This our final WitCH from the 2021 NHT Methods Exam 2. It is, in its own way, a masterpiece.
UPDATE (30/08/21)
This won’t be an epic like our previous update. There are a few things to say, however, plenty of straight out crap to detail, and there’s some non-crap (which is still crap). First, the non-crap.
The Critical Radius of Insulation
As commenters have noted, there is a surprising but well-known phenomenon (amongst insulation fanatics) that in certain contexts a thin layer of insulation around a pipe may increase the rate of heat transfer. This effect continues up to a maximum radius, known as the critical radius of insulation, after which a further thickening will decrease the rate of heat transfer. As well as simply being interesting, the critical radius can be of practical use, in situations where at least some heat transfer is desirable, in the insulation of electrical wires, for example.
The intuitive explanation for the phenomenon is pretty simple. The insulation reduces the heat loss from conduction but also increases the surface area, which increases the heat loss from convection. It is then a battle for supremacy between the two competing effects. Of course the detailed modelling, and determining whether or not the phenomenon will occur in a given scenario, is less straight-forward. Still, the standard model works out pretty nicely. The (dimensionless) rate of heat loss, Q, can be written in the form
![\[\color{blue}\boldsymbol{Q(R) = \frac1{\log R +\dfrac{C}{R}}}\,.\]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-592732c33740f37acb9f00146b395e85_l3.svg "Rendered by QuickLaTeX.com")
Here, R is the ratio of the total pipe radius (including the insulation) to the inner pipe radius, and C is a positive constant representing the conduction-convection battle (for the given inner pipe radius). Notice that in this framing R = 1 corresponds to zero insulation.
Now, by “using calculus”, as the Philistines like to say, or, by using CAS, as the Neanderthals like to do, it is easy to determine that the maximum of Q occurs at R = C. So, if C > 1 (convection wins) then C is the critical radius. And, if C < 1 (conduction wins) then the turning point of Q occurs outside the meaningful domain, the maximum of Q on this domain is at R = 1, and there is no critical radius.
It’s a nice model, and it seems like it could be the basis of an interesting and educational SAC (if such a thing were possible). However, …
CRAP
Why the writers made up their own “model” of heat transfer, God only knows. More to the point for an exam, to have a question framed around a surprising phenomenon and to not flag that surprising nature is pretty nasty. The other, extensive crap has been pretty well documented by the commenters, although the depth of one particular insanity appears to have gone unnoticed. In brief:
- The introduction is appallingly written. Describe the scenario and then introduce the function.
- Having the variable x be the total radius is gratuitously and probably deliberately confusing, and the choice of x to represent this radius is lazy and dumb. (The choice of total radius R as the variable in the standard model makes sense, above, because it is the ratio of the two radii that determines the behaviour.)
- That the fundamental quantity is a rate is easily missed. At least put the damn word in italics.
- Part (d)(i) is meaningless and completely insane. Indeed, the question is so insane, it is difficult to pull apart the insanity. The central insanity is that it is logically impossible to determine the function h1 from h. The exam question only gives a model for a pipe of inner radius 10mm, which can tells us absolutely nothing about the behaviour of a pipe of inner radius 20mm. (For a quick comparison with reality, the constant C in the standard model, above, is only constant for a given inner radius; if that radius changes then so does C.)
Update (22/09/21) The meaninglessness of the question hasn’t stopped VCAA pretending otherwise. The examination report simply, and invalidly, assumes that h1(x) = h(x/2).
- Part (e) is utterly absurd. The areas to be computed are absolutely meaningless.
Completely fucking bizarre.
I’m just going to follow the post for now…
Fractally awful.
An autopsy on this would make a perfect assessment task for my equally fractally awful ITE course.
Who makes this shit up?
Seriously: WTF?
It’s more of a WTF than even a WiTCH. What goes through the minds of the creators of these problems? Without going immediately into the mathematics, the setup is completely stupid. For one thing, for any pipe, the insulation surely is constant. Since the uninsulated pipe is of a fixed width (10mm) I would have thought it natural to have x be the depth of the insulation, rather than the radius of pipe+insulation.
The idea of insulation is generally to minimize heat loss, and hence finding the thickness of insulation that maximizes heat loss immediately makes a mockery of this being an “applied problem”. And given that any insulation is better than none, a model for heat loss which shows initial increasing heat loss for increasing insulation depth is surely a flawed model.
It might be argued that those arguments are irrelevant, in that this problem is to test mathematical skills; in which case, why not test those skills directly, rather than dressing them up in a fake and confusing “application”? It’s hard to know what skills are being tested here, other than patience, and putting up with bullshit. These are two important life skills for sure, but what a waste of time in a mathematics exam!
I initially thought that the heat loss should be strictly decreasing, but looking into it further, that counter-intuitively isn’t the case in real life! Apparently this is the “critical radius of insulation”. Clearly I’m not qualified to say what else is wrong with this question, then.
You’re quite right! Well, there I go shooting my mouth off without thinking first. It’s not the first time and (sigh) it probably won’t be the last. But thank you very much. Still a shit question, but.
Thanks edder. Without looking it up, I figured there was probably some such phenomenon, due to the increasing surface area, and the curve and model (?) were just too weird to not have some substance. I ‘m not sure this aspect of the question is defended so easily, however. As for the rest of the question, I think you’re more than qualified to criticise it.
Not even in my deepest acid-induced fever dream could I come up with this.
A few more specific comments:
1) That h(x) is a *rate* of heat loss is going to get lost amongst all the torturous wording. I wonder (we’ll never know, since VCAA don’t provide any information about NHT exam performance) how many students answered part a. by finding h'(0)? (And b ii. by using the second derivative)?
2) I’m still not completely sure I understand what part c is asking, but I think it just requires solving h(x) = h(10)???
3) For part d i, why bother trying to read this nonsense about “doubling the radius… etc” and just reverse engineer the transformation by comparing h(x) and hI(x)? Also requiring the domain here just reeks of a dirty trick.
4) For part e ii. again, why bother even trying to make the connection with part d? Just get CAS to compute the ratio of the definite integrals.
If you ignore the context (which is difficult), the repeated mention of 3 decimal places means to me that the examiners intended this to be a button-pushing exercise.
Which is itself more of a reason to ignore the context.
Even then…
What in the actual f$*k did I just read?
VCAA, get a grip. Trying to meander to wannabe engineers with pipe insulation questions does nothing whatsoever to your credibility.
Stick to mathematics. Please.
Oh wait, you can’t even do that right.
Move on then folks, nothing to see here.
Interesting – I will try to read the question today before commenting.
Reaction 1. Nice to have a genuine application of mathematics instead of the usual pretence.
Reaction 2. The candidate will need to wade through a lot of verbiage. Thinks: is that good or bad? Well that is a skill required in genuine applied mathematics so maybe it is a skill worth testing. If so the syllabus should reflect that. Something like “the ability to read an extended English description of a model and then press buttons in the correct order”.
Reaction 3. WHAT! conductivity increases as insulation thickens.
Reaction 4. Better check with mother Google. Oh yes the standard model does show that phenomenon. As Marty intimated it’s a balance between convection increasing and conduction decreasing. But the standard model gives the rate of heat loss in the form
![h(x) = 1/[k_1 log((x-k_2)/k_3)+k_4/(x-k_2)]](https://mathematicalcrap.com/wp-content/ql-cache/quicklatex.com-7a2054d351dd5f11be996498ab71b8de_l3.svg "Rendered by QuickLaTeX.com")
quite different from this question. So did they just invent a model? Are we back to the usual pretending?
Reaction 5. If the model was a genuine one, then the questions are the sort of things that one might ask. Testing that button pressing to the max. Question dii is difficult, both a dilation and a translation – more than one way to specify this.
NO What is this last one? What can such an integral possibly represent? Integrating over a range of thicknesses! Fell at the last hurdle.
Tom, I think d ii. is difficult only if you try to determine the transformation by deciphering the text. By comparing h(x) to h1(x), the transformation is just
. Then a student would just need to write their answer by *starting* with that transformation and then deriving the rule for h1(x), so as not to fall afoul of VCAA’s expectations for “show that” questions.
OK I see what you mean. So they have assumed that it is sufficient to change
to 
, giving their new formula, but that actually ignores the change of the pipe diameter that is hidden in the 5 value.
Can you see any meaning in part e ? I’m guessing they just wanted to test some automated integration button.