Thermal conductivity, 1.18

Geothermal Research Institute of the Dagestan Scientific Center of the Russian Acad ny of Sciences, Thermophysical Division, Makhachkala 367030, 

Chemical Engineering Depariment, Aristotle University, Thessaloniki 54124, Greece 

The thermal conductivity of a fluid measiues its propensity to dissipate energy, when disturbed from equiUbrimn by the imposition of a temperature gradient. For isotropic fluids the thermal conductivity, X, is defined by Fourier s law 

Pure water is one of the primary standard thermal conductivity reference liquids. The thermal conductivity value proposed by the Subcommittee on Transport Properties of the International Union for Pure and Applied Chemistry (lUPAC) at 298.15 Kand 0.101 325 MPa is (Nieto de Castro et al, 1986) 

The value of the thermal conductivity recommended by the International Association for the Properties of Water and Steam (lAPWS) is A = 0.6072 0.009 W m K (Kestin et al, 1984), which is in full agreement with the above lUPAC value. The temperature dependence of the thermal conductivity of water at atmospheric pressure in the temperature range 274 to 360 K, is given by the following recommended correlation (Nieto de Castro et al, 1986) 

The thermal conductivities of the SOFC components are all in the order of 1.5-2 W/(mK) [3]. These are low compared to 20 W/(mK) for stainless steel and 400 W/(mK) for copper. Therefore, heat dissipation could be a problem which must be considered in the SOFC design. This is particularly true for higher power density monolithic and planar SOFCs, but is less of a problem in the tubular design. 

Thermal conductivity is the rate of transfer of heat by conduction through a unit area across a unit thickness for a unit difference in temperature  

Thermal conductivity, as already stated, is the quantity of heat passing per unit time normally through unit area of a material of unit thickness for unit temperature difference between the faces. 

In the steady state, when the temperature at any point is constant with time, this parameter controls heat transfer as follows  

The thermal conductivity of solid rubbers is of the order of 0.1-0.2 W/m. deg, which is in the region of fairly low value where experimental errors due to heat loss are the greatest. 

FIGURE 9.13 Variation of specific heat with volatile matter yield. (From Baughman, G.L., Synthetic Fuels Data Handbook, 2nd edn., Cameron Engineers, Inc., Denver, CO, 1978.) 

However, the banding and bedding planes in coal (Chapters 1 and 4) can complicate the matter to such an extent that it is difficult, if not almost impossible, to determine a single value for the thermal conductivity of a particular coal. Nevertheless, it has been possible to draw certain conclusions from the data available. Thus, monolithic coal is considered to be a medium conductor of heat with the thermal conductivity of anthracite being on the order of 5-9 x 10 kcal/s -cm °C while the thermal conductivity of monolithic bituminous coal falls in the range of 4-7 x 10 kcal/s cm °C. For example, the thermal conductivity of pulverized coal is lower than that of the corresponding monolithic coal. For example, the thermal conductivity of pulverized bituminous coal falls into the range of 2.5-3.5 x 10 kcal/s cm °C. 

The thermal conductivity of coal generally increases with an increase in the apparent density of the coal as well as with volatile matter yield, ash yield, and temperature. In addition, the thermal conductivity of the coal parallel to the bedding plane appears to be higher than the thermal conductivity perpendicular to the bedding plane. 

There is little information about the influence of water on the thermal conductivity of coal, but since the thermal conductivity of water is markedly higher than that of coal (about three times) the thermal conductivity of coal could be expected to increase if water is present in the coal. 

Thermal conductivity (K) and thermal diffusivity (/c) measurements versus temperature or blend composition can be employed to reveal structural information. However, it is not as sensitive as other methods and relatively few studies have been reported on blends. Thermal diffusitivies of polymers are generally in the range of 10 cm /s. A relevant review of thermal conductivity of polymer blends has been reported by Tsutsumi [197], where PVME/PS, PVC/PCL, PMMA/PC and PVF2/PMMA blend data were reviewed. The thermal conductivity, K, and thermal diffusivity, K, have analogies with permeability, P, and diffusion coefficient, D, respectively. This analogy is the result of the similarities between Fourier s law and Fick s law  

The formalism employed for permeabihty relationships in the this section, such as the series model, parallel model. Maxwell s equation, and the equivalent box model (EBM) can be employed for thermal conductivity by replacing P with K. 

Thermal conductivity is one of the most demanding thermophysical properties but difficult to obtain experimentally, because on Earth thermogravitational convection exerts a major effect on heat transfer and it is almost impossible to suppress this effect. There are four methods to obtain the thermal conductivity of molten silicon. Historically, thermal conductivity has been estimated from the measurement of electrical conductivity x and applying the Wiedemann-Franz law, as shown in Eq. (4.2) [5, 8, 52, 53]. Thermal diffusivity was measured also by a laser flash method [7, 24, 54] and is converted into thermal conductivity using density p and mass heat capacity Although transient hot-wire and hot-disk methods assure direct 

Application of noncontact AC calorimetry using electromagnetic levitation under a strong DC magnetic field enables direct measurement of thermal conductivity melt fiow is suppressed and heat transfer is controlled only by conduction within a droplet, so that a droplet behaves as if it were solid [35, 36]. Furthermore, measurements are free from the sensor insulation coating. The thermal conductivity can be obtained from the ( )-co relation simultaneously with total hemispherical emissivity st, as described in Section 4.3. 

The thermal conductivity detector (TCD) operates on the principle that gases eluting from the column have thermal conductivities different from that of the carrier gas, which is usually helium. Present in the flow channel at the end of the column is a hot filament, hot because it has an electrical current passing through it. This filament is cooled to an equilibrium temperature by the flowing helium, but it is cooled differently by the mixture components as they elute, since their thermal conductivities are different from 

A flow-modulated design of the TCD has become popular. In this design, a single filament is used and the column effluent is alternated with the pure helium through the flow channel where the filament is located. This eliminates the need to use two matched filaments. 

The thermal conductivity detector is universal (detects everything) and nondestructive (can be used with preparative GC), but it does not detect very small concentrations, compared to other detectors. 

The thermal conductivity of cellular polymers has been thoroughly studied in heterogeneous materials [33,34] and plastic foams [25,35-38]. 

Heat transfer can be separated into its component parts as follows  

Compressive modulus Shear modulus Flexural modulus 

Most data on short columns Limited data 

Empirical, uncontrolled skin thickness, low foam density Law of mixtures, thick skins, high foam density Tentative, few data from many test procedures Very tentative, rule-of-thumb Very tentative, rule-of-thumb 

The thermal conductivity, X, of phosgene vapour has been calculated at standard pressure. At 15, 50 and 1(X) C the thermal conductivities are 9.13 x 10 3 i.07 x lO , and 1.29 x lO w m K , respectively [1198]. Measurement of the thermal conductivity of phosgene vapour at 0 C, using a calibrated hot wire apparatus, gave the value (7.61 0.11) X 10 3 w m K [ICI104], but is not consistent with the data given above when extrapolated to the lower temperature. Estimated values of X have been extended from 0 to 500 C corresponding to an increase from 8.0 x 10 to 4.6 x 10 3 w m K [715]. 

The thermal conductivity of adherends influences the temperature conditions in the glueline during adhesive curing. It plays a special role in the application of hot-melt adhesives on metals due to the quick solidification of the melt in the boundary layer zone and the possible impairment of the adhesion development. The thermal conductivity A is indicated in the dimension W/cm K (Watt per centimeter Kelvin). Values of certain materials  

The thermal conductivity is the parameter in Fourier s law that relates the flow of heat to the temperature gradient. Fourier s law is 

In specific terms, the thermal conductivity is a measure of a material s ability to conduct heat. The thermal conductivity of a polymer is the amount of heat conducted through a unit thickness of a material per unit of area and time, with a 

The thermal conductivity, n, of a substance is defined as the rate of heat transfer by conduction across a unit area, through a layer of unit thickness, under the influence of a unit temperature difference, the direction of heat transmission being normal to the reference area. Fourier s equation for steady conduction may be written as 

The SI unit for thermal conductivity is W m K although other units such as cal s cm °C and Btu h ft °F are still commonly encoxmtered. The conversion factors are  

The thermal conductivity of a crystalline solid can vary considerably according to the crystallographic direction, but very few directional values are available in the literature. Some overall values (Wm K ) for polycrystalline or non-crystalline substances include KCl (9.0), NaCl (7.5), KBr (3.8), NaBr (2.5), MgS04 7H20(2.5), ice (2.2), K2Cr20v(1.9), borosilicate glass (1.0), soda glass (0.7) and chalk (0.7). 

Temperature, C Water Thermal conductivity, k (Wm Acetone Benzene Methanol K-i) Ethanol CCL, 

An increase in the temperature of a liquid usually results in a slight decrease in thermal conductivity, but water is a notable exception to this generalization. Furthermore, water has a particularly high thermal conductivity compared with other pure liquids Table 2.6). There are relatively few values of thermal conductivity for solutions recorded in the literature and regrettably they are frequently conflicting. There is no generally reliable method of estimation. 

The thermal conductivity of a material is essentially a proportionality constant between the conductive heat flux and the temperature gradient driving the heat flux. The thermal conductivity of polymers is quite low, about two to three orders of magnitude lower than most metals. From a processing point of view, the low thermal conductivity creates some real problems. It very much limits the rate at which polymers can be heated and plasticated. In cooling, the low thermal conductivity can cause non-uniform cooling and shrinking. This can result in frozen-in stresses, deformation of the extrudate, delamination, shrink voids, etc. 

The thermal conductivity of amorphous polymers is relatively insensitive to temperature. Below the Tg, the thermal conductivity increases slightly with temperature above the Tg, it reduces slowly with temperature. The thermal conductivity above the Tg as a function of temperature can be approximated by [41]  

In most extrusion problems, however, the thermal conductivity of an amorphous polymer can be assumed to be independent of temperature. 

The thermal conductivity of semi-crystalline polymers is generally higher than amorphous polymers. Below the crystalline melting point, the thermal conductivity reduces with temperature above the melting point, it remains relatively constant. The thermal conductivity increases with density and, thus, with the level of crystallinity. The thermal conductivity at constant temperature as a function of density can generally be written as  

The change in thermal conductivity with temperature is relatively linear at temperatures above 0°C. The thermal conductivity as a function of temperature can be described with  

The thermal conductivity or k (i.e., the time rate of transfer of heat by conduction) of interstitial carbides is different from that of most other refi actory materials as k increases with increasing temperature as shown in Fig. 4.2.l l Typically, the mechanism of thermal conductivity involves two components electron thermal conductivity and phonon (lattice) conductivity kp. As shown in Fig. 4.3 (in this case for titanium carbide), k increases markedly with temperature. This behavior is believed to be the 

As can be seen in Table 4.2, the thermal conductivities of the Group rv carbides, nitrides, and borides are relatively close. They are also similar to those of the host metals and, from this standpoint, reflect the metallic character of these compounds. However, their conductivities are much lower than that of the best conductors such as Type II diamond (2000 W/m-K), silver (420 W/m-K), copper (385 W/m K), beryllium oxide (260 W/m-K), and aluminum nitride (220 W/m-K). 

The thermal conductivity of coal is especially important for gasification modeling purposes. It generally ranges between 0.2 and 0.8 W/(m K) for the organic fraction and is proportional to 

The following empirical equation is proposed to evaluate the thermal conductivity of coal A in W/(m-K) from the ash fraction A (wt%(wf)/100), moisture fraction W (wt%(ar)/100), and temperature in °C [2,10]. 

In situ thermal conductivity measurements have been used to determine the extent of the equilibrium in reaction (62) 

The thermal properties of conductivity and expansion are strongly influenced by the anisotropy of the graphite crystal. The thermal conductivity (fQ is the time rate of transfer of heat by conduction. In graphite, it occurs essentially by lattice vibration and is represented by the following relationship (Debye equation)  

C = specific heat per unit volume of the crystal V = speed of heat-transporting acoustic wave (phonon) 

In a polycrystalline materials, the waves (phonon, i.e., quantum of thermal energy) are scattered by crystallite boundaries, lattice defects, and other phonons. Little of this occurs in a perfect or near-perfect graphite crystal in the basal plane and, as a result, the factor L is high and thermal conductivity is high in the ab directions. However, in the direction perpendicular to the basal plane (c direction), the conductivity is approximately 200 times lower since the amplitude of the lattice vibration in that direction is considerably lower than in the ab directions. These differences in vibration amplitude in the various crystallographic directions of graphite are shown in Fig. 

The thermal conductivity in the cdirection is approximately 2.0 W/m-K and, in that direction, graphite is a good thermal insulator, comparable to phenolic plastic. 

The thermal conductivity of graphite decreases with temperature as shown in Fig. 3.1In the Debye equation (Eq. 1), K is directly proportional to the mean free path, L, which in turn is inversely proportional to temperature due to the increase in vibration amplitude of the thermally excited carbon atoms. L becomes the dominant Victor above room temperature, more than offsetting the increase in specific heat, Cp, shown in Fig. 3.8. 

The treatment of thermal conductivity parallels quite closely the treatment given for viscosity. Let us suppose that we have two parallel plates (Fig. VIII.3) at fixed temperatures Ti and T2, with a fluid between them. 

There will be a transport of energy from the hotter plate T2 to the colder plate Tij again by a mechanism of molecular collision. We can define a coefficient of thermal conductivity 12 for the fluid as the ratio 

Following the method already used, we select a plane at a distance Z from one of the two parallel surfaces (Z = 0 at Ti, Z = d at T2, T2 Ti) and choose an element of area AS in it. If we assume that the temperature varies linearly through the gas (Newtonian heat flow), the temperature gradient is a constant equal to T2 — Ti)/d (Fig. VIII.3). 

At a steady state, the flow of molecules through the area AS must be the same in both directions, while the net perpendicular component of momentum transported across AS must be independent of the location of 

These two conditions cannot be satisfied simultaneously without choosing distribution functions for molecular density and velocity which are functions of Z. To avoid such a choice, which would lead to an almost impossibly complicated calculation, let us rather make a crude but simpler calculation which leads to results of the correct order of magnitude and is in qualitative agreement with experiment. 

In the case of the thermal-conductivity, there are three main techniques those based on Equation (1) and those based on a transient application of it. Prior to about 1975, two forms of steady-state technique dominated the field parallel-plate devices, in which the temperature difference between two parallel disks either side of a fluid is measured when heat is generated in one plate, and concentric cylinder devices that apply the same technique in an obviously different geometry. In both cases, early work ignored the effects of convection. In more recent work, exemplified by the careful work in Amsterdam with parallel plates, and in Paris with concentric cylinders, the effects of convection have been investigated. Indeed, the parallel-plate cells employed in Amsterdam by van den Berg and his co-workers have the unique feature that, because the temperature difference imposed can be very small and the horizontal fluid layer very thin, it is possible to approach the critical point in a fluid or fluid mixture very closely (mK). 

The improvement in thermal conductivity of SR nanocomposites could lead to its many promising industrial applications, e.g., circuit boards in power electronics, thermal greases, elastomeric thermal pads, and phase change materials [185,186]. Therefore, many attempts at SR nanocomposites with higher thermal conductivity have been developed [39,124,130,179]. 

The thermal conductivity of PDMS is increased by 58%, 102%, and 123% in presence of 1.5 vol % of unmodified MWNTs, diphenyl-carbinol-functionalized MWNT (D-MWNTs) and D-MWNTs functionalized by silane, respectively [124]. The higher thermal conductivity of silane-modified MWNT/PDMS composites is attributed to 

Nanocomposites with good thermal conductivity (k) have potential applications in printed circuit boards. The thermal conductivity of connectors is in the order of 103 W/(m-K), while typical thermoplastics have k 0.1 W/(m K). A theoretical model was proposed by Nan et al. to predict the thermal conductivity of CNT-based composites (75,76). Theoretically, the thermal conductivity of the composites filled with 0.1 wt% MWNTs can be six times that of neat polymers. 

Cai et al. proposed a new route to reduce the interfacial phonon scattering (27). Semicrystalline PU dispersions were used as latex host to accommodate MWNTs following the colloidal physics 

When analyzing thermal processes, the thermal conductivity, k, is the most commonly used property that helps quantify the transport of heat through a material. By definition, energy is transported proportionally to the speed of sound. Accordingly, thermal conductivity 

Due to the increase in density upon solidification of semi-crystalline thermoplastics, the thermal conductivity is higher in the solid state than in the melt. In the melt state, however, the thermal conductivity of semi-crystallinepolymers reduces to that of amorphous polymers as can be seen in Fig. 2.2 [40], 

Furthermore, it is not surprising that the thermal conductivity of melts increases with hydrostatic pressure. This effect is clearly shown in Fig. 2.3 [19]. As long as thermosets are unfilled, their thermal conductivity is very similar to amorphous thermoplastics. Anisotropy in thermoplastic polymers also plays a significant role in the thermal conductivity. Highly drawn semi-crystalline polymer samples can have a much higher thermal conductivity as a result of the orientation of the polymer chains in the direction of the draw. 

The higher thermal conductivity of inorganic fillers increases the thermal conductivity of filled polymers. Nevertheless, a sharp decrease in thermal conductivity around the melting temperature of crystalline polymers can still be seen with filled materials. The effect of filler on thermal conductivity for PE-LD is shown in Fig. 2.5 [22], This figure shows the effect of fiber orientation as well as the effect of quartz powder on the thermal conductivity of low density polyethylene. 

There are various models available to compute the thermal conductivity of foamed or filled plastics [39,47, 51]. A rule of mixtures, suggested by Knappe [39], commonly used 

It is often quoted that the thermal conductivity of SiC is higher than that of copper at room temperature. There are even claims that it is better than any metal at room temperature [7]. The thermal conductivity of copper is 4.0 W/(cm-K) [8]. That of silver is 4.18 W/(cm-K) [8]. Values of the thermal conductivity as high as 5 W/(cm-K) have been measured by Slack [9] on highly perfect Lely platelets. 

More detailed studies have been made where the thermal conductivity in the different crystal directions have been determined for SiC (see Table 1.1 [5]). As can be seen, there is a dependence on the purity of the crystal as well as on the crystal direction. At the time of this study, the material was not of the high quality we see today, and more sophisticated techniques have been developed to measure thermal conductivity. Thus, as higher-quality material has been grown, values close to the theoretical values have been measured using the laser flash technique [10]. 

High-purity semi-insulating (SI) SiC material has the highest reported thermal conductivity with a value of 4.9 W/(cm-K). Lower values are measured for the doped crystals but they are all above 4 W/(cm-K) at room temperature [10]. 

The electronic contribution to the thermal conductivity, which is negligible for 

27 Thermal conductivity of nanometer-, submicrometer-, icrometer-sized ZnO heated from room temperature to 

Because ZnO thin films deposited on foreign substrates are used in different applications, thermal conductivities of the substrates would be of concern when designing device structures. A comparison of thermal conductivities for various templates used for ZnO growth is provided in Table 2.3. 

The basic method of determining the thermal conductivity is to place a sample of known dimensions in a temperature gradient and measure the rate of the resulting heat flow through it. Suitable apparatus consists of a disc of ice cream sandwiched between two plates made from a material of known thermal conductivity and placed in an insulated cylinder. One plate is heated and the other cooled to produce the temperature gradient, which is measured with thermocouples embedded in the plates and sample. 

There are even less data available for the thermal conductivity of hydrogen sulfide than there are for the viscosity. To the best of the authoi s knowledge, the only measured data for the thermal 

The DIPPR Data Book (Daubert et al., 1999) gives the following correlation for the low-pressure thermal conductivity of F S  

The form of the DIPPR equation was retained, but new coefficients were fit to the data of Barua et al. (1968). The resulting equation is  

This equation is also plotted on figure 2A.3 and demonstrates the improvement in the prediction of the thermal conductivity. 

Mukhopadhay. 1968. Thermal conductivity and rotational relaxation in some polar gases. J. Chem. Phys. 49 2422-2424. Daubert, T.E., R.P. Danner, H.M. Sibul, C.C. Stebbins, R.L. Rowley, W.V. Wilding, J.L. Oscarson, M.E. Adams, and T.L. Marshall. 1999. Physical 

Transient hot-wire and eo-axial cylinder methods are typically applied for measurement of the thermal eonductivity of supercritical fluids.  

The relationships between the equation of state at low pressure and the virial coefficient are  

Wong and Sandler has shown that the following mixing rule does satisfy the second virial coefficient equation  

The pressure correction to the thermal conductivity for a pure component or mixture at low pressure is given by  

The vapor molar volume, ean be obtained from the equation of state models as de-seribed above. 

Equation (1-1) is the defining equation for thermal conductivity. On the basis of this definition, experimental measurements may be made to determine the thermal conductivity of different materials. For gases at moderately low temperatures, analytical treatments in the kinetic theory of gases may be used to predict accurately the experimentally observed values. In some cases, theories are available for the prediction of thermal conductivities in liquids and solids, but in general, many open questions and concepts still need clarification where liquids and solids are concerned. 

The mechanism of thermal conduction in a gas is a simple one. We identify the kinetic energy of a molecule with its temperature thus, in a high-temperature region, the molecules have higher velocities than in some lower-temper- 

The physical mechanism of thermal-energy conduction in liquids is qualitatively the same as in gases however, the situation is considerably more complex because the molecules are more closely spaced and molecular force fields exert a strong influence on the energy exchange in the collision process. Thermal conductivities of some typical liquids are shown in Fig. 1-5. 

In the English system of units heat flow is expressed in British thermal units per hour (Btu/h), area in square feet, and temperature in degrees Fahrenheit. Thermal conductivity will then have units of Btu/h ft °F. 

The thermal conductivities of various insulating materials are also given in Appendix A. Some typical values are 0.038 W/m °C for glass wool and 0.78 W/m °C for window glass. At high temperatures, the energy transfer through 

For polycrystalline samples values of the total thermal conductivity X at 293 K, of the electronic component X(el) derived by the Friedemann-Franz law under the assumption that the Lorenz number L = 2.45x 10 V/K as for a degenerate electron gas, and of the lattice component X(lat) obtained by subtraction are, Zhuze et al. [15]  

A number of nonmetallic materials are called high thermal conductivity materials. The most notable of these is diamond, with a thermal conductivity of 2000 Wm K. All of the others have a dia-mond-Uke stracture, and include boron nitride, BN, and aluminium nitride, AIN (Table 15.2). 

Thermal conductivity is attributed mainly to the mobile electrons present and the vibration waves in the structure, phonons  

Mobile electrons make the greatest contribution, and so metals would be expected to show a much higher thermal conductivity than insulators. At the simplest level the electrons can be imagined as a free-electron gas, moving with a velocity that is higher at the hot end of the solid than at the cold end. (The same model was mentioned in Section 2.3.5, with respect to electrical conductivity.) The kinetic energy is gradually transferred to the cold end by collisions between the electrons themselves and with the atoms in the structure. The thermal conductivity increases as the number of free electrons increases. The model is successful in some ways. For example, it predicts that thermal 

The thermal conductivity depends on the mean free path of the phonons, which is the distance between collisions of the phonons in the stmcmre. A short mean free path correlates with a low thermal conductivity. Defects in a stmcmre drastically shorten the mean free path and reduce thermal conductivity significantly. An inherent problem with ceramic materials is that they are usually formed by sintering. This process naturally leads to the formation of many internal defects such as grain boundaries, pores and voids. Because of this, the thermal conductivity of sintered bodies is usually much lower than the intrinsic thermal conductivity. Fired-clay ceramics have very high porosity and a very low thermal conductivity. At 

Point defects can drastically lower the thermal conductivity of the important carbide and nitride high thermal conductivity ceramics. In this respect, oxygen, which is a common impurity, has been found to be very important. For example, silica (Si02), an impurity in silicon nitride (Si3N4), formed by oxidation at high temperatures in air, can react to produce substitutional defects and vacancies in the following way  

According to Fourier law, the scalar coefficient of thermal conductivity k rel s the thermal flux density Q [in erg/cm s] to the gradient of temperature Q = —kVT [units of k erg/cm.s.K], The corresponding thermal diffusion coefficient [in cm /s] includes density of substance p and heat capacitance Cp (at constant pressure) 

In anisotropic phases the magnitude of the thermal flux depends on the direction of gradient VT  

In the case of a uniaxial phase, the thermal conductivity tensor has a familiar form (7.25a) kjj = k dij + katiitij, where k = k — kx 0 for calamitic phases and ka 0 for discotic ones. 

The experimental methods used for the determination of thermal conductivity are described by Tsederberg (1965), who also lists values for many substances. The four-volume handbook by Yaws (1995-1999) is a useful source of thermal conductivity data for hydrocarbons and inorganic compounds. 

The thermal conductivity of a solid is determined by its form and structure, as well as composition. Values for the commonly used engineering materials are given in various handbooks. 

The data available in the literature up to 1973 have been reviewed by Jamieson et al. (1975). The Weber equation (Weber, 1880) can be used to make a rough estimate of the thermal conductivity of organic liquids for use in heat transfer calculations  

Bretsznajder (1971) gives a group contribution method for estimating the thermal conductivity of liquids. 

Approximate values for the thermal conductivity of pure gases, up to moderate pressures, can be estimated from values of the gas viscosity, using Eucken s equation (Eucken, 1912)  

By analogy with the thermal conduction equation for a gas, given by the kinetic theory of gases, we can write for the thermal conductivity, K, of a solid 

When a substance is transparent to visible light, such as single crystals of simple inorganic solids which do not contain transition metal ions, or some glasses, another significant component in the thermal conductivity is the transmission of photons in the infra-red region, which becomes more important with 

Substances containing a significant porosity also show an increasing photon transfer of energy as the temperature increases due to the superior transmissivity of infra-red photons through the pores over the surrounding solid. 

As described above, quantum restrictions limit the contribution of the free electrons in metals to the heat capacity to a very small effect. These same electrons dominate the thermal conduction of metals acting as efficient energy transfer media in metallic materials. The contribution of free electrons to thermal transport is very closely related to their role in the transport of electric current through a metal, and this major effect is described through the Wiedemann-Franz ratio which, in the Lorenz modification, states that 

Thus for silver these values at room temperature are 

Recently, Miiller-Plathe suggested a NEMD method [51] for calculation of thermal conductivity in atomic fluids that was subsequently adapted and applied 

Using the imposed heat flux method, we carried out NEMD simulations for the HMX melt at six temperatures (550 K - 800 K, in 50 K intervals) and atmospheric pressure. The simulation methodology was similar to the one described above for equilibrium MD simulations with a few exceptions. Each system contained 100 HMX molecules. The orthorhombic simulation box, extended in the z direction, was subdivided into 10 equal slabs with width of about 5.0 A and cross-sectional area of about 625.0 A2. The molecular center-of-mass velocities were exchanged every 500 fs (W=0.002 fs 1) for pairs of molecules belonging to the cold and hot slabs. This choice of the W was based on our previous experience with simulations of liquid n-butane and water [52], 

In Fig. 8 we show a comparison of the thermal conductivity for liquid HMX obtained from our NEMD simulations with measured values for crystalline HMX [54] as well as values used in combustion models for HMX [55]. Despite being weak, the temperature dependence of the thermal conductivity of liquid HMX is not featureless. The thermal conductivity exhibits a sharp drop in the temperature interval from the melting point (550 K) up to 650 K. At higher temperatures the thermal conductivity exhibits almost no temperature dependence. The predicted value at 550 K is consistent with the HMX crystal data [54]. The thermal conductivity used in some combustion models [55] agrees to within about 25% with our NEMD predictions over the entire temperature interval. 

We close our discussion of thermal conductivity with the observation that experimental determinations of this property at elevated temperatures have large error bars. Indeed, the two measurements of which we are aware differ by roughly 50%. (The experimental line in Fig. 8 is the recommended linear 

It is appropriate here to make some remarks on the physical foundations of thermal conductivity. The dependence of thermal conductivity on temperature has been experimentally recognized. However, there is no universal theory explaining this dependence. Gases, liquids, conducting and insulating solids can each be explained with somewhat different microscopic considerations. Although the text is on the continuum aspects of heat transfer, the following remarks are made for some appreciation of the microscopic aspects of thermal conductivity. 

For dilute gases, molecules are assumed to be independent from each other, and thermal conductivity is explained by means of kinetic theory, which analytically leads to k Tl z, Experimental results, however, indicate that for real gases 

The temperature dependence of thermal conductivity for liquids, metal alloys, and nonconducting solids is more complicated than those mentioned above. Because of these complexities, the temperature dependence of thermal conductivity for a number of materials, as illustrated in Fig. 1,11, does not show a uniform trend. Typical ranges for the thermal conductivity of these materials are given in Table 1.1, We now proceed to a discussion of the foundations of convective and radiative heat transfer. 

Unalloyed aluminium is an excellent heat conductor, with roughly 60% of the thermal conductivity of copper, the optimum performer among common metals. The thermal conductivity of aluminium alloys depends on their composition and metallurgical temper (Tables A.3.5 and A.3.9). 

As early as the end of the 19th century, this property led to replacing tin-plated copper with aluminium alloys in the manufacture of kitchen utensils, both for domestic and professional use. 

Whenever there is a problem related to heat exchange, the use of aluminium is always taken into consideration, under the condition, of course, that the medium is appropriate when liquid-liquid or liquid-gaseous exchange is envisioned. There are many applications of aluminium heat-exchangers cars, commercial vehicles, refrigerators, air conditioning, desalination of seawater, solar energy, coolers in electronic devices, etc. 

Kazeminejad [20] has described the construction of an apparatus for the measurement of thermal conductivity accordingly to ASTM C177 [21] and DIN 52612-2 [22]. He describes a method of determining thermal conductivity of insulating materials, based on a copper-coated printed circuit board. Thermal conductivity values are reported for pure PE, pure PC, and PE and PC mixed with conductive fillers such as aluminium powder and carbon black. 

Yu and co-workers [25] have reported the results of thermal conductivity measurements 

Dos Santos and Gregorio, Jr., [26] measured the thermal conductivity of PA-PMMA, 

Prociak [27] studied the effect of parameters such as the method of sample preparation, the temperature gradient and the average temperature of measurement on the thermal conductivity of rigid PU foams blown with hydrocarbons and hydrofluorocarbons (HFC). The thermal insulation properties of different cellular plastics, such as rigid and flexible PU foams and expanded PS, were compared. The thermal conductivity and thermal diffusivity of foams were correlated with the PU matrix structure to demonstrate the effect of cell anisotropy on the thermal insulation properties of the rigid foams blown with cyclopentane and HFC-365/227 (93 wt% pentafluorobutane/7wt% heptafluoropropane). 

Patton and co-workers [28] evaluated the ablation, mechanical and thermal properties of vapour grown carbon-fibre (VGCF Pyrograf in from Applied Sciences Inc)/phenolic resin (SC-1008 from Borden Chemical Inc) composites to determine the potential of using this material in solid rocket motor nozzles. Composite specimens with 

Tj(p=0) can be calculated by the low pressxne Chung-Lee-Starling model. The parameter p, is the reduced dipole moment given by  

The rate of heat transport in and through polymers is of great importance. For good thermal insulation the thermal conductivity has to be low. On the other hand, polymer processing requires that the polymer can be heated to the processing temperature and cooled to ambient temperature in a reasonable time. 

Thermal conductivity is the intensive property of a material that indicates its ability to conduct heat. For one-dimensional heat flow in the x-direction the steady state heat transfer can be described by Fourier s law of heat conduction  

thermal conductivity, X, is the heat flux transported through a material due to a 

For non-stationary heat conduction in a semi-infinite stationary medium the onedimensional transient heat conduction without heat production, we have next parabolic differential equation 

An adequate theory that might be used to predict accurately the thermal conductivity of polymeric melts or solids does not exist. Most of the theoretical or semi-theoretical 

Another transport property of molten salts for which there is a considerable amormt of data is their thermal conductivity Xth- In earher years the then available measuring techniques led to the conclusion that Xth increases mildly with increasing temperatures. More modem techniques, such as transient hot wire measurements, yield values of that diminish mildly and linearly with increasing temperatures. The scatter of values reported in the literature is large, however, and they have not been critically compiled so far. Gheribi et al. [277] provided an explicit model expression for 2, the required inputs for their model being the ionic radii, and the density, velocity of sound, heat capacity, and melting temperature of the salt. Table 3.21 shows the recommended predicted values in terms of the parameters of the linear temperature dependence  

The predicted values of Ath agree as weU as may be expected with modem experimental values. The values of at the corresponding temperature of 1.1 r , are also shown in Table 3.21, and those for the aUcali metal halides depend reciprocally on the molar mass of the salt Ath(l l m)/W m K 35/ (M/g mol ), but this cannot be generalized to other salts. A reciprocal dependence on the mass of the molten salts (a fractional power of it) was also noted by Cornwell [278] for 13 salts other than alkali metal halides at temperatures near T. DiGuilio and Teja [279] reviewed several models for the thermal conductivity of molten salts, and later Hossain et al. [280] employed essentially the same model as in [279] to obtain the reduced thermal conductivity 2th, but without a clear definition how is related to the measured thermal conductivity. 

The heat generated when work is done to the polyurethane must be removed or the material will become overheated. The thermal conductivity of polyurethane is poor compared to metals. Unless the product is properly designed, there can be failures in some polyurethane applications. 

Thermal conductivity is expressed in several different internationally recognized ways. One method of expressing thermal conductivity (A) is in terms of the heat flux under steady conditions per square meter for one meter of thickness of one degree Kelvin difference in temperature. Kelvin is a thermodynamic scale and is centigrade starting at absolute zero. 

Another method for expressing thermal conductivity is to determine the quantity of heat passing through a unit area of the material, when the temperature gradient (when measured across unit thickness in the direction of the heat flow) is unity (Smith, 1993). The thermal conductivity of a polyurethane is on the order of 0.1 to 0.3 W/m.K. Other references give a value of approximately 1.7 to 3.5 x 10 4 cal.-cm/sec. cm2.°C (Gallagher Corporation, 1994). 

The low thermal conductivity of polyurethanes must be taken into account in the design of parts. The efficient dissipation of the heat must be allowed for when any part is subject to vibration, flexing, or impact. 

The transport of heat in metallic materials depends on both electronic transport and lattice vibrations, phonon transport. A decrease in thermal conductivity at the transition temperature is identified with the reduced number of charge carriers as the superconducting electrons do not carry thermal energy. The specific heat and thermal conductivity data are important to determine the contribution of charge carriers to the superconductivity. The interpretation of the linear dependence of the specific heat data on temperature in terms of defects of the material suggests care in interpreting the thermal conductivity results to be described. 

Several techniques are available for thermal conductivity measurements, in the steady state technique a steady state thermal gradient is established with a known heat source and efficient heat sink. Since heat losses accompany this non-equilibrium measurement the thermal gradient is kept small and thus carefully calibrated thermometers and heat source must be used. A differential thermocouple technique and ac methods have been used. Wire connections to the sample can represent a perturbation to the measurement. Techniques with pulsed heat sources (including laser pulses) have been used in these cases the dynamic response interpretation is more complicated. 

Early results (67) for YBa2Cu307 showed an increase in k (thermal conductivity) from lower temperatures to about 60 K, a slight decrease to near Tc and a nearly constant value between 100-150 K. The low temperature thermal conductivity in single crystal Bi-Sr-Ca-Cu-O (2212 phase) has been measured by Zhu et al. (68) and at temperatures less than 1 K they obtain a fit k = 0.15 T2 W/mK which they note is similar in temperature dependence to Y-Ba-Cu-O and 

Glass 0.04 Other Non-Fibrous Materials Aluminum 205 Steel 16-60 Water 0.58 Ice 2.22 Snow 0.05-0.25 Air 0.024 Most non-fibrous materials are isotropic and only have one thermal conductivity value.  

Sources Morton, W.E., et. al., Physical Properties of Textile Fibres, Fourth Edition, Woodhead  

Publishing Limited, 2008. Warner, S.B., Fiber Science, Prentice Hall, 1995.  

From Table 17.2, it also is seen that caibon fibers are thermally conductive, but polymer and glass fibers are good thermal insulators. Among the polymer fibers shown in Table 17.2, Kevlar fibers have the highest thermal conductivities probably beeause of the elose sequence of benzene rings along the polymer main ehains. 

Where as heat capacity is a measure of energy, thermal diffiisivity is a measure of the rate at which energy is transmitted through a given plastic. In contrast, metals have values hundreds of times larger than those of plastics. Thermal diffiisivity determines plastics rate of change with time. Although this function depends on thermal conductivity, specific heat at constant pressure, and density, all of which vary with temperature, the thermal diffiisivity is relatively constant. 

The specific heat of amorphous plastics increases with temperature in an approximately linear fashion below and above Tg, but a steplike change occurs near the Tg. No such stepping occurs widi crystalline types. The high degree of die molecular order for crystalline TPs makes their values tend to be twice those of the amorphous types. The TSs has the highest values. To increase TC the usual approach is to add metallic fillers, glass fibers, foamed structure, or electrically insulating fillers such as alumina. 

In general, TC is low for plastics and the plastic s structure does not alter its value significandy. TC of plastics depends on several variables and cannot be reported as a single factor. But it is possible to ascertain the two principal dependencies of temperature and molecular orientation (MO). In fact, MO may vary within a product producing a variation in TC. It is important for the product designer and processor to recognize such a situation. Certain products require personal skill to estimate a part s performance under steady-state heat flow. 

These values are based on the coefficient of linear thermal-expansion (CLTE). It is the ratio between the change of a linear dimension to the original dimension of the material per unit change in temperature (per ASTM D 696 standard). It is generally given as cm/cm/C or in./in./F. Plastics CLTE behaviors vary because different plastics have 

Plastic products are often constrained from freely expanding or contracting by rigidly attaching them to another structure made of a material with a lower CLTE. When such composite structures are heated, the plastic component is placed in a state of compression and may buckle. When such composite structures are cooled, the plastic component is placed in a state of tension, which may cause the material to yield or crack. The precise level of stress in the plastic depends on the relative compliance of the component to which it is attached, and on assembly stress. 

dQ is the amount of heat flowing normal to the area A in time dt dQ/dt is called the heat flux. For a given area, the heat flow is proportional to the temperature gradient, -dT/dx. The proportionality constant k is called the thermal conductivity. 

Xj and X2 are two points lying on a line perpendicular to thickness. Ti and T2, respectively, are the temperatures at these two points. For heat flow along the radius of a hollow tube, the heat flux is given by  

In this equation, 1 is the length of fhe tube, T2 is the temperature at the end of an outer diameter, and Tj is the temperature at the end of the inner diameter lying along the same outer diameter. 

In conventional composites, the models of thermal conduction are based on the Fourier heat conduction theory. However, these models are not valid at the nanoscale due to the ballistic phonon transport and interfacial scattering. Chen et al. [31] have reviewed the status and progress of theoretical and experimental studies of thermal transport phenomena in nanostructures. We discuss here some theoretical and numerical efforts toward the prediction of thermal conductivity of nanopartide-polymer nanocomposites. 

One significant progress is the use of phonon Boltzmarm equation and MC simulation to study the thermal conductivity of nanocomposites with different nanopartides (e.g., nanowires and nanopartides). The effects of various factors have 

Effective medium theory (EMT) is commonly used to describe the microstructure-property relationships in heterogeneous materials and predict the effective physical properties. It has recently been revised to predict the thermal conduction of nanocomposites. For nanocomposites with nanopartides on the order of or smaller than the phonon mean free path, the interface density of nanopartides is a primary factor in determining the thermal conductivity. In graphite nanosheet polymer composites, the interfacial thermal resistance still plays a role in the overall thermal transport. However, the thermal conductivity depends strongly on the aspect ratio and on the orientation of graphite nanosheets. 

The heat conduction of whiskers relies mainly on lattice vibration, namely phonon vibration thus thermal conductive performance is excellent, whereas the heat conduction of polymer materials depends mainly on internal vibration between atoms, which makes the thermal conductive performance poor. If whiskers are filled into polymers, they overlap with each other in a matrix and form pathways of heat conduction. Therefore the thermal conductivity of the composite will be greatly increased. Zhou found that low filling content can effectively improve the thermal conductivity performance of epoxy resin. When the whisker content is only 10%, the thermal conductivity of the material increased by 3 times compared with the pure matrix. Li et al. found that the thermal conductivity of PP filled with 30% zinc oxide whiskers increased by 55.9% over pure PP. 

The usual polymers do not conduct electricity. Consequently, heat cannot likewise be transported by electrons in these polymers heat must 

At still lower temperatures, a plateau is reached at 5-15 K. A slower decrease with temperature is then observed until, below 0.5 K, the thermal conductivity becomes proportional to the square of the temperature. 

Above temperatures of 150 K, heat is essentially transported by intermolecular collisions. A decrease in the thermal conductivity above the glass transition temperature can be expected because of the increasingly loose arrangement of the molecules. The thermal conductivity above and below the glass transition temperature does not differ very much because the molecular packing above and below this temperature is also not very different. The thermal conductivity exhibits only a weak maximum at the glass transition temperature. 

On the other hand, the packing density of crystalline polymers changes drastically at the melting temperature. The decrease becomes stronger with 

Wunderlich and H. Bauer, Heat capacities of linear high polymers, Adv. Polym. Sci. 7, 151 (1970). 

In an isotropic medium, as for normal liquids, the Fourier equation holds  

In the case that X Cp, and p are independent of position and temperature, it follows from Eqs. (94) and (95) that 

A thermal diffusion equation is expressed, as in Pick s second law of diffusion, hy 

Data on high-temperature melts are still limited. Conventional methods are difficult to apply because of the high values of thermal conductivity. Other difficulties in measuring molten salts are their corrosiveness, high electrical conductivities, and the necessity of careful preparation. Special care should be taken to exclude convection errors, which are usually the most serious source of errors, even at room temperature. 

At temperatures above ca. 1000 K, heat transfer via radiation becomes significant, that is, the heat transfer can occur by optical energy waves (photons) as well as conduction (phonons), with the heat transfer equation expressed by 

Equation (8.3) is the basic form into which more complex situations often are cast. For example, 

Thermal conductivity is a fundamental property of substances that basically is obtained experimentally although some estimation methods also are available. It varies somewhat with temperature. In many heat transfer situations an average value over the prevailing temperature range often is adequate. When the variation is linear with 

TABLE 8.1. Thermal Conductivities of Some Metals Commonly Used in Heat Exchangers [kBtu / (hr)(sqft)(°F/ft)] 

Another assumption we made—that all collisions with the wall are elastic and transfer no energy because of the large mass difference—may strike you as quite strange if you think about it carefully. Heat transfer between gases and solids is readily observed. The air in your refrigerator will cool down food the air in your oven will heat it. 

If two plates with area a and separation d are maintained at temperatures 7) and T2, for a time t, the average heat flow q/t is given by 

K is called the thermal conductivity, which for air at STP is. 023 W/(m K). One way to reduce this energy flow is to decrease the pressure. Cryogenic liquids (such as liquid nitrogen, which boils at 77K) are commonly stored in Dewar flasks, which are double-walled containers with an evacuated region between the walls. 

For cylinders and spheres, A is a function of radial position (see Fig. 5-2) 2nrL and 4nr2, where L is the length of the cylinder. For constant k, Eq. (5-4) becomes 

Conduction with Resistances in Series A steady-state temperature profile in a planar composite wall, with three constant thermal conductivities and no source terms, is shown in Fig. 5-3a. The corresponding thermal circuit is given in Fig. 5-3b. The rate of heat transfer through each of the layers is the same. The total resistance is the sum of the individual resistances shown in Fig. 5-3b  

Additional resistances in the series may occur at the surfaces of the solid if they are in contact with a fluid. The rate of convective heat transfer, between a surface of area A and a fluid, is represented by Newton s law of cooling as 

5-3 Steady-state temperature profile in a composite wall with constant thermal conductivities k kg, and kc and no energy sources in the wall. The thermal circuit is shown in (b). The total resistance is the sum of the three resistances shown. 

The macroscopic phenomenological equation for heat flow is Fourier s law, by the mathematician Jean Baptiste Joseph Fourier (1768-1830). It appeared in his 1811 work, Theorie analytique de la chaleur (The analytic theory of heart). Fourier s theory of heat conduction entirely abandoned the caloric hypothesis, which had dominated eighteenth century ideas about heat. In Fourier s heat flow equation, the flow of heat (heat flux), q, is written as  

As discussed in Section 6.1, the number of independent components needing to be specified is reduced by the crystal symmetry. If, for example, the x axis is taken as the [1 0 0] direction, the y axis as the [0 1 0], and the z axis as the [1 0 0], a cubic crystal or a polycrystalline sample with a random crystallite orientation gives  

If a temperature gradient VTis directed along one of the principal axes of a crystal, say thex axis, what will be the angle between the vector q and the vector 9T/9x  

Henceforth, for simplicity during our discussions of the underlying physical basis for thermal conductivity, only cubic crystals or nontextured polycrystalline solids will be considered, for which case a single scalar quantity is sufficient. 

TABLE 8.1. Thermal Conductivities of Some Metals Commonly Used in Heat Exchangers 

Recently Isnardi (102a) has brought the process to a high degree of perfection and applied it, as did Langmuir 

The conduction of heat through solids occurs as a result of temperature gradients. In analogy to Pick s first law, the relationship between the heat flux and temperature gradients dTfdx is given by 

Describing the mechanisms of conduction in solids is not easy. Here only a brief qualitative sketch of some of the physical phenomena is given. In general, thermal energy in solids is transported by lattice vibrations, i.e. phonons, free electrons, and radiation. Given that the concentration of free electrons in ceramics is low and that most ceramics are not transparent, phonon mechanisms dominate and are the only ones discussed below. 

A situation not unlike the propagation of light or sound through a solid. 

By assuming the number of these thermal energy carriers to be and their average velocity it is reasonable to assume that, in analogy to the electrical conductivity equation of cr = n/iq, is given by 

Furthermore, the lack of long-range order in amorphous ceramics results in more phonon scattering than in crystalline solids and consequently leads to lower values of Ath- 

For mixtures of supercritical fluids with liquids, as the pressure increases and more of the supercritical fluid dissolves in the liquid phase the viscosity of the liquid phase decreases at constant temperature an increase in temperature reduces the viscosity at constant pressure. The viscosity of the vapor phase increases with increasing concentrations of the liquid component. At higher pressures, more of the liquid component dissolves into the supercritical phase and the viscosity increases, but not to the same extent as the drop in viscosity with increasing concentration of the supercritical fluid. 

Thermal conductivity is associated with the transport of energy at the molecular level and is thus intrinsically related to viscosity and heat capacity. For most gases in ambient conditions, thermal conductivity, X, is between 0.01 and 0.025 W m K .  

The temperature-dependent fiber mass fraction,/f, and resin mass fraction, are given by Eq. (4.22) and Eq. (4.23)  

The heat capacity is an indicator of how much heat has to be added to a unit mass of plastic in order to raise its temperature 1°C, and it is readily measured by ASTM tests. 

above about 150 K, heat is essentially transported by inter- 

Slade and L, T. Jenkins, eds., rechniques and Methods of Polymer Evaluation, M. Dekker, New York, Vol. 1 (Thermal Antlysis) 1966 Vol. 2 (Thermal Characterisation Techniques) 

Bartenev and Yu. V. Zelenev, eds.. Relaxation Phenomena in Polymers, Halsted, New York, 1974. 

Relaxation in polymers, in Macromolecular Science (Vol. 8 of Physical Chemistry Series 2) (C. E. H. Bawn, ed.), MTP International Review of Science, 1975, p. 1. 

Calorimetric studies of state and transitions in solid high polymers, Fortschr. Hochpolym. Forschg. 2, 221 (1960). 

Details of the apparatus for performing this measurement to ASTM C177 [19] and DIN 52612 [20] standards are given in Table 16.1. 

Dos Santos and Gregorio [38] measured the thermal conductivity of Nylon 6,6, PP, PMM A, rigid polyvinyl chloride (PVC), and polyurethane (PU) foam. 

Dos Santos and Gregorio [48] measnred the thermal conductivity of polyamide 6,6, polymethyl methacrylate, rigid polyvinyl chloride ether, and polyurethane foam. 

Various methods have been described for the determination of thermal conductivity. Capillarity has been used to measure the thermal conductivity of low-density polyethylene, high-density polyethylene, and polypropylene at various temperatures and pressure [50]. A transient plane source technique has been applied in a study of the dependence of the effective thermal conductivity and thermal diffusivity of polymer composites [51]. 

Results of measurements on high-density arc-cast and zone-melted ThN at temperatures of 80 to 375 K by the guarded longitudinal heat flow technique of Moore et al. [34] have been 

Values for a, )8, y, and a, b, c, as well as average heat capacity information are provided in the literature.  

Solution. Note that 1.0 Btu/lb °F is equivalent to 1.0 cal/g °C. This also applies on a mole basis, i.e., 

The heat capacity can be converted from units of cal/g °C to Btu/lb °F using appropriate conversion factors. 

Thermal conductivity of methanol = 0.0512 cal/m s °C (at 60°F) Convert the thermal conductivity to English units. 

Note that the usual engineering notation for thermal conductivity is k. 

When a material is heated varies its molecular state behavior, increasing its movement. That is, the molecules out of their state of rest and acquire inertia or kinetic movement caused by the temperature rise. 

The thermal conductivity (A) characterizing the amoimt of heat needed per m so that crossing over the unit of time, 1 m to obtain a homogeneous material 1 °C difference in temperature between the two faces. The thermal conductivity is expressed in units of W/(m K) or J/(s m °C). 

In the literature, there are some models proposed to study the thermal conductivity as a function of temperature [28]. Although initially was Debye who proposed the idea in 1917 to consider the phonons as particles, it was not until the early twenty-first century when Srivastava conceptualized a model of thermal conductivity in solids. Srivastava s model is general enough to explain the solid thermal conductivity across the range of temperatures from 0 °C to about the melting temperature. 

But as we have seen, masonry materials can have a mixed behavior vitreous and crystalline, the determination of the conductivity and specific heat curves is inclined to the crystalline nature of the materials. According to Eurocode 1996-1-2 2005 values of thermal expansion ( ), specific heat (c), and thermal conductivity (A) of ceramic materials subjected to high temperatures should be taken by testing in a database data (test) or the National Annex of each country. 

This is not a major concern in the case of plastic packages because they already exhibit values of / ja between 30 and 100°C W .  

Many equations have been proposed for the transport properties of two-phase systems and in-depth details of the existing models are discussed elsewhere [4]. Noticing that virtually all the early theories neglected the effects of the particle shape, their packing density, and the possible formation of anisotropic clusters, Lewis and Nielsen modified the Halpin-Tsai equation for the elastic modulus of composite materials by incorporating the maximum volume fraction of filler cpm while still maintaining a continuous matrix phase [33,34]. Transposed to thermal conductivity Lewis and Nielsen s equation becomes 

Being inexpensive and providing an excellent shear strength, alumina is broadly used to formulate thermally conductive adhesives. The best solventless epoxy adhesives contain about 70% of aluminium oxide and give thermal conductivities in the range of 1.4-1.7 W m These values are 8-10 times greater than for the 

Another work provides numerical examples using epoxy and silicone resins with thermal conductivity 0.15-0.2 Wm [45]. The addition of 0.25 volume 

The conclusion that can be drawn from these experiments is that the use of highly priced fillers such as diamond powder does not improve the thermal conductivity better than less expensive materials such as aluminium nitride, boron nitride, boron carbide, or sdicon carbide. Within certain limits, the higher the X value of the filler particles, the higher the thermal conductivity of the adhesives with respect to the X/Xp ratio that exhibits a favourable optimized value at about 100. This means that fillers with a thermal conductivity in the range 

Similarly, the heat flux per unit area, QJA At is calculated as 

Once the temperature gradient and the heat flux are known, the thermal conductivity is subsequenfly calculated. 

Another approach to calculate thermal conductivity is equilibrium molecular dynamics (EMD) [125] that uses the Green-Kubo relation derived from linear response theory to extract thermal conductivity from heat current correlation functions. The thermal conductivity X is calculated by integrating the time autocorrelation function of the heat flux vector and is given by 

J(t) is the heat flux vector at time t and is defined as 

j represents the short-range van der Waals force and real part of the Ewald-Coulomb force (calculated within certain cutoff distance). In addition, it also includes forces due to bonded interaction terms such as bond stretching, angle bending, and so on. On the other hand, tensor S represents the forces due to electrostatic interactions beyond the cutoff distance. 

Concept Check 19.1 (a) Explain why a brass lid ring on a glass canning jar loosens when 

For steady-state heat flow, dependence of heat flux on the thermal conductivity and the temperature gradient 

Thermal conduction is the phenomenon by which heat is transported from high- to low-temperature regions of a substance. The property that characterizes the ability of a material to transfer heat is the thermal conductivity. It is best defined in terms of the expression 

The units of q and k are W/m (Btu/ft -h) and W/m-K (Btu/ft-h °F), respectively. Equation 19.5 is valid only for steady-state heat flow—that is, for situations in which the heat flux does not change with time. The minus sign in the expression indicates that the direction of heat flow is from hot to cold, or down the temperature gradient. 

Equation 19.5 is similar in form to Pick s first law (Equation 5.2) for steady-state diffusion. For these expressions, k is analogous to the diffusion coefficient D, and the temperature gradient is analogous to the concentration gradient, dC/dx. 

Utot 8.03 0 freeO.25% Ctot.9.85% freel 07 o [534, 1095] Data taken from graph Same 

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The chromatogram can finally be used as the series of bands or zones of components or the components can be eluted successively and then detected by various means (e.g. thermal conductivity, flame ionization, electron capture detectors, or the bands can be examined chemically). If the detection is non-destructive, preparative scale chromatography can separate measurable and useful quantities of components. The final detection stage can be coupled to a mass spectrometer (GCMS) and to a computer for final identification. 

Changes in thermal conductivity, e.g. carbon dioxide in flue gas. 

Thermal conductivity is expressed in W/(m K) and measures the ease in which heat is transmitted through a thin layer of material. Conductivity of liquids, written as A, decreases in an essentially linear manner between the triple point and the boiling point temperatures. Beyond a reduced temperature of 0.8, the relationship is not at all linear. For estimation of conductivity we will distinguish two cases < )... 

The frequency correlation lowers environmental disturbances. The correlation provides an output proportional to the content of Aa at the reference signal fundamental frequency, the phase conelation gives the sign of Aa. Where the stress gradients are very steep in materials of high thermal conductivity being loaded at low frequencies, the SPATE signals are attenuated and a correction factor has to be introduced to take into account this effect. 

This definition is in terms of a pool of liquid of depth h, where z is distance normal to the surface and ti and k are the liquid viscosity and thermal diffusivity, respectively [58]. (Thermal diffusivity is defined as the coefficient of thermal conductivity divided by density and by heat capacity per unit mass.) The critical Ma value for a system to show Marangoni instability is around 50-100. 

Alternatively, gas chromatography may be used Fig. XVII-5 shows a schematic readout of the thermal conductivity detector, the areas under the peaks giving the amount adsorbed or desorbed. 

The increases in melting point and boiling point arise because of increased attraction between the free atoms these forces of attraction are van der Waal s forces (p. 47) and they increase with increase of size. These forces are at their weakest between helium atoms, and helium approaches most closely to the ideal gas liquid helium has some notable characteristics, for example it expands on cooling and has very high thermal conductivity. 

It is common practice to omit the second summation on the right hand side of (11.118) on the groiands that it is small compared with the contribution of the conductive flux, which appears on the left hand side. However, this may not be so If the reactions are rapid and the thermal conductivity of the pellet material is low. One should, therefore, at least be aware of the approximation involved in the fona of the enthalpy balance most commonly seen in the literature. 

Equations (12.13) and (12.14) may be approximated by rather simple equations in most conditions of physical interest. This is possible because of the relatively large value of the thermal conductivity of the solid matrix, which has two important consequences. First, the conductive enthalpy flux, represented by the second term on the left hand side of... 

Einstein relationships hold for other transport properties, e.g. the shear viscosity, the bu viscosity and the thermal conductivity. For example, the shear viscosity t] is given by ... 

The thermal conductivity of polymeric fluids is very low and hence the main heat transport mechanism in polymer processing flows is convection (i.e. corresponds to very high Peclet numbers the Peclet number is defined as pcUUk which represents the ratio of convective to conductive energy transport). As emphasized before, numerical simulation of convection-dominated transport phenomena by the standard Galerkin method in a fixed (i.e. Eulerian) framework gives unstable and oscillatory results and cannot be used. 

Pure silver has a brilliant white metallic luster. It is a little harder than gold and is very ductile and malleable, being exceeded only by gold and perhaps palladium. Pure silver has the highest electrical and thermal conductivity of all metals, and possesses the lowest contact resistance. It is stable in pure air and water, but tarnishes when exposed to ozone, hydrogen sulfide, or air containing sulfur. The alloys of silver are important. 

It is a white crystalline, brittle metal with a pinkish tinge. It occurs native. Bismuth is the most diamagnetic of all metals, and the thermal conductivity is lower than any metal, except mercury. It has a high electrical resistance, and has the highest Hall effect of any metal (i.e., greatest increase in electrical resistance when placed in a magnetic field). 

The electronic configuration for an element s ground state (Table 4.1) is a shorthand representation giving the number of electrons (superscript) found in each of the allowed sublevels (s, p, d, f) above a noble gas core (indicated by brackets). In addition, values for the thermal conductivity, the electrical resistance, and the coefficient of linear thermal expansion are included. 

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