## Put in spin-half, and transverse waves and polarization fall out.

2007-05-01

See also the other posts in this category. Reading them in the order posted might be a good idea.

In the last of seven lectures on “The Character of Physical Law” which Richard Feynman gave at Cornell University in 1964, he said,

One of the most important things in this “guess — compute consequences — compare with experiment” business is to know when you are right. It is possible to know when you are right way ahead of checking all the consequences. You can recognize truth by its beauty and simplicity. It is always easy when you have made a guess, and done two or three little calculations to make sure that it is not obviously wrong, to know that it is right. When you get it right, it is obvious that it is right — at least if you have any experience — because what happens it that more comes out than goes in. Your guess is, in fact, that something is very simple. If you cannot see immediately that it is wrong, and it is simpler than it was before, then it is right.

Archimerged has been improving his “guess” for over 10 years now, and once again, it looks good. This time it looks a lot better than it ever did before. So, once again, he tries to explain it.

Ark still holds the opinion that the key to everything is the principal null directions (eigenvectors) of the Weyl tensor.  These were discovered after Einstein’s death, and Ark also holds the opinion that Einstein would have seen their significance.  Note that specifying the principal null directions puts strong constraints on the metric, since the Weyl tensor is calculated from the metric, and the principal null directions specify a set of equations which the Weyl tensor must satisfy.

In general relativity, you start by specifying the position and momentum of the matter and energy (using the metric in this specification). The “field equations” simply specify that the Einstein tensor calculated from the metric must equal a constant times the stress-energy-momentum tensor, also calculated from the metric. They are horribly complicated because the metric occurs on both sides, and you can’t solve for it.

Ark notes that Einstein said that the left-hand-side of his equation (the Einstein tensor) was like marble, while the right-hand-side was just an ad-hoc addition, like a cheap plywood annex to a marble building. So he feels no particular loyalty to the field equations of general relativity. Instead, he proposes a different set of field equations. Instead of specifying the stress-energy tensor in advance and trying to find a matching metric, he specifies the principal null vector fields and tries to find matching metrics. Then, given a metric, he thinks he can calculate the stress-energy-momentum tensor to find the mass and momentum distribution.

Each principal null vector field is tangent to a space-filling family (a congruence) of curves, so to specify one of these vector fields is to divide spacetime up into disjoint one-dimensional light-like curves. As a guess to be confirmed, Ark assumes that there are only two distinct fields, in particular, the four fields are a pair of identical twins. A metric which yields these fields is called Petrov type D.

So, what do the principal null curves have to do with particles? Well, if you specify a reference frame (three space axes and one time axis), then at a given time, each curve corresponds to a single point in three dimensional space. So light up selected points in a 3D display and watch as they move along the null curves.

Ark knows how to lay out these curves for a pair of spin-half fermions exchanging spin-one bosons, and he plans to make an animation and post it on YouTube with a link to this blog. The amazing thing is that once you satisfy the spin-half condition (go around twice to get to the starting point), the lighted points moving in space from one fermion to the other take the form of a transverse wave. Things like polarization also fall out. Ark says more came out than he put in.

A fermion is like a sphere rotating in 3D space, with threads attached to every point. The spin-half condition specifies that the sphere must rotate continuously, yet the threads are bundled into a rope with the other end fixed to a point, and they never get twisted or tangled.

Ark actually tried it with four ribbons attached at one end to a tin can and the other to a table. Two ribbons were taped to opposite sides of the top, and two to the bottom. He raised the can above all ribbons and rotated it one turn, and the ribbons got twisted. Then he lowered the can below all the ribbons and continued to rotate it a second turn in the same direction. All the ribbons came untwisted. This is literal spin-half. Ark is pretty sure a picture like this is in MTW Gravitation, but he didn’t quite grasp it the first (or second) time around.

If instead of moving the can up and down, you move the bundle of ribbons so the bundle starts below the can and after one rotation of the can, the bundle has revolved half way and is at the top, and after two rotations, the bundle is back to the bottom, the bundle is describing a cylinder. You can do this with one set of ribbons and two cans. Both cans rotate. The bundle of ribbons revolves about the axis connecting the two cans (which is perpendicular to each can’s axis), but none of the ribbons twist, and the bundle as a whole does not twist or rotate about its own axis.

Now for the finale. Everything described so far is in 3D space. But the ribbons are supposed to define the path of light. They do. In the animation Ark is going to figure out how to make real soon now, a point of light starts at the sphere surface and moves along the ribbon until it comes to the other sphere, where it vanishes. The ribbons are always straight (after they enter the bundle), and the points of light move along the path which the ribbon described at the time the point of light first appeared. So these points move straight from one end of the bundle to the other. But at the next instant, after the first point has moved away from the sphere, the next point starts a little higher. When the spheres have rotated one full turn, the bundle reaches the top of both spheres and starts back down. If the bundle is outlining a cylinder, then the points of light form a helix wrapping around the cylinder. Viewed from the side, it looks like a sine wave. This is circular polarization. If the bundle is moving almost up and down, it is vertical polarization.

For an encore, Ark will exhibit anti-particles. Obviously, if we mark the north pole of a rotating sphere, and say particles rotate clockwise as viewed from the north pole, while anti-particles rotate counterclockwise, we can tell an upside-down particle from an anti-particle. But Ark’s particles don’t seem to have any difference between the north and south poles. But actually, since the arrangement of ribbons in the bundle is fixed, we can put the north pole’s ribbon on the outside of the bundle, and the south pole’s ribbon at the center.

Next time, Ark hopes to explain why a pair of particles move apart, while a particle and an anti-particle move together. And why high frequency is high energy, etc., etc.

## Photology and Photological spaces

2007-05-01

See also the other posts in this category. Reading them in the order posted might be a good idea.

Archimerged finds that it is a bad idea to use well-known terms when subtle changes in the rules invalidate a lot of what people know about the terms. Since he is changing a bunch of rules, he needs some new terms. The systems in an ensemble (i.e., the universes in a multiverse) will be sets of points with certain additional structure.

Ark’s approach starts with a topological space, i.e., a set of points and a topology, which is a set of subsets of the space known as the open sets. He includes a manifold structure so as to be able to label the points. Next, instead of adding a metric, he adds the most abstract form of time and causality. Ark has decided to call a topological space with this added structure a photological space. Photology is an obsolete word for optics, and it seems appropriate.

So, a photological space is a topological space together with a photology, an ordered set (B, H, T, N) of four relations. (A relation is a set of ordered pairs of points. We denote the fact that the pair (p, q) is in the relation R by writing pRq.)  The four relations of a  are

• the before / after relation denoted bBa or aAb (when point b is before a and a is after b),
• the here relation denoted pHq (when p and q can be considered to be in the same place at different times),
• the there (or elsewhere) relation denoted pTq (when p and q can be considered to be in different places at the same time),
• and the neither here nor there (or null) relation denoted pNq (when p and q can neither be in the same place at different times, nor in different places at the same time, but are on a path light could follow).

Relative to a given point p, all sufficiently nearby points q must satisfy exactly one of pHq, pTq, or pNq. Points which satisfy pHq or pNq must also satisfy either pAq and qBp, or pBq and qAp.

A photology (B, H, T, N) has a close connection to an equivalence class of conformally equivalent metrics, and Ark’s approach uses a manifold together with a photology instead a manifold together with a metric.

This machinery is not really necessary, but it serves to emphasize the difference between this approach and standard general relativity. Ark used to say “all conformally equivalent metrics are really the same thing.”  See Wald, General Relativity, p. 445: conformally equivalent spaces have identical causal structure, and their Weyl tensors are equal.

## Back to foundations of physics for a while…

2007-05-01

It may be possible to blow Archimerged’s cover given the information herein. If you succeed, please do not publish your results. Ark likes to toil in anonymity, and believes the basic credit for the work described herein belongs to Einstein and others, and would rather not receive the blame if it fails completely.

See also the other posts in this category. Reading them in the order posted might be a good idea.

When Archimerged was in college (the first time), he never knew what he wanted to do, and kept switching majors. Starting in engineering school, he would have studied four semesters of physics, with the last one quantum mechanics. But he switched to anthropology (physical anthro actually taught evolution in detail) and so always felt deficient in quantum mechanics. Within a year or so he dropped out and worked in software for around 10 years, but managed to finish his undergraduate work, transferring to a nearby university just before his freshman credits would have become non-transferable.

Anyway, over the years he has learned quite a lot about quantum physics and related subjects. Intertwined in those years, he went to grad school and got a PhD in chemistry (related to bio-molecules) but by the time he finished he was more interested in solving the big physics problem: unifying gravity with quantum physics. At the beginning of grad school, he had to take undergraduate physical chemistry, not having taken it before. The derivation of the Maxwell-Boltzmann distribution was very interesting, and he considered ensembles to be very fundamental to the nature of the universe.

His approach has always been ensemble-based. At first, he thought of considering every possible pattern of dots on a display, varying in time. Any one sequence of frames makes a movie of sorts, but essentially all of them are just “snow.” But if you add some rules for how the dots start out and how they can change from frame to frame, more meaningful movies might emerge. He felt (and still feels) that the world is like this: particles move about according to certain laws (including the law of gravity, or maybe only the law of gravity) in a given “system” (where an ensemble is a set of systems), and the “real” universe we see is just one of these systems. Not the best of all possible worlds, but the most likely.

When he read a popular account of the “Many Worlds” interpretation of quantum mechanics, he thought that’s right, but why this branching stuff? Eliminate the branching, just start with all possible distributions of particles and all ways for them to move.

So he said, particles just move around according to the law of gravity, nothing else, and that is enough. But, what is the law of gravity? So he studied general relativity. A book club happened to offer an introductory package including Wald’s General Relativity, and he studied the first chapters in great detail. He formed the opinion that exact solutions are the only thing of interest. He also formed the opinion that you start with the metric and calculate the curvature and the energy tensor, rather than starting with the energy tensor and trying to find a metric. While reading Gravitation (usually called MTW after the authors), page 901, he formed the opinion that the principal null geodesics of exact solutions are very important.

It’s now been almost 20 years from the start and 10 years since Ark first ran across the metrics, and Ark has finally figured out a way to use certain simple vacuum metrics as electrons and photons. It really works! (see also blog motto). So he will try to explain it in the following posts.

## Big surprise — Archimerged has bipolar disorder

2007-05-01

Archimerged sometimes remarked to colleagues that he probably was bipolar, but was always very receptive to any suggestion that he wasn’t, and happy to forget about the possibility. And when under diagnosis for depression, he somehow never mentioned the behaviors which would suggest bipolar.  They were embarrassing, so he never brought them up. Or something like that.  And so, he isn’t going to list all of the ridiculous projects he has worked on for a while and then abandoned…  But he did usually learn something from them.

But a drug specific for bipolar disorder, Lamictal (lamotrigine), has worked well for him for several months. Diagnosis by asking questions depends on the subjects understanding the questions in the same way, and on answering correctly. Archimerged thinks that the operational diagnosis (especially when treatment is in the form of drugs) is whether the drugs work or not.  This one seems to be working and doesn’t have unpleasant side-effects.

## How can you be an Aspie and not know the weekday of any date?

2006-09-08

Archimerged has decided to start blogging again on more subjects. Today he gives his version of Conway’s “Doomsday” algorithm. (Google for Conway Doomsday to find other treatments, or see the Wikipedia article). He also talks a little about Asperger’s syndrome and mood disorders and his dog down at the bottom of the post. Since he uses a slightly different definition of Doomsday, he has decided to rename it. Googling various possibilities (Fixedday, Fixday, Lastday, Lassday, Lasday, Postday, Markday, Seedday, etc.) for a name without too many alternate meanings, he has just now thought of Keyday, which seems superior to all of the above.

The lead paragraphs in the Wikipedia article are actually very good (the rest is too):

For any year, Doomsday is the day of the week on which the last day of February falls. It is also the day of the week of 4/4, 6/6, 8/8, 10/10 and 12/12, as well as 5/9, 9/5, 7/11, and 11/7.

It is a convenient way of characterizing each of the 7 possible calendars for the months March – December. Only for the calendar of January and February is a further distinction between common year and leap year needed. Alternatively the fixed connection with the Doomsday of the previous year is used for these two months.

Anyway, it isn’t hard to figure the weekday of any date in your head if you learn the technique a step at a time. Archimerged recommends this order:

First, learn that Conway’s Doomsday is the same particular weekday every week of any given calendar year: the last day of February and every 7 days before and after. Archimerged’s Keyday always falls on February 27, one or two days later on the last day of February, and every 7 days up to and including February 27 of the next year. So Keyday always falls on December 12, January 9, February 6 (note the progression due to December and January both having 31 days), while Doomsdays in January and February depend on whether it’s leap year or not. But Doomsday is the same weekday all year, while you have to remember that in January and February Keyday is the same weekday as last year’s Doomsday. So Archimerged writes of the Doomsday of a year, not the Keyday, because any given Keyday applies to the March to February year.

The Keyday rule makes it clear that in regular years, Keyday advances by one weekday on February 28, while in leap years, Keyday advances by two weekdays on February 29. February 27 was Keyday, and February 28 or 29 is also Keyday — the new year always advances Keyday by one or two weekdays on the last of February. Doomsday advances on January 1 so January 2 is not a Doomsday. A year always has 52 Doomsdays and 53 Keydays.

The key to Doomsday and Keyday is that they always fall on the same day of the month in March through December. Conway gives this mnemonic: “He works nine to five at the Seven-Eleven.” Doomsday falls on 4/4, 6/6, 8/8, 10/10, 12/12 (and Keyday on 1/9 and 2/6), and on 9/5 and 7/11 as well as 5/9 and 11/7. Doomsday falls on 1/10 or 1/11 and on 2/7 or 2/8 depending on leap year. (Contrary to popular belief, and with disrespect only to politicians and other opportunistic greedy pigs, 9/11 was not a Doomsday). That covers all months except March, and you know Doomsday is the last day of February so it must fall on March 7, 14, 21, and 28. Doomsday also falls at a fixed offset from certain holidays: July 4 is a Doomsday, Christmas is the day before Doomsday, New Year’s day is the day before Keyday, and Veteran’s day 11/11 is always 4 days after Doomsday 11/7 (duh). The other U.S. Federal holidays always fall on Monday (M. L. King day, President’s day, Labor day, Columbus day, Veteran’s day) or Thursday (Thanksgiving day). Archimerged remembers the magazine insert (probably Parade) in the Sunday paper which urged people to write to Congress asking for holidays to fall on Monday.

The first Keyday of the month is August 1, January and May 2, October 3, April and July 4, September and December 5, February and June 6, March and November 7. August and October are the only months which do not share a Keyday with another month. This analysis comes out different for Doomsday and depends on leap year.

Next, learn to add and subtract small numbers from weekdays to get weekdays. This is useful even if you don’t finish the rest. Start with 3 (no counting on your fingers!) and work up to adding and subtracting 1 to 31. When you’re done you know instantly that Wednesday plus 31 is Wednesday plus 3, Saturday; Wednesday minus 31 is always Sunday; Sunday plus 13 is the same as Sunday minus 1, etc., etc. Archimerged believes this is superior to using a code for days like 0 for Sunday, 1 for Monday, adding numbers, taking the remainder after dividing by 7, and translating back to weekdays. (They ought to teach this number plus weekday gives weekday addition table in elementary school).

Third, in order to be able to use this knowledge, memorize the current year’s Doomsday. In 2006, Doomsday is Tuesday. Let this settle for a while and practice figuring weekdays for dates of this year.

After you get good at the current year, learn this bit of trivia: the last day of February, 1900, was Wednesday the 28th, and the last day of February, 2000, was Tuesday the 29th. Conway gives “We in dis day” as a mnemonic for 1900, since most people alive today were born in the 20th century.

Some other day, learn that the last day of February is Tuesday the 29th in 2000, 2400, 2800, etc. and in Rome also 1600 (but not most other places). Also the last day of February is Wednesday the 28th in 1900, 2300, etc., Friday the 28th in 1800, 2200, etc., and Sunday the 28th in 1700 (in Rome), 2100, etc. The last day of February never falls on Monday, Thursday, or Saturday on century years. It is for this reason that over the long haul, the 13th of any Gregorian month is more likely to be Friday than any other day.

On yet some other day, figure out the rule for Julian century years and memorize the dates when different countries switched. (You are an Aspie, aren’t you?)

But before learning all the other centuries, learn to figure what weekday Doomsday falls on for any given year of the century. The most obvious approach is to note that every year adds one weekday and every leap year adds an additional weekday. Therefore, add the two digit year to the century Doomsday (you did learn to add numbers to weekdays, right?). Then correct for leap years by adding the number of leap years since the century year (and not counting the century year, which as purists know is the last year of the previous century anyway). That is, given year C*100+Y, add Y + [Y/4] to Doomsday of C*100, where [4/4] = [5/4] = [6/4] = [7/4] = 1, etc. This applies for for all C, but Doomsday of the century year depends on the calendar (Gregorian or Julian).

Writing D(’00) for the century year Doomsday and using addition of numbers to weekdays, the general rule is: for Y in 0 .. 99,

• D(’00+Y) = D(’00) + Y + [Y/4].

A shortcut is to figure the number of dozens in the two-digit year. Add the number of dozens plus the remaining years plus 1 or 2 leap days to the century’s Doomsday.

For Y in 0 … 11, D(’00+Y) = D(’00) + Y + [Y/4], and

• D(’12+Y) = D(’00+Y) + 1,
• D(’24+Y) = D(’00+Y) + 2,
• D(’36+Y) = D(’00+Y) + 3,
• D(’48+Y) = D(’00+Y) – 3,
• D(’60+Y) = D(’00+Y) – 2,
• D(’72+Y) = D(’00+Y) – 1,
• D(’84+Y) = D(’00+Y), and
• D(’96+Y) = D(’00+Y) + 1 (for Y in 0 … 3 only).

A different shortcut is to first take the remainder after division by 28. Subtract 28, 56, or 84 from Y and then figure Doomsday for the remainder. If you are very good at doing the years ’00 to ’27, this lets you extend your skill to the rest of the century at the expense of an extra subtraction.

For Y in 0 … 27, D(’00+Y) = D(’00) + Y + [Y/4], and

• D(’28+Y) = D(’00+Y),
• D(’56+Y) = D(’00+Y), and
• D(’84+Y) = D(’00+Y) (for Y in 0 … 15 only).

To cover the first 28 years, the dozens rule expands as follows:For Y in 0 … 3,

• D(’00+Y) = D(’00) + Y (because [Y/4] is 0),
• D(’12+Y) = D(’00) + Y + 1 (because [Y/4] is 0),
• D(’24+Y) = D(’00) + Y + 2 (because [Y/4] is 0).

For Y in 4 … 7,

• D(’00+Y) = D(’00) + Y + 1 (because [Y/4] is 1),
• D(’12+Y) = D(’00) + Y + 2 (because [Y/4] is 1).

For Y in 8 … 12,

• D(’00+Y) = D(’00) + Y + 2 (because [Y/4] is 2),
• D(’12+Y) = D(’00) + Y + 3 (because [Y/4] is 2).

There is also a completely different method: you can learn which years of a century have the same Doomsday as the century year, i.e., the years ’00 + Y where Y + [Y / 4] is a multiple of 7. The pattern repeats every 28 years, of course.

Here is the complete table, including the leap years ’12, ’40, ’68, and ’96 in which the century Doomsday falls on February 28 while Doomsday of course falls on February 29.

• D(’00) = D(’28) = D(’56) = D(’84) = D(’00),
• D(’06) = D(’34) = D(’62) = D(’90) = D(’00),
• D(’12) = D(’40) = D(’68) = D(’96) = D(’00) + 1,
• D(’17) = D(’45) = D(’73) = D(’00),
• D(’23) = D(’51) = D(’79) = D(’00).

To use the above facts, memorize the list of years. Given year Y, find the largest year T in the table less than Y. Figure Y – T, which cannot be more than six. Figure how many of the years T+1, T+2, … Y are leap years. This can be zero, one, or two. Then,

• D(Y) = D(’00) + (one if T is ’12, ’40, ’68, or ’96) + (Y-T) + (the number of leap years in T+1 … Y).

With practice, this may be a little faster than adding the number of dozens, the number of years left over, and the number of leap years left over, especially when Y = T. For example, Y = ’51 is in the table so D(’51) = D(’00). Using dozens, ’48 is 4 dozens past ’00, and ’51 is 3 years and no leap years past ’48, giving D(’51) = D(’00) + 4 + 3 = D(’00).

So, why did Archimerged spend most of a day writing this? Just like the girl with the spoons in that documentary, practicing a skill like this is soothing. Archimerged suffers from some kind of mood disorder in addition to believing he has Asperger’s syndrome even if the mental health people he has talked to don’t see how he meets the DSM critera. Antidepressants help but have not really fixed it after several years of use. This suggests that the diagnosis of major depression is wrong. A long history of unfinished projects may suggest a new diagnosis of some kind of bipolar disorder, but the treatment there may be worse than the disease. It can be very hard to ignore mood and act strictly logically, and mood disorder means it is harder than usual, but drugs are still quite imprecise tools and Archimerged frankly doesn’t trust them.

(Archimerged gives up on second person seeing as his dog is male…) Why do people and animals do what they do? Mood must have a lot to do with it. People can use abstract thought to adjust what mood leads them to do, while animals can’t. I observe my dog remembering: when he has encountered a situation before, he remembers what happened. I believe this directly controls his mood. For a long time he didn’t connect taking the leash down from the hook and going out. I was trying also to always say “out,” but sometimes I would say “out” but then not be able to go immediately. But I never took the leash down until I was actually ready. He might go down the steps and get stranded. (He can’t climb back up due to patella malformation). Then he feels bad, and howls. So I think when the circumstances match up, he remembers how he felt after going down, and stays at the top of the steps. Now when I say “out” he doesn’t really believe me until I take the leash down.

So Archimerged looks at this and say the dog’s brain recongnizes a pattern including many factors. The result is a mood (excitement or lonliness) and a prediction of future events (being taken out or being stranded at the bottom of the stairs). Archimerged thinks people’s brains work this way too. In a mood disorder, the pattern recognition responds inaccurately causing strong mood effects that do not match reality. Maybe there is a way to counteract these responses without using drugs that have side effects like tardive dyskinesia or lithium poisoning.

## Sidetracked again, working on gnucash

2006-07-16

Archimerged has decided it is time to do the accounting and get automatic downloads of financial information set up. Gnucash version 2.0 has just been released, with the conversion to Gnome 2 finally finished. So he it trying out the online banking interface. It doesn’t work yet, but there are segfaults which can be fixed, and now that the upheaval is over, he has some patches to merge and submit… Hopefully back to energy design soon.

## Heat pipelines in Death Valley

2006-07-05

Archimerged has done some calculations regarding how much gravitational energy is available to a turbine installed in the liquid return pipe of a heat pipeline running 2 km up a mountain.  It seems that there is about 54 kW gravitational power for every MW heat power carried up the 2km mountain by a propane heat pipeline.  This is about 10 kW gravitational power for every liter per second propane liquid flowing down the mountain.  The propane vapor flowing upward needs a larger pipeline, perhaps 15 times larger than the liquid pipeline.  The liquid return pipeline must be very well insulated.  The vapor pipeline does not need as much insulation, but calculations are needed for that.

## Nearly stationary bubbles don’t separate easily during horizontal motion

2006-07-04

Archimerged has found some data for air bubble behavior. There is a graph of bubble size vs velocity on page 67 of this pdf file. Very slow bubble rise is possible for 100 micron bubbles. The problem is those bubbles also don’t rise after leaving the bubble pump and trompe. It becomes clear that either some mechanism to trigger bubble separation is needed, or a large separation chamber is needed at the top of the bubble pump and the bottom of the trompe. But at least, if the separation chamber is very shallow and wide, the forward flow velocity need not drop much while the distance the bubble needs to rise to reach the surface can be just a few cm. Bubbles rising at 10 cm/s would reach the surface in under a second, meaning the separation chamber needs to be under ten meters long. Inconvenient but not impossible.

The ratio of water speed to bubble rise speed would apparently be a lower bound on inefficiency: if the bubbles rise in still water at 1% of the speed of the water, e.g. 0.1 m/s vs. 10 m/s, then in the bubble pump the water rises at 10 m/s while the bubbles rise at 10.1 m/s and in the trompe, the water descends at 10 m/s while the bubbles descend at 9.9 m/s. Then the bubble pump and the trompe both lose 1% efficiency, leading to at most 98% efficiency, while break even would be around 95%.

So maybe we need slower than 1 in 100. Bubbles 0.2 mm in diameter rise at about 2 cm/s. With water moving 10 m/s, that is 2 in 1000, and the bubble speed accounts for an efficiency loss of 0.4%. The separation chamber must spread out so widely that that the water is only 2 cm thick and it must be 10 meters long. Actually, that might make it wider than it is long, depending on the volume. And it has to be well insulated, and the separation chamber must be at around atmospheric pressure: the bubbles have to separate before the water descends to the heat exchanger.

At the top of the bubble pump tower, that is kind of inconvenient, to say the least. Some more clever ideas are needed…

It begins to sound like the bubble pump is made of fairly small diameter pipe, say 10 cm, and multiple pipes are used to increase the power.  Maybe we are back with the hyperbolic shaped tubing which tilts to nearly horizontal at the top.  This allows the water to continue at constant speed say 10 m/s (need to check the hydrodynamic drag to pick actual pipe sizes and water speeds) while the vertical velocity component slows to zero. Note well, the bubbles are confined by the top wall of the tilted pipe so their rise is slowed.  Also they start getting separated before reaching the top of the bubble pump.

## Let’s try something solid once again: the WaterAirWheel

2006-06-29

Archimerged has not given up on submerged air containers, but he is down on air entrained in water. Unless some polymer could form a skin around the bubbles which somehow anchors the bubble to the water flow. Maybe a long hydrophobic chain connected to a longer hydrophilic chain, like soap except instead of a single charged acid group, it has long polarized tails extending into the bulk water. But forget that for the moment. Today he is thinking about big wheels and chains of buckets pulling cold air down a column of cold water while expanding hot air in another chain of buckets pulls on the wheel while moving up through a column of hot water.

Imagine swinging buckets on a completely submerged water wheel, a WaterAirWheel, so to speak. That gets rid of the chains but requires a very big wheel. Looking broadside onto the counter-clockwise turning wheel, the left side is in a deep pool of cold water and the right side is in a deep pool of hot water. The two pools are separated by the hub of the wheel and a wall of stationary insulation which fits closely to the wheel to prevent water flow. The wheel and buckets are coated with insulation so heat doesn’t flow into the wheel on the hot side or out of the wheel on the cold side. The buckets are carefully designed so that no cold water or air is carried over to the hot side at the bottom, and no hot water or air is carried over to the cold side at the top. Unlike the TrombePump, the WaterAirWheel doesn’t need to repeatedly heat and cool the water, but it does need a long flexible partition which rubs against the walls of the wheel to keep water from flowing from one tank to the other.

On the hot side, hot compressed air bubbles up into buckets near the bottom of the wheel, and escapes at the top when the bucket tips upward. On the cold side, cold atmospheric pressure air is captured at the top. The buckets are shaped so the compressed cold air escapes upward into a fixed inverted collector just before reaching the bottom of the wheel and the insulated wall between the cold and hot tanks. A solid cylinder fills the buckets at the bottom, displacing cold compressed air and water as the bucket envelops the cylinder and swings around it. As the bucket pulls away from the cylinder, hot water and some hot compressed air flows into the bucket which is now tilted to hold the air in as it rises.

That’s rather complicated and very big. On second thought, Archimerged begins to like chains of buckets with idler wheels at top and bottom, and a large gear transmitting force from the hot idler to the cold idler. That reduces the size of the water tanks and allows them to be arbitrarily deep with only a linear increase in cost instead of a quadratic increase. The width and breadth of the tanks is constant with increasing depth, instead of constant breadth with width equal to depth. Since this still has a (relatively) big wheel, Archimerged figures he can still call this modified version a WaterAirWheel.

In the cold tank, buckets carry cold air downward, compressing it, and return full of water. In the hot tank, they descend full of water but return full of expanding hot air.

A large countercurrent heat exchanger cools hot atmospheric pressure air at constant pressure while warming cold high pressure air at constant pressure. The hot heat source keeps the hot water hot, and the cold heat sink keeps the cold water cold.  The machine can turn an additional load at the wheel, or it can admit additional atmospheric pressure air and output high pressure cold or hot air.

## What goes down must come up

2006-06-27

Archimerged has been thinking about why he believes the TrombePump should efficiently convert hot compressed air into an equal volume and a greater number of moles of cold compressed air. He assumes that the countercurrent heat exchanger can be made to operate efficiently so that not much heat flows from the hot source to the cold sink without heating expanding air, and that turbulence can be reduced to tolerable levels. The question is about how bubbles that don’t even look like they can push water ahead of them should raise water.

Well, the bubbles don’t push water up. The water is moving and will tend to continue to move. In the steady state, the volume of air in the bubble pump equals the volume in the trompe and the hydrostatic forces exactly balance. Adding a little air to the bubble pump will reduce the weight of water above and cause an imbalance of forces so that water in the bubble pump moves a little faster. And adding more air to the trompe will similarly reduce the weight of water in the trompe and reduce the downward force, slowing the watter a little.

From another viewpoint, the water which goes down the trompe has nowhere to go but back up the bubble pump, and vice-versa. We aren’t raising any more water than we let down. The mass of water balances.

If too much compressed air is released into the bubble pump, the excess energy goes into accelerating the water. If too much air goes into the trompe, excess energy to compress it comes from the kinetic energy of the flowing water, which must slow down because there is more hydrostatic pressure coming down the bubble pump than down the trompe and the net force is backwards, slowing the water.

So, Archimerged concludes that the apparent ability of water to flow around the bubbles is not a problem. The bubbles don’t push the water. Gravity pulls the water downward as far as it can go, as usual.

Revised:  Archimerged has thought about this some more, and realized that he was wrong.  Gravity pulls water down in both tubes, with the flow in the trompe and against the flow in the bubble pump.  Bubbles will tend to rise due to water flowing from along side to underneath them.  This slows the descent in the trompe and speeds the ascent in the pump.  Thus, to keep the same average density in both tubes, more gas flow by volume is required in the pump than in the trompe.

So Archimerged is now looking for more information about how bubbles behave.  Do tiny bubbles move more slowly?  What is the effect of detergent in the water?  Does fast moving water reduce the effect or leave it alone?