Page 703 - Bird R.B. Transport phenomena

§22.3 Correlation of Binary Transfer Coefficients in One Phase 683

In computing x j, we have assumed ideal gas behavior, equilibrium at the interface, and com-
A
plete insolubility of air in water. The mean mole fraction, x f, of the water vapor is sufficiently
A
small that it can be neglected in evaluating the physical properties at the film conditions:
с = 3.88 X 10~ 5 g-moles/cm 3
p = 1.12xlO" g/cm 3
3
4
fi = 1.91 X 10" g/cm • s (from Table 1.1-1)
ЯЬ = 0.292 cm7s (from Eq. 17.2-1)
АВ
( _ M \ __L91X1O- 4
S r = = 0.58
(1.12 x 10" )(0.292)
3
3
(0.05X215X1.12 X 10" )
Re = = 63
1.91 X 10~ 4
When these values are used in Eq. 22.3-21 we get
Sh w = 2 + 0.60(63) (0.58) 1/3 = 5.96 (22.3-29)
1/2
and the mean mass transfer coefficient is then
5
, сЯЬ АВ (3.88 X 10~ )(0.292)
k xm = — Sh w = — (5.96)
3
= 1.35 X 10~ g-mol/s • cm 2 (22.3-30)
Then from Eq. 22.3-26 the evaporation rate is found to be
2
02
3
W A0 = (1.35 X 10- )(тг)(0.05) ^ ^ 7 024 ^
= 2.70X10" g-mole/s (22.3-31)
7
3
This result corresponds to a decrease of 1.23 X 10 cm/s in the drop diameter and indicates
that a drop of this size will fall a considerable distance before it evaporates completely.
In this example, for simplicity, the velocity and surface temperature of the drop were
given. In general, these conditions must be calculated from momentum and energy balances,
as discussed in Problem 22B.1.

EXAMPLE 22.3-2 We next turn to a problem for which the analogy between heat and mass transfer leads to a sur-
prisingly simple and useful, if approximate result. The system, shown in Fig. 22.3-2, is a pair of
The Wet and Dry Bulb thermometers, one of which is covered with a cylindrical wick kept saturated with water. The
Psychrometer wick will cool by evaporation into the moving air stream and for steady operation will ap-
proach an asymptotic value known as the wet bulb temperature. The bare thermometer, on the
other hand, will tend to approach the actual temperature of the approaching air, and this value
is called the dry bulb temperature. Develop an expression for determining the humidity of the air
from the wet and dry bulb temperature readings neglecting radiation and assuming that the re-
placement of the evaporating water has no significant effect on the wet bulb temperature mea-
surement. In Problem 22B.2 we will see how radiation can be taken into account.

SOLUTION For simplicity, we assume that the fluid velocity is high enough that the thermometer read-
ings are unaffected by radiation and by heat conduction along the thermometer stems, but
not so high that viscous dissipation heating effects become significant. These assumptions are
usually satisfactory for glass thermometers and for gas velocities of 30 to 100 ft/s. The dry
bulb temperature is then the same as the temperature Т of the approaching gas, and the wet
ж
bulb temperature is the same as the temperature T of the outside of the wick.
o
Let species A be water and species В be air. An energy balance is made on a system that
contains a length L of the wick (the distance between planes 1 and 2 in the figure). The rate of