Author: Dr. Sean Moran
Despite the title of this article, there is no answer to be found here. The question which I set those students has no right answer. There are, however, an infinite number of wrong answers, and a number of good ones. This is the first thing which the students don’t get. They know how to use calculus or iteration to get a right answer. They don’t know how to get a right enough range of answers, and pick from them, like engineers do. It never occurred to any of them to simply graph out the possibilities and note how broad the peak is.
Mathematics is not the foundation of chemical engineering, it is just another tool in our box.Sean Moran
Next, let’s take a look at their second weakness, converting words into formulae. You can’t use rigorous techniques without a formula to work with. Anyone who has taught chemical engineering design must have noticed that there’s a sticking point early on which can require a lot of spoon-feeding because even “senior students” have no familiarity with vagueness.
My question is deliberately vague, but not so vague as to be meaningless. The question was :
“What is the optimal height to diameter ratio for a tank for liquid with a 5 m3 capacity made of 3 mm thick stainless steel?”
But what do I mean by “optimal”? What is a “tank”? How big is “5”?
Without getting too philosophical, we can allow “optimal” to mean “least capital cost”, though this is often not the case in professional life. This assumption should be stated in our calculation if we are going to be like real engineers. Annotation of assumptions is important in engineering calculations.
A “tank” might have a dictionary definition of “usually large receptacle for holding, transporting, or storing liquids (such as water or fuel)”, but that’s not quite how engineers understand its implications. Most pertinently in this case, a tank is neither a paddling pool, nor a vertical pipe. There are mathematically valid height to diameter ratios which contain the necessary volume, but don’t look like a tank. The “perfect” tank in the engineer’s mind might well vary between sectors, but its height to diameter ratio (or diameter to height) might tend towards the “golden ratio“. I’m not suggesting that they calculate this, just that, that’s what will look “rightest” to them. The actual dimensions will have other constraints in the real world.
So that’s a couple of keywords addressed. How about the one number in the question? Did you notice that it’s just that, one number? How big is “5”? Anywhere between 4.5 and 5.4 (ish). In real life a client who specified a 5 m3 tank and got a 4.5 m3 tank would have got what he asked for. Of course, they might not appreciate your hair-splitting, and you’d probably be best to give them the 5.0 m3 working volume tank they probably meant to ask for. Students are, however, blind to implied precision, (and have no client to annoy). I give them a question with numbers to one decimal place, and they give me in return every figure on the calculator display.
So the right-looking answer is probably a tank with around 5 m3 liquid capacity, with let’s say between 2:1 and 1:2 aspect ratio. I’m ignoring a great many real world considerations. I haven’t said if it is cylindrical or spherical, and if cylindrical, if is has a top, or a bottom. There is no allowance there for freeboard, agitator or aeration holdup, minimum and maximum working depths, NPSH, load bearing capacity of the soil, land values, headroom, stock steel plate sizes, and the aesthetics of non-engineers.
That’s just some general ones off the top of my head. Neither have I stated what’s in the tank. If it’s water at ambient temperature, why bother with stainless? Plastic is a lot cheaper. If it’s mercury at 200 °C, is 3 mm of stainless steel going to be strong enough? I could go on. Let’s get back to our simple exercise for newbies. What’s the most we can expect of them?
The tools at their disposal are calculus, iteration and graphical solution. Well, I know at least that they have passed exams requiring use of these techniques with good grades. However, 99% of maybe 500 students at a top university could not think of a way to apply any of these three techniques to come up with an answer. Every single one of the 1% who could see a way to do it used calculus, despite them all having access to a laptop in the classroom which would allow rapid graphical or iterative solutions. (I seem to remember that every single one of those students got a first class degree later. The top 1% of students are top at everything, just as the top 1% of professors are.) But I digress.
What we do, requires the application of a whole non-mathematical toolkit, involving amongst other things verbal comprehension, framing the right question in the right way, choosing the right tool, having an idea of what the right answer looks like, having a good idea of the limitations of the tool we chose, product knowledge, the requirements of other disciplines, spatial intelligence, aesthetics etc.Sean Moran
It was suggested in a comment on a previous post on this subject that all engineers do what these students did, use calculus to determine the right height and diameter of tanks. I don’t think that this is right in general and I’m certain it’s not how I do it. If you use MS Excel to produce a graphical representation of all sensible answers, you will see that there are quite broad maxima (or minima, depending on what you plot). Just to the right or left of the “right answer” are “almost as right” answers which costs £1 more. But in the real world, you are not designing or fabricating the tank. Tanks which neatly use available plate sizes will generate less waste and be a little bit cheaper (and greener). Manufacturers’ standard tanks will be way cheaper than specials. Real world price curves are not continuous, they are stepped. The students are not expected to know this. The very brightest might however tell me that the right answer is X plus or minus Y, a range set by the implied precision of the question. (It’s not “X.0” plus or minus “Y.0”) It turns out that this tends to overlap quite a bit with the professional engineer’s intuitively “right” answers.
So, to return to my original assertion, mathematics is not the foundation of chemical engineering, it is just another tool in our box. What we do, requires the application of a whole non-mathematical toolkit, involving amongst other things verbal comprehension, framing the right question in the right way, choosing the right tool, having an idea of what the right answer looks like, having a good idea of the limitations of the tool we chose, product knowledge, the requirements of other disciplines, spatial intelligence, aesthetics etc.
The author originally wrote this article on LinkedIn. The article was reproduced here with the permission of Author.