(224g) On the Violation of Laws of Thermodynamics by Laws of Heat Conduction

There are six reasons to seek a generalized Fourier's law of heat conduction. The reasons are the violation of Onsager's law of microscopic reversibility, Landau's observation that light has the speediest velocity and heat velocity has to be no greater than the speed of light, singularties found in the model solution of Fourier model for transient heat conduction to represent flux and temperature, empirical nature of the development of Fourier's law, Casimir limit at nanoscales and overprediction of theory to experiment in a number of widley used industrial processes such as fludized bed combustion, CPU overheating, chromotography, PAGE, polyacrylamide gel electrophoresis, laser heating of semi-conductors, etc. Maxwell and later Cattaneo (1948) & Vernotte (1958) hypothesized the hyperbolic heat conduction law.

Early investigators like Taitel (1972) solved the transient hyperbolic heat conduction equation for a finite slab subject to a constant wall temperature boundary condition and found that in the model solution the temperature inside the slab was found to be greater than the boundary temperature. This is referred to as the "temperature overshoot" problem. Later Bai and Lavine (1995) and Barletta and Zanchini (1997) discuss a violation of second law of thermodynamics by the Cattaneo & Vernotte Law.

Sharma (2005,2006) showed that the damped wave conductin and relaxation when solved for a finite slab subject to a constant wall boundary condition using the FINAL condition in time as the fourth condition leads to well bounded solutions. These solutions are not in any violation of Clausius' inequality.

Further in this study it is shown that the law of heat conduction does not violate the law of thermodynamics. But adequate care must be taken to define the boundary and time conditions such as the initial condition. For instance, consider a long rod with a temperature dependant heat source. Even when Fourier's law of heat conduction is used at steady state a critical region can be found beyond which there would not be any further heat transfer. After this critical length there can be negative temperature or "temperature undershoot". Can this be concluded as violation of the third law of thermodynamics ? i.e., does Fourier's law of heat condution violate third law of thermodynamics ? With the extensive use of Fourier's law of heat conduction and how it is taught at universities this cannot be. Upon furhter investigation it can be seen that the boundary condition at x = l, T = 0 K cannot be specified. In fact x = lc where lc is a critical length has to be calculated beyond which there would be any heat transfer.

In the same fashion, the damped wave conduction and relaxation equation has to be used with care. The time taken to steady state can be calculated for the case when the finite slab is subject to a constant wall temperature.

So in the reports of the "temperature overshoot" care must be taken in defining the fourth time condition. This can be at steady state or what have you. But it cannot be done indiscriminately. For a second order hyperbolic PDE with second order with respect to both time and space four conditions have to be specified. These cannot be any set of conditions. They have to be consistent with the laws of thermodynamics and heat transfer.