Page 283 - Bird R.B. Transport phenomena

§9.1 Fourier's Law of Heat Conduction (Molecular Energy Transport) 267

ture gradient, or, to put it pictorially, "heat slides downhill on the temperature versus
distance graph/' Actually Eq. 9.1-2 is not really a "law" of nature, but rather a sugges-
tion, which has proven to be a very useful empiricism. However, it does have a theoreti-
cal basis, as discussed in Appendix D.
If the temperature varies in all three directions, then we can write an equation like
Eq. 9.1-2 for each of the coordinate directions:

If each of these equations is multiplied by the appropriate unit vector and the equations
are then added, we get

q = -kVT (9.1-6)

which is the three-dimensional form of Fourier's law. This equation describes the molec-
ular transport of heat in isotropic media. By "isotropic" we mean that the material has
no preferred direction, so that heat is conducted with the same thermal conductivity к in
all directions.
Some solids, such as single noncubic crystals, fibrous materials, and laminates, are
3
anisotropic. For such substances one has to replace Eq. 9.1-6 by
q=-[K-VT] (9.1-7)

in which к is a symmetric second-order tensor called the thermal conductivity tensor.
Thus, the heat flux vector does not point in the same direction as the temperature gra-
dient. For polymeric liquids in the shearing flow v (y, t), the thermal conductivity may
x
increase above the equilibrium value by 20% in the x direction and decrease by 10% in
the z direction. Anisotropic heat conduction in packed beds is discussed briefly in §9.6.

1
J. B. Fourier, Theorie analytique de la chaleur, CEuvres de Fourier, Gauthier-Villars et Fils, Paris (1822).
(Baron) Jean-Baptiste-Joseph Fourier (pronounced "Foo-ree-ay") (1768-1830) was not only a brilliant
mathematician and the originator of the Fourier series and the Fourier transform, but also famous as an
Egyptologist and a political figure (he was prefect of the province of Isere).
2
Some authors prefer to write Eq. 9.1-2 in the form

q j = hk < 9 л 2 а )
' ~ % '
in which J is the "mechanical equivalent of heat," which displays explicitly the conversion of thermal
c
units into mechanical units. For example, in the c.g.s. system one would use the following units: q [ = ]
xi
2
erg/cm • s,k [ = ] cal/cm • s • C, T [ = ] С, у [ = ] cm, and J [=] erg/caL We will not use Eq. 9.1-2a in this
e
book.
Although polymeric liquids at rest are isotropic, kinetic theory suggests that when they are
3
flowing the heat conduction is anisotropic [see B. H. A. A. van den Brule, Rheol. Ada, 28, 257-266 (1989);
and C. F. Curtiss and R. B. Bird, Advances in Polymer Science, 25,1-101 (1996)]. Experimental
measurements for shear and elongational flows have been reported by D. C. Venerus, J. D. Schieber,
H. Iddir, J. D. Guzman, and A. W. Broerman, Phys. Rev. Letters, 82, 366-369 (1999); A. W. Broerman,
D. С Venerus, and J. D. Schieber, /. Chem. Phys., Ill, 6965-6969 (1999); H. Iddir, D. C. Venerus, and
J. D. Schieber, AIChE Journal, 46, 610-615 (2000). For oriented polymer solids, enhanced thermal conductiv-
ity in the direction of orientation has been measured by B. Poulaert, J.-C. Chielens, C. Vandenhaende,
J.-P. Issi, and R. Legras, Polymer Comm., 31,148-151 (1989). In connection with the bead spring models of
polymer thermal conductivity, it has been shown by R. B. Bird, C. F. Curtiss, and K. J. Beers [Rheol. Ada,
36, 269-276 (1997)] that the predicted thermal conductivity is exceedingly sensitive to the form of the
potential energy used for describing the springs.