Fourier heat-conduction

Non-uniform temperature distribution in a reactor assumed model based on the Fourier heat conduction in an isotropic medium equality of temperatures of the medium and the surroundings assumed at the boundary critical values of Frank-Kamenetskii number given. 

Common Dimensionless Groups and Their Relationships 507 for example, the Fourier heat-conduction equation qi = —XVt will be replaced by ... 

Yuen, W. W. and Lee, S. C. (1989) Non-Fourier Heat Conduction in a Semi-Infinite Solid Subjected to Oscillatory Surface Thermal Disturb ices, Journal of Heat Transfer, Vol. Ill, pp. 178-181. 

Tan, Z. M., and Yang, W.-J. (1997) Non-Fourier Heat Conduction in a Thin Film Subjected to a Sudden Temperature Change on Two Sides, Journal of Non-Equilibrium Thermodynamics, Vol. 22, pp. 75-87. 

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). 

It is important to emphasize that the mathematical constraint imposed by coordinate invariance addresses only the selection of an allowable form of a constitutive equation, given the physical assumption, based on an educated guess, that there is a linear relationship between q and VO. Whether the resulting constitutive equation captures the behavior of any real material is really a question of whether the physical assumption of linearity is an adequate approximation. In fact, in the generalized Fourier heat conduction model, Eq. (2-65), there are several additional physical assumptions that must be satisfied, besides linearity between q and V0 ... 

Another feature of the present theory is that it provides a formalism for deducing a complete mathematical representation of a phenomenon. Such a representation consists, typically, of (1) Balance equations for extensive properties (such as the "equations of change" for mass, energy and entropy) (2) Thermokinematic functions of state (such as pv = RT, for simple perfect gases) (3) Thermokinetic functions of state (such as the Fourier heat conduction equation = -k(T,p)VT) and (4) The auxiliary conditions (i.e., boundary and/or initial conditions). The balances are pertinent to all problems covered by the theory, although their formulation may differ from one problem to another. Any set of... 

The Darcy law for liquid, the Darcy law for gas, the Pick law for the relative velocity of vapour, and the Fourier heat conduction law, respectively, get the form... 

The thermal diffusivity can also be measured directly by employing transient heat conduction. The basic differential equation (Fourier heat conduction equation) governing heat conduction in isotropic bodies is used in this method. A rectangular copper box filled with grain is placed in an ice bath (0°C), and the temperature at its center is recorded [44]. The solution of the Fourier equation for the temperature at the center of a slab is used ... 

To demonstrate the development of the integral transform pair in a practical way, consider the Fickian diffusion or Fourier heat conduction problem in a slab object (Fig. 11.2)... 

The meshless methods have found their applications in various fluid mechanics and solid mechanics as well as multidisciplinary problems. They have also been successfully used for multiphysics problems such as the use of magnetohydrodynamics (MHD) to control turbulence and the study of non-Fourier heat conduction. 

The well-known dual-phase-lag heat conduction model introduces time delays to account for the responses among the heat flux vector, the temperature gradient and the energy transport. The dual-phase-lag heat conduction model has been used to interpret the non-Fourier heat conduction phenomena. The onedimensional dual-phase-lag constitutive equation relating heat flux to temperature gradient is expressed as (Xu, 2011 Zhou et al., 2009)... 

For the design and analysis of fixed-bed catalytic reactors as well as the determination of catalyst efficiency under nonisothermal conditions, the effective thermal conductivity of the porous pellet must be known. A collection of thermal conductivity data of solids published by the Thermophysical Properties Research Centre at Purdue University [ ] shows "a disparity in data probably greater than that of any other physical property ". Some of these differences naturally can be explained, as no two samples of solids, especially porous catalysts, can be made completely identical. However, the main reason is that the assumed boundary conditions for the Fourier heat conduction equation... 

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. 

Fourier heat conduction equation for an isotropic material ... 

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