Tag Archives: Thermodynamics

Soil Loss – Grantham

Lane-Emden Equation

The Lane-Emden equation is a second order differential equation which is used to model a spherically symmetric gas cloud, eg a stellar interior. Although it is a considerably simplified model (eg, no rotation), it still provides a good starting point. Traditionally the LE equation is written in terms of a linear independent variable, say x, which represents the relative radius of the gas cloud. The variable x goes from 0 (center of cloud) to 1 (outer boundary of cloud). With the assumption of a particular type of model of the gas cloud’s thermodynamics — called a polytropic model — one ends up with the LE equation. The model is characterized by a parameter n, called the polytropic index (n is a non-negative real number, not just an integer) which determines a function f (which depends upon the choice of polytropic index n), which in turn determines the density profile of the cloud, and thereby its other profiles, eg its mass profile.

It turns out, however, that the function f is an even function, and its power series expansion, for instance, involves only even powers of x. For instance, f(x) equals 1 – (1/6)*x^2 + (n/120)*x^4 + … and so on, with terms for x^6, x^8 etc. So one asks, whether it might be useful to write the LE equation (a second order differential equation) in terms not of independent variable x, but in terms of independent variable s = x^2. The variable s is the relative surface area of a spherical shell of relative radius x. At x=1 (outer boundary of the gas cloud), the value of s=1, ie relative area of that outer shell, is also 1.

There is a possibility for getting some physical insight from such a rewrite of the LE equation. The physics which is being modeled involves transport of energy and force between nested shells, and such transport may be conceptually more meaningful if it considered as a function of relative area rather than relative radius.

I have prepared a short pdf (2 pages) which describes such a rewrite of the Lane-Emden equation. Nothing new there; I’m sure such a rewrite has been done by others, as the LE equation has been a subject of study for over a century, and there is a considerable literature. Still, it may be of interest to others who are working their way into the details of the Lane-Emden equation and the stellar interior models for which it is a starting place. I give a few references, including to Chandrasekhar’s classic book on Stellar Interiors, which is still a better explanation than some of the more recent books, as it illustrates some motivations. And also a reference to a very interesting little paper by Klaus Rohe, who used Python to calculate rational expressions for the first 15 coefficients of a power series representation of the LE equation (ie, up to the x^28 term), as expressions in the polytropic index n. His paper is at Arxiv 1409.2008 if you want to go there directly.

Best wishes,
Ken Roberts
07-Mar-2015

Link to pdf file mentioned above, with my rewrite of Lane-Emden equation using relative area…
https://lasi2.wordpress.com/wp-content/uploads/2015/03/lane-emden-rewrite.pdf

Thermoelectric Cooler

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I recently had an opportunity to test a thermoelectric cooler, and am very happy with its operation. It maintains the contents of the refrigeration compartment at 7.0 degrees C when the room temperature is 20.1 degrees C. A difference of 13.1 degrees C. That is better than the stated capability of the cooler; the brochure says it can cool its contents 10 degrees C.

For the past couple of years, I’ve been involved with a thermoelectric materials working group, colleagues who are trying to develop improved thermoelectric materials. My particular role has been to work on certain mathematical and computational tasks which arise during our activities. Mostly, it’s quite abstract. I don’t usually encounter actual materials or thermoelectric modules. So, when I do get a chance to fiddle with some practical device, even in an “amateur” way, it’s rather an enjoyable diversion.

This particular cooler is intended to plug into a car’s 12 Volt DC socket (aka cigarette lighter). There is also an adapter, so that it can be run from the AC wall outlet when the car is not in use. For instance, when travelling, the cooler can be brought into the motel room overnight and run from the wall outlet.

One of the advantages of a thermoelectric cooler is that it is essentially vibrationless. There is a fan for this cooler, but it does not rely upon a compressor. Vibrationless coolers are particularly desirable for instance as wine cabinets, because vibration can disturb the sediments in the wine.

There is rapid advance in thermoelectric materials. But we are still at the infancy of this field. New materials and material preparation improvments are discovered frequently. The design configuation of thermoelectric modules in applications is still very rudimentary. Some of the subtleties, such as focusing of thermal radiation, have not been explored very much at all. I contrast the field of thermoelectricity, to the field of radio electronics. We are at the equivalent of the “crystal receiver” stage.

There is a tremendous opportunity to do interesting and useful work. There are only perhaps some 10,000 people working in the field, and the potential for energy efficiency improvements is such that the world will probably have 100,000 to 1 million practical engineers and other skills involved a decade or two hence. If one is looking for interesting work, over a long duration of increasingly sophisticated applications, consider thermoelectricity!

Best wishes,
Ken Roberts
27-Jun-2014

Charlier – Stellar Statistics

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C.V.L. Charlier’s book on stellar statistics is an enjoyable read. He writes clearly. The book is online, via Project Gutenberg, and is about 50 pages. The URL is below.

Charlier in 1911 proposed the word “siriometer” for interstellar distances. One siriometer is 1 million astronomical units, or approximately twice the distance from Sol to Sirius. Alternatively, 1 siriometer is 15.79 lightyears, or a parallax of 0.206265 arseconds. That is, a parsec is 0.206065 siriometer, and a siriometer is 4.848 parsecs.

Charlier notes (section 4) that Seeliger (and other German astronomers) had long used another distance measure, the Siriusweite which is defined as a parallax of 0.200 arcseconds. The idea of using (twice the) Sol-Sirius distance as a baseline, one variant or another, has a long pedigree. I think one reason the parsec (a term introduced in 1912) caught on is that it is brief — two syllables instead of five.

Another remark of Charlier, also in section 4, is upon the inadvisability of defining a length unit (the Siriusweite, or the parsec) in terms of an angle, particularly an angle which corresponds to the harmonic mean distance of a star, not to its arithmetic mean distance. That remark has merit. We are “saved” from that confusion nowadays because we have redefined the parsec in terms of a distance, a certain number of meters. Astrometric observations, such as by the Gaia space observatory, make appropriate corrections to their measurements of angular displacement. The angle vs distance confusion is not unlike the question of temperature T versus inverse temperature 1/T in thermodynamics, mentioned in my earlier post about Peter Atkins’ book, “Four Laws that Drive the Universe”.

In section 19, Charlier describes the technique used to prepare the original Bonner Durchmusterung catalogue of stars. Fascinating to read! Quite similar to Gaia’s method. A wide field of observation, 6 degrees, with all stars of 9th magnitude or brighter passing through a 1-degree-wide strip of declination noted, manually recording time, declination and magnitude as the stars passed a central hour line. One person called out the declinations and magnitude, another (in a lighted adjacent room) recorded that info with time. They could attain a data rate as high as 30 transits per minute. The original BD catalog was prepared by Argelander, over 625 observing nights in 1852-1859, and contained 324,198 stars. What an achievement!

It’s interesting, when reading an astronomy book from a century ago, to reflect upon the many changes which have happened, improvements in instruments and in concepts. That time distance may be 3 generations, but it is only 2 professional careers.

Relevant links:

The Project Gutenberg page for reading Charlier’s book, “Lectures on Stellar Statistics”, 1921:
http://www.gutenberg.org/ebooks/22157

The Wikipedia page for Siriometer:
http://en.wikipedia.org/wiki/Siriometer

The motivation to read Charlier came via the Project Gaia DPAC newsletter number 05, July-2009:
http://www.cosmos.esa.int/web/gaia/dpac/newsletter

My earlier post about Peter Atkins’s book:

Four Laws that Drive the Universe

Best wishes,
Ken Roberts
27-Jan-2014

Four Laws that Drive the Universe

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More Peter Atkins reading — this one, “Four Laws that Drive the Universe”, about the laws of thermodynamics. I particularly liked chapters 4 (Gibbs free energy, and ATP-ADP cycle in biology), and chapter 5 (negative temperature). His is a chatty book, clearly written with vivid descriptions of ideas. I found it online but it is also available in hardcover, and I’ll probably shell out for a personal copy for long term reference.

Atkins makes a good case that we should be using $\beta={1/T}$ , inverse temperature, as our basic variable, not temperature T. There is no such thing, in nature, as “absolute zero”, which is easily understood when we think of zero temperature corresponding to infinite inverse temperature beta. Moreover, in chapter 5, he shows how the transition between states with an inversion of energy levels (like a population of excited atoms) and states without the inversion, is without discontinuities if graphed vs inverse temperature — see his figure 22.

I wonder if someone has tried writing up the field of thermodynamics, with all its associated equations and parameters such as heat capacity, using inverse temperature instead of temperature as one of the basic variables. That might lead to some insights. Perhaps comparable to the insights gained from considering electromagnetic waves via frequency instead of via wavelength.

Have fun reading!

Ken Roberts
14-Jan-2014