Page 472 - Bird R.B. Transport phenomena

452 Chapter 14 Interphase Transport in Nonisothermal Systems

gravitational and viscous forces; (iv) sensible heat changes, may be assumed to be well insulated. The rate of liquid
C dT, in the condensate film are unimportant compared to flow through the tank has a negligible effect on the flow
p
the latent heat transferred through it; and (v) the heat flux pattern in the tank.
is very nearly normal to the wall surface. Develop a general form of dimensionless heat transfer
(a) Recall from §2.2 that the average velocity of a film of correlation for the tank corresponding to the correlation
2
constant thickness 8 is (v ) = pg8 /3fi. Assume that this re- for tube flow in §14.3. Choose the following reference
z
lation is valid for any value of z. quantities: reference length, D, the impeller diameter; ref-
(b) Write the energy equation for the film, neglecting film erence velocity, ND, where N is the rate of shaft rotation in 2
2
curvature and convection. Show that the heat flux through revolutions per unit time; reference pressure, pN D ,
the film toward the cold surface is where p is the fluid density.
- T 14D.1. Heat transfer from an oblate ellipsoid of revolu-
~q xl = k\ o (14C.1-1) tion. Systems of this sort are best described in oblate ellip-
(c) As the film proceeds down the wall, it picks up addi- soidal coordinates (f, 77, фУ for which
tional material by the condensation process. In this f = constant describes oblate ellipsoids (0 < f < 00)
process, heat is liberated to the extent of AH vap per unit 77 = constant describes hyperboloids of revolution
mass of material that undergoes the change in state. Show (0 < 77 < if)
that equating the heat liberation by condensation with the
heat flowing through the film in a segment dz of the film ф = constant describes half planes (0 < ф < 2тг)
leads to Note that f = £ 0 c a n describe oblate ellipsoids, with 0 = 0
f
being a limiting case of the two-sided disk, and the limit as
pbH d((v )8) =
vap z £ 0 —> 00 being a sphere. In this problem we investigate the
corresponding two limiting values of the Nusselt number.
(d) Insert the expression for the average velocity from (a)
into Eq. 14C.1-2 and integrate from z = 0 to z = L to obtain (a) First use Eq. A.7-13 to get the scale factors from the re-
(4k{T - T ) LY /4 lation between oblate ellipsoidal coordinates and Carte-
8(L) = d 0 M (14C.1-3) sian coordinates:
V
x = a cosh f sin 77 cos ф (14D.1-1)
(e) Use the definition of the heat transfer coefficient and у = a cosh f sin 77 sin ф (14D.1-2)
the result in (d) to obtain Eq. 14.7-5. z = a sinh f cos 77 (14D.1-3)
(f) Show that Eqs. 14.7-4 and 5 are equivalent for the con-
ditions of this problem. in which a is one-half the distance between the foci. Show
that
14C.2. Heat transfer correlations for agitated tanks (Fig. h^ = h = aVcosh 2 f - inS (14D.1-4)
s
v
14C.2), A liquid of essentially constant physical properties к = a cosh f sin 77 (14D.1-5)
is being continuously heated by passage through an agi- ф
tated tank, as shown in the accompanying figure. Heat is Equations A.7-13 and 14 can then be used to get any of the
supplied by condensation of steam on the outer wall of the V-operations that are needed.
tank. The thermal resistance of the condensate film and the (b) Next obtain the temperature profile outside of an
tank wall may be considered small compared to that of oblate ellipsoid with surface temperature T , which is em-
o
the fluid in the tank, and the unjacketed portion of the tank bedded in an infinite medium with the temperature T x far
from the ellipsoid. Let 0 = (T - T )/(T X - T ) be a dimen-
O
o
sionless temperature, and show that Laplace's equation
describing the heat conduction exterior to the ellipsoid is
1 дв 1 0 (14D.1-6)
2
г-*- Liquid out fl (cosh 2 £ - sin rj) — cosh f
2
Steam in —*- ,
Steam jacket Steam jacket

Liquid in 1 For a discussion of oblate ellipsoidal coordinates, see
P. Moon and D. E. Spencer, Field Theory Handbook, Springer, Berlin
Condensate out (1961), pp. 31-34. See also J. Happel and H. Brenner, Low Reynolds
Number Hydrodynamics, Prentice-Hall, Englewood Cliffs, N J.
Fig. 14C.2. Continuous heating of a liquid in an agitated (1965), pp. 512-516; note that their scale factors are the reciprocals
tank. of those defined in this book.