Page 301 - Bird R.B. Transport phenomena

§9.8 Work Associated with Molecular Motions 285

7

V

V
dS

Fig. 9.8=1. Three mutually perpendicular surface elements of area dS at point P along with
the stress vectors ъ , тг, тт acting on these surfaces. In the first figure, the rate at which
2
у
х
work is done by the fluid on the minus side of dS on the fluid on the plus side of dS is then
(TT. • v)dS = [тг • v] dS. Similar expressions hold for the surface elements perpendicular to
x
A
the other two coordinate axes.
Table 1.2-1). Since the fluid is moving with a velocity v, the rate at which work is done
by the minus fluid on the plus fluid is Ы х • v)dS. Similar expressions may be written for
the work done across the other two surface elements. When written out in component
form, these rate of work expressions, per unit area, become
• V) = + + = Ы (9.8-1)
Ы х 7T XX V X TT xy V y 7T XZ V Z • V] x
• V) = + TTyyVy + = [ТТ (9.8-2)
(iT y TT yx V x 7T yz V z • V] y
• V) = + 7T zy Vy + = [тГ v (9.8-3)
(ТТ 2 7T ZX V X 7T ZZ V Z • ] 2
When these scalar components are multiplied by the unit vectors and added, we get the
"rate of doing work vector per unit area/' and we can call this, for short, the work flux:

[тт • v] = 6 V (TT V • v) + • v) + 6 (тг 2 • v) (9.8-4)
2
Furthermore, the rate of doing work across a unit area of surface with orientation given
by the unit vector n is (n • [IT • v]).
Equations 9.8-1 to 9.8-4 are easily written for cylindrical coordinates by replacing
x, y, z by г, 0, z and, for spherical coordinates by replacing x, y, z by г, в, ф.
We now define, for later use, the combined energy flux vector e as follows:

e = (Ipv 2 + pLOv + [тг • v] + q (9.8-5)
The e vector is the sum of (a) the convective energy flux, (b) the rate of doing work (per
unit area) by molecular mechanisms, and (c) the rate of transporting heat (per unit area)
by molecular mechanisms. All the terms in Eq. 9.8-5 have the same sign convention, so
that e is the energy transport in the positive x direction per unit area per unit time.
x
The total molecular stress tensor тг can now be split into two parts: IT = pb + т
so that [тг • v] = pv + [т • v]. The term pv can then be combined with the internal
energy term pUv to give an enthalpy term pUv + pv = p(U + (p/p))v = p(U + pV)v =
pHv, so that
e = (\pv 2 + pH)v + [T • v] + q (9.8-6)
We shall usually use the e vector in this form. For a surface element dS of orientation n,
the quantity (n e e) gives the convective energy flux, the heat flux, and the work flux
across the surface element dS from the negative side to the positive side of dS.