Page 391 - Bird R.B. Transport phenomena

Problems 373

The first term on the right side is the rate of entropy production associated with heat trans-
port, and the second is the rate of entropy production resulting from momentum transport.
Equation 11 D.I-4 is the starting point for the thermodynamic study of the irreversible
processes in a pure fluid.
(d) What conclusions can be drawn when Newton's law of viscosity and Fourier's law of
heat conduction are inserted into Eq. 11 D.I-4?
11D.2. Viscous heating in laminar tube flow.
(a) Continue the analysis begun in Problem 11B.2—namely, that of finding the temperature
profiles in a Newtonian fluid flowing in a circular tube at a speed sufficiently high that vis-
cous heating effects are important. Assume that the velocity profile at the inlet (z = 0) is fully
developed, and that the inlet temperature is uniform over the cross section. Assume all physi-
cal properties to be constant.
(b) Repeat the analysis for a power law non-Newtonian viscosity. 16
11D.3. Derivation of the energy equation using integral theorems. In §11.1 the energy equation is
derived by accounting for the energy changes occurring in a small rectangular volume ele-
ment Ax Ay Az.
(a) Repeat the derivation using an arbitrary volume element V with a fixed boundary S by
following the procedure outlined in Problem 3D.1. Begin by writing the law of conservation
of energy as

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4-1 (pU + \pv ) dV= - f (n • e)dS + f (v • g) dV (11D.3-1)
at J v J J v
Then use the Gauss divergence theorem to convert the surface integral into a volume integral,
and obtain Eq. 11.1-6.
(b) Do the analogous derivation for a moving "blob" of fluid.

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R. B. Bird, Soc. Plastics Engrs. Journal, 11, 35-40 (1955).