Heat Transfer Handbook part 14 pps

2.3.2 Behavior of Thermophysical Properties of Solids The thermal conductivity and specific heat are defined above as continuum properties.. As the temperature approaches absolute zero, bo

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ρm c p dT

Grouping the material properties, the thermal diffusivity is defined asαD = λ/ρ m c p.

Thus, the important thermophysical properties are αD , λ, ρ m, and c p In general,

these properties can be functions of direction, deformation, and temperature Some crystalline elements, such as carbon, bismuth, and tin, have anisotropic thermal con-ductivities Some polymers develop anisotropy after finite deformation (Choy et al., 1978; Broerman et al., 1999; Ortt et al., 2000) The temperature dependence ofα

is sometimes less strong than that ofλ, which can simplify analytical solutions of

conduction problems (Ozisik, 1980) For analyses oftransient heat transfer,αDis the

important parameter, while for analyses of steady heat transfer and boundary condi-tions oftransient analyses,λ is required

Equation (2.58) is nonphysical because it predicts an infinite speed ofpropagation oftemperature change, that is, a temperature change in one part ofthe body causes

an immediate change in temperature throughout the body Substituting the Maxwell–

Cattaneo equation for Fourier’s equation yields

1

αD

dT

1

wherec has dimensions ofvelocity Ifc is approximated as the speed ofsound in the

body, then for good conductors the ratio of the coefficients isαD /c = 10−11s (Parrott

and Stuckes, 1975) Thus, except in rare circumstances, finite propagation speeds are important only for very short times Joseph and Preziosi (1989, 1990) provide an extensive review ofstudies examining Maxwell–Cattaneo conduction

In general, analyses ofthermal conduction assume that materials are rigid and incompressible This is not strictly so Relaxing this assumption requires the addition ofthe simultaneous solution ofthe balance oflinear momentum and the addition of the stress work term (see, e.g., Day, 1985) The linear coefficient of thermal expansion may be defined asµ = (L − L0)/ [L0(T − T0)], where L is the length ofthe solid

at its new temperatureT , and L0 is its length at the reference temperatureT0 At room temperature,µ typically ranges from 0.6 × 10−6°C−1 for silicon carbide to

500× 10−6°C−1 for rubber (Brown, 1967) The volumetric coefficient of thermal

expansionµv = (1 + µ)3≈ (1 + 3µ) relates the specific volume at T to the specific

volume atT0

2.3.2 Behavior of Thermophysical Properties of Solids

The thermal conductivity and specific heat are defined above as continuum properties

Some indication oftheir behavior may be found by considering the molecular nature ofmaterials In solids, thermal transport properties result from molecular vibrations, and in electrical conductors, from electron transport The vibrations of the molecules

in a crystal lattice may be analyzed as harmonic oscillators and quantized (see, e.g.,

Reif, 1965 or Brown, 1967) These quanta are called phonons, and heat conduction

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can be pictured as the diffusion of phonons from a hotter region to a colder one

Transport ofelectrons dominates that ofphonons in metals at moderate temperatures, which explains why their thermal conductivities are typically larger than those of dielectrics Typical values ofthermal conductivity for metallic elements at 300 K range from 23 W/m· K for zirconium to 427 W/m · K for silver, whereas typical

values for dielectrics run from 0.12 W/m· K for paper to almost 3.0 W/m · K f or

granite and marble (Baehr and Stephan, 1998) Diamond is anomalous in that it has

a thermal conductivity as high as 2310 W/m· K at 300 K (Ho et al., 1974)

As the temperature approaches absolute zero, both the thermal conductivity and the specific heat tend toward zero, in accordance with the third law ofthermodynam-ics In dielectrics, the change in conductivity decreases as 1/T3, while in metals it decreases as 1/T , owing to the transport ofelectrons As the material warms, the

conductivity usually reaches a maximum and then decreases with increasing temper-ature The trend at higher temperatures is not universal, however, and the thermal conductivity may still increase with temperature for some materials that melt or de-compose before the maximum is reached The molar specific heat of many simple materials reaches approximately 3R at moderate temperatures, in accordance with the observations ofDulong and Petit (see, e.g., Brown, 1967; Reif, 1965) Modeling ofmaterials such as glasses and amorphous polymers is less complete (Kittel, 1996)

Dashora (1994) examined the temperature dependence ofthe thermal diffusivity of elastomers, and Eiermann (1966) discussed a resistive network model ofheat con-duction in amorphous polymers

Some material systems, such as heterogeneous polymers (Bigg, 1995), biological tissues (Chato, 1985), and composite materials (Dowding et al., 1996) have been modeled using apparent thermal properties In fact, these material systems are com-posed ofmaterials with different densities, thermal conductivities, and specific heats

Prediction ofthe performance ofmaterial systems that include continuous fibers of dispersed particles or voids within a matrix material is difficult Measurement of ap-parent properties may then be more expedient and accurate for a given system Cooper and Trezek (1971), for example, proposed correlations for the apparent conductivity

of biological soft tissue as functions of the mass fractions of water, protein, and fat

Even so, tabulated and correlated values ofthe thermophysical properties ofbiolog-ical materials must be used with caution because ofpotential anisotropy, specimen-to-specimen variation, and changes due to denaturation ofprotein during heating

2.3.3 Property Values of Solid Materials

Unlike gases and liquids, there is no standard, reference-quality computer package for the calculation ofthermophysical properties ofsolids Perhaps the most extensive compilation of solid properties comes from the Center for Information and Numerical Data Synthesis and Analysis (CINDAS) at Purdue University, which was established

by Yeram S Touloukian (1981) as the Thermophysical Properties Research Center (TPRC) The reference materials produced by this group have been used extensively

as a resource for this part ofthe chapter Tables oftypical values for metallic alloys and nonmetallic solids have been taken from the introductory text by Bejan (1993)

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Tables 2.8, 2.9, and 2.10 list the properties ofsolids that have been grouped into categories ofsolid elements, solid metallic alloys, and miscellaneous nonmetallic solids Unlike those offluids, the properties ofsolids vary little with changes in pressure, and thus the tables in this section neglect the effects of pressure on solid property values

The thermal conductivities ofelements are sensitive to even minute levels ofim-purity, especially at low temperatures, although the purity ofthe samples measured

is often omitted in reports of measurements Even near room temperature, values quoted by different sources may vary by 15% or more The values of thermal con-ductivity for all elements recommended by Ho et al (1974), listed in Table 2.8, were derived from critical and comprehensive evaluations by CINDAS of the data available

in the literature up to the early 1970s Accuracy for the thermal conductivity values may vary from 2 to 20%, depending on the purity and temperature levels The reader should refer to the original work for detailed information if requirements for accuracy are high

2.3.4 Measuring Thermophysical Properties of Solids

Measuring thermal conductivity and thermal diffusivity requires the development ofexperimental approximations ofboundary value problems Carslaw and Jaeger (1959) provide analytical solutions to many classical boundary value problems that have been useful in the measurement of thermal transport properties Reviews of methods for measuring thermal transport properties may be found in Maglic et al

(1984, 1992), Shirtliffe and Tye (1985), Parrott and Stuckes (1975), and Jakob (1955)

The American Society for Testing and Materials (ASTM) has established several standards for measuring transport properties, and the National Institute of Standards and Technology (NIST) can provide some standard reference materials (SRMs) As noted above, sources oferror include impurities in the sample and variables omitted from the analysis, such as deformation of polymers (Choy et al., 1978; Greig and Sahota, 1978; Doss and Wright, 2000) New methods ofmeasurement are constantly being developed to overcome such sources oferror, and the results are published as appropriate

Thermal Conductivity Direct measurement ofthermal conductivity has tradi-tionally used steady-state methods For materials ofmoderate to high thermal con-ductivity (∼10 to 500 W/m · K), axial heat flow, radial heat flow, and direct electrical

heating methods are often used (Maglic et al., 1984) Materials of lower thermal con-ductivity are most commonly tested in using the guarded hot plate (thermal insulation materials) or hot wire methods; the latter is a transient method These methods can provide high accuracy and simple data reduction but require a relatively long time to reach steady state This reduces their suitability for measuring properties of a material that may change during measurement, such as biological tissues Thermal conductiv-ity is sometimes determined via indirect methods by measuring the diffusivconductiv-ity and using the density and specific heat to calculate the conductivity The 3ω technique

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TABLE 2.8 Thermophysical Properties of Solid Elementsa

Aluminum (Al)

Antimony (Sb)

Beryllium (Be)

Bismuth (Bi)

(continued)

TABLE 2.8 Thermophysical Properties of Solid Elementsa (Continued)

Bismuth (Bi) (Continued)

Boron (B)

Cadmium (Cd)

Calcium (Ca)

Carbon (C)

Cerium (Ce)

Cesium (Cs)

Chromium (Cr)

(continued)

TABLE 2.8 Thermophysical Properties of Solid Elementsa (Continued)

Cobalt (Co)

Copper (Cu)

Gold (Au)

Iron (Fe)

Lead (Pb)

Lithium (Li)

Magnesium (Mg)

Manganese (Mn)

(continued)

TABLE 2.8 Thermophysical Properties of Solid Elementsa (Continued)

Nickel (Ni)

Platinum (Pt)

Potassium (K)

Silicon (Si)

Silver (Ag)

Sodium (Na)

Tantalum (Ta)

Tin (Sn)

(continued)