Algebra. No, that's not a former French colony in Northern Africa, but with me it may as well be. Many of you know that I do somewhat technical work, and therefore I need to use quite a bit of math every day. Recently I assigned myself a project that was considerably more taxing than my limited math skills are used to dealing with.

The major process at our plant in Stuart depends upon a consistent ratio mixture of wood fiber to resin. In order to maintain that consistency I have a scale that constantly monitors the pounds per minute feedrate of wood fiber into the mixing process, that scale then sends a signal to the resin pump to regulate it's speed varying the volume of resin delivered accordingly. This very simple control loop has worked fairly well until now, but I wanted a more precise regulation of resin flow. Besides, each time the ratio needed to be changed it required me to manually make an adjustment on the pump's controller, then physically test the resulting mixture to see if the ratio was correct. That was not very convenient if it occurred at 2:00am.

I took a long hard look at the current installation, and what I wanted to accomplish. I soon realized that my goal could be achieved using the same equipment I already have on hand.

Currently all motors and hydraulic systems and even warning lights and bells are controlled by a PLC, or Programmable Logic Controller. A PLC is essentially a small, yet powerful computer that I program using basic Boolean logic. Boolean logic is logic in it's most basic form, if some certain set of conditions are currently true, then this must happen, if not true then another action must take place. The PLC also has some very strong math capabilities built in, so it's perfect for the calculations I need to perform.

I decided I wanted to read the signal from the scale and write it into the PLC, giving me a number representing the current wood fiber feedrate in pounds per minute. I can also create a fairly simple circuit that will send a variable voltage to the PLC that can be read as a number that represents the wood to resin ratio of the operator's choice. I can then use that information to determine what the resin flow from the pump should be and regulate it's speed accordingly.

That's where I ran smack dab into the algebra wall. For simplicity I'll use nice fat round numbers. Let's assume that wood is currently flowing into the system at 100 pounds per minute and the operator has chosen a ratio of 10:1, or 100 pounds per minute of wood fiber to 10 pounds per minute of resin. So far, it's very simple and any 5th grader can do the math. But. There always seems to be a but, doesn't there? But, only 65% of the resin mixture is resin, the remaining 35% that is water should not be used in calculating the 10:1 ratio.

In other words, if I use the simple formula of wood pounds per minute (WPPM) times 10% equals resin pounds per minute (RPPM), and again assuming that WPPM=100, then RPPM=10. But that's not right. Instead I have to calculate how much resin mixture I need to deliver so that 65% of the resin flow equals 10 pounds per minute.

One would at first assume that I should simply increase my initial calculation by 35% in order to discount the water. OK, that would give me a RPPM of 13.5 pounds per minute resin flow. But to double check that, let's take a look at 65% of 13.5 and we get 8.78 pounds per minute instead of the 10ppm that we need, much too low. So how does one calculate the required resin flow in order to achieve my goal of 65% of the final resin flow being equal to 10% of the wood flow? I have no idea. Thanks to Excel I am able to solve the problem by brute force, and I came up with the following algorithm that works.

RPPM=WPPM*1.5385

This works across the board for a resin with 35% water content. But what if we decide to further dilute the resin mixture and I need to adjust for that. Suppose the resin mixture was 60/40. I can again use Excel to force the answer, but I know there is an equation that will work in every case. I want to learn that equation so I can program it into the PLC and keep me from having to do math or play with Excel at 2:00am.

If you've made it this far into this post, you are either a total geek, or you have way too much time on your hands, or you will read almost anything. Either way that makes you a perfect candidate to help me with this problem. So help.

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