Page 390 - Bird R.B. Transport phenomena

372 Chapter 11 The Equations of Change for Nonisothermal Systems

(a) Verify the dimensional consistency of each interfacial balance equation.
(b) Under what conditions are v and v 11 equal?
1
(c) Show how the balance equations simplify when phases I and II are two pure immiscible
liquids.
(d) Show how the balance equations simplify when one phase is a solid.

11C.7. Effect of surface-tension gradients on a falling film.
(a) Repeat the determination of the shear-stress and velocity distributions of Example 2.1-1 in
the presence of a small temperature gradient dT/dz in the direction of flow. Assume that this
temperature gradient produces a constant surface-tension gradient dcr/dz = A but has no
other effect on system physical properties. Note that this surface-tension gradient will pro-
duce a shear stress at the free surface of the film (see Problem 11C.6) and, hence, will require a
nonzero velocity gradient there. Once again, postulate a stable, nonrippling, laminar film.
(b) Calculate the film thickness as a function of the net downward flow rate and discuss the
physical significance of the result.
2
ряд 2 cos /3 Г /v\ l A8 ( x\
Answer: (a) r xz = pgxcosfi + A;v z = ™ 0<> h _ Ш + ^ 1 - f
11D.1. Equation of change for entropy. This problem is an introduction to the thermodynamics of
irreversible processes. A treatment of multicomponent mixtures is given in §§24.1 and 2.
(a) Write an entropy balance for the fixed volume element Ax Ay Az. Let s be the entropy flux
vector, measured with respect to the fluid velocity vector v. Further, let the rate of entropy pro-
duction per unit volume be designated by g . Show that when the volume element Ax Ay Az is
s
allowed to become vanishingly small, one finally obtains an equation of change for entropy in ei-
ther of the following two forms: 11

jpS= -(V • pSv) - (V • s) + g s (11D.1-1)
t

in which S is the entropy per unit mass.
(b) If one assumes that the thermodynamic quantities can be defined locally in a nonequilib-
rium situation, then U can be related to S and V according to the thermodynamic relation
dU = TdS - pdV. Combine this relation with Eq. 11.2-2 to get
Щ ~\ ' " \ (HD.1-3)
9 = ( V q ) ( T : W )

12 15
(c) The local entropy flux is equal to the local energy flux divided by the local temperature " ;
that is, s = q/T. Once this relation between s and q is recognized, we can compare Eqs. 11 D.I-2
and 3 to get the following expression for the rate of entropy production per unit volume:
gs = ~ 2 (q • VT) - \ (T:VV) (11D.1-4)
г

11
G. A. J. Jaumann, Sitzungsber. der Math.-Naturwiss. Klasse der Kaiserlichen Akad. der Wissenschaften
(Wien), 102, Abt. Ha, 385-530 (1911).
12
Carl Henry Eckart (1902-1973), vice-chancellor of the University of California at San Diego
(1965-1969), made fundamental contributions to quantum mechanics, geophysical hydrodynamics,
and the thermodynamics of irreversible processes; his key contributions to transport phenomena are
in С. Н. Eckart, Phys. Rev., 58, 267-268, 269-275 (1940).
13 C. F. Curtiss and J. O. Hirschfelder, /. Chem. Phys., 18,171-173 (1950).
14 J. G. Kirkwood and B. L. Crawford, Jr., /. Phys. Chem. 56,1048-1051 (1952).
ь S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland, Amsterdam (1962).