Coefficient of linear thermal expansion

[russkey]

My old physics textbook states that thermal expansion is is directly proportional to the change in temperature for relatively small changes in temperature (less than 100C˚ or so). It also hints that the coefficient depends somewhat on the temperature. It does not, however give any details of the non-linearities. Given that a steam engine's parts might be machined somewhat above room temperature (especially with proper coolant), and run at temperatures over 100C˚, I'm thinking that the old theoretical linear expansion equation (dL = alpha * L_0 * dT) might not quite apply. All of the text books on my shelf seem to be less than helpful on this, though. Can anybody shed some light on this for me? Or perhaps point me to a resource that does provide more detailed information on the topic? All of the search results I've found so far have stuck to the classical linear expansion equation.
Al

[fredrosse]

The ASME Code, Section II, Subpart 2 Part D Gives properties for hundreds of materials, including thermal expansion as a function of temperature from 70˚F to as high as 1650˚F (for materials rated for such a high temperature).

The instantaneous linear value for Group 1 steels as a function of temperature are:
Temp, ˚F, = 100, 200,300,400,500,600 with the corresponding Coefficient of Linear Expansion per F˚, x 10^6 = 6.6, 7.0, 7.3, 7.7, 8.0, 8.3

A second order polynomial approximating these values is :

Y = -1.1071E-6*(Temp˚F)^2+4.1664E-3*(Temp˚F)+6.198

You can integrate this function over the range of interest to get the results you are after. As a matter of fact, the change is nearly linear, so you could represent the non-constant expansion coefficient as a linear function of temperature and probably have sufficient accuracy for your evaluation.

Buy the way, it is refreshing to see your use of C˚ rather than ˚C when referring to an indeterminate temperature change. So many make no distinction between C˚ and ˚C (or F˚ and ˚F).

You should join another forum, which gives access to all kinds of similar questions and answers, the Engineering Tips Forum. Be careful of the answers you get on any forum, as many times the people who respond only know half the answer, or are very much off-base in understanding the issue. This Steamboating Forum is very good with respect to this, but some technical forums need close scrutiny of the answers.

[barts]

Note that the impact of this difference is quite small for hobby scale steamboats, and most of the parts that are working in close proximity (such that the small size change could be important) are subject to roughly the same temperatures..... Any design where this was critical would probably be much more affected by warm up transient effects.

One area I might consider this is machining the bore of a uniflow cylinder; in operation, the ends of the cylinder are typical close to inlet steam temps, while the center is much cooler as it more closely approximates exhaust conditions. The piston can be assumed to be somewhere in between.

In my case w/ a 5" bore cylinder, the ends might be at 450 F and the middle at 175 F, and the piston at 300. Let's use a value of 7.3 x 10^-6 and see what we get....

At the ends: 5 * (450-300)*7.3 x 10^6 = .0054" more clearance
In the center 5 * (300-175)*7.3 x 10^6 = .0046" less clearance

Now, this might be ignored w/ sufficient piston clearance, but the .010" flex might be hard on the rings. Since I don't have a CNC lathe to generate a bell-shaped bore, if I wanted to do anything about this I could either heat the ends with propane torches while the cylinder spins slowly in the lathe, or chill the center of the cylinder w/ liquid nitrogen, or some combination of heating and cooling so as to achieve 275 F difference in temp.

Now, how much difference would using the more accurate numbers make?

We'd take the coefficients as 6.9 at the middle, and 7.9 at the ends.... and we'd see about .0005" more expansion at the ends and the same decrease in the middle - probably not enough to matter.

- Bart