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# What is process linearity?

October 29, 2010

Dear Control Engineering: Recently you were talking about linearity in instrumentation devices. What is process linearity?

Process linearity is similar in that it looks at the relationships between elements of a process and the output. Does output change in direct proportion to a change in input.

Let’s say you want to boil a gallon of water to make spaghetti, so you put a big pot on the stove. For the sake of this discussion, you have a thermometer in the pot so you can read the water temperature, and there is a calibrated knob on the stove that allows you to see how much gas you are sending to the burner.

After the first 10 minutes, you see that burning 5 units of gas per minute has raised the temperature of the water by 40 °F. If you double the amount of gas, will the temperature rise twice as fast? Probably not. Here’s why:

The fire will be burning higher, but all that heat will not necessarily go into the pot. Some will simply wash up the sides and heat the room. Just because there is more heat does not mean the pot is able to absorb it. Also, as the water gets hotter, the overall heat transfer characteristics will begin to change and more heat will escape out of the top of the pot as the difference in the temperature of the water and ambient air becomes greater.

If you really get carried away, you could try and try to graph the relationship between gas consumption and temperature change. You would have to consider it at all sorts of conditions, with cold water vs. hot water, high vs. low burner, different ambient temperatures, etc. It’s not a simple relationship and won’t likely be very linear when you figure it all out.

In a typical chemical process application, there are many such relationships and the linearities are not necessarily working together. That’s why some processes are difficult to regulate. You may want to increase production by forcing more feedstock into a reactor, but that doesn’t make the reactor any bigger and the required reaction may not be complete.

As a practical matter, most processes are designed to operate within a relatively narrow range where behaviors are predictable. An oil refinery doesn’t normally turn a unit down to 40% because demand has decreased temporarily. It probably can’t operate at that level, nor can it operate at 200%.

A well designed process unit tries to bring all the different elements together so they are all operating in their sweet spots at the desired production level. This kind of plant should be relatively easy to control. A poorly designed plant often has various elements mis-matched such that one is at its upper limit and another is at its lower limit. These end up working at cross purposes and make for an unstable environment. The art is being able to identify those issues and correct them.

Peter Welander, pwelander@cfemedia.com