gravitational settling of an aerosol particle

According to equation 9.42 in Seinfeld & Pandis (2006), settling velocity of a spherical particle is

\[ v_t = \frac{1}{18} \frac{D_p^2 \rho_p g C_c}{\mu} \]

settl_vel <- function(Dp, # particle diameter [micron]
               rhop=1,    # particle density [g/cm^3]
               Cc         # slip correction factor [-]
               ) {
  g  <- 980.7        # gravity accel. [cm/s^2]
  mu <- 1.72*10^(-4) # atm.viscosity [g/(cm*s)]
  Dp_cm <- Dp/10000  # particle diameter [cm]
  v_t <- 1/18 * (Dp_cm^2*rhop*g*Cc) / mu # settling velocity [cm/s]
  return(v_t)
}

For a 10 micron diameter, with C_c=1.016 (Tab.9.3, Seinfeld & Pandis 2006)

settl_vel(Dp=10, Cc=1.016) * 3600  # [cm/h]

[1] 1158.594

We can express the slip correction factor as a linear approximation to the empirical values (see Tab.9.3)

slip_corr_fact <- function(Dp) {
  approxfun(x = c(0.001,0.002,0.005,0.01,0.02,0.05,0.1,0.2,0.5,1,2,5,10,20,50,100),
            y = c(216,108,43.6,22.2,11.4,4.95,2.85,1.865,1.326,1.164,1.082,1.032,1.016,1.008,1.003,1.0016))(Dp)
}

Let’s plot settling velocities as a function of diameter and density. Tipical values for density are in the range 0.1-1.7 g/cm³ (Rissler et al, 2014).

# data for lines
density <- c(0.1,0.5,1,1.7)
diameter <- c(0.01,0.02,0.05,0.1,0.2,0.5,1,2,5,10,20)
library(dplyr)
df <- expand.grid(diameter=diameter, density=density) %>%
  mutate(velocity=settl_vel(Dp = diameter, 
                            rhop = density, 
                            Cc = slip_corr_fact(diameter)),
         density=as.factor(density),
         velocity=velocity*3600  # cm/h
         )
# data for points
density <- c(0.1,1.7)
diameter <- c(0.1, 1,10)
df2 <- expand.grid(diameter=diameter, density=density) %>%
  mutate(velocity=settl_vel(Dp = diameter, 
                            rhop = density, 
                            Cc = slip_corr_fact(diameter)),
         density=as.factor(density),
         velocity=velocity*3600  # cm/h
         )
library(ggplot2)
ggplot(data=df, aes(x=diameter, y=velocity, col=density, group=density)) +
  geom_line(data=df) +
  labs(x=expression(D[p]~(mu*m)), y=expression(v[t]~(cm/h))) +
  scale_x_log10(breaks=c(0.01,0.1,1,10), labels=c(0.01,0.1,1,10)) +
  scale_y_log10(breaks=c(0.1,1,10,100,1000), labels=c(0.1,1,10,100,1000)) +
  theme_bw() +
  geom_text(data=df2, aes(label=signif(velocity,2)), col="black")