Heat conduction, law

The equation for the conservation of energy is similar to that for mass conservation. The equation is obtained following similar steps as the diffusion equation starting from the equation for the conservation of energy, combining it with the constitutive heat conduction law (Fourier s law), which is similar to Pick s law (in fact. Pick s law was proposed by analogy to Fourier s law), the following heat conduction equation (Equation 3-1 lb) is derived ... 

The local heat flux from the surface to the fluid at any x value may be computed by Fourier s heat conduction law,... 

Although this simplified version of Fourier s heat conduction law is well known to be an accurate constitutive model for many real gases, liquids, and solids, it is important to keep in mind that, in the absence of empirical data, it is no more than an educated guess, based on a series of assumptions about material behavior that one cannot guarantee ahead of time to be satisfied by any real material. This status is typical of all constitutive equations in continuum mechanics, except for the relatively few that have been derived by means of a molecular theory. 

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... 

In particular, if the heat flux vector is given by Fourier s heat conduction law, q = kV T, then the components are... 

Fourier s Law of Heat Conduction. The heat-transfer rate,, per unit area,, in units of W/m (Btu/(ft -h)) transferred by conduction is directly proportional to the normal temperature gradient ... 

The Tube Wall Tubular heat exchangers are built using a number of circular (or noncircular) tubes thus, the heat-transfer rate across tubular walls, following Fourier s law of heat conduction, becomes... 

Thermal conductivity describes the ease with which conductive heat can flow through a vapor, hquid, or sohd layer of a substance. It is defined as the proportionahty constant in Fourier s law of heat conduction in units of energy length/time area temperature e.g., W/m K. 

Conduction takes place at a solid, liquid, or vapor boundary through the collisions of molecules, without mass transfer taking place. The process of heat conduction is analogous to that of electrical conduction, and similar concepts and calculation methods apply. The thermal conductivity of matter is a physical property and is its ability to conduct heat. Thermal conduction is a function of both the temperature and the properties of the material. The system is often considered as being homogeneous, and the thermal conductivity is considered constant. Thermal conductivity, A, W m, is defined using Fourier s law. 

A simple case of heat conduction is a plate of finite thickness but infinite in other directions. If the temperature is constant around the plate, the material is assumed to have a constant thermal conductivity. In this case the linear temperature distribution and the heat flow through the plate is easy to determine from Fourier s law (Eq. (4.154)). 

Equations (18) and (19) are valid too, in hydrod5mamic regions for calculating surface temperature, if we assume that viscous heating concentrates on the middle layer of lubricating films and temperature varies linearly across the film [20]. TheFourier law of heat conduction gives rise to the following expressions ... 

For the heat conduction equation with a heat source depending on the temperature in accordance with the law... 

The relationship between the diffusional flux, i.e., the molar flow rate per unit area, and concentration gradient was first postulated by Pick [116], based upon analogy to heat conduction Fourier [121] and electrical conduction (Ohm), and later extended using a number of different approaches, including irreversible thermodynamics [92] and kinetic theory [162], Pick s law states that the diffusion flux is proportional to the concentration gradient through... 

This section deals with problems involving diffusion and heat conduction. Both diffusion and heat conduction are described by similar forms of equation. Pick s law for diffusion has already been met in Sec. 1.2.2 and the similarity of this to Pourier s law for heat conduction is apparent. 

This result for the most likely change in moment is equivalent to Fourier s law of heat conduction. To see this take note of the fact that in the steady state the total rate of change of moment is zero, E = 0, so that the internal change is... 

Fourier s law of heat conduction, reservoirs, second entropy, 63-64 Fourier transform ... 

Fick first recognized the analogy among diffusion, heat conduction, and electrical conduction and described diffusion on a quantitative basis by adopting the mathematical equations of Fourier s law for heat conduction or Ohm s law for electrical conduction [1], Fick s first law relates flux of a solute to its concentration gradient, employing a constant of proportionality called a diffusion coefficient or diffu-sivity ... 

The first step in the process is to relate heat flow to a temperature gradient, just as a diffusive flux can be related to a concentration gradient. The fundamental law of heat conduction was proposed by Jean Fourier in 1807 and relates the heat flux (q) to the temperature gradient ... 

To use Fourier s law of heat conduction, a thermal balance must first be constructed. The energy balance is performed over a thin element of the material, x to x + Ax in a rectangular coordinate system. The energy balance is shown in equation 13 ... 

Heat conduction in one dimensional systems Fourier law, chaos, and heat control... 

In this paper we give a brief review of the relation between microscopic dynamical properties and the Fourier law of heat conduction as well as the connection between anomalous conduction and anomalous diffusion. We then discuss the possibility to control the heat flow. 

Heat conductivity has been studied by placing the end particles in contact with two thermal reservoirs at different temperatures (see (Casati et al, 2005) for details)and then integrating the equations of motion. Numerical results (Casati et al, 2005) demonstrated that, in the small uj regime, the heat conductivity is system size dependent, while at large uj, when the system becomes almost fully chaotic, the heat conductivity becomes independent of the system size (if the size is large enough). This means that Fourier law is obeyed in the chaotic regime. 

Indeed, numerical results in (Li et al, 2003) show that in the irrational case (when the ratio 6/ir and 4>/ir are irrational numbers) the system in Fig 3 exhibits normal diffusion and the heat conduction obeys the Fourier law. In the rational case instead, the system shows a superdiffusive behavior, (a2) = 2Dt1178 (Li et al, 2003)and the heat conductivity diverges with the system size as jy0.25 o.oi ... 

Numerical experiments have shown that in many one dimensional systems with total momentum conservation, the heat conduction does not obey the Fourier law and the heat conductivity depends on the system size. For example, in the so-called FPU model, k IP, with (3 = 2/5, and if the transverse motion is introduced, / = 1/3. Moreover, in the billiard gas channels (with conserved total momentum), the value of P differs from model to model(Li and Wang, 2003). The question is whether one can relate / to the dynamical and statistical properties of the system. 

This relation connects heat conduction and diffusion, quantitatively. As expected, normal diffusion (a =1)corresponds to the size-independent (/ = 0) heat conduction obeying the Fourier law. Moreover, a ballistic motion (a = 2) implies that the thermal conductivity is proportional to... 

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