TRANSFER: BASICS OF MASS AND HEAT TRANSFER - Biomass as a Sustainable Energy Source for the Future - page 116

20 K, one obtains t d = 0.054 s.
b. Solving the droplet temperature equation
When the droplet and air temperatures change with time, an explicit analytic
solution in general is not available, but a numerical solution can be obtained.
By solving the aforementioned equations until

For T s =T boil −

the droplet diameter

reaches 0, we can obtain the droplet lifetime.

s −1 ,

For T s,i =T boil − 20 = 331 K, v s =10m the droplet lifetime t d =

0.0204 s, and for
T s,i =T boil − s −1 , the droplet lifetime t d = 0.0209 s.

A detailed study of this model and a sensitivity analysis of the results are
presented as project at the end of this chapter. Results of such a project may
look as depicted in Figures 4.1 and 4.2.

10 = 341 K, v s =10m

Due to the balance between heating and heat extraction for evaporation, liquid dro-
plets usually do not reach the boiling point when they are heated. Instead, the temper-
ature evolves toward the wet-bulb temperature. An estimation of the wet-bulb
temperature can be obtained by setting the left-hand side of Equation (4.44) to zero.
When droplets are situated in a very hot carrier gas, they will reach the wet-bulb

350

340

320

300

0 5 10

Time (ms)

300

0 10 20 30 40 50

Time (ms)

100

50

0

0 10 20 30 40 50

Time (ms)

FIGURE 4.1 Droplet temperature and diameter versus time at a slip velocity between droplet
and gas, v s of 0 m s −1 . Initial droplet temperatures: 301 K (dark gray), 321 K (light gray), and

341 K (black). Initial droplet diameters: 100 μ m (solid lines) and 50 μ m (dashed) lines

(surrounding gas temperature T b = 800 K).