This our final WitCH from the 2021 NHT Methods Exam 2. It is, in its own way, a masterpiece.
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
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, …
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.