## Abstract

As was discussed in Chapter 2, heat transfer is a dissipative phenomenon, and if it represents the only source of dissipation the second law requires that, at steady state, the scalar product of the heat flux and the temperature gradient should be nonpositive:
[We shall see in more detail in Section 7.5 that the second law reduces to the requirement in equation (7.1.1) only for steady-state phenomena. In other words, for unsteady-state phenomena, the second law does not forbid heat to flow in the direction of increasing temperature, if only for short intervals of time.]

$$\mathrm{\mathbf{q}\cdot grad}\;T\;\leq \;0$$

(7.1.1)

## Keywords

Heat Flux Constitutive Equation Entropy Production Diffusive Flux Obtuse Angle
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## Literature

- In this chapter, most of the dissipative phenomena which have been considered belong to the class usually identified with transport phenomena, the driving force being the gradient of some potential such as temperature. The classical textbook for these is R. B. Bird, W. E. Stewart, and E. N. Lightfoot,
*Transport Phenomena*, Wiley, New York (1960).Google Scholar - However, it should be borne in mind that all phenomena of evolution of internal state variables (such as, typically, chemical reactions) are also dissipative, although no gradients of any quantity are required for such processes. The TIP literature does not clearly distinguish between these two types of dissipative phenomena.Google Scholar
- Data showing that in multicomponent mixtures the mobility matrix is not invariably symmetrical are presented by P. J. Dunlap and L. J. Gosting,
*J. Am. Chem. Soc.***77**, 5238 (1955).CrossRefGoogle Scholar - The possibility that a relaxation time may appear in the constitutive equation for the heat flux, and the consequences thereof, have been discussed by R. Ocone and G. Astarita,
*AIChE J.***33**, 423 (1987); the idea goes back to the work ofCrossRefGoogle Scholar - Relaxation in heat flux in Maxwellian gases was first analyzed by J. C. Maxwell himself,
*Scientific Papers*, Cambridge University Press (1890). See in this regard also E. Ikenberry and C. Truesdell,*J. Ratl. Mech. Anal.***5**, 1 (1956) andGoogle Scholar - The possibility that, on sudden reversal of the direction of shearing, a Maxwellian gas may exhibit for a short time a tangential stress pointing in a direction opposite to that of shearing is discussed by C. Truesdell,
*Rational Thermodynamics*, Springer-Verlag, Berlin (1984).CrossRefGoogle Scholar - The fact that, if the kinetic energies of diffusion are not neglected, the diffusion equations become hyperbolic and hence discontinuities propagate at finite speeds is discussed by I. Muller and P. Villaggio,
*Meccanica***11**, 191 (1976).CrossRefGoogle Scholar - The literature on TIP is very abundant; it is generally difficult to read and overwhelmingly confusing. The best presentation is given by K. Denbigh,
*The Thermodynamics of the Steady State*, Methuen, London (1951). This is also the only presentation to clarify that, if at all, the theory is applicable only to steady-state phenomena.Google Scholar - The earliest, and possibly still today the clearest discussion of coupling is the analysis of the thermoelectric effect presented by Lord Kelvin,
*Mathematical and Physical Papers*, Cambridge University Press, London (1922), Vol. 1, p. 232.Google Scholar - The viewpoint expressed in this chapter follows that of C. A. Truesdell,
*Rational Thermodynamics*, 2nd Ed., Springer-Verlag, New York (1985).Google Scholar - For a discussion of the difficulties related to the Onsager reciprocal relationships, see B. D. Coleman and C. A. Truesdell,
*J. Chem. Phys.***33**, 28 (1960).CrossRefGoogle Scholar - While the symmetry relations are usually regarded as following from the so-called principle of microscopic reversibility, some authors regard them as a postulate of the theory; see, e.g., E. A. Desloge,
*Thermal Physics*, Holt, Rinehart, and Wilson, New York (1968). From this viewpoint, the question of proof does not arise; however, the question of the validity of the postulate becomes crucial.Google Scholar - The question of the relationship between the Helmholtz-Korteweg variational principle and entropy production is discussed by G. Astarita,
*J. Non-Newtonian Fluid Mech.***2**, 343 (1977) and**13**, 223 (1983).CrossRefGoogle Scholar - The talandic theory of the thermodynamics of biological systems is discussed in detail by B. C. Goodwin,
*Temporal Organization in Cells*, Academic Press, New York (1963).Google Scholar

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