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 called transport 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 temperature, 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  

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. The gradient of a scalar function /(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 VT  

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 

Assume that the concentration of substance number 2 is represented by the function 

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The macroscopic description of nonequilibrium states of fluid systems requires independent variables to specify the extent to which the system deviates from equilibrium and dependent variables to express the rates of processes. 

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