Page 300 - Bird R.B. Transport phenomena

284 Chapter 9 Thermal Conductivity and the Mechanisms of Energy Transport

velocity is v. The volume rate of flow across the surface element dS perpendicular to the
x-axis is v dS. The rate at which energy is being swept across the same surface element is
x
then
(W + Wv dS (9.7-1)
x
P
2
in which \pv 2 = \p{v 2 x + v 2 + v ) is the kinetic energy per unit volume, and pU is the inter-
nal energy per unit volume.
The definition of the internal energy in a nonequilibrium situation requires some
care. From the continuum point of view, the internal energy at position r and time t is as-
sumed to be the same function of the local, instantaneous density and temperature that
one would have at equilibrium. From the molecular point of view, the internal energy
consists of the sum of the kinetic energies of all the constituent atoms (relative to the
flow velocity v), the intramolecular potential energies, and the intermolecular energies,
within a small region about the point r at time t.
Recall that, in the discussion of molecular collisions in §0.3, we found it convenient
to regard the energy of a colliding pair of molecules to be the sum of the kinetic energies
referred to the center of mass of the molecule plus the intramolecular potential energy of
the molecule. Here also we split the energy of the fluid (regarded as a continuum) into
kinetic energy associated with the bulk fluid motion and the internal energy associated
with the kinetic energy of the molecules with respect to the flow velocity and the intra-
and intermolecular potential energies.
We can write expressions similar to Eq. 9.7-1 for the rate at which energy is being
swept through the surface elements perpendicular to the y- and z-axes. If we now multi-
ply each of the three expressions by the corresponding unit vector and add, we then get,
after division by dS,

(W + рШ у + (W + pil)b v + {\pv 2 + pU)b v = (W + plfiv (9.7-2)
х х y y z z
and this quantity is called the convective energy flux vector. To get the convective energy
flux across aunit surface whose normal unit vector is n, we form the dot product
(n • dpv 2 + pU)v). It is understood that this is the flux from the negative side of the sur-
face to the positive side. Compare this with the convective momentum flux in Fig. 1.7-2.

§9.8 WORK ASSOCIATED WITH MOLECULAR MOTIONS
Presently we will be concerned with applying the law of conservation of energy to
"shells" (as in the shell balances in Chapter 10) or to small elements of volume fixed in
space (to develop the equation of change for energy in §11.1). The law of conservation of
energy for an open flow system is an extension of the first law of classical thermodynam-
ics (for a closed system at rest). In the latter we state that the change in internal energy is
equal to the amount of heat added to the system plus the amount of work done on the
system. For flow systems we shall need to account for the heat added to the system (by
molecular motions and by bulk fluid motion) and also for the work done on the system
by the molecular motions. Therefore it is appropriate that we develop here the expres-
sion for the rate of work done by the molecular motions.
First we recall that, when a force F acts on a body and causes it to move through a
distance dr, the work done is dW = (F - dr). Then the rate of doing work is dW/dt =
(F • dt/dt) = (F • v)—that is, the dot product of the force times the velocity. We now
apply this formula to the three perpendicular planes at a point P in space shown in
Fig. 9.8-1.
First we consider the surface element perpendicular to the x-axis. The fluid on the
minus side of the surface exerts a force rs dS on the fluid that is on the plus side (see
x