Page 383 - Bird R.B. Transport phenomena

Problems 365

11B.6. Transpiration cooling in a planar system. Two large flat porous horizontal plates are sepa-
rated by a relatively small distance L. The upper plate at у = L is at temperature T , and the
L
lower one at у = 0 is to be maintained at a lower temperature T . To reduce the amount of
o
heat that must be removed from the lower plate, an ideal gas at T is blown upward through
o
both plates at a steady rate. Develop an expression for the temperature distribution and the
amount of heat q that must be removed from the cold plate per unit area as a function of the
0
fluid properties and gas flow rate. Use the abbreviation ф = pC v L/k.
p y
ефуя _ Ф k(T L -T 0 )( ф
T-T L е
Answer: = ; q =
T -T L
0
11B.7. Reduction of evaporation losses by transpiration (Fig. 11B.7). It is proposed to reduce the
rate of evaporation of liquefied oxygen in small containers by taking advantage of transpira-
tion. To do this, the liquid is to be stored in a spherical container surrounded by a spherical
shell of a porous insulating material as shown in the figure. A thin space is to be left between
the container and insulation, and the opening in the insulation is to be stoppered. In opera-
tion, the evaporating oxygen is to leave the container proper, move through the gas space,
and then flow uniformly out through the porous insulation.
Calculate the rate of heat gain and evaporation loss from a tank 1 ft in diameter cov-
ered with a shell of insulation 6 in. thick under the following conditions with and without
transpiration.
Temperature of liquid oxygen -297°F
Temperature of outer surface of insulation 30°F
Effective thermal conductivity of insulation 0.02 Btu/hr • ft • F
Heat of evaporation of oxygen 91.7 Btu/lb
Average C p of O flowing through insulation 0.22 Btu/lb • F
2
Neglect the thermal resistance of the liquid oxygen, container wall, and gas space, and ne-
glect heat losses through the stopper. Assume the particles of insulation to be in local thermal
equilibrium with the gas.
Answers: 82 Btu/hr without transpiration; 61 Btu/hr with transpiration
11B.8. Temperature distribution in an embedded sphere. A sphere of radius R and thermal conduc-
tivity к is embedded in an infinite solid of thermal conductivity k . The center of the sphere is
0
л
located at the origin of coordinates, and there is a constant temperature gradient A in the posi-
tive z direction far from the sphere. The temperature at the center of the sphere is T°.
The steady-state temperature distributions in the sphere T } and in the surrounding
medium T have been shown to be: 3
o
T,(r,0) - T° = -; u —\Ar cos в r<R (11В.8-1)

Tank wall
Gas space

Porous
insulation

Fig. 11B.7. Use of transpiration to reduce the
evaporation rate.

3 L. D. Landau and E. M. Lifshitz, Fluid Mechanics, 2nd edition, Pergamon Press, Oxford (1987), p. 199.