I recommend reading the above title the same way that Michelango screams "I love being a turtle!" at the end of the original Teenage Mutant Ninja Turtles movie. Not that this is relevant at all...
Anyways, today I was approached by a coworker (also a chemical engineer) for input on a problem he had been thinking about in the back of his head for the past couple days and couldn't come up with a solution for. Basically, he was trying to estimate the rate at which we consume Argon during operations. Nothing all that exciting.
The trouble was that for the Argon system, everything is measured in PSI. This is a annoying, since the Argon system is supplied by a big tank which keeps the stuff at liquid temperatures (around 85 K). Typically, pressures become a little less meaningful as a unit of measurement when there's a phase change going on (read as, we were pretty much like, "oh, shit, how do we account for that?").
So, we started just chatting about it. You know, throwing out all those classic chemical engineering buzz words we all learned in school, hoping that the other one would know what the words actually meant and could magically write down an equation that would just solve all our problems. Examples would include such gems as "pressure differentials", "saturated vapor pressure", "vaporization energy", and "thermodynamics" (okay, we both had at least an okay idea of what those last two mean). To make a long story short, we agreed that the problem seemed a lot harder than it should be, and I promised to try to come up with a way to get a approximation for it.
So, I started thinking about it. Around this point, it occurred to me that, well, we have this Argon system and it's only about 30 seconds of my time to walk over and use it. So, maybe I should stop trying to come up with ways to predict the rate of vaporization based upon the pressure differential that the boiling liquid is experiencing (and assuming there's no heat input since obviously our 85 K tank is well insulated) and just see how long it takes to fill up a tank.
But, wait! That's too easy. I didn't spend four years studying all sorts of complicated chemistry, math, and physics applications to just hook up some pipes, turn a valve and look at my watch.
So, how do I make it more complicated? Well, remember, everything we measure is in PSI. So, how do you define a tank as "full"? Well, the simple thing to do is just pick a goal PSI and fill it to there. Then, since Argon is about as close to an ideal gas as anyone could possibly ask for, just use that law to figure out how much we actually put in. Simple.
But still too simple. That would just require me to hook up some pipes, turn a valve, look at my watch, and type some numbers in my calculator. I didn't go to school for four years to be able to do that, either.
Here's where I finally got to bring in some serious chemical engineering. I pointed out (to me) that the tank has a resistance to increasing the pressure. The closer we are to the final tank pressure set point, the weaker the driving force is to actually bringing the tank to that set point. The result is that if I were to just fill up a tank to a specified pressure, I would significantly underestimate the true capabilities of the system. This would be especially problematic during tank purges, when the valves are left open so the tank stays at atmospheric pressure (resulting in a constant maximum driving force).
There we go. Now we have a problem worthy of the attention of a chemical engineer. The fact that it took a significant amount of the chemical engineer's attention to actually create the problem is irrelevant.
So, the end result? I had to make a simplifying assumption and stated that the molar flowrate is linearly dependent on the pressure differential between the tank and the pressure set point of the valve (which I have no way of justifying but intuitively makes sense, especially since the evaporation rate of the liquid Argon is almost certainly not going to be a limiting factor). From there, I built a simple differential equation from the above assumption and solved it with respect to the number of moles in the tank. With these, I just needed the actual time values, so that I could plug them in and determine the slope of the molar flowrate vs pressure differential graph.
So, I did have to eventually go out and just fill up a tank, but now it was with the goal of determining the rate constant, which is way cooler than a normal constant. In case you're curious, the difference between just filling up a tank and using that value vs. my analysis was around a factor of 2 (which is why we chemical engineers make the big bucks).
I love being a chemical engineer!
That is all.
PS - We just needed a reasonable value for approximation purposes, so I'm definitely not interested in the far better approximations that actually deal with the fact that there is a phase changed involved, thank you very much.