Optimal iteration and its application to some problems in aerosol science and particle dynamics

Figures & data

Figure 1. Cunningham correction, C, and its asymptotic value for low mobility diameter, d_mob, and X ≡ d_mob/C(d_mob), as functions of the mobility diameter. C is dimensionless, units for X are nm.

Figure 2. Percent differences from actual values of mobility diameter, d_mob, of the successive iterations using schemes given by Equation (3)(3) $$d_{\text{mob},i+1}=X\cdot C\left(d_{\text{mob},i}\right).$$(3) , black; Equation (4)(4) $$d_{\text{mob},i+1}={\left[d_{\text{mob},i}\cdot X\cdot C\left(d_{\text{mob},i}\right)\right]}^{\frac{1}{2}}.$$(4) , red; and Equation (5)(5) $$d_{\text{mob},i+1}={\left\{d_{\text{mob},i}\cdot X^2\cdot {\left[C\left(d_{\text{mob},i}\right)\right]}^2\right\}}^{\frac{1}{3}}.$$(5) , blue, as a function of mobility diameter.

Figure 3. Convergence ratio, R_ε, given by Equation (6)(6) $$R_{\epsilon }\equiv \underset{i\to \infty }{\lim }\frac{{\epsilon }_{i+1}}{{\epsilon }_i}={\left[\left(\frac{x}{f}\right)\cdot \left(\frac{df}{dx}\right)\right]|}_{x_{\infty }}={\frac{d\hspace{0.17em}\text{ln}\hspace{0.17em}f}{d\hspace{0.17em}\text{ln}\hspace{0.17em}x}|}_{x_{\infty }}$$(6) evaluated for the iteration schemes given by Equations (3)–(5), as functions of the mobility diameter.

Figure 4. Geometric illustration of iteration procedure, for f ′(x) < 0 (top panel) and f ′(x) > 0 (bottom panel).

Figure 5. Percent differences from actual values of aerodynamic diameter, d_aero, of the successive iterations using schemes given by Equation (12)(12) $$d_{\text{aero},i+1}={\left[\frac{Y}{C\left(d_{\text{aero},i}\right)}\right]}^{\frac{1}{2}}.$$(12) , black; Equation (13)(13) $$d_{\text{aero},i+1}=\frac{Y}{d_{\text{aero},i}\cdot C\left(d_{\text{aero},i}\right)}$$(13) , red; and Equation (14)(14) $$d_{\text{aero},i+1}=\frac{Y^n}{{d_{\text{aero},i}}^{2n-1}\cdot C^n\left(d_{\text{aero},i}\right)};.$$(14) with n = 2/3, blue, as a function of aerodynamic diameter.

Figure 6. Convergence ratio, R_ε, given by Equation (15)(15) $$R_{\epsilon }=-\left(2n-1\right)-n\cdot \frac{d\mathit{ln}C}{d\mathit{ln}{d}_{\text{aero}}}=1-n\cdot \left(2+\frac{d\mathit{ln}C}{d\mathit{ln}{d}_{\text{aero}}}\right),$$(15) evaluated for the iteration schemes given by Equations (12)–(13), and Equation (14)(14) $$d_{\text{aero},i+1}=\frac{Y^n}{{d_{\text{aero},i}}^{2n-1}\cdot C^n\left(d_{\text{aero},i}\right)};.$$(14) with n = 2/3, as functions of the aerodynamic diameter.

Figure 7. Drag coefficient, C_D, with asymptotic limits for small and large Reynolds number, Re, and quantities Re²⋅C_D and Re/C_D, as functions of Re.

Figure 8. Logarithmic derivative of drag coefficient, C_D, with respect to Reynolds number, Re, and convergence ratio, R_ε, for Equations (19)(19) $${\mathit{Re}}_{i+1}={\left\{\frac{Y^2}{{\mathit{Re}}_i\cdot {\left[C_D\left({\mathit{Re}}_i\right)\right]}^2}\right\}}^{\frac{1}{3}}.$$(19) and (20)(20) $${\mathit{Re}}_{i+1}={\left\{{\mathit{Re}}_i\cdot X^2\cdot {\left[C_D\left({\mathit{Re}}_i\right)\right]}^2\right\}}^{\frac{1}{3}}.$$(20) , as functions of Re.