Question 4 of 5: Material Selection by Thermal Properties

Question 4: Material Selection by Thermal Properties (25 marks: (a) 4, (b) 4, (c) 3, (d) 4, (e) 4, (f) 6)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given.

Find. Identify, and justify, the single best candidate material for each of the six applications (a)–(f).

Approach. Each application is governed by a different combination of the three raw properties: steady-state shielding needs the lowest thermal conductivity $k$; short-burst shielding and fast/slow thermal response need the thermal diffusivity $\alpha=k/(\rho C_p)$, which sets how quickly a temperature disturbance penetrates a given thickness ($\tau\sim L^2/\alpha$); rapid surface heat extraction from a contacting melt needs the thermal effusivity $e=\sqrt{k\rho C_p}$, which sets how fast a material's own surface temperature responds to an imposed flux; and pure heat storage needs the largest specific heat $C_p$. Step 1 tabulates $\alpha$ and $e$ for all nine candidates once, then each part is answered by picking the extremum of the relevant column (subject to any stated "cheap"/"economical" constraint).

1. Derived thermal-response properties. $\alpha=k/(\rho C_p)$ and $e=\sqrt{k\rho C_p}$, with $\rho$ converted to kg/m³:

   Material	$\alpha$ (m²/s)	$e=\sqrt{k\rho C_p}$ (W·s$^{1/2}$/m²·K)
   Aluminum	$9.61\times10^{-5}$	24,275
   Copper	$1.15\times10^{-4}$	37,055
   Gold	$1.26\times10^{-4}$	28,135
   Silver	$1.73\times10^{-4}$	32,315
   Diamond	$1.28\times10^{-3}$	64,918
   Graphite	$3.94\times10^{-5}$	10,039
   Lime	$6.23\times10^{-6}$	6,208
   Silica	$9.41\times10^{-7}$	1,546
   Alumina	$1.22\times10^{-5}$	11,143

(a) Steady-state minimum-flux shield. Steady conduction gives $q''=k\Delta T/L$, so the flux is minimized purely by the lowest $k$ — Silica ($k=1.5$ W/m·K, an order of magnitude below every other candidate).

(b) Short-burst, long-timescale shield. The characteristic time for a heat pulse to penetrate a shield of fixed thickness scales as $\tau\sim L^2/\alpha$; a long penetration time (protection through a brief pulse) needs the lowest thermal diffusivity — again Silica ($\alpha=9.4\times10^{-7}$ m²/s, the lowest of all nine, over six times lower than the next candidate, Lime). Silica wins both (a) and (b) here because it is simultaneously the poorest conductor and (with an unremarkable $\rho C_p$) the slowest diffuser of heat among the set.

(c) Cheap, fast-response sensor (not diamond). Rapid response needs the highest $\alpha$; excluding diamond ($\alpha=1.28\times10^{-3}$, the highest, but disallowed), the next-highest are Silver and Gold — both precious metals, unsuitable for a "cheap" sensor. Among commodity metals, Copper gives the best affordable diffusivity ($\alpha=1.15\times10^{-4}$ m²/s), noticeably ahead of Aluminum, and is the practical choice for a low-cost, fast-responding thermocouple sheath or contact sensor.

(d) Light heat reservoir, most heat per degree C per unit weight. "Per unit weight" is literally the mass-specific heat capacity $C_p$; the maximum in the table is Aluminum ($C_p=917$ J/kg·K), which is also the lightest metal listed ($\rho=2.70$ g/cm³), reinforcing the "light" requirement.

(e) Heat sink minimizing $\Delta T$ for a given flux. Steady conduction again gives $\Delta T=q''L/k$, minimized by the highest $k$ — Diamond ($k=2320$ W/m·K, roughly 5.5× Silver, the next-best conductor). Diamond heat spreaders are used industrially in exactly this role (high-power laser diode and RF device packaging) despite cost, since (e) states no economic constraint.

(f) Economical melt-spinning chill wheel. Extracting heat quickly from a liquid metal that momentarily contacts the wheel surface is a transient-contact problem, governed by thermal effusivity $e=\sqrt{k\rho C_p}$ (the property that sets the interface temperature/flux when two bodies of different $e$ touch). Excluding Diamond, Silver and Gold as too costly for an "economically viable" high-volume wheel, the best remaining effusivity is Copper ($e\approx37{,}055$, ahead of Aluminum at 24,275) — consistent with copper (and copper alloys) being the material actually used for melt-spinning/chill-wheel substrates in industrial rapid-solidification processing.