Design to exploit thermal properties 263

12.6 Design to exploit thermal properties 263

expanded yet—there has not been time for the heat to diffuse into it. That means there is a temperature gradient, and even though the material has just one expan- sion coefficient, there are different strains in different places. Since the surface is stuck to the interior, it is constrained, so thermal stresses appear in it just as they did in the thin film of Figure 12.12. In brittle materials, these stresses can cause fracture. The ability of a material to resist this, its thermal shock resistance, ∆T s (units: K or °C), is the maximum sudden change of temperature to which such a material can be subjected without damage.

Even if the stresses do not cause failure, temperature gradients lead to distor- tion. Which properties minimize this? Figure 12.15 shows the temperature pro- file in the quenched body at successive times τ 1 1 ,τ 2 2 ...τ 5 5 . A cooling or heating front propagates inwards from the surface by the process of heat diffusion dis- cussed earlier—and while it is happening there is a temperature gradient and consequent thermal stress. If the material remains elastic, then the stresses fade away when the temperature is finally uniform—though the component may by then have cracked. If the material partially yields, the stresses never go away, even when the temperature profile has smoothed out, because the surface has yielded but the center has not. This residual stress is a major problem in metal processing, particularly welding, causing distortion and failure in service.

The important new variable here is time—it is this that distinguishes the con- tours in Figure 12.15. The material property that determines this time is the thermal diffusivity a that we encountered earlier:

with units of m 2 /s. The bigger the conductivity, λ, the faster the diffusivity, a, but why is ρ C p on the bottom? If this quantity, the heat capacity per unit vol- ume, is large, a lot of heat has to diffuse in or out of unit volume to change the temperature much. The more that has to diffuse, the longer it takes to do so, so that the diffusivity a is reduced. If you solve transient heat flow problems (of which this is an example), a general result emerges: it is that the characteristic distance x (in meters) that heat diffuses in a time τ (in seconds) is

x ≈ 2τ a (12.15) This is a particularly useful little formula, and although approximate (because

the constant of proportionality, here 兹苶 2, actually depends somewhat on the geometry of the body into which the heat is diffusing), it is perfectly adequate for first estimates of heating and cooling times.

Thermal distortion due to thermal gradients is a problem in precision equip- ment in which heat is generated by the electronics, motors, actuators or sensors that are necessary for its operation: different parts of the equipment are at

264 Chapter 12 Agitated atoms: materials and heat

different temperatures and so expand by different amounts. The answer is to make the equipment from materials with low expansion α (because this mini- mizes the expansion difference in a given temperature gradient) and with high conductivity λ (because this spreads the heat further, reducing the steepness of the gradient). It can be shown that the best choice is that of materials with high values of the ratio λ/α, just as our argument suggests.

The α–λ chart of Figure 12.4 guides selection for this too. The diagonal con- tours show the quantity λ/α. Materials with the most attractive values lie towards the bottom right. Copper, aluminum, silicon, aluminum nitride and tungsten are good—they distort the least; stainless steel and titanium are much less good.