Nanoaerosol adhesion efficiency Calculate the adhesion efficiency using the DMT model with the following properties of a glass fiber filter for NaCl nanoparticles 1–100 nm when filter fiber diameter df = 5 μm.

Question

Nanoaerosol adhesion efficiency
Calculate the adhesion efficiency using the DMT model with the following properties of a glass fiber filter for NaCl nanoparticles 1–100 nm when filter fiber diameter [latex]d_{f}[/latex] = 5 μm.

Accepted Answer

For salt particles and glass fiber filter
[latex]H_{p}=7 × 10^{-20}[/latex] J
[latex]H_{f}=8.5 × 10^{-20}[/latex] J
[latex]K_{p}=0.75 imes 10^{-11}\frac{m^{2}}{N}[/latex]
[latex]K_{f}=0.443 imes 10^{-11}\frac{m^{2}}{N}[/latex]

Then the Hamaker constant
[latex]H=(H_{p}H_{f})^{1/2}=\sqrt{7 imes 8.5} imes 10^{-20}J=7.71 imes 10^{-20}J[/latex]

And the specific adhesion energy is
[latex]\Delta \gamma =\frac{A_{H}}{12\pi Z^{2}_{e}}=\frac{7.71 imes 10^{-20}J}{12\pi (0.4 imes 10^{-9})^{2} m^{2}}=0.0128 J/m^{2}[/latex]

The composite Young’s modulus of bodies with the mechanical constants of [latex]K_{p}[/latex] and [latex]K_{f}[/latex]
[latex]Y^{\ast}=\frac{4}{3\pi}\left(\frac{1}{K_{p} + K_{f}} ight)=\frac{4}{3\pi}\left(\frac{1}{0.75 imes 10^{-11} + 0.443 imes 10^{-11}} ight)[/latex]
[latex]=3.51 × 10^{11}(Pa)[/latex]

The characteristic radius of two bodies is
[latex]R^{\ast}=\frac{1}{2}\left(\frac{1}{d_{p}}+\frac{1}{d_{f}} ight)^{-1}[/latex]

The impact contact area is determined by
[latex]a=\left[\frac{R^{\ast}}{Y^{\ast}} (2\Delta \gamma \pi R^{\ast }) ight]^{1/3}[/latex]

Then the adhesion energy is calculated using
[latex]E_{ad}=Δγπa^{2}[/latex]

Then the critical speed and impact speed are calculated using
[latex]v_{cr}=\sqrt{\frac{12E_{ad}}{\pi ho _{p}d^{3}_{p}e^{2}}}[/latex]; [latex]\bar{v}_{im}=\sqrt{\frac{48kT}{\pi ^{2} ho _{p}d^{3}_{p}}}[/latex]

Then the adhesion efficiency is determined by
[latex]z=\frac{2}{\sqrt{\pi}}\frac{v_{cr}}{\bar{v}_{im}}[/latex]; [latex]\eta _{ad}= ext{erf}(z)-\frac{2z}{\sqrt{\pi}}\exp [-z^{2}][/latex]

In calculation using spread sheet, the error function is approximated with
[latex] ext{erf}(z)\approx 1-\frac{1}{(1 + a_{1}z + a_{2}z^{2} + a_{3}z^{3} + a_{4}z^{4})^{4}}[/latex]

where [latex]a_{1} = 0.278393, a_{2} = 0.230389, a_{3} = 0.000972, and a_{4} = 0.078108.[/latex]

The results are shown in Fig. 13.7 in terms of penetration efficiency.

Question 13.3: Nanoaerosol adhesion efficiency Calculate the adhesion effic...

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