Thermal Conductivity

10 Key excerpts on "Thermal Conductivity"

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  Incropera's Principles of Heat and Mass Transfer

  • Frank P. Incropera, David P. DeWitt, Theodore L. Bergman, Adrienne S. Lavine(Authors)
  • 2017(Publication Date)
  • Wiley

    (Publisher)

  As illustrated in Figure 2.4, the Thermal Conductivity of a solid may be more than four orders of magnitude larger than that of a gas. This trend is due largely to differences in intermo- lecular spacing for the two states. The Solid State In the modern view of materials, a solid may be comprised of free elec- trons and atoms bound in a periodic arrangement called the lattice. Accordingly, transport of thermal energy may be due to two effects: the migration of free electrons and lattice vibra- tional waves. When viewed as a particle-like phenomenon, the lattice vibration quanta are termed phonons. In pure metals, the electron contribution to conduction heat transfer domi- nates, whereas in nonconductors and semiconductors, the phonon contribution is dominant. Kinetic theory yields the following expression for the Thermal Conductivity [1]: k Cc 1 3 mfp λ = (2.7) For conducting materials such as metals, C ≡ C e is the electron specific heat per unit vol- ume, c is the mean electron velocity, and λ mfp ≡ λ e is the electron mean free path, which is defined as the average distance traveled by an electron before it collides with either an imperfection in the material or with a phonon. In nonconducting solids, C ≡ C ph is the pho- non specific heat, c is the average speed of sound, and λ mfp ≡ λ ph is the phonon mean free 0.1 0.01 1 10 100 1000 Thermal Conductivity (W/m • K) GASES INSULATION SYSTEMS LIQUIDS NONMETALLIC SOLIDS ALLOYS PURE METALS Silver Zinc Nickel Aluminum Oxides Ice Plastics Fibers Foams Oils Water Mercury Hydrogen Carbon dioxide FIGURE 2.4 Range of Thermal Conductivity for various states of matter at normal temperatures and pressure. 64 Chapter 2 ■ Introduction to Conduction path, which again is determined by collisions with imperfections or other phonons. In all cases, the Thermal Conductivity increases as the mean free path of the energy carriers (elec- trons or phonons) is increased.

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  Handbook of Polymer Testing

  Physical Methods

  • Roger Brown(Author)
  • 1999(Publication Date)
  • CRC Press

    (Publisher)

  1 Introduction 24 Thermal Properties David Hands Consultant, Sutton Farm, Shrewsbury, England This chapter deals with the measurement of Thermal Conductivity, thermal diffusivity, and specific heat. Other properties that are sometimes included under the umbrella term thermal properties are dealt with in other parts of this volume. In most cases it does not matter whether the sample is a rubber or a plastic, the experimental techniques are the same. 2 General Theory 2.1 Equation of Conduction of Heat For the flow of heat in one direction, the heat flux luis related to the temperature gradient ae;ax by Fourier's law ae lu = -K-(1) ax where K is the Thermal Conductivity. The minus sign indicates that the heat flows in the opposite direction to the temperature gradient. The form of Eq. I implies that heat con-duction is a random process. If energy were propagated without scattering, then the heat flow would depend on the temperature difference between the end faces of the specimen instead of the temperature gradient [1]. The general equation from which the time-dependent temperature distribution may be calculated is obtained from Eq. I and the equation of continuity aJu ae -= -pc-ax at (2) 597 598 Hands where pis the density, cis the specific heat, i.e., the heat capacity per unit mass, and tis the time. The equation of continuity is an expression of the conservation of energy. The heat flux can be eliminated between Eqs. I and 2 to give a 2 e 1 aK (ae) 2 1 ae ax 2 + J( ae ax = ~ at (3) where a= K/ pc is the thermal diffusivity. Eq. 3 is the equation of conduction of heat, in the absence of heat generation and convection, for heat flow in one direction. If the conductivity is independent of temperature it reduces to a 2 e 1 ae ax 2 -~at (4) which is the equation usually referred to. It has been shown that for rubbers, and for plastics except at melting transitions, the conductivity term in Eq. 3 is very small, and Eq. 4 is adequate for most heat flow calculations [2] .

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  Thermal Management Materials for Electronic Packaging

  Preparation, Characterization, and Devices

  • Xingyou Tian(Author)
  • 2023(Publication Date)
  • Wiley-VCH

    (Publisher)

  The Thermal Conductivity of object usually varies with temperature. When the temperature range of the objective is small, the Thermal Conductivity is linearly related to the temperature: k = k 0 (1 + bT) (1.8) where k 0 is the Thermal Conductivity under a certain reference state and b is the con- stant determined experimentally. Due to the difference of heat conduction mecha- nism, the Thermal Conductivity of object in different forms is quite different. The heat conduction of objects is the collision and transfer of microscopic particles, includ- ing the thermal movement of molecules, the phonon movement formed by lattice vibration, and the movement of free electrons. Generally, the Thermal Conductivity of solid is the highest, while that of gas is the lowest. 1.2 Heat Conduction Differential Equation and Finite Solution 1.2.1 Heat Conduction Differential Equation Figure 1.3 shows a cube element with side lengths of dx, dy, dz. Here, the density , specific heat C p , and Thermal Conductivity k are constants. According to the law of conservation of energy, the sum of the net heat flowing into the cube unit in a certain time ΔQ i and the heat generated by the cube unit itself ΔQ p is equal to the increase in the enthalpy ΔE of the infinitesimal cube. 4 1 Physical Basis of Thermal Conduction –k ∂ ∂y ∂y dy dx dz T + ∂T –k ∂ ∂x ∂x dz dx dz T + ∂T – k ∂ ∂z ∂z dz dx dy T + ∂T –k ∂ ∂z dx dz –k ∂ ∂x dx dz –k ∂ ∂y dx dz Figure 1.3 Cube element. During d time, the total net inflow heat in the x-, y-, z-directions is ΔQ i = k (  2 T x 2 +  2 T y 2 +  2 T z 2 ) dx dy dz d (1.9) The heat generated by the heat source in the cube unit in time d is ΔQ p = q 1 dx dy dz d (1.10) where q 1 is the calorific value of the heat source per unit time and volume, and the unit is W/m 3 .

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  Process Heat Transfer

  Principles, Applications and Rules of Thumb

  • Thomas Lestina, Robert W. Serth(Authors)
  • 2010(Publication Date)
  • Academic Press

    (Publisher)

  The theory was based on the results of experiments similar to that illustrated in Figure 1.1 in which one side of a rectangular solid is held at temperature T 1 , while the opposite side is held at a lower temperature, T 2 . The other four sides are insulated so that heat can flow only in the x -direction. For a given material, it is found that the rate, q x , at which heat (thermal energy) is transferred from the hot side to the cold side is proportional to the cross-sectional area, A , across which the heat flows; the temperature difference, T 1 − T 2 ; and inversely proportional to the thickness, B , of the material. That is: q x ∝ A ( T 1 − T 2 ) B Writing this relationship as an equality, we have: q x = k A ( T 1 − T 2 ) B (1.1) T 2 q x x Insulated Insulated Insulated B q x T 1 Figure 1.1 One-dimensional heat conduction in a solid. HEAT CONDUCTION 1/3 The constant of proportionality, k , is called the Thermal Conductivity. Equation (1.1) is also applicable to heat conduction in liquids and gases. However, when temperature differences exist in fluids, con-vection currents tend to be set up, so that heat is generally not transferred solely by the mechanism of conduction. The Thermal Conductivity is a property of the material and, as such, it is not really a constant, but rather it depends on the thermodynamic state of the material, i.e., on the temperature and pressure of the material. However, for solids, liquids, and low-pressure gases, the pressure dependence is usually negligible. The temperature dependence also tends to be fairly weak, so that it is often acceptable to treat k as a constant, particularly if the temperature difference is moderate. When the temperature dependence must be taken into account, a linear function is often adequate, particularly for solids. In this case, k = a + bT (1.2) where a and b are constants. Thermal conductivities of a number of materials are given in Appendices 1.A–1.E.

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  Handbook of Superconductivity

  • Charles K. Poole, Horacio A. Farach, Richard J. Creswick(Authors)
  • 1999(Publication Date)
  • Academic Press

    (Publisher)

  Fig. 10.3. Experimental setup to measure Thermal Conductivity using the longitudinal steady-state method. C. Thermal Conductivity 499 b. Normal State Heat in a solid is carried by two distinct entities, free carriers, ~%, and quantized lattice vibrations called phonons, top. The total Thermal Conductivity, to, is then K -- Ke -[- Kp. (23) The relative magnitude of the two contributions in Eq. (23) can be used to classify solids in a way analogous to the magnitude of the electrical resistivity, Fig. 10.4. Each heat-conducting channel (carriers and phonons) is subject to relaxation mechanisms (scattering) that ensure the stationary nature of the heat-conducting process. Charge carriers are scattered by phonons yielding the thermal resistivity contribution We_p, by defects yielding We_d, and by other charge carriers, We_ e. Phonons can scatter on free carriers, Wp_e, on defects, Wp_ d and in interactions with other phonons, Wp_p. According to Matthiessen's rule, the scattering processes within each channel are additive, yielding We - 1/• e - We_ p -[- We_ d -'[- We_ e (24) Wp -- 1/tCp - Wp_ e + Wp_ d + Wp_p. (25) Depending on the carrier density, the density of defects, and the temperature range, the magnitude and the temperature dependence of each term in Eqs. (24) and (25) can be evaluated (see, for example, Klemens, 1965). For instance, a Fig. 10.4. / METAL SEMICONDUCTOR ]~e = K~ A K~p __~ KZ e K:p small / Main contributions to the Thermal Conductivity of various solids. 500 Chapter 10: Thermal Properties normal metal (ic ~ Ice) can be modeled by the following expression with the temperature dependence W e = 1/tc e -aT 2 + b/T, (26) where the first term on the RHS stands for the cartier scattering by phonons and the second term represents the interaction of carriers with static lattice defects. Carrier-cartier interaction is usually small and is neglected in Eq.

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  Fluid Mechanics

  Volume 6

  • L D Landau, E. M. Lifshitz(Authors)
  • 2013(Publication Date)
  • Pergamon

    (Publisher)

  CHAPTER V

  THERMAL CONDUCTION IN FLUIDS

  Publisher Summary

  This chapter discusses the thermal conduction in fluids. In a viscous fluid, the law of conservation of energy holds, that is, the change per unit time in the total energy of the fluid in any volume must be equal to the total flux of energy through the surface bounding that volume. If the temperature of the fluid is not constant throughout its volume, there will be a transfer of heat by thermal conduction. This signifies the direct molecular transfer of energy from points where the temperature is high to those where it is low. It does not involve macroscopic motion and occurs even in a fluid at rest. The processes of heat transfer in a fluid are more complex than those in solids, because the fluid may be in motion. A heated body immersed in a moving fluid cools considerably more rapidly than one in a fluid at rest, where the heat transfer is accomplished only by conduction. The temperature differences in the fluid are so small that its physical properties may be supposed independent of temperature but are, at the same time, so large that one can neglect in comparison with them the temperature changes caused by the heat from the energy dissipation by internal friction.

  §49 The general equation of heat transfer

  IT has been mentioned at the end of §2 that a complete system of equations of fluid dynamics must contain five equations. For a fluid in which processes of thermal conduction and internal friction occur, one of these equations is, as before, the equation of continuity, and Euler’s equations are replaced by the Navier–Stokes equations. The fifth equation for an ideal fluid is the equation of conservation of entropy (2.6)

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  Elements of Heat Transfer

  • Ethirajan Rathakrishnan(Author)
  • 2012(Publication Date)
  • CRC Press

    (Publisher)

  From the above discussions it is evident that, Summary 19 1. A perfect gas must be both thermally and calorically perfect. 2. A perfect gas must satisfy both thermal equation of state ; p = ρ R T and caloric equations of state ; c p = ( ∂h/∂T ) p , c v = ( ∂u/∂T ) v . 3. A calorically perfect gas must be thermally perfect, but a thermally perfect gas need not be calorically perfect. That is, thermal perfectness is a prerequisite for caloric perfectness. 4. For a thermally perfect gas, c p = c p ( T ) and c v = c v ( T ); that is, both c p and c v are functions of temperature. But even though the specific heats c p and c v vary with temperature, their ratio, γ becomes a constant and independent of temperature, that is, γ = constant = γ ( T ). 5. For a calorically perfect gas, c p , c v as well as γ are constants and inde-pendent of temperature. 1.9 Summary Heat transfer is the science of energy in transit due to a temperature difference. In a heat transfer process, Heat flow = Thermal potential difference Thermal resistance • Conduction is an energy transfer process from more energetic particles of a substance to the adjacent, less energetic ones as a result of the interaction between the particles. • Convection is the mode of heat transfer between a solid surface and the adjacent liquid or gas that is in motion. • Radiation is the heat transfer mode in which the energy is emitted by matter in the form of electromagnetic waves (or photons) as a result of the changes in the electronic configurations of the atoms or molecules. Any physical quantity can be characterized by dimensions . The magni-tudes assigned to the dimensions are called units . The law of dimensional homogeneity states that, an analytically derived equation representing a phys-ical phenomenon must be valid for all systems of units . A system is an identified quantity of matter or an identified region in space chosen for a study. The region outside the system is called surrounding .

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  Thermodynamics and Heat Power

  • Irving Granet, Maurice Bluestein(Authors)
  • 2014(Publication Date)
  • CRC Press

    (Publisher)

  The negative sign has been included in Equation 11.2 to indicate a positive heat flow in the increasing x direction, which is the direction of decreasing temperature. The conductivity, k , is usually found to be a function of temperature, but for moderate tem-peratures and temperature differences, it can be taken to be a constant. If we now rewrite Equation 11.2 in more general terms for one-dimensional conduction, we have  Q kA t x = -∆ ∆ (11.3) Equation 11.3 is called Fourier’s law of heat conduction in one dimension in honor of the noted French physicist Joseph Fourier. In this equation, the heat transfer rate  Q is expressed in usual English engineering units as Btu/h, the normal area A is expressed in ft. 2 , the temperature difference Δ t is in °F, and the length Δ x is in ft., giving us k in units of Btu/(h) (ft. 2 ) (°F/ft.). This unit of k is often written as Btu/(h·ft.·°F). The nomenclature of Equation 11.3 is shown in Figure 11.3. Table 11.1 gives the thermal conductivities of some solids at room temperature, and Figure 11.4 shows the variation of Thermal Conductivity as a function of temperature for many materials. A table of the Thermal Conductivity of some building and insulating equipment is given in Appendix 2, Table A.6. In SI units, k is W/m·°C. The foregoing development was based on observable events in a hypothetical experi-ment. The conduction of heat can also be visualized as occurring as the transfer of energy by more active molecules at a higher temperature colliding with less active molecules at a lower temperature. Gases have longer molecular spacings and consistently exhibit much 527 Heat Transfer lower thermal conductivities than liquids. Because of the complex structure of solids, some have high values of k while others have low values of k . However, for pure crystalline metals, which are good electrical conductors, there are a large number of free electrons in the lattice structure that also makes them good thermal conductors.

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  Treatise on Geophysics, Volume 2

  Mineral Physics

  • G David Price(Author)
  • 2010(Publication Date)
  • Elsevier

    (Publisher)

  19 Properties of Rocks and Minerals – Thermal Conductivity of the Earth A. M. Hofmeister and J. M. Branlund , Washington University, St. Louis, MO, USA M. Pertermann , Rice University, Houston, TX, USA ª 2009 Elsevier B.V. All rights reserved. 19.1 Introduction 544 19.1.1 The Importance of Thermal Conductivity to Geophysics 544 19.1.2 Types of Thermal Transport and Justification for Omitting Metals 545 19.1.3 History of Mineral Physics Efforts and the Current State of Affairs 546 19.1.4 Scope of the Present Chapter 546 19.2 Theory of Heat Flow in Electrically Insulating Solid Matter 547 19.2.1 Phonon Scattering 547 19.2.1.1 Acoustic models of lattice heat transport 547 19.2.1.2 The damped harmonic oscillator (DHO) – phonon gas model 549 19.2.1.3 Bulk sound model for pressure derivative s 550 19.2.2 Radiative Transport in Partially Transparent Materials (Insulators) 550 19.2.2.1 Distinguishing direct from diffusive radiative transport on the basis of frequency-dependent attenuation 550 19.2.2.2 Spectroscopic models for diffusive radiative transport inside the Earth 552 19.2.2.3 Understanding the general behavior of k rad,dif from asymptotic limits 554 19.2.2.4 An approximate formula that connects k rad with concentration 554 19.3 Experimental Methods for the Lattice Contribution 555 19.3.1 Conventional Techniques Involving Multiple Physical Contacts 555 19.3.2 Methods Using a Single Physical Contact 556 19.3.3 Contact-Free, Laser-Flash Analysis 557 19.3.4 Additional Contact-Free (Optical) Techniques 559 19.4 The Database on Lattice Transport for Mantle Materials 559 19.4.1 Evaluation of Methodologies, Based Primarily on Results for Olivine 559 19.4.1.1 Ambient conditions 559 19.4.1.2 Elevated temperature 564 19.4.1.3 Elevated pressure 566 19.4.2 Laser-Flash Data on Various Minerals 566 19.4.2.1 Effect of chemical composition and hydration on room temperature values 567 19.4.2.2 Comparison of D ( T ) for dense oxides and silicates 568 19.4.3 Comparison of the Room Temperature Lattice Contribution to Theoretical Models and Estimation of D and k for Some High-Pressure Phases 568 19.4.3.1

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  Progress in Refrigeration Science and Technology

  Proceedings of the XIth International Congress of Refrigeration, Munich, 1963

  • Sam Stuart(Author)
  • 2014(Publication Date)
  • Pergamon

    (Publisher)

  III. Commission 2 Transfer of heat. Thermal properties of materials. Instrumentation. Insulating materials· Transmission de chaleur· Propriétés thermiques des matériaux· Instruments de mesures. Matériaux isolants. SESSIONS : Problems of Insulation Problèmes d'isolation Thermodynamics Thermodynamique Heat Transfer Transmission de chaleur This page intentionally left blank Problems of Insulation Problèmes d'isolation II-2 Heat Transfer by Natural Convection in Porous Insulants Transmission de chaleur par convection naturelle dans les isolants poreux G. MARTIN and G. G. HASELDEN Department of Chemical Engineering, University of Leeds, Leeds, England SOMMAIRE. On décrit un appareil dans lequel on peut mesurer la conductibilité thermique renforcée d'un isolant poreux 3 provenant des courants de convection naturelle qui s'y produisent. L'influence de la convection est augmentée par l'élévation de la pression. On indique des ré-sultats pour de la laine de laitier tassée à une densité de 145 kg/m* à une température moyenne de — 81° C et à S pressions. On décrit une méthode de prévision de l'influence de la convection. Une section à travers l'isolation est divisée en un réseau dans chapue carré duquel les bilans thermiques et massignes sont résolus successivement par une calcultatrice digitale jusqu'à ce qu'on obtienne le point de convergence. L'influence de la convection déterminée expérimentalement est plus élevée que celle prévue^ ce qui montre qu'il peut exister des courants multiples. INTRODUCTION In most cases in which a temperature gradient exists in a granular or fibrous insulating material there will be a tendency for bulk convection currents to occur within the insulant leading to enhanced heat transmission. Whether the effect is significant will depend on the porosity of the insulant and size factors. The presence of these currents has been demon-strated by a number of workers [1, 2, 3].

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