the barrier. This thin film of fluid is difficult to quantify, its characteristics depending upon complex conditions of turbulence and viscosity, but when dealing with thin high-conductance barriers it can sometimes be quite significant.

Intensive-property representation

The previous conductance equations written in terms of extensive properties, can be reformulated in terms of intensive properties.

Ideally, the formulae for conductance should produce a quantity with dimensions independent of distance, like Ohm's Law for electrical resistance: $R=V/I\,\!$, and conductance: $G=I/V\,\!$.

From the electrical formula: $R=\rho x/A\,\!$, where ρ is resistivity, x = length, A cross sectional area, we have $G=kA/x\,\!$, where G is conductance, k is conductivity, x = length, A = cross sectional area.

For Heat,

${\big .}U={\frac {kA}{\Delta x}},\quad$

where U is the conductance.

Fourier's law can also be stated as:

${\big .}Q=U\,\Delta T\quad$

analogous to Ohm's law: $I=V/R\,\!$ or $I=VG.\,\!$

The reciprocal of conductance is resistance, R, given by:

${\big .}R={\frac {\,\Delta T}{Q}},\quad$

analogous to Ohm's law: $R=V/I.\,\!$

The sum of conductances in series is still correct.

Cylinders

Conduction through cylinders can be calculated when variables such as the internal radius $r_{1}$, the external radius $r_{2}$, and the length denoted as $\ell$.

The temperature difference between the inner and outer wall can be expressed as $T_{2}-T_{1}$.

The area of the heat flow: $A_{r}=2\pi r\ell$

When Fourier’s equation is applied:

$Q=-kA_{r}{\frac {\mathrm {d} T}{\mathrm {d} r}}=-2k\pi r\ell {\frac {\mathrm {d} T}{\mathrm {d} r}}$

Rearranged:

$Q\int _{r_{1}}^{r_{2}}{\frac {1}{r}}\mathrm {d} r=-2k\pi L\int _{r_{1}}^{r_{2}}\mathrm {d} T$

Therefore the rate of heat transfer is

$Q=2k\pi \ell {\frac {T_{1}-T_{2}}{\ln r_{2}-\ln r_{1}}}$

The thermal resistance is

$R_{c}={\frac {\Delta T}{Q}}={\frac {\ln r_{2}-\ln r_{1}}{2\pi k\ell }}$

And $Q=2\pi k\ell r_{m}{\frac {T_{1}-T_{2}}{r_{2}-r_{1}}}$, where $r_{m}={\frac {r_{2}-r_{1}}{\ln r_{2}-\ln r_{1}}}$ and it is important to note that this is the log-mean radius.

See also

• Heat
• Thermal conductivity
• Heat flux
• Heat transfer
  • Convection
  • Convection diffusion equation
  • Thermal radiation
  • Thermal contact conductance
• Heat pipe
• Fick's law of diffusion
• Relativistic heat conduction
• Churchill-Bernstein Equation

References

• Dehghani, F 2007, CHNG2801 – Conservation and Transport Processes: Course Notes, University of Sydney, Sydney

External links

• Newton's Law of Cooling by Jeff Bryant based on a program by Stephen Wolfram, Wolfram Demonstrations Project.
• When Will My Turkey Be Done?is an example of applied heat conduction equations similar to Newton's Law of Cooling which predict the cooking time of turkeys and other roasts.