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I am reading a text about gas nuclei I encountered the following formula:

$$r = \alpha + \beta \left( \frac{T}{P} \right)^\frac{1}{3} + \kappa \left( \frac{T}{P} \right)^\frac{2}{3}$$

$r$ is the critical radius of the nucleus, while $T$ and $P$ are the current temperature and pressure respectively.

I have no idea what $\alpha$, $\beta$ or $\kappa$ are. Also, I don't even known the name of such formula and the principle behind it.

Can anybody explain what $\alpha$, $\beta$ or $\kappa$ are and what the underlying principle is?

The text is REDUCED GRADIENT BUBBLE MODEL, page 12 equation 61.


The problem is that I'm not able to find the values of the three constants. If I use the values from the first three rows of the table below the equation:

  • $T = 293 K, P = 13 fsw, r = 2.10$
  • $T = 293 K, P = 33 fsw, r = 1.36$
  • $T = 293 K, P = 53 fsw, r = 1.34$

I get (approximately) $\alpha = 4.395, \beta = -3.260, \kappa = 0.866$. However such values do not work for e.g. $P = 273 fsw$. If I use my constants, I get $r = 1.966$, while the table reports $r = 0.80$.

Also, the table shows that $r$ decreases for $P \in [13, 273]$, my function increases approximately at $P = 73$ (where my $r$ is 1.403 against 1.32).

I think $\alpha$, $\beta$ and $\kappa$ must be functions of pressure. The Wikipedia page about Equation of State shows $\alpha$ and $\kappa$ as functions of pressure and temperature, but there are no traces of $\beta$.

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It would help it you could tell us what the title of this text is. My first thought is that alpha, beta, and kappa are empirical coefficients, with no particular meaning otherwise. –  Alan Rominger Dec 4 '12 at 15:03
Comment to the question(v6): It would be great if OP (or someone else) could create a more substantive title. –  Qmechanic Dec 5 '12 at 14:59
@Qmechanic: As shown in my answer below, it is an ordinary virial equation of state truncated at quadratic order and applied to bubbles (spherical symmetry) with reparametrization to critical radius. I do not have edit privileges. –  juanrga Dec 6 '12 at 12:15
@juanrga: My comment refers to a previous version(v6) of the title. The title is okay now(v8). –  Qmechanic Dec 6 '12 at 12:20
@Qmechanic: Yes, it is ok now, but when I submitted my comment I saw still the old title "What's the meaning of this equation?". –  juanrga Dec 6 '12 at 13:03

3 Answers 3

up vote 2 down vote accepted

Setting $Y \equiv (T/P)^\frac{1}{3}$ the expression looks as a virial equation of state in a power series

$$r = \alpha + \beta Y + \kappa Y^2$$

with $\alpha$, $\beta$, and $\kappa$ the coefficients on the expansion. The meaning of the coefficients is the usual: e.g., $\alpha$ is the radius when $Y=0$...

EDIT: I have found a paper by the same authors confirming that the above is a virial equation of state truncated to quadratic order.

They start from the ordinary virial equation of state at arbitrary order and substitute volume for systems with spherical symmetry $V=(4\pi/3) r^3$. Then take the cubic root and obtain (I am using their own notation in this paper)

$$r = \sum_{i=0}\beta_i \left( \frac{nRT}{P} \right)^{i/3}$$

Next, they truncate the expansion to quadratic order

$$r = \beta_0 + \beta_1 \left( \frac{nRT}{P} \right)^{1/3} + \beta_2 \left( \frac{nRT}{P} \right)^{2/3}$$

re-parametrize from total bubble radius to excitation radius and rewrite the expression by introducing the $nR$ factors into new $\alpha$, $\beta$, and $\kappa$, which they again name "virial coeficients".

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-1: The linked text says it is valid for all $T$, that is generally not true for power series. Do you have any proof for your claim? –  Bernhard Dec 5 '12 at 11:39
@Bernhard: Thank you for the negative point, but you are completely wrong. It is trivial to see that the above is a virial equation truncated up to quadratic order, but in any case I have found a paper by the same authors confirming what I said in my answer. I am editing it. –  juanrga Dec 6 '12 at 11:40
Thanks for the additional information. Makes things clear! –  Bernhard Dec 6 '12 at 12:14
@Bernhard: Thank you very much by undo the negative point! –  juanrga Dec 6 '12 at 13:11
That's beautiful! I haven't tried finding the betas (to see if everything fits) however it makes much more sense now. Many thanks also for providing the link to the paper! –  user16538 Dec 7 '12 at 18:41

As stated in your text, $\alpha, \beta$, and $\kappa$ are equation-of-state constants, or in otherwords are constants based on the pressure, temperature, volume, etc of the substance being dealt with. Wikipedia has some good background information and examples on how some similar values are calculated.

I also came across this article out of CalTech that has a bit more information about critical radii as it pertains to bubble dynamics.

Hope this helps!

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Without reading the entire paper, it looks like the authors are exploring, among other things, the growth of gas bubbles resulting from decompression in the context of underwater diving.

More specifically, an underwater diver experiences an additional atmosphere of pressure for each 34 feet she descends below the surface of the water. When its time to swim back to the surface, the diver must be very careful not to ascend too quickly or else she might experience Decompression Sickness. Essentially, dissolved gases in the body can start to bubble out as the pressure decreases, causing potentially serious health issues.

The process of forming bubbles is called "nucleation". The equation you asked about gives the minimum nuclei radius such that the bubble will grow in size as the pressure is decreased from $P$ at temperature $T$. According to the paper you provided, the greek letters are "Equation of State skin constants" which probably means (as AlanSE said) they are experimentally determined coefficients...nothing profound.

The equation is useful because one must have information about bubble growth in order to figure out a safe rate of ascent to the surface. The paper contains additional modeling that further develops the analysis of safe ascent.

It does not seem that the authors have named the formula.

EDIT Sorry, I think I missed the intent of your question. You probably already know what I wrote! Anyway, since you're trying to find the skin constants I put the values from Table 5 into MS Excel and did a second order polynomial fit at $T=293K$ and got ${\alpha}=0.458$, ${\beta}=0.465$, ${\kappa}=0.034$. The R-Squared value is $0.90$. I'm not sure how accurate you need this to be...if you know the temperature/depth profile of the water you are modeling then we could improve the fit. I know you'd prefer to have the constants as functions of temperature and pressure but unfortunately I couldn't find that info. Maybe someone else will have more success doing the research. Hopefully this rough estimate helps somewhat.

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