Representative sampling
Causes for loss
- Aspiration effects
- Particle deposition
- Coagulation, condensation, evaporation
- Inhomogeneities in the aerosol cloud
- Inhomogeneities in the inlet
- Clogging of the inlet
- Particle resuspension
Efficiencies
Aspiration efficiency
$η_\mathrm{asp} = \frac{c_\mathrm{in}}{c_\mathrm{aerosol}}$
Efficiencies
Transmission efficiency
$η_\mathrm{trans} ≡ \frac{c_\mathrm{out}}{c_\mathrm{in}}$
Efficiencies
Inlet efficiency
$η_\mathrm{inlet} ≡ η_\mathrm{asp} η_\mathrm{trans}$
-
Inlet efficiency depends on
- Particle size (or St)
- Flow conditions
Sampling in a flow
Isokinetic
Sampling in a flow
Super-isokinetic
Sampling in a flow
Sub-isokinetic
Isoaxial aspiration efficiency
Rader & Marple, 1987:
$η_\mathrm{asp} = 1+\left(\frac{u_0}{u} - 1\right) \left(1 - \frac{1}{1+3.77 \mathrm{St}^{0.883}}\right)$
-
Maximum error 5 % if:
- $0.005 < \mathrm{St} < 10$
- $0.5 < \frac{u_0}{u} < 5$
- Note: $\mathrm{St} ≡ \frac{τu_0}{d}$
Isoaxial aspiration efficiency
$η_\mathrm{asp} = 1+\left(\color{red}{\frac{u_0}{u}} - 1\right) \left(1 - \frac{1}{1+3.77 \color{red}{\mathrm{St}}^{0.883}}\right)$
Anisoaxial sampling
Sampling in calm air
\begin{equation}
η_\mathrm{asp} = \color{red}{\frac{v_{ts}}{u}} \cos{\color{red}{φ}} + \exp\left(-\frac{4\color{red}{\mathrm{St}}^{\left(1+\sqrt{\color{red}{\frac{v_{ts}}{u}}}\right)}}{1+2\color{red}{\mathrm{St}}}\right)
\end{equation}
Ref: Grinshpun, Willeke & Kalatoor, 1993
Transmission efficiency
$η_\mathrm{trans} ≡ \frac{c_\mathrm{out}}{c_\mathrm{in}}$
Transmission losses
Loss due to inertia
Transmission losses
Loss due to gravity
Transmission losses
Loss due to brownian motion
Transmission losses
Loss due to turbophoresis
Transmission losses
Loss due to thermophoresis
Transmission losses
Loss due to electrophoresis
Dilution
$DR = \frac{Q_t}{Q_s} = \frac{Q_d + Q_s}{Q_s} = \frac{Q_t}{Q_t-Q_d}$
Measure $Q_t$ and $Q_s$ ($DR = \frac{Q_t}{Q_s}$)
\begin{eqnarray}
(Δ DR)^2 &=& \left(\frac{∂ DR}{∂Q_t}\right)^2 (ΔQ_t)^2 + \left(\frac{∂ DR}{∂Q_s}\right)^2 (ΔQ_s)^2\\
&=& \frac{1}{Q_s^2} (ΔQ_t)^2 + \frac{Q_t^2}{Q_s^4} (ΔQ_t)^2\\
&=& \frac{Q_t^2}{Q_s^2} \left(\frac{ΔQ_t}{Q_t}\right)^2 + \frac{Q_t^2}{Q_s^2} \left(\frac{ΔQ_s}{Q_s}\right)^2
\end{eqnarray}
$⇒ Δ_\% DR = \sqrt{\left(\Delta_\% Q_t\right)^2 + \left(\Delta_\% Q_s\right)^2}$
Measure $Q_d$ and $Q_s$ ($DR = \frac{Q_d + Q_s}{Q_s}$)
\begin{eqnarray}
(Δ DR)^2 &=& \frac{1}{Q_s^2} (ΔQ_d)^2 + \frac{Q_d^2}{Q_s^4} (ΔQ_s)^2\\
&=& \frac{Q_d^2}{Q_s^2} (Δ_\%Q_d)^2 + \frac{Q_d^2}{Q_s^2} (Δ_\%Q_s)^2\\
&=& (DR-1)^2\left((Δ_\%Q_d)^2 + (Δ_\%Q_s)^2\right)\\
\end{eqnarray}
$⇒ Δ_\% DR = \left(1-\frac{1}{DR}\right)\sqrt{(\Delta_\% Q_d)^2 + (\Delta_\% Q_s)^2}$
Measure $Q_d$ and $Q_t$ ($DR = \frac{Q_t}{Q_t-Q_d}$)
\begin{eqnarray}
(Δ DR)^2 &=& \frac{Q_t^2}{(Q_t - Q_d)^4} (ΔQ_d)^2 + \frac{Q_d^2}{(Q_t - Q_d)^4} (ΔQ_t)^2\\
&=& \left(\frac{Q_d}{Q_t-Q_d}DR\right)^2\left((Δ_\%Q_d)^2 + (Δ_\%Q_t)^2\right)\\
&=& ((DR-1)DR)^2\left((Δ_\%Q_d)^2 + (Δ_\%Q_t)^2\right)\\
\end{eqnarray}
$⇒ Δ_\% DR = \left(DR-1\right)\sqrt{(\Delta_\% Q_d)^2 + (\Delta_\% Q_t)^2}$
Dilution ratio error