M E 433
Professor John M. Cimbala
Lecture 33
Today, we will:
•
Continue to discuss particle cleaning by raindrops and spray chambers
•
Discuss the Calvert and Englund model for single drop grade removal efficiency
•
Discuss Brownian motion and its effect on single drop collection grade efficiency
Last time, we developed an expression for the single drop grade removal efficiency Ed (Dp )
of a spherical raindrop of diameter Dc (radius Rc) falling down at
speed U0, and encountering dusty air containing particles of diameter
Rc Raindrop
Dp. We defined r1 as the limiting trajectory radius, such that all
particles of diameter Dp within this radius are collected by the
raindrop, and all particles outside of this radius are not collected, but
Limiting
rather miss hitting the raindrop altogether. Based on the geometry,
trajectory
we derived:
2
r1
E D = r R
d
( p) ( 1
c
)
Now the problem reduces to generating an expression for r1.
U0
Example: Single-drop collection grade efficiency for one particle diameter
Given: Raindrops of diameter 200 microns are falling through dusty air at STP
(ρ = 1.184 kg/m3, µ = 1.849 × 10-5 kg/(m s)). Consider aerosol particles of unit density
(ρp = 1000 kg/m3), and of diameter 5 microns.
To do: Calculate the single-drop collection grade efficiency Ed(Dp) as a percentage to three
significant digits for these particles.
Solution:
• First, calculate Vt,c = U0 for the raindrops due to gravimetric settling [must iterate using
drag coefficient as a function of Reynolds number, etc. as we have done previously].
I did the calculations for 200 micron particles, and got U0 = 0.700464 m/s.
• Next, calculate Vt,p for the particles in the same way [iterate again]. For 5 µm particles, I
did the calculations and got Vt,p = 0.00076057 m/s.
• Complete the problem using the Calvert-Englund model for raindrop collection:
2
ρ p − ρ ) D p 2 (U 0 − Vt , p )
(
 r1  
Stk

Ed=
,
( Dp ) =
 
 where Stk =
0.35
R
Stk
18
µ
D
+


c
 c
2
Example: Single-drop collection grade efficiency as a function of particle diameter
Given: A polydisperse aerosol at STP (ρ = 1.184 kg/m3, µ = 1.849 × 10-5 kg/(m s)) is
cleaned by raindrops of diameter 200 microns. The particles are of unit density (ρp = 1000
kg/m3). The raindrop (the “collector” falls at terminal gravitational settling velocity Vt,c =
0.700464 m/s. For convenience, I have pre-calculated the terminal gravitational settling
velocity Vt,p for various particle diameters. Assume that particles are absorbed by the raindrop
when they hit the raindrop, and are therefore removed from the air.
To do: For each diameter, calculate the single-drop collection grade efficiency Ed(Dp) as a
percentage to three significant digits for each particle diameter Dp in the table below.
Solution:
Table created in Excel for various particle diameters:
Vt (m/s)
Ed (Dp) (%)
Dp (µm)
Vt (m/s)
2 0.000127646 1.149675069
Dp (µm)
3 0.000279763 4.524218465
25 0.018468719
4 0.000490739 10.52314578
30 0.026513567
7 0.001476776 35.38805555
40 0.046775902
8.5 0.002168354 46.76422259
50 0.072295146
10 0.002992228 56.15821262
70 0.13687402
15 0.006692553 75.69004041
85 0.194258109
20 0.01185533 84.99538738
100 0.25629326
Ed (Dp) (%)
89.89263573
92.73095237
95.6846146
97.09435181
98.33191736
98.73657951
98.95794227