Thermal conductivity tensor

The effective thermal conductivity tensor depends on the transformation vec-... 

Fffective thermal conductivity tensor Kinetic coefficients Flectric potential... 

Izawa K, Yamaguchi H, Sasaki T, Matsuda Y (2002) Superconducting gap stmcture of /c-(BEDT-TTF)2Cu(NCS)2 probed by thermal conductivity tensor. Phys Rev Lett 88 27002/1 ... 

This means that the resistivity and thermal conductivity tensors are no longer symmetric. For example,... 

The thermal conductivity tensor may likewise be split into symmetric and antisymmetric parts, with expansions in powers of B as in eqs. (35) and (36). But Z is not necessarily a symmetric tensor at B = 0, and so the expansion of the antisymmetric part of Z in an equation like eq. (36) is not applicable. Instead,... 

The heat flux in the direction of a coordinate axis depends on the temperature gradients in the direction of all the coordinate axes. The thermal conductivity A in Fourier s law is a tensor. As we can prove with methods of thermodynamics for irreversible processes [3.3], the thermal conductivity tensor is symmetrical As = A-s. It consists of six components. Out of these, for certain crystals a few or several agree with each other or disappear. 

Example 3.4 Quartz crystals have different thermal conductivities along the directions of the individual coordinate axes. The thermal conductivity tensor is given by... 

Here, K is a second-order tensor that is known as the thermal conductivity tensor, and the constitutive equation is known as the generalized Fourier heat conduction model for the surface heat flux vector q. The minus sign in (2-65) is a matter of convention the components of K are assumed to be positive whereas a positive heat flux is defined as going from regions of high temperature toward regions of low temperature (that is, in the direction of —V0). 

The fluid is assumed to be homogeneous. The form of the relationship between the heat flux q and the temperature gradient V0 is the same at all points. Furthermore, the only dependence of q on position x is due to the possible dependence of the so-called thermal conductivity tensor K on the thermodynamic state variables (say, p and 9) or the dependence of V0 on spatial position. 

In the two-medium treatment of the single-phase flow and heat transfer through porous media, no local thermal equilibrium is assumed between the fluid and solid phases, but it is assumed that each phase is continuous and represented with an appropriate effective total thermal conductivity. Then the thermal coupling between the phases is approached either by the examination of the microstructure (for simple geometries) or by empiricism. When empiricism is applied, simple two-equation (or two-medium) models that contain a modeling parameter hsf (called the interfacial convective heat transfer coefficient) are used. As is shown in the following sections, only those empirical treatments that contain not only As/but also the appropriate effective thermal conductivity tensors (for both phases) and the dispersion tensor (in the fluid-phase equation) are expected to give reasonably accurate predictions. 

Then we expect a functional relationship for the effective thermal conductivity tensor, of the form... 

Equation (1) is the equation of conservation of heat, where heat conduction is assumed to be the only mechanism of heat transport. In this equation, K j is the thermal conductivity tensor (W/m / C), p is the density of the bulk medium (kg/m ), C is the bulk specific heat of the medium (J/kg/ °C) and q accounts for distributed heat generation in the poro-elastic medium (W/m ). 

As the same approach, the homogenized thermal conductivity tensor A",j can be written as follows. 

A finite element simulator derived from the Equation (1) and (3) is used. In the equations the parameters of disposal panel such as permeability tensor and thermal conductivity tensor have to be decided. To specify the parameters, the homogenization theory (Ene Polisevski (1987)), Kyoya Terada(2001)) is applied. 

In order to compute the thermal conductivity tensor kj, the multitude of walls shown in Fig. 3.7 can be regarded as a network of thermal resistors. On this basis, the components of the thermal conductivity tensor are derived as... 

Time constant for Hookean dumbbell model Time constants for Rouse chain model Solvent contnbution to thermal conductivity Tensor virial multiplied by 2 Momentum space distribution function Integration variable in Taylor series Stress tensor (momentum flux tensor) External force contribution to stress tensor Kinetic contribution to stress tensor Intramolecular contribution to stress tensor Intermolecular contribution to stress tensor Fluid density... 

There are also major experimental challenges in this field. Systematic studies of the effects of velocity fields on the difliisivity and thermal conductivity tensors would be most welcome. Regrettably there are virtually no experimental data on the various cross effects for which theoretical formulas are now available. Perhaps this review will stimulate some laboratory studies of these various phenomena. 

Thermal conductivity The thermal conductivity tensor is a directional quantity depending on the orientation of temperature gradient and heat current. At low temperatures its anisotropy may contain important information on the relative position of nodal points or lines with respect to crystal axes. For example, this has proved decisive in the identification of the E2u SC order parameter of UPts (sect. 4.1). The uniaxial thermal conductivity tensor in the unitary scattering limit for a spherical Fermi surface is explicitly given (Norman and Hirschfeld, 1996 Kiibert and Hirschfeld, 1998 Machida et al., 1999 Graf et al., 2000 Wu and Joynt, 2002) as... 

Three-dimensional tensor of the thermal conductivity Tensor which, when multiplied by the temperature gradient vector according to the rules of matrix multiplication, gives the heat flux density vector, i.e., heat flux density and temperature gradient must not have the same orientation, which is the case in media with an anisotropic thermal conductivity Thiolate An organic molecule terminated with an functionality. An example is 4-fluorophenylthiolate, FCsIUS ... 

Here kij is the thermal-conductivity tensor. Note that other notations are also used in the literature, e.g. k = k or K =k. 

Van den Brule proposed a connection between the thermal conductivity tensor and the total stress tensor as follows ... 

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