# Scaling for horizonal media mills

What do manufacturers of milling equipment use as scaling factors, and what are their primary concerns with scale up? Equipment manufacturers calculate the amount of grinding energy required to achieve a specified level of particle size reduction and the resultant dispersion of the ground particles. This number is known as the specific energy requirement (Espec). However, several other variables are involved in the scale up, including the following.

- Premixing
- Energy input
- Flow velocity
Hydraulic packing

Viscosity relationship – is the material very high in viscosity?

Media size

Media separation

Viscosity

Media size

- Temperature control
Cooled surface area

- Materials of construction

## Premixing

The dispersion is initially made using a high-speed disperser in the lab, which is a controlled environment where the ingredients are carefully measured and added slowly. The vessel-to-blade ratio is perfect and the dispersion time is lengthy to get the best result. There is solid technical information about this process since it has been modeled; therefore, there is no reason why this process should present any problems in scale up. However, in the plant there are many instances where the premix is not nearly at the level of dispersion as the lab scale. This could be due to factors such as hastiness because of production pressures, or using a low blade tip speed so the dispersion will not splash out of the tank.Some manufacturers make a premix at very low viscosity so that the pigment is easily incorporated into the vehicle. When all the pigment is in, the viscosity is so low that when the disperser speed is increased, the material splashes out of the tank. This results in a very low shear rate and poor initial deagglomeration. The coating is, therefore, a very coarse initial particle size, with agglomerates up to over 2 millimeters in particle size. This causes wear and tear on the pumps and the feed end of the mill, settling of the pigment in the tank since it is not agitated during milling, and possibly blocking the mill screen because of poor start procedures with the mill. This type of situation is why premixing systems are available that use vacuum feeding of powders into scraped wall, enclosed tanks. These systems allow very high viscosity dispersion without dust and in-plant environmental problems.

Another premixing issue from mill manufacturers’ perspective is premix supplied for lab testing. Although it might be more convenient for both the coatings and mill manufacturers to supply the premix directly from the plant, this means that the coating has been sitting around for at least a week before milling. Three possibilities can result. First, the solids to be dispersed are more effectively wet out and can then be easily milled. Second, the solids have settled very hard in the drum and are more agglomerated than they would be in the plant. Third, for waterborne coatings, when the coating is dispersed at the plant, there is a high degree of entrained air in the dispersion. Air in the dispersion causes many problems in the mill, such as high temperature due to poor heat transfer, inefficient grinding due to micro bubbles inhibiting the contact of the grinding media and thixotropic viscosity preventing efficient bead separation. If this dispersion sits around for a week or so, it will deaerate. The dispersion might have been milled efficiently at the lab, but when it is scaled up to the plant and the premix is brought right to the mills, there will be a problem with viscosity, heat transfer, and grinding efficiency.

## Power Consumption

The most accurate method to scale up from lab tests is to look at the Espec. Research in Europe has examined the effect of particle size reduction related to the specific energy input, but this concept has not caught on in North America. Essentially, the power consumption on the mill and the mass flow rate need to be recorded. The typical values recorded are the kilowatt input and the mass flow rate in tons per hour. By dividing the kilowatt input by the mass flow rate, the specific energy input is given in kilowatt-hours per ton, the Espec. A slightly more accurate method is to use a totaling kilowatt-hour meter. The mass of the entire batch processed through the mill is known, even in a ‘continuous’ process. By dividing the total kilowatt hours used by the mass of product processed, kilowatt hours per ton can be determined. The grinding and dispersion process is an energy process. There is no real magic involved in this process, but there are several unknown factors. However, if the specific energy requirement for the product on the lab scale is calculated and used to scale up to the production mill, there will be no problem in achieving a realistic production number at the desired quality level.Following is an example of how this calculation can be performed. Netzsch has installed on its laboratory equipment kilowatt meters. The “no-load” kilowatt draw is measured for each machine. This means that measurements are made at various agitator speeds of the kilowatt draw without any grinding media in the machine. This is the baseline. When a test is run, the mass flow rate is measured in kilograms per minute and converted to tons per hour. The kilowatt draw is measured, and the agitator speed is recorded. The no-load kilowatt draw for the agitator speed used is subtracted from the gross kilowatt draw. This net kilowatt draw is divided by the mass flow rate in tons per hour. (In some industries the mass flow rate is calculated in dry tons per hour, looking at the energy used to grind the solid material; for the coatings industry this value is not crucial). Knowing the quality level achieved for each sample at a known Espec, we can scale up to the larger production machine. Again the no-load kilowatt draw for the production machine has been determined. The no-load kilowatt draw can be subtracted from the total installed power of the production machine. This net available kilowatt draw is then divided by the Espec for the product. The resulting tons per hour is the potential production rate from the machine for the required quality.

The idea of using Espec as a scale up factor vs. residence time is shown in Figure 1. This figure examines the expected throughput rate based on a direct scale up factor based on installed power vs. mill volume. The figure shows that for a given mill volume, the expected flow rate by directly multiplying the ratio of the lab mill to the production mill volume is tremendously higher than using a scale up factor based on the installed horsepower. But, as stated above, grinding and dispersion is an energy process. For this, there are high-speed dissolvers with motors available — big ones for big tanks, little ones for little tanks. If the energy required for dispersion were free, then a little motor in a big tank would be used. How can users expect to scale up by residence time, a volumetric calculation, when there is not the amount of power installed on the machine required for this process? Another question to be asked is “who started scaling up by residence time in the first place?” For further discussion, the expected residence time based on the Espec calculation should be examined. Using our scale up factors from Figure 1, the flow through the mill in residence time can be correlated. For example, if a one-minute residence time is used as a baseline on the one-liter mill, the flow rate would be about half the volume of the mill per minute (based on volume taken up by beads). So, theoretically, a 500-liter mill would give about 250 liters per minute flow rate with a direct scale up. However, experience shows that a flow rate of about 80 L/min. would be more realistic.

## Flow Velocity

What is happening in the larger mill is the age-old concept of hydraulic packing. It is a problem that manufacturers address with a fundamental principle of countering the hydraulic force of the product flow with their media separation system. The cause of hydraulic packing in a mill is related to various factors: viscosity, tip speed, media density and size, and the flow velocity.Flow velocity in a pipe is calcuated by dividing the volumetric flow by the open area of the pipe, i.e., cubic millimeters per second flow divided by the square millimeters open surface area of the pipe. As mills become larger, the same length to diameter ratio (L/D) remains, but the diametric surface area increases exponentially. Figure 2 shows the effect of the increasing mill size. The L/D value is fairly constant, but the flow velocity for a 1-minute residence time is increasing dramatically. So to scale up, the flow velocity (i.e., if the mill condition results in hydraulic packing) needs to be considered. But if the Espec is used, this variable is nearly eliminated. This is due to the fact that the so-called residence time for the production mill will be longer than the lab mill.

Another argument for scaling up by Espec has to do with residence time distribution and the flow velocity. Figure 2 shows the flow velocity through a mill at a given residence time. The flow velocity is calculated by the annular volume created by subtracting the volume of the shaft as a solid cylinder from the volume of the vessel. As mill size increases, this velocity increases if the residence time is kept constant. What this translates into is the higher linear velocity causing hydraulic packing. This is also an increase in flow at the separation system and a requirement for better bead separation. If the flow velocity based on the residence time calculated from an Espec scale up is examined, it appears that the linear velocity through the mill is constant.

## Temperature Control

A perceived problem for a bead mill should now be addressed. As bead mills increase in size, the cooled surface area to volume decreases. This is a common comment made by customers and a concern because in principle there is not enough cooled surface area to remove all of the heat generated. But, in reality, there is. Figure 3 shows that the curve for the ratio of the available cooled surface area to mill volume is decreasing as mill volume increases. However, the curve for the available cooled surface area to installed horsepower (or kilowatts) is more or less flat, i.e., the same cooled surface area is available for the lab mill as the production mill, and there is enough capacity to remove the heat generated. As mills are designed larger in size, the cooling efficiency of the machine is also increased by such things as baffles or cooling spirals where only flooded pipe was used on the lab mills. So, in theory, heat should not be a problem in the larger production mill.## Tip Speed

When scaling up, it is a known rule of thumb that the tip speed must be equal. However what usually occurs is that the tip speed on the production mill must be increased to increase the power input. This generally occurs due to the use of the radius of the agitator disc as the basis for the calculation. However, a more sound and scalable result might be obtained by using the shaft surface area in the bead mill and the torque applied to the mill. Area is formed by the rotational surface area of the disc or plane formed by the pins. This area is calculated by the radius of the spacer between the discs and the radius of the agitator disc or the base diameter of a shaft and the overall length formed by the pins.This rotational surface area may have different forms, depending on the type of mill used; for example, the surface area of the discs for a disc mill would be calculated not including the spacers, but in the case of a pin mill, the surface area of the solid shaft the pins are attached to would be an important number. These values are available from the bead mill manufacturer. This value should provide a better scaling value because of the principle of shear stress on a shaft.

The torque applied to the shaft should be calculated first. This is done by multiplying the kilowatts (convert horsepower to kilowatts by multiplying horsepower by 0.746) used by 60,000 and dividing by the shaft rotational speed in revolutions per minute giving torque (T) in Newton-meters. Torque is the force moved over a distance; in this case, the distance is the radius of the shaft. Therefore, the force on the shaft is the torque divided by the radius of the shaft (in meters).

The shear stress on the agitator shaft discs can now be calculated by dividing it by the rotational surface area. Using this shear stress value and knowing the surface area available on the production mill, we can multiply the values and get the force required on the production mill. Multiply this value by the production mill shaft diameter for the torque required from the mill. Divide the available power on the production mill (kilowatts) by the torque, multiply by 60,000 and the required shaft speed has been determined.

Figure 4 shows the difference in this calculation obtained when running a lab mill at full speed. Based on the surface area scale up, there is a requirement for a faster agitator speed in the mills in a mid range size, while the larger (and newer design) production mills (over 100 liters) are more geometrically sound in their design. This calculation has been proven empirically over the years; many times it has been required to increase the speed of a mill to compensate for a low power input. Using this calculation (or a mill with a variable speed drive) should reduce some of the scale up changes required at mill start up.

Eventually, after this type of scale up has been made a few times, users might find that the production mill is producing more than expected, i.e., based on Espec higher grinding efficiency is achieved. This condition is experienced mostly in high-flow multiple-pass operations. When this occurs, a question of how much energy is required for grinding and dispersion and how much energy is wasted should be determined. This is a difficult factor to predict and requires more fundamental research comparing production equipment to lab scale equipment and this does not always occur. Therefore, it is still safe to assume that the grinding energy required in the lab mill is the true energy required and that the same level is required in the production mill. It is a further benefit if less energy is required in the production mill.