Page 363 - Bird R.B. Transport phenomena

§11.4 Use of the Equations of Change to Solve Steady-State Problems 345

Porous spherical shells Fig. 11.4-1. Transpiration cooling. The
inner sphere is being cooled by means
of a refrigeration coil to maintain its
temperature at T . When air is blown
K
outward, as shown, less refrigeration is
required.

Air flow out

SOLUTION We postulate that for this system v = Si>(r), T = T(r), and & = ). The equation of continuity
r
r
in spherical coordinates then becomes
U (11.4-21)
dr
This equation can be integrated to give

2 (11.4-22)
4тг
Y pV y = COnst. =
Here w is the radial mass flow rate of the gas.
Y
The r-component of the equation of motion in spherical coordinates is, from Eq. B.6-7,
dv d d
r
(11.4-23)
The viscosity term drops out because of Eq. 11.4-21. Integration of Eq. 11.4-23 then gives
U-ff (11.4-24)
L

Hence the modified pressure 2P increases with r, but only very slightly for the low gas veloc-
ity assumed here.
The energy equation in terms of the temperature, in spherical coordinates, is, according to
Eq. B.9-3,
C dT ^d_( dT\
p V = 2 k r2 (11.4-25)
' dr r dr\ dr j
Here we have used Eq. 11.2-8, for which we assume that the thermal conductivity is constant,
the pressure is constant, and there is no viscous dissipation—all reasonable assumptions for
the problem at hand.
When Eq. 11.4-22 for the velocity distribution is used for v r in Eq. 11.4-25, we obtain the
following differential equation for the temperature distribution T(r) in the gas between the
two shells:
dT _ 4TT* d l 2 d T (11.4-26)
dr 7n r dr\ dr
w r C/