Page 295 - Bird R.B. Transport phenomena

§9.4 Theory of Thermal Conductivity of Liquids 279

§9.4 THEORY OF THERMAL CONDUCTIVITY OF LIQUIDS
A very detailed kinetic theory for the thermal conductivity of monatomic liquids was
developed a half-century ago, 1 but it has not yet been possible to implement it for prac-
tical calculations. As a result we have to use rough theories or empirical estimation
methods. 2
We choose to discuss here Bridgman's simple theory 3 of energy transport in pure
liquids. He assumed that the molecules are arranged in a cubic lattice, with a center-
1/3
to-center spacing given by (У/Д/) , in which V/N is the volume per molecule. He
further assumed energy to be transferred from one lattice plane to the next at the
sonic velocity v s for the given fluid. The development is based on a reinterpretation of
Eq. 9.3-11 of the rigid-sphere gas theory:

к = ЬСупХ = рС \щ\а (9.4-1)
у
The heat capacity at constant volume of a monatomic liquid is about the same as for a
solid at high temperature, which is given by theJDulong and Petit formula 4 C v = 3(к/ш).
The mean molecular speed in the у direction, | u \, is replaced by the sonic velocity v . The
y s
distance a that the energy travels between two successive collisions is taken to be the lat-
1/3
tice spacing (V7N) . Making these substitutions in Eq. 9.4-1 gives
к = 3(N/V) 2/3 KV (9.4-2)
S
which is Bridgman's equation. Experimental data show good agreement with Eq. 9.4-2,
even for polyatomic liquids, but the numerical coefficient is somewhat too high. Better
agreement is obtained if the coefficient is changed to 2.80:

к = 2.80(N/V) 2/3 KV S (9.4-3) 5

This equation is limited to densities well above the critical density, because of the tacit
assumption that each molecule oscillates in a "cage" formed by its nearest neighbors.
The success of this equation for polyatomic fluids seems to imply that the energy trans-
fer in collisions of polyatomic molecules is incomplete, since the heat capacity used here,
C v = 3(к/т), is less than the heat capacities of polyatomic liquids.
The velocity of low-frequency sound is given (see Problem 11C.1) by

The quantity (др/др) т may be obtained from isothermal compressibility measurements
or from an equation of state, and (C /C ) is very nearly unity for liquids, except near the
p
v
critical point.

1
J. H. Irving and J. G. Kirkwood, /. Chem. Phys., 18, 817-829 (1950). This theory has been extended
to polymeric liquids by C. F. Curtiss and R. B. Bird, /. Chem. Phys., 107, 5254-5267 (1997).
2
R. C. Reid, J. M. Prausnitz, and В. Е. Poling, The Properties of Gases and Liquids, McGraw-Hill,
New York (1987); L. Riedel, Chemie-Ing.-Techn., 27, 209-213 (1955).
3
P. W. Bridgman, Proc. Am. Acad. Arts and Sci., 59,141-169 (1923). Bridgman's equation is often
misquoted, because he gave it in terms of a little-known gas constant equal to |к.
4
This empirical equation has been justified, and extended, by A. Einstein [Ann. Phys. [4], 22,
180-190 (1907)] and P. Debye [Ann. Phys., [4] 39, 789-839 (1912)].
5
Equation 9.4-3 is in approximate agreement with a formula derived by R. E. Powell,
W. E. Roseveare, and H. Eyring, Ind. Eng. Chem., 33,430-435 (1941).