442 10 Transport Processes
10.1 The Macroscopic Description of
Nonequilibrium States
In earlier chapters we were able to calculate changes in thermodynamic state functions
for nonequilibrium processes that began with equilibrium or metastable states and
ended with equilibrium states. In this chapter we present a nonthermodynamic analysis
of three nonequilibrium processes: heat conduction, diffusion, and viscous flow. These
processes are calledtransport processes, since in each case some quantity is transported
from one location to another. We will discuss only systems that do not deviate too
strongly from equilibrium, excluding turbulent flow, shock waves, supersonic flow,
and the like.
In order to discuss the macroscopic nonequilibrium state of a fluid system, we will
first assume that we can use intensive thermodynamic variables such as the temper-
ature, pressure, density, concentrations, and chemical potentials. In order to justify
this assumption we visualize the following process: A small portion of the system is
suddenly removed from the system and allowed to relax adiabatically to equilibrium
at fixed volume. Once equilibrium is reached, intensive thermodynamic variables are
well defined and can be measured. The measured values are assigned to a point inside
the volume originally occupied by this portion of the system and to the time at which
the subsystem was removed. We imagine that this procedure is performed repeatedly
at different times and different locations in the system. Interpolation procedures are
carried out to obtain smooth functions of position and time to represent the temperature,
pressure, and concentrations:TT(x,y,z,t)T(r,t) (10.1-1)
PP(x,y,z,t)P(r,t) (10.1-2)
cici(x,y,z,t)ci(r,t)(i1, 2,...,s) (10.1-3)whereciis the molar concentration of substancei. The position vector with components
x,y, andzis denoted byr.
The values of these intensive variables at a point are not sufficient for a complete
description of the nonequilibrium state at that point. We also need a measure of how
strongly the variables depend on position. We use the gradients of these variables for
this purpose. Thegradientof a scalar functionf(x,y,z) is defined by Eq. (B-43) of
Appendix B. It is a vector derivative that points in the direction of the most rapid
increase of the function and has a magnitude equal to the derivative with respect to
distance in that direction. The gradient of the temperature is denoted by∇T:∇Ti∂T
∂x+j∂T
∂y+k∂T
∂z(10.1-4)
wherei,j, andkare unit vectors in the directions of thex,y, andzaxes. Thegra-
dient operator∇is sometimes called “del.” The concentration gradient of substance
numberiis∇cii∂ci
∂x+j∂ci
∂y+k∂ci
∂z(10.1-5)
Gradients of other variables are defined in the same way. Gradients can also be expressed
in other coordinate systems. Appendix B contains the formulas for the gradient in
spherical polar coordinates and cylindrical polar coordinates. When possible we will
discuss cases in which variables depend on only one Cartesian coordinate, so that only