Page 389 - Bird R.B. Transport phenomena

Problems 371
(a) For the "true" system we know that at a large distance L from the system (i.e., L » R'),
the temperature field will be given by a slight modification of Eq. 11B.8-2, provided that the
tiny occluded spheres are very "dilute" in the true system:

T (r, в) - Г = 1 - n ^ ~ *j° ( у ) \Ar cos в (11С.5-1)
0
|_ k } + 2/c 0 \ / J
Explain carefully how this result is obtained.
(b) Next, for the "equivalent system," we can write from Eq. 11B.8-2
fce
fc
feTV
Г(г, в) - Г = Tl - , "~ ,° feT cos в (ПС.5-2)
0
Zk
L ^eff + 0 \ / / J
3
(c) Next derive the relation nR = cf>R' , in which ф is the volume fraction of the occlusions in
3
the "true system."
7
(d) Equate the right sides of Eqs. 11C.5-1 and 2 to get Maxwell's equation in Eq. 9.6-1.
11C.6. Interfacial boundary conditions. Consider a nonisothermal interfacial surface S(t) be-
tween pure phases I and II in a nonisothermal system. The phases may consist of two im-
miscible fluids (so that no material crosses S(0), or two different pure phases of a single
substance (between which mass may be interchanged by condensation, evaporation, freez-
ing, or melting). Let n be the local unit normal to S(t) directed into phase I. A superscript I
1
or II will be used for values along S in each phase, and a superscript s for values in the in-
terface itself. The usual interfacial boundary conditions on tangential velocity v t and tem-
perature T on S are
v) = v) 1 (no slip) (11C.6-1)
T 1 = T" (continuity of temperature) (11C.6-2)
In addition, the following simplified conservation equations are suggested 8 for surfactant-free
interfaces:
Interfacial mass balance
s
s
(n 1 • {pW - v ) - p"(v" - V ))) = О (11С.6-3)
Interfacial momentum balance
11
[
1
l
n 1 (p - f) + (pV 2 - A 112 ) + o-Q- + ±)\ + [n • |T - т )] = -VV (ИС.6-4)
Interfacial internal energy balance
l
m
11
11
s
s
1
(n 1 • p V - v ))[(H ~ H ) + \{v n - v )] + (n • {q 1 - q )) = o-(V • v ) (11C.6-5)
s
The momentum balance of Eq. 3C.5-1 has been extended here to include the surface gradient
W of the interfacial tension; the resulting tangential force gives rise to a variety of interfacial
flow phenomena, known as Marangoni effects? ™ Equation 11C.6-5 is obtained in the manner
1
of §11.2, from total and mechanical energy balances on S, neglecting interfacial excess energy
s
IF, heat flux q , and viscous dissipation T :W); fuller results are given elsewhere. 8
S
(
7
J. C. Maxwell, A Treatise on Electricity and Magnetism, Vol. 1, Oxford University Press (1891,
reprinted 1998), §314.
J. C. Slattery, Advanced Transport Phenomena, Cambridge University Press (1999), pp. 58, 435; more
8
complete conditions are given in Ref. 8.
9
C. G. M. Marangoni, Ann. Phys. (Poggendorf), 3, 337-354 (1871); C. V. Sternling and L. E. Scriven,
AIChE Journal, 5, 514-523 (1959).
10
D. A. Edwards, H. Brenner, and D. T. Wasan, Interfacial Transport Processes and Rheology,
Butterworth-Heinemann, Stoneham, Mass. (1991).