The overall process can be divided into three major portions: (i) the film flow beneath the bell, (ii) the formation of ligaments, leading to their disintegration and the resulting formation of droplets, and finally (iii) the flight phase with subsequent droplet impact on the solid or liquid surfaces. The used rotary bell atomizer injects the liquid to the inner side of the spinning bell through a centered channel in the rotating shaft. A thin film of liquid is formed due to strong centrifugal forces, which then emerges from the bell edge, either by directly forming droplets, as well as ligaments or different forms of lamellae that consecutively disintegrate into droplets. The desirable type of droplet formation for most technical processes is the ligament disintegration, as it leads to defined droplet size distributions with narrow span. For smooth-edged bells, the ligament properties (ligament spacing, initial diameter) are a result of flow instabilities that induce specific flow patterns depending on liquid properties and process parameters. This naturally leads to a partially chaotic behavior, due to small disturbances in the film. A defined variation of this ligament buildup is realized by the use of serrated bells. Small triangular channels are cut into the edge of the bell, which force the liquid film to split into a defined number of rivulets, which consecutively discharge as ligaments. In contrast to smooth-edged bells, these serrations will maintain a constant number of emerging ligaments during the variation of process parameters within the limits of the ligament disintegration mode. The discharge velocity and ligament diameter will change according to volumetric flow rate, bell dimension, and rotational speed.
The droplet formation during bell atomization is a result of disintegrating paint ligaments. Therefore, it is necessary to identify the physical parameters that act at this location correctly for the dimensional analysis. The initial diameter of the ligaments \(d_{{\mathrm{Lig}},0}\) varies weakly with relation to varying process conditions and is approximately of the order of the serration cross-sectional diameter \(d_{\mathrm{Ser}}\). The process and material parameters considered in the dimensional analysis are defined at the bell edge and, therefore, primarily influence the length of the ligament emerging. In ambient air, the ligament breaks down into droplets, the average size of which is dependent on a critical ligament diameter that can depend on the initial ligament diameter \(d_{{\mathrm{Lig}},0}\) and on the ligament length \(l_{\mathrm{Lig}}\). The ligament length and the droplet diameter both can be used as target values for the dimensional analysis.
The major geometrical dimensions are given by
\(d_{\mathrm{p}}\) as the droplet diameter,
\(d_{\mathrm{Ser}}\) as the characteristic measure of the serration (e.g., hydraulic diameter),
\(l_{\mathrm{Lig}}\) as ligament length, and
R as bell radius. The process parameters are the bell speed
\(\omega\) and the liquid volumetric flow rate
\({\dot{V}}\), which is implemented as an axial ligament flow velocity
\(c_{\mathrm{ax}}\) using equation (
1). Under the assumption of an ideal liquid distribution between the entirety of
N different equilateral triangular serrations with a flat rivulet and an initially undisturbed rotationally symmetric ligament, we obtain
$$\begin{aligned} c_{\mathrm{ax}}= \frac{4{\dot{V}}}{{\sqrt{3}\cdot d^2_{\mathrm{Ser}}}\cdot N}. \end{aligned}$$
(1)
This simplified solution yields a good estimate of the initial average velocity inside the ligament, yet does not take into account disturbances of the free surface inside the serration rivulet or local velocity distributions, as it is based on an integral balance. However, the axial velocity for the disturbed surface is directly proportional to
\(c_{\mathrm{ax}}\); therefore, the dependency on
\({\dot{V}}\) is valid for the product of power laws. For constant ambient conditions and moderate bell speed, both gas viscosity
\(\eta _{\mathrm{g}}\) and gas density
\(\rho _{\mathrm{g}}\) are treated as constant. Surface tension
\(\sigma\) and liquid density
\(\rho _{\mathrm{l}}\) are assumed to be constant. The rheological behavior of the liquid phase is characterized by a shear and an elongational relaxation time (
\(\lambda _{\mathrm{s}},\lambda _{\mathrm{e}}\)) and by shear and elongational viscosities (
\(\eta _{\mathrm{s}},\eta _{\mathrm{e}}\)) at a constant given shear (
\({\dot{\gamma }} = 3\times 10^5\,{\mathrm{s}}^{-1}\)) and strain rate (
\({\dot{\varepsilon }}\)). Both relaxation times of viscoelastic liquids can be obtained experimentally as described by Macosko.
12 The amount of glass flakes of density
\(\rho _{\mathrm{p}}\) defines the solid content of the liquid phase
\(\phi\), which serves as an additional dimensionless group. Applying Buckingham’s
\(\Pi\) theorem for the given system, a set of dimensionless groups can be derived that yields the general form
$$\begin{aligned} \Pi _{\sum } = f(d_{\mathrm{p}},d_{\mathrm{Ser}},l_{\mathrm{Lig}},R,\omega ,c_{\mathrm{ax}},\sigma ,\lambda _{\mathrm{e}},\lambda _{\mathrm{s}},\eta _{\mathrm{e}},\eta _{\mathrm{s}},\eta _{\mathrm{g}},\rho _{\mathrm{l}},\rho _{\mathrm{g}}, \rho _{\mathrm{g}},\phi ). \end{aligned}$$
(2)
However, the approach of using a defined elongational viscosity for the definition of dimensionless groups is problematic. Typically
\(\eta _{\mathrm{e}}\) is obtained from the CaBER experiment (
\({\dot{\varepsilon }}_{\mathrm{max}}= 100\,{\mathrm{s}}^{-1}\)) and, therefore, calculated using the surface tension
\(\sigma\) and the rate of deformation as shown in Stelter et al.
13 Based upon the rate of deformation, these values may vary by several orders of magnitude. Recent work of Kuhnhenn et al.
6 provides a solution for a defined rivulet flow emerging from a cylindrical bore. According to the mathematical definition of the strain rate, we expect strain rates ranging from
\({\dot{\varepsilon }}= 1000{-}7000\,{\mathrm{s}}^{-1}\) for the used rotary atomizer. Therefore, the CaBER device does not cover the strain rates during high-speed bell atomization. For the given case, we do not have a precise way of calculating the strain rate
\({\dot{\varepsilon }}\), as the exact position of the free surface and, therefore, the exact flow field is unknown. Accordingly, we neglect the elongational viscosity
\(\eta _{\mathrm{e}}\) and obtain twelve dimensionless groups to characterize the general breakup process at constant ambient conditions (
\(T = {\mathrm{const.}}, \varphi = {\mathrm{const.}}, p_0 = {\mathrm{const.}}\)), i.e.,
$$\begin{aligned} \Pi _{\sum }= & {} g\left( \Pi _{1}, \dots , \Pi _{12}\right) , \end{aligned}$$
(3)
$$\begin{aligned} \Pi _{\sum }= & {} g\left( \frac{l_{\mathrm{Lig}}}{d_{\mathrm{Ser}}},\frac{d_{\mathrm{p}}}{d_{\mathrm{Ser}}},\frac{\rho _{\mathrm{g}}}{\rho _{\mathrm{l}}}, \frac{\eta _{\mathrm{g}}}{\eta _{\mathrm{s}}},\frac{d_{\mathrm{Ser}}}{R} , \frac{\lambda _{\mathrm{e}}}{\lambda _{\mathrm{s}}},\frac{c_{\mathrm{ax}}}{\omega \cdot R} ,\frac{c_{\mathrm{ax}} \cdot \rho _{\mathrm{l}} \cdot d_{\mathrm{Ser}}}{\eta _{\mathrm{s}}} ,\frac{{c^2_{\mathrm{ax}}} \cdot \rho _{\mathrm{l}} \cdot d_{\mathrm{Ser}}}{\sigma },\right. \nonumber \\&\left. \lambda _{\mathrm{e}} \cdot \left( \frac{ \sigma }{\rho _{\mathrm{l}} \cdot d^3_{\mathrm{Ser}}}\right) ^{0.5}, \frac{\rho _{\mathrm{p}}}{\rho _{\mathrm{l}}}, \phi \right) . \end{aligned}$$
(4)
Here
\(\Pi _{1}{-}\Pi _{6}\) and
\(\Pi _{11}\) are ratios of parameters of equal dimensions,
\(\Pi _{7}\) is the ratio of axial liquid velocity inside the ligament and tangential velocity at the bell edge and, therefore, defining the stretching characteristics of the ligament flow.
\(\Pi _{8}\) is a liquid-related Reynolds number inside the ligament at its origin (Re
\(_{{\mathrm{ax,l}}}\)), formed with the constant shear viscosity at the bell edge and axial ligament flow velocity.
\(\Pi _{9}\) is a liquid-related Weber number inside the ligament (We
\(_{{\mathrm{ax,l}}}\)), formed with the axial ligament flow velocity and
\(\Pi _{10}\) is a Deborah number formed with a characteristic breakup time. Both the Reynolds and the Weber number can be formed as corresponding gas-related dimensionless groups. The Deborah number can be formed with different characteristic timescales relative to a specific relaxation time. For the given case, the capillary breakup time
\(t_{\mathrm{cap}}\) is chosen as the liquid-specific timescale.
7 Hence, we obtain
$$\begin{aligned} t_{\mathrm{cap}}= & {} \left( \frac{\rho _{\mathrm{l}} \cdot {d^3_{\mathrm{Ser}}}}{\sigma }\right) ^{0.5}, \end{aligned}$$
(5)
$$\begin{aligned} \Pi _{10}= & {} \frac{\lambda _{\mathrm{e}}}{t_{\mathrm{cap}}} = {\mathrm{De}}_{\mathrm{cap}}. \end{aligned}$$
(6)
Dynamic and viscous timescales can be obtained through dimensional analysis by choosing the corresponding basis. The ideal timescale for the ligament breakup is the actual breakup time
\(t_{\mathrm{Lig}} = l_{\mathrm{Lig}} \cdot {c_{\mathrm{ax}}}^{-1}\), which is a function of the axial velocity
\(c_{\mathrm{ax}}\) and one of the two target parameters
\(l_{\mathrm{Lig}}\) and, therefore, not available as a variable. For the majority of technically used paint liquids
\(\Pi _{6} \rightarrow 0\), the shear relaxation time (
\(\lambda _{\mathrm{s}} \approx 10^2{-}2 \times 10^4 \,{\mathrm{s}}\)) is larger by several orders of magnitude compared to the elongational relaxation time (
\(\lambda _{\mathrm{e}} \approx 10^{-2}{-}5 \times 10^{-1}\,{\mathrm{s}}\)) as described in Oswald and Willenbacher.
9 Additionally, for these liquids
\(t_{\mathrm{Lig}} \ll \lambda _{\mathrm{s}}\) holds, therefore, we neglect the influence of shear relaxation on ligament breakup.