Page 342 - Bird R.B. Transport phenomena

326 Chapter 10 Shell Energy Balances and Temperature Distributions in Solids and Laminar Flow

10B.11. Free convection with temperature-dependent viscosity. Rework the problem in §10.9, tak-
ing into account the variation of viscosity with temperature. Assume that the "fluidity" (reci-
procal of viscosity) is the following linear function of the temperature

Use the dimensionless quantities y, v , and Gr defined in §10.9 and in addition
z

b = kB,AT and P = Ъг -г 1 + ,
n
and show that the differential equation for the velocity distribution is

Follow the procedure in §10.9, discarding terms containing the second and higher powers of
bp. Show that this leads to P = ^ Grb^ and finally:

v z = I Gr[(f - y) - ^bykf ' D(5y 2 - 1)] (10B.11-5)
Sketch the result to show how the velocity profile becomes skewed because of the tempera-
ture-dependent viscosity.
10B.12. Heat conduction with temperature-dependent thermal conductivity (Fig. 10B.12). The
curved surfaces and the end surfaces (both shaded in the figure) of the solid in the shape of a
half-cylindrical shell are insulated. The surface 0 = 0, of area (r 2 - r^)L, is maintained at tem-
perature T , and the surface at в = тт, also of area (r - r )L, is kept at temperature T .
2
o
n
x
The thermal conductivity of the solid varies linearly with temperature from k at T = T o
Q
to К at T = 7V
(a) Find the steady-state temperature distribution.
(b) Find the total heat flow through the surface at в = 0.
10B.13. Flow reactor with exponentially temperature-dependent source. Formulate the function
F(@) of Eq. 10.5-1 for a zero-order reaction with the temperature dependence
S c = Ke~ E/RT (10B.13-1)
in which К and E are constants, and R is the gas constant. Then insert F(S) into Eqs. 10.5-15
through 20 and solve for the dimensionless temperature profile with k neglected.
ze((
10B.14. Evaporation loss from an oxygen tank.
(a) Liquefied gases are sometimes stored in well-insulated spherical containers vented to the
atmosphere. Develop an expression for the steady-state heat transfer rate through the walls of
such a container, with the radii of the inner and outer walls being r and r respectively and
0 x

-r -
2
Fig. 10B.12. Tangential heat conduction in an annular shell.