# Scale-Up of Topical Products: Math Matters

December 1, 2021

Proper scale up of a 1 kg batch to a 1,000+ kg batch takes engineering effort to execute properly.  Why is this so difficult?  The short answer is because of the math.  The physical forces on a semi-solid product while manufacturing a 1 kg laboratory batch vs. that of a 1000 kg + commercial batch are dramatically different.  Addition rates, heating rates, and cooling rates typically take longer, and mixing forces are also different.  Let’s look at the math.

## Heating and Cooling

Semi-solid products are typically heated and cooled during processing from the outside in.  Jacketed vessels or water baths are used to regulate temperature.  If the only source of heating and cooling are the vessel walls, then identifying the thermo-regulation differences can be calculated using simple geometry.1  A representative laboratory vessel for a 1 kg batch may be 1 ft tall and have a 0.5-foot diameter.  This representative vessel has a surface area to volume ration of ~9.  While a commercial vessel with a diameter of 4 feet and is 4 feet tall has a surface area to volume ratio of 1.5.  With this calculation alone, the small vessel will heat and cool 6 times faster than the larger vessel.  Imagine something on the bench taking 30 minutes to cool.  This same exercise could take up to 3 hours at larger scale.

## Mixing Efficiencies

This reduced efficiency also applies to mixing dynamics.2  During homogenization of a 1 kg batch the manufactured material can cycle through a laboratory scale mixing head as many as 250 times in 5 minutes.  A scaled up homogenizer (10 horsepower motor, flowing at 1100 liters per minute) for a 1000 kg batch will only have ~5.5 turn overs in the same 5 minutes.  The larger scale homogenization is only 2% as efficient as the laboratory system.  And yet this is the standard scale homogenizer to use for commercial manufacturing.

## Mixing Speeds

With the decrease in large scale efficiencies there may be the temptation to try and increase mixing speeds to make up for the loss.  This needs to be done with caution.3  Consider a tip (blade) speed calculation.  Tip speed measures how far a point on the outermost edge of the disperser blade travels in a given amount of time.  Similar revolutions per minute (rpm) of mixers can have large tip speed differences when the mixer blades get larger.  A 2.5 inch sawtooth blade running at 350 rpm has a tip speed of 230 feet per second (fps), while a large scale 10 inch sawtooth blade at the same 350 rpm has a tip speed of 1100 fps.  If any of the materials being manufactured are shear sensitive, the tip speed of the mixer was just increased by over four-fold while running at the same speed.

Proper understanding of the physical forces involved and the required techniques and equipment is critical to successful scale up exercises.  Scaling up a product is not simply making something bigger.  As the forces on your product change, so should your approach to manufacturing.  Experimentation and experience are critical to a successful scale up campaign.

## References:

1. F. Incropera & D. DeWitt. 2011. Fundamentals of Heat and Mass Transfer (7th ed.). New York: J. Wiley.
2. 2. Kehn. June 9, 2021. “6 things process engineers should consider when scaling up (or down) mixing processes.”
3. C. A. Challener. April 23, 2013. “Optimizing the Selection of Mixing Equipment.”