And the total conductivity the sum the two contributions

METALS

As noted in Table 17.1, linear coefficients of thermal expansion for some of the common metals range between about 5 � 10�6and 25 � 10�6(�C)�1. For some applications, a high degree of dimensional stability with temperature fluctuations is essential. This has resulted in the development of a family of iron-nickel and iron-nickel-cobalt alloys that have � l values on the order of 1 � 10�6(�C)�1. One such alloy, tradename of Kovar (Table 17.1), has been designed to have expansion characteristics close to those of borosilicate (or Pyrex) glass; when joined to Pyrex and subjected to temperature variations, thermal stresses and possible fracture at the junction are avoided. Kovar and two other low-expansion alloys (Invar and Super-Invar) which have small � l values are also included in Table 17.1.

Some polymeric materials experience very large thermal expansions upon heating as indicated by coefficients that range from approximately 50 � 10�6to 400 � 10�6 (�C)�1. The highest � l values are found in linear and branched polymers because the secondary intermolecular bonds are weak, and there is a minimum of crosslink-ing. With increased crosslinking, the magnitude of the expansion coefficient dimin-ishes; the lowest coefficients are found in the thermosetting network polymers such as phenol-formaldehyde, in which the bonding is almost entirely covalent.

17.4 THERMAL CONDUCTIVITY

The units of q and k are W/m2(Btu/ft2-h) and W/m-K (Btu/ft-h-�F), respec-tively. Equation 17.5 is valid only for steady-state heat flow, that is, for situations in which the heat flux does not change with time. Also, the minus sign in the expression indicates that the direction of heat flow is from hot to cold, or down the temperature gradient.

where k l and k e represent the lattice vibration and electron thermal conductivities, respectively; usually one or the other predominates. The thermal energy associated with phonons or lattice waves is transported in the direction of their motion. The k l contribution results from a net movement of phonons from high- to low-temperature regions of a body across which a temperature gradient exists.

Free or conducting electrons participate in electronic thermal conduction.To the free electrons in a hot region of the specimen is imparted a gain in kinetic energy. They then migrate to colder areas, where some of this kinetic energy is transferred to the atoms themselves (as vibrational energy) as a consequence of collisions with phonons or other imperfections in the crystal. The relative contribu-tion of k e to the total thermal conductivity increases with increasing free electron concentrations, since more electrons are available to participate in this heat transfer-ence process.

where � is the electrical conductivity, T is the absolute temperature, and L is a constant. The theoretical value of L, 2.44 � 10�8�-W/(K)2, should be independent of temperature and the same for all metals if the heat energy is transported entirely by free electrons. Included in Table 17.1 are the experimental L values for these several metals; note that the agreement between these and the theoretical value is quite reasonable (well within a factor of 2).