# Do the Maths!

Article by Jimmy Hunter BSc CEng CSci

## Johannesburg & Umbogintwini, South Africa, 1975

### Using mathematical modelling to cool a bead mill

AFTER graduating, many chemical engineers (including myself) had the fond idea that equations have been written for all problems in chemical engineering, and the formulae merely need to be looked up, for example in Perry's Chemical EngineersHandbook. It came as a bit of a shock that there are many areas of chemical engineering which had not been covered, or at least not well. Although we've been taught how to formulate equations and to solve them, many chemical engineers still lack the confidence to tackle a problem head on and develop the maths required.

This article looks at a problem that could not have been solved without developing differential equations and solving them. The result was a very successful design and a highly satisfactory commercial outcome for our company.

## The problem

South African chemicals group AECI was planning to move into the fast-growing 'flowables' market for various agrochemicals. A promising product was flowable Atrazine, a weedkiller. The proposed manufacturing method was to mill the solid Atrazine crystals in water to a particle size where the slurry behaved as a stable suspension. The proposed method of milling the product was to use a bead mill.

A sketch of the bead mill layout is given in Figure 1. The feed slurry is pumped into the main chamber, which is filled with glass beads, and contains an agitator with polyurethane discs spaced along a rotating shaft. The grinding chamber is enclosed in a cooling chamber fitted with a cooling water inlet and outlet. The slurry flows through the grinding chamber, where the solid feed is ground between the rotating glass beads.

Figure 1: Schematic of bead mill

The bead mill here was the research unit, which was used to develop the process and produce product samples. Temperature probes (not shown) were fitted to provide the inlet and outlet temperature of the slurry and product. The intention was to pass the slurry through the grinding chamber twice, to get the particle size of the product low enough.

## Bizarre behaviours

The two chemists who were doing this preliminary work were puzzled by the “bizarre” behaviour of the system. For example, the temperature of the product went from about 20°C to 40°C after a single pass, but after a second pass it was still at 40°C. The temperature of the product was critical, since Atrazine flowables decompose above around 45°C.

Puzzled by this behaviour, we decided to produce a mathematical model to explain the bead mill process. This was not just an academic exercise. We had to be sure of the design of the full-scale version of the process. The plan was to install a large-scale bead mill at our Umbogintwini factory on the Natal South coast, which is sub-tropical. Since the ambient temperature could often be as high as 30°C, and the humidity was usually close to 100%, cooling water temperatures are often as high as 28–30°C. We had to be sure that we could cool the large-scale unit enough to keep the product temperature below 45°C.

The commercial bead mill had already been ordered, and was due for shipment within a few weeks. We had to know whether a chiller unit would be needed, and what its capacity and minimum temperature should be. The bead mill supplier was not worried, but all of its previous supplies had been in Continental Europe, where cooling water temperatures are seldom above 15°C. We were much less relaxed.

## From theory...

The development of the model was presented in our paper in Industrial and Engineering Chemistry.1 However, the difficulty of solving the differential equations was not mentioned in the paper. I struggled to solve them for many hours. However, our boss, Colin Schlesinger, took one look and said: “Aha! Linear differential equations!” They were linear, but a special form (ie Bernoulli). We soon solved them, and suddenly we understood the behaviour of the system so we could make sensible decisions about the best way to operate the bead mill.

The model gave us a number of important insights. One surprise is that, with counter-current flow of the feed and coolant, it would be possible to exceed the maximum temperature within the mill, without realising that it had happened, since the outlet temperature could still be less than 45°C. We worked out that, without coolant, the temperature of the product would reach 268°C! We calculated the overall heat transfer coefficient by an iterative process, using a BASIC program. (Remember BASIC?). We found, not surprisingly, that the heat transfer coefficient was the same for co-current and counter-current flow at the same flowrate. Finally, the model showed that we needed refrigeration of the coolant to achieve a product temperature below 45°C.

Efficient cooling was thus essential, and for good control it would be necessary to operate with co-current flow. The cooling jacket was poorly designed, since the coolant velocity was very low. We re-designed the jacket with a fitted spiral to speed up the coolant flow through the jacket. One of our technicians, Peter, flew to Umbogintwini, armed with the stainless steel spiral which he welded into position in the cooling chamber. At first, it was too tight a fit, so he turned the spiral in a lathe to make it fit.

## ...To practice

The results were outstanding! We could operate the bead mill at double the design feed rate without exceeding the 45°C maximum product temperature. Our competitor, part of a global petrochemical company, could not match either our output rate or quality, so the product was a great commercial success.

We briefly considered patenting the spiral insert, but decided against it. The bead mill market was too small and the turnover would not be enough to pay the patent costs. Our bead mill supplier soon copied our new design, except it used a polyurethane spiral instead of a stainless steel one. However, by the time our competitor installed the new design, we had captured the lion's share of the market.

This project raised some important points about the training and development of chemical engineers. We were lucky enough to work in a department which created an environment in which we were trusted and encouraged to develop our chemical engineering skills, with a healthy scepticism towards 'expert' suppliers. My message to aspiring young chemical engineers is “back yourself” (but make sure you get someone to check your calculations).

### References

1. Baskir,  CI, Hunter, JB, and Schlesinger, CB, "Evaluation of the Heat Transfer Characteristics of a Bead Mill with the Aid of a Mathematical Model", Industrial & Engineering Chemistry Process Design and Development, 1978, 17 (3), 318-320.

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