Effect of the finite width of the temperature transition in diffusive condensation particle counters

Figures & data

Figure 1. Functions solving (21–24) for k from 0 (top) to 10 (bottom). The dotted curve immediately below the top line is F₁(η) given in (9(9) $$F_1\left(\eta \right)=\frac{\left(6+{\eta }^3\right)\mathrm{\Gamma }\left[1/3,{\eta }^3/9\right]-3^{4/3}\mathrm{ }\eta e^{-{\eta }^3/9}\mathrm{ }}{6\Gamma (1/3)},$$(9) ), providing the solution (5(5) $$T=T_s-x\beta F_1(\eta )\mathrm{ }\text{for}\mathrm{ }0<x<\Delta$$(5) ) to the thermal problem in terms of the similarity variable (6(6) $$\eta =y{a}^{1/3}/{(\alpha x)}^{1/3}.$$(6) ).

Figure 2. Comparison between the experimental vapor pressure curve for butanol (continuous gray curve) and two exponential fits: one linear in T (lower dotted) another quadratic (upper dashed, indistinguishable from the p_v(T) data).

Table A1. $F_k\left(x\right)={[A}_k\left(x\right)\mathrm{\Gamma }\left[1/3,x^3/9\right]-x{B}_k(x)\mathrm{ }exp(-\frac{x^3}{9})3^{-2/3}]/\mathrm{\Gamma }[1/3].$

k	Ak(x)	Bk(x)
0	1	0
1	$1+x^3/6$	$3/2$
2	$1+\left(30x^3+x^6\right)/90$	$(24+x^3)/10$
3	$1+(1080x^3+72x^6+x^9)/2160$	$(738+66x^3+x^6)/240$
4	$1+(47520x^3+4752x^6+132x^9+x^{12})/71280$	$(28728+4050x^3+126x^6+x^9)/7920$
5	$1+\frac{2494800x^3+332640x^6+13860x^9+210x^{12}+x^{15}}{2993760}$	$\frac{1364040+266220x^3+12690x^6+204x^9+x^{12}}{332640 }$
6	$1+\frac{152681760x^3+25446960x^6+1413720x^9+32130x^{12}+306x^{15}+x^{18}}{152681760}$	$\frac{76651920+19266660x^3+1246320x^6+30384x^9+300x^{12}+x^{15}}{16964640 }$
7	$1+\frac{1}{9160905600}(10687723200x^3+2137544640x^6+148440600x^9+4498200x^{12}+64260x^{15}+420x^{18}+x^{21})$	$\frac{4981752720+1539881280x^3+126398880x^6+4148280x^9+61830x^{12}+414x^{15}+x^{18}}{1017878400 }$
8	$1+\frac{1}{632102486400}(842803315200x^3+196654106880x^6+16387842240x^9+620751600x^{12}+11823840x^{15}+115920x^{18}+552x^{21}+x^{24})$	$\frac{1}{70233609600 }(367847101440+135418106880x^3+13513443840x^6+559120320x^9+11175840x^{12}+112698x^{15}+546x^{18}+x^{21})$
9	$1+\frac{1}{49303993939200}(73955990908800x^3+19721597575680x^6+1917377542080x^9+87153524640x^{12}+2075083920x^{15}+27125280x^{18}+193752x^{21}+702x^{24}+x^{27})$	$\frac{1}{5478221548800 }(30427717703040+13030432934400x^3+153448047040x^6+76772836800x^9+1928324880x^{12}+26023788x^{15}+189630x^{18}+696x^{21}+x^{24})$
10	$1+\frac{1}{4289447472710400}(7149079121184000x^3+2144723736355200x^6+238302637372800x^9+12637261072800x^{12}+361064602080x^{15}+5899748400x^{18}+56188080x^{21}+305370x^{24}+870x^{27}+x^{30})$	$\frac{1}{476605274745600}(2787798587212800+1363968851414400x^3+185445982656000x^6+10899799447680x^9+330122459520x^{12}+5588295300x^{15}+54432000x^{18}+300240x^{21}+864x^{24}+x^{27})$

Note: Polynomials A_k(x) and B_k(x) may also be obtained for arbitrary k by explicit formulae for A_k(x) or recurrence relations for B_k(x) (Stolzenburg 2020). A Mathematica program for computing these eigenfunctions is included in the supplementary information.

Figure 3. Saturation ratio (26) versus dimensionless variables z, Y for an exponential vapor pressure and Le = 1. The insulator region corresponds to z < z_o=2.264. The early part of the constant wall temperature region (z_o < z < 5) is also included. As one moves along the ridge, at small Y (and z) S_max is close to unity. At larger Y (or z) the top of the ridge becomes flat, resulting in a constant asymptote for S_max(Y). Recall that z=z_o(x/Δ).

Figure 4. Comparison of the calculated initial conditions at z=z_o for the vapor concentration (top) and the temperature (bottom) with the linear combination (32) of two F₀(λs) functions (dashed). s is a generic variable for either η or η_D.

Table 1. Optimal parameters to fit the initial value functions (first column) according to (32).

	b	λ1	λ2
[T(zo,η)-Ts)/(Tc-Ts)	3.15916	2.35353	1.17383
[N(zo, ηD)-1)/[Nc-1]	5.43733	1.91207	1.09323

Figure 5. (a) Calculated axial variation of the saturation ratio S(z) for Le = 1 at various wall distances Y (indicated in the legend by its right asymptote, from bottom to top). (b) S_max versus Y for Le = 1 (bottom points, taken as the maxima for each Y value in (a)) and Le = 2.5 (top curve). Notice the minimal dependence of the boundary layer thickness on Le.

Stolzenburg, M. R. 2020. Comments on “Effect of the finite width of the temperature transition in diffusive condensation particle counters” by J. Fernandez de la Mora, Aerosol Sci. Tech., this issue.

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