Table 2.

Scaling laws for the prediction of droplet diameter in a single EHDA system

Droplet diameter	Current	Dimensionless parameters	References
${\mathrm{D}}_{\mathrm{d}}~{\mathrm{d}}_{\mathrm{j}}~{\mathrm{R}}^{\ast }={(\frac{\rho {\mathrm{Q}}^2}{\gamma })}^{1/3}$ when η ≫ 1	I=f(ε)(γQK/ε)1/2	$\eta ={\left(\frac{\rho \text{KQ}}{\gamma {\epsilon }_0\epsilon }\right)}^{1/2}$ (0.51 ≤ η ≤ 2.01)	(de la Mora and Loscertales, 1994)
${\mathrm{D}}_{\mathrm{d}}~{\mathrm{d}}_{\mathrm{j}}~{\mathrm{r}}^{\ast }={(\frac{\mathrm{Q}\epsilon {\epsilon }_0}{\mathrm{K}})}^{1/3}$ when η ~ 1
${\mathrm{D}}_{\mathrm{d}}=k_{\mathrm{d}}{\mathrm{Q}}^{\frac{1}{2}}{\left(\frac{\rho {\epsilon }_0}{\gamma \mathrm{K}}\right)}^{\frac{1}{6}}$ when β ~ 1	$\mathrm{I}={\mathrm{k}}_{\mathrm{I}}{\left(\frac{\text{QK}{\epsilon }_0\gamma }{\rho }\right)}^{1/4}$	$\beta =\frac{\epsilon }{{\epsilon }_0}$	(Gañán-Calvo., 1994)
${\mathrm{D}}_{\mathrm{d}}={\mathrm{k}}_{\mathrm{d}}^{\prime }{\beta }^{1/6}{\left(\frac{\mathrm{Q}{\epsilon }_0}{\mathrm{K}}\right)}^{\frac{1}{3}}$ when β ≫ 1	$\mathrm{I}={\mathrm{k}}_{\mathrm{I}}^{\prime }{\left(\frac{\text{QK}\gamma }{{\beta }^{1/2}}\right)}^{1/2}$
$\frac{\mathrm{d}}{{(\epsilon -1)}^{\frac{1}{3}}{\mathrm{d}}_0}=1.6{\left(\mathrm{Q}/\left({(\epsilon -1)}^{\frac{1}{2}}{\mathrm{Q}}_0\right)\right)}^{\frac{1}{3}}-1.0$	$\mathrm{I}=6.2{\left(\frac{\mathrm{Q}}{{\mathrm{Q}}_0{(\beta -1)}^{1/2}}\right)}^{1/2}$	${\delta }_{\mu }{\delta }^{\frac{1}{3}}={\left(\frac{{\epsilon }_0^2{\gamma }^3}{{\mathrm{K}}^2{\mu }^3\mathrm{Q}}\right)}^{\frac{1}{3}}$	(Gañán-Calvo, et al., 1997)
${\delta }_{\mu }{\delta }^{\frac{1}{3}}\ll 1$		Q0=γε0/ρK
$\frac{\mathrm{d}}{{\mathrm{d}}_0}=1.2{(\frac{\mathrm{Q}}{{\mathrm{Q}}_0})}^{1/2}-0.3$	$\mathrm{I}/{\mathrm{I}}_0=11.0{\left(\frac{\mathrm{Q}}{{\mathrm{Q}}_0}\right)}^{1/4}-5.0$	I0=(ε0γ2/ρ)1/2
${\delta }_{\mu }{\delta }^{\frac{1}{3}}\gg 1$		${\mathrm{d}}_0={(\gamma {\epsilon }_0^2/\rho {\mathrm{K}}^2)}^{1/3}$
Dd=G(κ)r*	I = f(κ)(γKQ/κ)1/2	κ = dielectric constant r* = (Qεε0/K)1/3	(Chen and Pui, 1997)
G(κ)=−10.9κ−6/5+4.08 κ−1/3	f(κ) = −499−0.21κ+ 157κ1/6+336κ
$\begin{array}{l}\mathrm{d}=2\times 1.89{\mathrm{R}}_0{\mathrm{f}}_{\mathrm{b}}\\ =3.78{\pi }^{-2/3}{\mathrm{Q}}^{1/2}{\left(\frac{\rho {\epsilon }_0}{\gamma \mathrm{K}}\right)}^{1/6}{\mathrm{f}}_{\mathrm{b}}\end{array}$	$\mathrm{I}=4.25{\left(\frac{\gamma \text{KQ}}{ln{\left(\frac{\mathrm{Q}}{{\mathrm{Q}}_0}\right)}^{1/2}}\right)}^{1/2}$	Q0=ρK/γε0	(Gañán-Calvo, 1997)
$\frac{\mathrm{d}}{{\mathrm{d}}_0}={\mathrm{k}}_{\mathrm{d}}{\left(\frac{\mathrm{Q}}{{\mathrm{Q}}_0}\right)}^{1/2}$	$\frac{\mathrm{I}}{{\mathrm{I}}_0}={\mathrm{k}}_{\mathrm{I}}{\left(\frac{\mathrm{Q}}{{\mathrm{Q}}_0}\right)}^{1/2}$	${\mathrm{d}}_0={\left(\frac{\gamma {\epsilon }_0^2}{{\pi }^2\rho {\mathrm{K}}^2}\right)}^{1/3}$	(Gañán-Calvo, 1997)
kd: Constant; It depends on needle-to-electrode potential difference as well as on the needle radius.	k1: Constant; It depends on needle-to-electrode potential difference as well as on the needle radius.	${\mathrm{I}}_0=\left(\frac{\gamma {\epsilon }_0^{1/2}}{{\rho }^{1/2}}\right)$
${\mathrm{d}}_{\mathrm{d}}~{\left(\frac{\rho {\epsilon }_0{\mathrm{Q}}^3}{\gamma \mathrm{K}}\right)}^{1/6}$	I=(γKQ)1/2	The velocity profile in the jet is assumed to be flat	(Hartman et al., 2000)
${\mathrm{D}}_{\mathrm{d}}~\mathrm{d}={\left(\frac{\rho {\epsilon }_0{\mathrm{Q}}^3}{\gamma \mathrm{K}}\right)}^{1/6}$	I=(γKQ)1/2	Inertia and electrostatic suction scaling *: ${\alpha }_{\mathrm{p}}\gg {\alpha }_{\mu }^{1/4}$ and αp/(β−1)≫1	(Gañán-Calvo, 2004)
${\mathrm{D}}_{\mathrm{d}}~\mathrm{d}={\left(\frac{\rho {\epsilon }_0{\mathrm{Q}}^3}{\gamma \mathrm{K}}\right)}^{1/6}$	$\mathrm{I}={\left(\frac{\rho K^2{Q}^2}{(\beta -1){\epsilon }_0}\right)}^{1/2}$	Inertia and polarization forces scaling: $1\gg \frac{{\alpha }_{\rho }}{(\beta -1)}\gg \frac{{\alpha }_{\mu }}{{(\beta -1)}^2}$
${\mathrm{D}}_{\mathrm{d}}~\mathrm{d}={\left(\frac{\mu {\epsilon }_0^2{\mathrm{Q}}^3}{\gamma {\mathrm{K}}^2}\right)}^{1/8}$	I=(γKQ)1/2	Viscous force and electrostatic suction scaling: ${\alpha }_{\mathrm{p}}\ll {\alpha }_{\mu }^{1/4}$ and αμ/(β−1)4≫1
${\mathrm{D}}_{\mathrm{d}}~\mathrm{d}={\left(\frac{\mu \mathrm{Q}}{(\beta -1)\gamma }\right)}^{1/2}$	$\mathrm{I}={\left(\frac{{\mu }^2{\mathrm{K}}^3{\mathrm{Q}}^2}{{(\beta -1)}^4{\gamma }^2{\epsilon }_0^2}\right)}^{1/2}$	Viscous force and polarization force scaling: $\frac{{\alpha }_{\rho }}{(\beta -1)}\ll \frac{{\alpha }_{\mu }}{{(\beta -1)}^2}\ll 1$

${\alpha }_{\rho }=\frac{\rho KQ}{\gamma {\epsilon }_0};{\alpha }_{\mu }=\frac{{\mu }^3{K}^{2}Q}{{\gamma }^3{\epsilon }_0^2}$;β is the liquid polarity parameter.