The rotating cup has been used for atomization of liquid for the last 70 years, for numerous purposes like spray drying, heating and cooling, combustion and crystallization. In the early years, it was used in domestic heating applications 1. Later, this principle was applied in agriculture, fire extinguishing and industrial painting. The rotating bell atomizer (RBA) is now a very popular device used in automotive coatings. In an RBA, the liquid coating is fed into a rotating cup, shaped like a bell. It then flows towards the edge of the cup, gains its rotating speed and forms spray drops at the edge of the cup. Shaping air is an optional measure used to control the spray. The combination of centrifugal force, angular speed of the liquid layer and droplets, shaping air flow rate, air velocity, and electrostatic charge of the liquid affect the size and velocity of the droplets, from the edge of the cup until they strike the surface 2. Considerable evaporation is expected during the flight of the droplets. These phenomena (droplet formation and evaporation) affect the efficiency with which coatings are transferred to the target and the final finish on the surface.
Although various researchers strived to understand the basic principle of the mechanism for the disintegration of liquid occurring on and after the edge of the cup 2,3,4,5, a detailed drop size and evaporation investigation, leading to empirical formulae using data from an industrial-scale application is reported here.
2 Literature Review
The lifetime of a droplet created by atomization of a liquid can be divided in two parts: the formation of droplets and the in-flight change before reaching the target surface. When a rotating cup is used for the atomization of a liquid, the liquid is fed into the interior of the rotating cup. Friction occurs between the liquid and the rotating cup, which results in the liquid achieving the same angular velocity as the cup. Centrifugal force causes the liquid to flow toward the edge of the cup. The length of the liquid front increases with radial distance, so by conservation of mass, the thickness of the liquid front must decrease 2. The movement of liquid through the rotating cup is shown in Figure 1.
Shaping Air Flow
Shaping Air Flow
Figure 1: Liquid and shaping air flow through a RBA
The breakup of the liquid layer into droplets is a complex phenomenon which occurs within a very short distance. The thin film of liquid faces several internal and external forces when it is separated from the edge of the cup 6. Pressure, surface tension, and centrifugal force are the driving forces. Viscosity and initial angular velocity contribute to the formation of droplets. Several investigators observed that the liquid sheet forms an aerodynamic pattern that looks like a wave at the edge of the cup 6,7, 8. The wavy liquid breaks into a succession of circumferential ligaments before disintegration occurs. Disintegration is believed to happen when the wavy film reach critical amplitude. The length (circumferential direction) of the ligaments is one half of the wave length of the liquid sheet. Upon further propagation of these unstable ligaments, they contract by surface tension and break circumferentially up into liquid droplets 6, 9. The formation of droplets from the liquid sheets is shown in Figure 2.
Figure 2: Formation of droplets from liquid sheets (adopted from 6)
The physical model of in-flight characteristics of droplets was described by several researchers 4,5,10. Regardless of the type of liquid, all of them evaporate when in a gaseous medium. The rate of this evaporation depends upon the ambient temperature and the vapor pressure of the liquid. When a water-borne coating is used, water is the solvent and other dissolved or suspended materials are the solute. For solvent-borne coatings, organic liquids are the solvents. For any practical spray application, the solute has a lower vapor pressure than the solvent 11 and hence the solvent evaporates more rapidly than the solute. When most of the solvent (50% or more) is evaporated, any co-solvents start evaporating, but at a slower rate 11.
When a water droplet travels through the air, it is subject to evaporation and hence a decrease in size. The droplet also cools down due to evaporation and a saturated vapor layer surrounds the droplet. At this point, the drop has reached its wet bulb temperature. The temperature of the droplet is then lower than the surrounding air temperature, which causes heat flow towards the droplet, and evaporation starts again 4,5,11. Thus, the rate of evaporation is not constant and the process is not steady, so the time required for evaporation can be divided into two steps. The first is the time span from the formation of droplets to the time when droplets reach their wet bulb temperature. The second is a period of unsteady evaporation. The processes continue until the droplet reaches the target surface, or the solvent and co-solvents completely evaporate.
The initial angular velocity of the droplet degrades rapidly due to air friction 11. If any downward axial air velocity is present, the droplet quickly loses its radial and tangential components until only the vertical component remains, due to gravity and drag by the ambient air. After a certain time, the droplets simply have a constant axial velocity, which is called the sedimentation velocity. Rayleigh was the first mathematician to investigate the properties of water droplet formation using a jet. He established a mathematical model to describe the disintegration of water jets projected into the air. He was the first to identify that the water jets form a thin layer or sheet, before forming droplets 13.
Hinze and Milborn did an initial investigation of the atomization of liquids using a rotating cup. They divided the occurrence of atomization into three possible “stages”, depending on the supply rate of the liquid. In the first stage, a liquid torus is formed at a very low flow rate, which eventually deforms by centrifugal forces and drops are formed. If the flow is increased, the second stage kicks in, in which the torus becomes ligaments of liquid. Increasing the flow rate increases the thickness and number of ligaments, which at their end become droplets due to disturbances caused by external forces. A further increase in flow rate results in the maximum thickness of ligaments where a film is extended from the edge of the cup, which later forms circumferential ligaments that break down into droplets. In this third stage, the thickness of the liquid film just outside and inside of the edge of the cup is practically the same. They used this assumption to calculate the thickness of the liquid layer, which was later used by most researchers to calculate the diameter of the droplets 2.
Dombrowski and Johns investigated the aerodynamic instability and disintegration of viscous liquid sheets 6. They discussed the characteristic change of the liquid layer after it is ejected from the rotating cup and acted on by the surrounding atmosphere. Though different operating conditions affect the method of disintegration of the liquid layer into droplets, the major cause of the disintegration is the interaction of liquid layers with surrounding conditions and the forces that work on them. They considered only a Newtonian fluid where the liquid sheet disintegrates to droplets. During the growth of aerodynamic waves, forces caused by gas pressure, surface tension, liquid inertia and viscosity of the liquid sheet were considered in deriving the initial drop size equation. Dombrowski and Johns established an equation between droplet diameter and fluid ligament diameter based on fluid properties. The relation holds true for viscous fluids which form droplets as in Stage III of the zoning chart by Hinze and Milborn 2. The relation is described by the following equation:
It is noted that since the physical properties for liquid and gas (air) are constant, the mean droplet diameter depends upon the mean liquid velocity in the air and K1, which differs according to the type and size of the rotating cup and the flow rate of the liquid. The value of K1 can be determined for each specific bell type and is constant for a rotary (and associated mean liquid) velocity 6.
A thorough investigation of the formation of a liquid sheet in a rotating cup was performed by Fraser et al. 7,14,16. They considered the shaping air flow rate in their series of studies. Formation of a uniform liquid sheet is important for a uniform spray of liquids over a wide range of operating conditions. In their first study 7 they showed the importance of an uninterrupted supply of liquids into the rotating cup wall for the production of uniform liquid sheets. In their second study 14, they showed the influence of various factors: rotating cup dimension, speed, flow rate, and viscosity on the thickness of liquid films. The sheet thickness increased with an increase in the liquid flow rate and liquid viscosity. The sheet thickness decreased with an increase in rotating cup speed and cup diameter. In their third study 16, they include shaping air flow to increase the air-liquid contact, which was believed to decrease the liquid sheet thickness at higher air flow rates. Shaping air flow ensures better energy transfer efficiency and finer droplet size, of which the latter is important in creating a better finish on the coated surface. The same concept is true for the relation between droplet diameter, rotating cup speed and flow rate. By increasing the shaping air flow rate from 100 to 400 ft3/s, a sharp decrease in mean droplet diameter was found at all liquid flow rates. However, any further increase of shaping air flow rate had little or no effect on droplet diameter.
Dominick considered Weber number (We) and critical flow number to postulate a non- dimensional correlation with the Sauter mean diameter of droplets for both Newtonian and non-Newtonian fluids in a high-speed rotary bell atomizer. They found that Sauter mean diameter increased with decreasing bell speed for both the fluids. However, no considerable effect on Sauter mean diameter for a change of Weber number or critical flow number was found 16.
A detailed investigation of the kinetics and evaporation of in-flight water droplets was described by Holterman 11. He described the physical phenomenon which occurs during the flight path of a droplet. He proposed that if the decrease of droplet size is relatively small and the evaporation rate does not change much, the classic equation for change in droplet diameter by Williamson and Threadgill 10 can be re-written as:
Hence, Equation 3 can be re-written as
Di Domenico and Henshaw 17 used a full factorial DOE matrix to find a relationship between the operating parameters: flow rate, bell speed, shaping air flow rate, electrostatic potential with the appearance of cured paint. Prior to their investigation, it was believed that small particles experience greater evaporation in flight (compared to large droplets) ensuing lower solvent content, greater viscosity during surface contact and hence are less amenable to surface-tension-induced leveling 3. However, they found contradictory results, where drier films had fewer wrinkles. They postulated that, surface-tension-gradient-driven solvent flow results in an uneven thickness of the coating resin, so that after the solvent has evaporated, the waviness in the film is higher. Thus, films from a drier spray would end up flatter. Ray and Henshaw 20 confirmed that decreased flow rate and increased bell speed decreased the spray droplet size and they measured more in-flight evaporation of solvent from smaller droplets. Accordingly, greater in-flight evaporation causes drier films, which have less solvent-driven migration and results in a smoother finish, especially in the lower wavelengths of surface undulation (