Fouriers Law of Heat Conduction

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. 

The Fourier law of heat conduction relates the flux of heat q per unit area, as a result of a temperature gradient, such that... 

Notice the similarity between Eq. (11-1) and the Fourier law of heat conduction. 

A physical interpretation of Equation (35) is possible if one notes that it is mathematically analogous to Fourier law of heat conduction. The constant factor in the right-hand side plays the role of thermal conductivity, and the local incident radiation GA(r) plays the role of temperature. In that sense, differences in the latter variable among neighboring regions in the medium drive the diffusion of radiation toward the less radiated zone. Note that the more positive the asymmetry parameter, the higher the conductivity that is, forward scattering accelerates radiation diffusion while backscatter-ing retards it. 

It is important to emphasize that thermodynamic force Xq is a vector, whereas Xq is its Cartesian component corresponding to the Cartesian coordinate i of heat flux Jq. The centuries old practice states the well known relationships between heat fluxes and temperature gradients, which are expressed by the Fourier law of heat conduction... 

The second fact that is implicit in macroscopic or continuum laws is the idea of local thermodynamic equilibrium. For example, when we write the Fourier law of heat conduction, it is inherently assumed that one can define a temperature at any point in space. This is a rather severe assumption since temperature can be defined only under thermodynamic equilibrium. The question that we might ask is the following. If there is thermodynamic equilibrium in a system, then why should there be any net transport of energy Thus, we implicitly resort to the concept of local thermodynamic equilibrium, where we assume that thermodynamic equilibrium can be defined over a volume which is much smaller than the overall size of the system. What happens when the size of the object becomes on the order of this volume Obviously, the macroscopic or continuum theories break down and new laws based on nonequilibrium thermodynamics need to be formulated. This chapter focuses on developing more generalized theories of transport which can be used for nonequilibrium conditions. This involves going to the root of the macroscopic or continuum theories. 

Now consider the next larger length and timescales or , and x or xr. When L , r and t x, xr, transport is ballistic in nature and local thermodynamic equilibrium cannot be defined. This transport is nonlocal in space. One has to resort to time-averaged statistical particle transport equations. On the other hand, if L , , and t x, xr, then approximations of local thermodynamic equilibrium can be assumed over space although time-dependent terms cannot be averaged. The nonlocality is in time but not in space. When both L , r and t x, xr, statistical transport equations in full form should be used and no spatial or temporal averages can be made. Finally, when both L , , and t x, xr, local thermodynamic equilibrium can be applied over space and time leading to macroscopic transport laws such as the Fourier law of heat conduction. 

This is the Fourier law of heat conduction with the thermal conductivity being k = Cvi 3. Note that we have not made any assumption of the type of energy carrier and, hence, this is a universal law for all energy carriers. The only assumption made is that of local thermodynamic equilibrium such that the energy density u at any location is a function of the local temperature. 

The first term containing /0 drops out since the integral over all the directions becomes zero. Equation 8.18 is the Fourier law of heat conduction with the integral being the thermal conductivity k. If one assumes that the relaxation time and velocity are independent of particle energy, then the integral becomes... 

Kinetic theory is introduced and developed as the initial step toward understanding microscopic transport phenomena. It is used to develop relations for the thermal conductivity which are compared to experimental measurements for a variety of solids. Next, it is shown that if the time- or length scale of the phenomena are on the order of those for scattering, kinetic theory cannot be used but instead Boltzmann transport theory should be used. It was shown that the Boltzmann transport equation (BTE) is fundamental since it forms the basis for a vast variety of transport laws such as the Fourier law of heat conduction, Ohm s law of electrical conduction, and hyperbolic heat conduction equation. In addition, for an ensemble of particles for which the particle number is conserved, such as in molecules, electrons, holes, and so forth, the BTE forms the basis for mass, momentum, and energy conservation equa-... 

Now the rate of heat flow or the heat current Jq = dQ/dt is given by the laws of heat conduction. For example according to the Fourier law of heat conduction, Jq = a(r 1 — T2), in which a is the coefficient of heat conductivity. Note that the thermod3mamic flow Jq is driven by the thermodynamic force F = (1/7 2 — 1/Ti). For the rate of entropy production we have from (3.5.2) that... 

For anisotropic solids the heat conductivity k is a tensor of the second rank. The empirical Fourier law of heat conduction is then written as... 

Such a system reaches a state with stationary temperature distribution and a uniform heat flow J. (A stationary temperature T x) implies that the heat flow is uniform otherwise there will be an accumulation or depletion of heat, resulting in a time-dependent temperature.) The evolution of the temperature distribution can be obtained explicitly by using the Fourier laws of heat conduction ... 

Thermal Conductivity. The Fourier law of heat conduction is applicable to steady-state heat transfer in polymers. For unidirectional, rectilinear heat fiow, it leads to the well-known relationship between heat fiux and temperature gradient... 

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