$2\mathrm{.}3$ Sampling losses in tubes $($after Nazaroff $2008$ and Friedlander $1977)$

Outside air is delivered to the instruments of an air monitoring station through a D $=2$ cm $($insidediameter$)$ copper tube at volumetric flow rate of Q $=20$ L$/$min$.$ The tube is horizontally oriented and is L $=5$ m long. Answer the following questions to evaluate the transmission efficiency of particles through this tube. Consider unit density particles with a size range of $0\mathrm{.}01$m $$ dp $<mtext\; id="EditorElement\_3593743"></mtext><mn\; id="EditorElement\_3593744">1</mn><mn\; id="EditorElement\_3593745">0</mn><mtext\; id="EditorElement\_3593746"></mtext><mi\; id="EditorElement\_3593747"></mi>$m$.$ Deposition occurs by means of Brownian diffusion and gravitational settling.

$($a$)$ If the Reynolds number $($Re$)$ based on tube diameter is less than $2100,$ then the flow in the boundary layer along the pipe walls will remain laminar. What is the Reynolds number forthis system?

$($b$)$ For laminar flow, the air velocity profile evolves from uniform at the inlet to parabolic over a distance of approximately $0\mathrm{.}04$ D Re$.$ Over what fraction of the length of this tube is the flow undeveloped $($i$.$e$.,$ not yet fully parabolic$)?$

$($c$)$ Use the approach developed in lecture, based on magnitude analysis, to predict the particlepenetration efficiency as a function of diameter. $($Equations are reproduced below.$)$

$($d$)$ Repeat the analysis of part $($c$),$ using the results from the papers by D$.$ B$.$ Ingham$($Journal ofAerosol Science, $6,125,1975)$ and J$.$ Pich $($Aerosol Science, $3,351,1972)$ to evaluate the penetration efficiency. $($Equations are reproduced below.$)$

$($e$)$ Plot your results from $($c$)$ and $($d$).$ How well do the results of the magnitude Equations:

a$.$ Results of magnitude analysis for particle penetration through a tube:

$P_d=(1-\sqrt[\phantom{2}]{\frac{2L{D}_b}{Q}}{)}^2,($only valid if $2L\frac{D_b}{Q}1$; otherwise set $P_d=0)$

$L=$ tube length; $D=$ tube diameter; $v_g=$ gravitational settling velocity; $Q=$ volume flow rate

through tube; $D_b=$ Brownian diffusivity of particles

b$.$ Results for gravitational penetration, from Pich $(1972)$

$P_s=1-\frac{2}{}\{2\sqrt[\phantom{2}]{1-{}^{\frac{2}{3}}}-{}^{\frac{1}{3}}\sqrt[\phantom{2}]{1-{}^{\frac{2}{3}}}+arcsin({}^{\frac{1}{3}})\}$ valid $if<mn\; id="EditorElement\_3592487">1</mn>$; otherwise $P_s=0$

$=\frac{3LD{v}_g}{16Q}$

c$.$ Results for Brownian diffusion, from Ingham $(1975)$

$P_d=0\mathrm{.}819$exp$(-14\mathrm{.}63)+0\mathrm{.}0976$exp$(-89\mathrm{.}22)+0\mathrm{.}0325$exp$(-228)+0\mathrm{.}0509$exp$(-125\mathrm{.}9{}^{\frac{2}{3}})$

$=\frac{D_{b}L}{4Q}$analysis $($c$)$compare with the more detailed analysis results $($d$)?$