# Basics of heat conduction

Heat conduction is the process of heat transfer across a stationary medium, either solid or fluid. The distinction between conduction and convection is that there is no bulk motion of heat transfer media in conduction. Heat conduction is an atomic or molecular activity that transfers thermal energy from a region with higher temperature to a region with lower temperature. The mechanism of heat conduction is different for different substances. For an electrically nonconducting solid, conduction is attributed to atomic activity in the form of lattice vibrational waves (or phonons). The mechanism of conduction in an electrically conducting solid is a combination of lattice vibration and translational motion of electrons. Heat conduction in a liquid or gas is due to the random motion and interaction of the molecules.

Since heat conduction always occurs in an incompressible substance and the velocity is zero in the conducting media, the energy equation (2.92) becomes

$\rho {c_p}\frac{{\partial T}}{{\partial t}} = - \nabla \cdot {\mathbf{q''}} + q'''\qquad \qquad(1)$

where q''' is the internal heat source (W / m3) that can be caused by electrical heating, chemical reaction heat, blood perfusion (for bioheat transfer), or coupling between electron and phonon (for ultrafast heat transfer in metal). For most engineering problems, the heat transfer rate is related to the temperature gradient by Fourier’s law:

${\mathbf{q''}} = - {\mathbf{k}} \cdot \nabla T \qquad \qquad(2)$

where the negative sign on the right-hand side means that the heat flux vector, ${\mathbf{q''}}$, is in the opposite direction of the temperature gradient, $\nabla T$. For anisotropic material, the thermal conductivity is a tensor of the second order that has nine components:

${\mathbf{k}} = \left[ {\begin{array}{*{20}{c}} {{k_{xx}}} & {{k_{xy}}} & {{k_{xz}}} \\ {{k_{yx}}} & {{k_{yy}}} & {{k_{yz}}} \\ {{k_{zx}}} & {{k_{zy}}} & {{k_{zz}}} \\ \end{array}} \right] \qquad \qquad(3)$

where kxy = kyx,kxz = kzx,andkyz = kzy according to the reciprocity relation derived from the Onsagar’s principle of thermodynamics of irreversible processes (Ozisk, 1993). The dot product of a second order tensor, k, and a vector, $\nabla T$, results in a vector, ${\mathbf{q''}}$. The energy equation for an anisotropic material can be obtained by substituting eq. (2) into eq. (1), i.e.,

$\rho {c_p}\frac{{\partial T}}{{\partial t}} = \nabla \cdot ({\mathbf{k}} \cdot \nabla T) + q''' \qquad \qquad(4)$

In a Cartesian coordinate system, eq. (4) becomes

$\rho {c_p}\frac{{\partial T}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {{k_{xx}}\frac{{\partial T}}{{\partial x}} + {k_{xy}}\frac{{\partial T}}{{\partial y}} + {k_{xz}}\frac{{\partial T}}{{\partial z}}} \right) + \frac{\partial }{{\partial y}}\left( {{k_{yx}}\frac{{\partial T}}{{\partial x}} + {k_{yy}}\frac{{\partial T}}{{\partial y}} + {k_{yz}}\frac{{\partial T}}{{\partial z}}} \right)$

${\rm{ + }}\frac{\partial }{{\partial z}}\left( {{k_{zx}}\frac{{\partial T}}{{\partial x}} + {k_{zy}}\frac{{\partial T}}{{\partial y}} + {k_{zz}}\frac{{\partial T}}{{\partial z}}} \right) + q''' \qquad \qquad(5)$

For the case that all non-diagonal components in eq. (3) are absent, the anisotropic material becomes orthotropic material and the energy equation for heat conduction becomes

$\rho {c_p}\frac{{\partial T}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {{k_{xx}}\frac{{\partial T}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {{k_{yy}}\frac{{\partial T}}{{\partial y}}} \right){\rm{ + }}\frac{\partial }{{\partial z}}\left( {{k_{zz}}\frac{{\partial T}}{{\partial z}}} \right) + q''' \qquad \qquad(6)$

Heat conduction in an orthotropic material is much easier to compute than that for the anisotropic material because cross-derivatives of the space variables are absent. If the thermal conductivities in the three directions are the same (kxx = kyy = kzz = k), the material becomes isotropic and the thermal conductivity becomes a scalar. Fourier’s law for an isotropic material is

${\mathbf{q''}} = - k\nabla T \qquad \qquad(7)$

and the energy equation (1) and (6) are simplified as

$\rho {c_p}\frac{{\partial T}}{{\partial t}} = \nabla \cdot (k\nabla T) + q''' \qquad \qquad(8)$
$\rho {c_p}\frac{{\partial T}}{{\partial t}} = \frac{\partial }{{\partial x}}\left( {k\frac{{\partial T}}{{\partial x}}} \right) + \frac{\partial }{{\partial y}}\left( {k\frac{{\partial T}}{{\partial y}}} \right){\rm{ + }}\frac{\partial }{{\partial z}}\left( {k\frac{{\partial T}}{{\partial z}}} \right) + q''' \qquad \qquad(9)$
Typical range of thermal conductivities

While eqs. (5), (6) and (9) are given in the Cartesian coordinate system, the corresponding equations in the cylindrical and spherical coordinate systems can be obtained by using the $\nabla$ operator in these coordinate systems.

Thermal conductivity is a very important thermophysical property of a material because the heat flux increases with increasing thermal conductivity under a prescribed temperature gradient. Since the interaction between molecules in the solid is strongest among the three phases, the thermal conductivity for the solid is usually higher than that in the liquid. For the same reason, the thermal conductivity for the gas is usually lowest among the three phases. The thermal conductivity for the metal is higher than that for a non-conducting solid because of the additional carrier – free electrons – of heat in metal. The typical ranges of thermal conductivities for various materials are shown in the figure on the right. It can be seen that the thermal conductivities for different materials can cross four orders of magnitude. Thermal conductivity is usually a function of temperature. The thermal conductivity for metal increases with increasing temperature, but it decreases with increasing temperature for gases.

The energy equation or the boundary condition is said to be linear if there are no products of unknown variable (temperature) or its derivative in the energy equation or boundary condition. For example, eq. (8) or (9) is nonlinear because product of ka function of temperature – and the derivative of temperature appeared. When the thermal conductivity is independent from temperature, eq. (8) and (9) become

$\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} = {\nabla ^2}T + \frac{{q'''}}{k} \qquad \qquad(10)$
$\frac{1}{\alpha }\frac{{\partial T}}{{\partial t}} = \frac{{{\partial ^2}T}}{{\partial {x^2}}} + \frac{{{\partial ^2}T}}{{\partial {y^2}}} + \frac{{{\partial ^2}T}}{{\partial {z^2}}} + \frac{{q'''}}{k} \qquad \qquad(11)$

where

$\alpha = \frac{k}{{\rho {c_p}}} \qquad \qquad(12)$

is the thermal diffusivity – another thermophysical property. Equations (10) and (11) are linear, because they do not contain products of T or its derivative.

Furthermore, a linear energy equation or boundary condition is said to be homogeneous if the temperature T can be replaced by CT, where C is a nonzero constant, and the equation is still satisfied. Thus, eq. (10) or (11) is non-homogeneous due to the presence of the internal heat generation term. They become linear homogeneous when there is no internal heat generation.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Ozisik, M.N., 1993, Heat Conduction, 2nd ed., Wiley-Interscience, New York.