February 13, 2005

Scaling Problems 1 

Scaling is the whole process of making something bigger (scaling up) or smaller (scaling down). I know quite well this process regarding chemical plants and similar systems, but similar considerations can be applied to almost any system.
The main difference is that chemical plants usually need to be scaled up, from a laboratory reactor used for R&D and associated devices to a full-scale industrial plant. A lab reactor will produce just a few grams per day of products (up to say 100 g, but as usual there's a lot of variability), while an industrial plant can go up to 1000 t per day (if I remember correctly, the biggest sulphuric acid plant produces something like 2000 t/day): this obviously means that much more reagents and products (and heating and cooling fluids etc) must be processed in the same time.

So why it's scaling up so problematic? There are several factors involved, but let's start with the inescapable ones: geometry and physics.
Scaling up a system almost invariably requires to increase the size and thus volume of its parts, simply because bigger volumes are needed to contain and process greater amounts of matter. What happens when solids increase in volume? Let's consider three simple solids: a cube of side l, a sphere of radius r and cylinder of radius R and heigth h, and calculate their surface area-to-volume ratios (A/V) - why this particular ratio is useful, will be clear later.

Cube: A/V = 6/l
Sphere: A/V = 3/r
Cylinder: A/V = 2*(1/R + 1/h)

This ratio is very simple for sphere and cube, and just a bit more complex for the cylinder. However, in all three cases it's evident that the A/V ratio is inversely proportional to the dimensions of the solid. In other words, bigger solids have less surface area per unit of volume. This is a natural property of objects, and cannot be ignored.

Another issue to take into consideration for industrial/chemical plants is heat transfer. Heat transfer is the process of heat passing from a hot body (or fluid) to a colder one, like hot combustion gases ceding ceding heat to water that vaporizes in the process. But especially in chemical reactors, heat must be carried away to avoid overheating; the worst case is an uncontrolled accumulation of heat, that will cause a rapid temperature increase with damage to the system and possibly an explosion. It is thus obvious that accumulation of heat must be avoided.
Heat transfer is a superficial process - that is, the flow of heat (kJ/s) is proportional to the area of the surface through which the transfer occurs (it is also proprtional to the temperature drop and the nature of materials involved). But we have just estabilished that bigger solids have less surface area per volume unit, so one might expect that heat transfer will be more problematic for bigger components.

A nd in fact it is: R&D usually starts with small tubular reactors, of 5-15 mm of external diameter: their A/V ratio is high and, and consequently heat transfer processes usually do not represent a concern and these reactors work happily without any internal cooling systems (it would also be a hell of a job to put any kind of cooling tubes inside a tube of 5 mm internal diameter...); eventually the effluents are cooled in a simple water heat exchanger (two concentrical tubes) downstream the reactor.

But industrial reactors are big cylinders of many cubic meters volume: their surface area is small compared to their volume, and an external cooling system (like a water jacket) often is not enough to avoid local overeheating and internal cooling coils or other devices become necessary. The technology to design and build those is well known, but nevertheless they may alter the flow inside the reacor, and surely add weight, complexity and cost to the plant.

One or more following posts will deal with other cases of scaling problems.


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