Thermal-expansion coefficients, convection as function of T

[Jerry Friedman]

I have two questions on fine points of thermal physics.

1. Is an isotropic solid's volume-expansion coefficient beta exactly equal to 3 times its linear-expansion coefficient alpha, or is it only approximately equal? Some textbooks (Fischbane, Walker, Young) say the relation is exact, while others (Giancoli, Katz) say it's approximate. It's also possible to find books that say it's approximate, such as https://books.google.com/books?id=SwsNbiMDqzcC&pg=PA100 . In poking around the Web, I can't find anything that gives sufficiently precise experimental data or that gives anything but the simple argument about differentiating a cubic function--which I don't see anything wrong with, so why wouldn't it be exact. (Technically these constants are defined only in the limit of small temperature changes, right?)

2. Someone told me that the rate of heat transfer by convection typically saturates as Delta T increases, but didn't have any further information. I can't imagine why it would. Does anyone know anything about that?

I'd appreciate any information or any sources, especially those that don't require access to a university library.

 

[Bystander]

Does (1 + α)³ = 1 + 3α? Or, something else?

 

[jtbell]

Some textbooks (Fischbane, Walker, Young) say the relation is exact

I would be surprised if they say that. What is the exact wording?

Does (1 + α)3 = 1 + 3α? Or, something else?

If α is very small, then α² is very very small and α³ is very very very small...

 

[Chestermiller]

If $\alpha$ is defined as $\alpha = \frac{1}{L}\frac{dL}{dT}$ and $\beta$ is defined as $\beta=\frac{1}{V}\frac{dV}{dT}$, then the relationship $\beta=3\alpha$ is exact.

If $\alpha$ is defined as $\alpha=\frac{1}{L_0}\frac{L-L_0}{T-T_0}$ and $\beta$ is defined as $\beta=\frac{1}{V_0}\frac{V-V_0}{T-T_0}$, then the relationship $\beta=3\alpha$ is not exact.

My understanding is that the strictly correct way to define these coefficients is by the former, and not the latter. The latter is only used for beginners who are just being introduced to these concepts.

Chet

 

[Jerry Friedman]

I would be surprised if they say that. What is the exact wording?

If α is very small, then α² is very very small and α³ is very very very small...

Thanks. Why would you be surprised?

Fishbane (correct spelling, sorry), Gasiorowicz, and Thornton: "We can show that β = 3α by considering a cube of volume V = L³." Then they differentiate.

Young and Freedman differentiate first, then say, "This is consistent with the infinitesimal form of Eq. (17.8), dV = βV₀, only if β = 3α." (The last equation is displayed and numbered 17.9.)

Walker: "The coefficients of volume expansion and linear expansion for a solid are related by β = 3α." The equation is displayed, 18-11.

 

[Jerry Friedman]

If $\alpha$ is defined as $\alpha = \frac{1}{L}\frac{dL}{dT}$ and $\beta$ is defined as $\beta=\frac{1}{V}\frac{dV}{dT}$, then the relationship $\beta=3\alpha$ is exact.

If $\alpha$ is defined as $\alpha=\frac{1}{L_0}\frac{L-L_0}{T-T_0}$ and $\beta$ is defined as $\beta=\frac{1}{V_0}\frac{V-V_0}{T-T_0}$, then the relationship $\beta=3\alpha$ is not exact.

My understanding is that the strictly correct way to define these coefficients is by the former, and not the latter. The latter is only used for beginners who are just being introduced to these concepts.

Chet

Thank you! That seems to be the answer, as both Giancoli and Katz define the coefficients in terms of $\Delta$'s, not derivatives. I should have noticed.

 

[jtbell]

If α\alpha is defined as α=1LdLdT\alpha = \frac{1}{L}\frac{dL}{dT} and β\beta is defined as β=1VdVdT\beta=\frac{1}{V}\frac{dV}{dT}, then the relationship β=3α\beta=3\alpha is exact.

If α\alpha is defined as α=1L0L−L0T−T0\alpha=\frac{1}{L_0}\frac{L-L_0}{T-T_0} and β\beta is defined as β=1V0V−V0T−T0\beta=\frac{1}{V_0}\frac{V-V_0}{T-T_0}, then the relationship β=3α\beta=3\alpha is not exact.

Good catch! The textbooks that I've used (at least the introductory ones) must have all used the second definition.