Why is it hot at the equator, cold at the poles?

Here’s a somewhat mathematical look at why it is hotter at the equator that at the poles. This is high school or basic college level science, using trigonometry (pre calculus), a slight step beyond the basic statement that the sun hits down more directly at the equator than at the poles. That’s the kid’s explanation, but we can understand better if we add a little math.

Solar radiation hits Detroit or any other non-equator point at an angle, As a result, less radiation power hits per square meter of land.

Lets use the diagram at right and trigonometry to compare the amount of sun-energy that falls on a square meter of land at the equator (0 latitude) and in a city at 42.5 N latitude (Detroit, Boston, and Rome are at this latitude). In each case, let’s consider high-noon on March 21 or September 20. These are the two equinox days, the only days each year when the day and night are equal length, and the only times when it is easy to calculate the angle of the sun as it deviates from the vertical by exactly the latitude on the days and times.

More specifically the equator is zero latitude, so on the equator at high-noon on the equinox, the sun will shine from directly overhead, or 0° from the vertical. Since the sun’s power in space is 1050 W/m2, every square meter of equator can expect to receive 1050 W of sun-energy, less the amount reflected off clouds and dust, or scattered off or air molecules (air scattering is what makes the sky blue). Further north, Detroit, Boston, and Rome sit at 42.5 latitude. At noon on March 31 the sun will strike earth at 42.5° from the vertical as shown in the lower figure above. From trigonometry, you can see that each square meter of these cities will receive cos 42.5 as much power as a square meter at the equator, except for any difference in clouds, dust, etc. Without clouds etc. that would be 1050 cos 42.5 = 774 W. Less sun power hits per square meter because each square meter is tilted. Earlier and later in the day, each spot will get less sunlight than at noon, but the proportion is the same, at least on one of the equinox days.

To calculate the likely temperature in Detroit, Boston, or Rome, I will use a simple energy balance. Ignoring heat storage in the earth for now, we will say that the heat in equals the heat out. We now ignore heat transfer by way of winds and rain, and approximate to say that the heat out leaves by black-body radiation alone, radiating into the extreme cold of space. This is not a very bad approximation since Black body radiation is the main temperature removal mechanism in most situations where large distances are involved. I’ve discussed black body radiation previously; the amount of energy radiated is proportional to luminosity, and to T4, where T is the temperature as measured in an absolute temperature scale, Kelvin or Rankin. Based on this, and assuming that the luminosity of the earth is the same in Detroit as at the equator,

T Detroit / Tequator  = √√ cos 42.5 = .927

I’ll now calculate the actual temperatures. For American convenience, I’ll choose to calculation in the Rankin Temperature scale, the absolute Fahrenheit scale. In this scale, 100°F = 560°R, 0°F = 460°R, and the temperature of space is 0°R as a good approximation. If the average temperature of the equator = 100°F = 38°C = 560°R, we calculate that the average temperature of Detroit, Boston, or Rome will be about .927 x 560 = 519°R = 59°F (15°C). This is not a bad prediction, given the assumptions. We can expect the temperature will be somewhat lower at night as there is no light, but the temperature will not fall to zero as there is retained heat from the day. The same reason, retained heat, explains why it is warmer will be warmer in these cities on September 20 than on March 31.

In the summer, these cities will be warmer because they are in the northern hemisphere, and the north pole is tilted 23°. At the height of summer (June 21) at high noon, the sun will shine on Detroit at an angle of 42.5 – 23° = 19.5° from the vertical. The difference in angle is why these cities are warmer on that day than on March 21. The equator will be cooler on that day (June 21) than on March 21 since the sun’s rays will strike the equator at 23° from the vertical on that day. These  temperature differences are behind the formation of tornadoes and hurricanes, with a tornado season in the US centering on May to July.

When looking at the poles, we find a curious problem in guessing what the average temperature will be. At noon on the equinox, the sun comes in horizontally, or at 90°from the vertical. We thus expect there is no warming power at all this day, and none for the six months of winter either. At first glance, you’d think the temperature at the poles would be zero, at least for six months of the year. It isn’t zero because there is retained heat from the summer, but still it makes for a more-difficult calculation.

To figure an average temperature of the poles, lets remember that during the 6 month summer the sun shines for 24 hours per day, and that the angle of the sun will be as high as 23° from the horizon, or 67° from the vertical for all 24 hours. Let’s assume that the retained heat from the summer is what keeps the temperature from falling too low in the winter and calculate the temperature at an .

Let’s assume that the sun comes in at the equivalent of 25° for the sun during the 6 month “day” of the polar summer. I don’t look at equinox, but rather the solar day, and note that the heating angle stays fixed through each 24 hour day during the summer, and does not decrease in the morning or as the afternoon wears on. Based on this angle, we expect that

TPole / Tequator  = √√ cos 65° = .806

TPole = .806 x 560°R = 452°R = -8°F (-22°C).

This, as it happens is 4° colder than the average temperature at the north pole, but not bad, given the assumptions. Maybe winds and water currents account for the difference. Of course there is a large temperature difference at the pole between the fall equinox and the spring equinox, but that’s to be expected. The average is, -4°F, about the temperature at night in Detroit in the winter.

One last thing, one that might be unexpected, is that temperature at the south pole is lower than at the north pole, on average -44°F. The main reason for this is that the snow on south pole is quite deep — more than 1 1/2 miles deep, with some rock underneath. As I showed elsewhere, we expect that, temperatures are lower at high altitude. Data collected from cores through the 1 1/2 mile deep snow suggest (to me) chaotic temperature change, with long ice ages, and brief (6000 year) periods of warm. The ice ages seem far worse than global warming.

Dr. Robert Buxbaum, December 30, 2017

The Addams Family did well on Broadway, in the movies, and on TV, but got predictably bad reviews in all three forms. Ordinary people like it; critics did not. Something I like about the series that critics didn’t appreciate is that Gomez is the only positive father character I can think of since the days of “Father knows best”.

Gomez is sexual, and sensual; a pursuer and lover, but not a predator.

In most family shows the father isn’t present at all, or if he appears, he’s violent or and idiot. He’s in prison, or in trouble with the law, regularly insulted by his wife and neighbors, in comedies, he’s sexually ambiguous, insulted by his children, and often insulted by talking pets too. In Star Wars, the only father figures are Vader, a distant menace, and Luke who’s just distant. In American shows, the parents are often shown as divorced; the children are reared by the mother with help of a nanny, a grandparent, or a butler. In Japanese works, I hardly see a parent. By contrast, Gomez is present, center stage. He’s not only involved, he’s the respected leader of his clan. If he’s odd, it’s the odd of a devoted father and husband who comfortable with himself and does not care to impress others. It’s the outsiders, the visitors, who we find have family problems, generally caused by a desire to look perfect.

Gomez is hot-blooded, sexual and sensual, but he’s not a predator, or violent. He’s loved by his wife, happy with his children, happy with his life, and happy with himself. As best we can tell, he’s on good-enough terms with the milk man, the newspaper boy, and the law. Though not a stick-in-the-mud, he’s on excellent terms with the rest of the Addams clan, and he’s good with the servants: Lurch, Thing, and for a while a gorilla who served as maid (none too well). One could do worse than to admire a person who maintains a balance like that between the personal, the family, the servants, and the community.

On a personal level, Gomez is honest, kind, generous, loving, and involved. He has hobbies, and his hobbies are manly: fencing, chess, dancing, stocks, music, and yoga. He reads the newspaper and smokes a cigar, but is not addicted to either. He plays with model trains too, an activity he shares with his son. Father and son enjoy blowing up the train — it’s something kids used to do in the era of firecrackers. Gomez is not ashamed to do it, and approves when his son does.

Gomez Addams, in the Addams Family Musical, gives advice and comfort to his daughter who is going through a rough stretch of relationship with a young man, and sings that he’s happy and sad.

Other TV and movie dads have less – attractive hobbies: football watching and beer-drinking, primarily. Han Solo is a smuggler, though he does not seem to need the money. TV dads take little interest in their kids, and their kids return the favor. To the extent that TV dads take an interest, it’s to disapprove, George Costanza’s dad, for example. Gomez is actively interested and is asked for advice regularly. In the video below, he provides touching comfort and advice to his daughter while acknowledging her pain, and telling her how proud he is of her. Kids need to hear that from a dad. No other TV dad gives approval like this; virtually no other male does. They are there as props, I think, for strong females and strong children.

The things that critics dislike, or don’t understand, as best I can tell, is the humor, based as it is on danger and dance. Critics hate humor in general (How many “best picture” Oscars go to comedies?)  Critics fear pointless danger, and disapproval, and law suits, and second-hand-smoke. They are the guardians of correct thinking — just the thinking that Gomez and the show ridicules. Gomez lives happily in the real world of today, but courts danger, death, law suits. He smokes and dances and does not worry what the neighbors think. He tries dangerous things and does not always succeed, but then lets his kids try the same. He dances with enthusiasm. I find his dancing and fearlessness healthier than the over-protective self-sacrifice that critics seem to favor in heroes. To the extent that they tolerate fictional violence, they require the hero to swooping, protecting others at danger to themselves only, while the others look on (or don’t). The normal people are presented as cautious, fearful, and passive. Cold, in a word, and we raise kids to be the same. Cold fear is a paralyzing thing in children and adults; it often brings about the very damage that one tries to avoid.

Gomez is hot: active, happy, and fearless. This heat.– this passion — is what makes Gomez a better male role model than Batman, say. Batman is just miserable, or the current versions are. Ms. Frizzle (magic school bus) is the only other TV character who is happy to let others take risks, but Ms Frizzle is female. Gomez’s thinks the best of those who come to visit, but we see they usually don’t deserve it. Sometimes they do, and this provides touching moments. Gomez is true to his wife and passionate, most others are not. Gomez kisses his wife, dances with her, and compliments her. Outsiders don’t dance, and snap at their wives; they are motivated by money, status, and acceptability. Gomez is motivated by life itself (and death). The outsiders fear anything dangerous or strange; they are cold inside and suffer as a result. Gomez is hot-blooded and alive: as a lover, a dancer, a fencer, a stock trader, an animal trainer, and a collector. He is the only father with a mustache, a sign of particular masculinity — virtually the only man with a mustache.

Gomez has a quiet, polite and decent side too, but it’s a gallant version, a masculine heterosexual version. He’s virtually the only decent man who enjoys life, or for that matter is shown to kiss his wife with more than a peck. In TV or movies, when you see a decent, sensitive, or polite man, he is asexual or homosexual. He is generally unmarried, sometimes divorced, and almost always sad — searching for himself. I’m not sure such people are positive role models for the a-sexual, but they don’t present a lifestyle most would want to follow. Gomez is decent, happy, and motivated; he loves his life and loves his wife, even to death, and kisses with abandon. My advice: be alive like Gomez, don’t be like the dead, cold, visitors and critics.

Robert Buxbaum, December 22, 2017.  Some years ago, I gave advice to my daughter as she turned 16. I’ve also written about Superman, Hamilton, and Burr; about Military heroes and Jack Kelly.

Hydrogen permeation rates in Inconel, Hastelloy and stainless steels.

Some 20 years ago, I published a graph of the permeation rate for hydrogen in several metals at low pressure, See the graph here, but I didn’t include stainless steel in the graph.

Hydrogen permeation in clean SS-304; four research groups’ data.

One reason I did not include stainless steel was there were many stainless steels and the hydrogen permeation rates were different, especially so between austenitic (FCC) steels and ferritic steels (BCC). Another issue was oxidation. All stainless steels are oxidized, and it affect H2 permeation a lot. You can decrease the hydrogen permeation rate significantly by oxidation, or by surface nitriding, etc (my company will even provide this service). Yet another issue is cold work. When  an austenitic stainless steel is worked — rolled or drawn — some Austinite (FCC) material transforms to Martisite (a sort of stretched BCC). Even a small amount of martisite causes an order of magnitude difference in the permeation rate, as shown below. For better or worse, after 20 years, I’m now ready to address H2 in stainless steel, or as ready as I’m likely to be.

Hydrogen permeation in SS 340 and SS 321. Cold work affects H2 permeation more than the difference between 304 and 321; Sun Xiukui, Xu Jian, and Li Yiyi, 1989

The first graph I’d like to present, above, is a combination of four research groups’ data for hydrogen transport in clean SS 304, the most common stainless steel in use today. SS 304 is a ductile, austenitic (FCC), work hardening, steel of classic 18-8 composition (18% Cr, 8% Ni). It shares the same basic composition with SS 316, SS 321 and 304L only differing in minor components. The data from four research groups shows a lot of scatter: a factor of 5 variation at high temperature, 1000 K (727 °C), and almost two orders of magnitude variation (factor of 50) at room temperature, 13°C. Pressure is not a factor in creating the scatter, as all of these studies were done with 1 atm, 100 kPa hydrogen transporting to vacuum.

The two likely reasons for the variation are differences in the oxide coat, and differences in the amount of cold work. It is possible these are the same explanation, as a martensitic phase might increase H2 permeation by introducing flaws into the oxide coat. As the graph at left shows, working these alloys causes more differences in H2 permeation than any difference between alloys, or at least between SS 304 and SS 321. A good equation for the permeation behavior of SS 304 is:

P (mol/m.s.Pa1/2) = 1.1 x10-6 exp (-8200/T).      (H2 in SS-304)

Because of the song influence of cold work and oxidation, I’m of the opinion that I get a slightly different, and better equation if I add in permeation data from three other 18-8 stainless steels:

P (mol/m.s.Pa1/2) = 4.75 x10-7 exp (-7880/T).     (H2 in annealed SS-304, SS-316, SS-321)

Hydrogen permeation through several common stainless steels, as well as Inocnel and Hastelloy

Though this result is about half of the previous at high temperature, I would trust it better, at least for annealed SS-304, and also for any annealed austenitic stainless steel. Just as an experiment, I decided to add a few nickel and cobalt alloys to the mix, and chose to add data for inconel 600, 625, and 718; for kovar; for Hastelloy, and for Fe-5%Si-5%Ge, and SS4130. At left, I pilot all of these on one graph along with data for the common stainless steels. To my eyes the scatter in the H2 permeation rates is indistinguishable from that SS 304 above or in the mixed 18-8 steels (data not shown). Including these materials to the plot decreases the standard deviation a bit to a factor of 2 at 1000°K and a factor of 4 at 13°C. Making a least-square analysis of the data, I find the following equation for permeation in all common FCC stainless steels, plus Inconels, Hastelloys and Kovar:

P (mol/m.s.Pa1/2) = 4.3 x10-7 exp (-7850/T).

This equation is near-identical to the equation above for mixed, 18-8 stainless steel. I would trust it for annealed or low carbon metal (SS-304L) to a factor of 2 accuracy at high temperatures, or a factor of 4 at low temperatures. Low carbon reduces the tendency to form Martinsite. You can not use any of these equations for hydrogen in ferritic (BCC) alloys as the rates are different, but this is as good as you’re likely to get for basic austenitc stainless and related materials. If you are interested in the effect of cold work, here is a good reference. If you are bothered by the square-root of pressure driving force, it’s a result of entropy: hydrogen travels in stainless steel as dislocated H atoms and the dissociation H2 –> 2 H leads to the square root.

Robert Buxbaum, December 17, 2017. My business, REB Research, makes hydrogen generators and purifiers; we sell getters; we consult on hydrogen-related issues, and will (if you like) provide oxide (and similar) permeation barriers.

Change home air filters 3 times per year

Energy efficient furnaces use a surprisingly large amount of electricity to blow the air around your house. Part of the problem is the pressure drop of the ducts, but quite a lot of energy is lost bowing air through the dust filter. An energy-saving idea: replace the filter on your furnace twice a year or more. Another idea, you don’t have to use the fanciest of filters. Dirty filters provide a lot of back-pressure especially when they are dirty.

I built a water manometer, see diagram below to measure the pressure drop through my furnace filters. The pressure drop is measured from the difference in the height of the water column shown. Each inch of water is 0.04 psi or 275 Pa. Using this pressure difference and the flow rating of the furnace, I calculated the amount of power lost by the following formula:

W = Q ∆P/ µ.

Here W is the amount of power use, Watts, Q is flow rate m3/s, ∆P = the pressure drop in Pa, and µ is the efficiency of the motor and blower, typically about 50%.

With clean filters (two different brands), I measured 1/8″ and 1/4″ of water column, or a pressure drop of 0.005 and 0.01 psi, depending on the filter. The “better the filter”, that is the higher the MERV rating, the higher the pressure drop. I also measured the pressure drop through a 6 month old filter and found it to be 1/2″ of water, or 0.02 psi or 140 Pa. Multiplying this by the amount of air moved, 1000 cfm =  25 m3 per minute or 0.42 m3/s, and dividing by the efficiency, I calculate a power use of 118 W. That is 0.118 kWh/hr. or 2.8 kWh/day.

The water manometer I used to measure the pressure drop through the filter of my furnace. I stuck two copper tubes into the furnace, and attached a plastic tube half filled with water between the copper tubes. Pressure was measured from the difference in the water level in the plastic tube. Each 1″ of water is 280 Pa or 0.04psi.

At the above rate of power use and a cost of electricity of 11¢/kWhr, I find it would cost me an extra 4 KWhr or about 31¢/day to pump air through my dirty-ish filter; that’s \$113/year. The cost through a clean filter would be about half this, suggesting that for every year of filter use I spend an average of \$57t where t is the use life of the filter.

To calculate the ideal time to change filters I set up the following formula for the total cost per year \$, including cost per year spent on filters (at \$5/ filter), and the pressure-induced electric cost:

\$ = 5/t + 57 t.

The shorter the life of the filter, t, the more I spend on filters, but the less on electricity. I now use calculus to find the filter life that produces the minimum \$, and determine that \$ is a minimum at a filter life t = √5/57 = .30 years.  The upshot, then, if you filters are like mine, you should change your three times a year, or so; every 3.6 months to be super-exact. For what it’s worth, I buy MERV 5 filters at Ace or Home Depot. If I bought more expensive filters, the optimal change time would likely be once or twice per year. I figure that, unless you are very allergic or make electronics in your basement you don’t need a filter with MERV rating higher than 8 or so.

I’ve mentioned in a previous essay/post that dust starts out mostly as dead skin cells. Over time dust mites eat the skin, some pretty nasty stuff. Most folks are allergic to the mites, but I’m not convinced that the filter on your furnace dies much to isolate you from them since the mites, etc tend to hang out in your bed and clothes (a charming thought, I know).

Old fashioned, octopus furnace. Free convection.

The previous house I had, had no filter on the furnace (and no blower). I noticed no difference in my tendency to cough or itch. That furnace relied for circulation on the tendency for hot air to rise. That is, “free convection” circulated air through the home and furnace by way of “Octopus” ducts. If you wonder what a furnace like that looks like here’s a picture.

I calculate that a 10 foot column of air that is 30°C warmer than that in a house will have a buoyancy of about 0.00055 psi (1/8″ of water). That’s enough pressure to drive circulation through my home, and might have even driven air through a clean, low MERV dust filter. The furnace didn’t use any more gas than a modern furnace would, as best I could tell, since I was able to adjust the damper easily (I could see the flame). It used no electricity except for the thermostat control, and the overall cost was lower than for my current, high-efficiency furnace with its electrical blower and forced convection.

Robert E. Buxbaum, December 7, 2017. I ran for water commissioner, and post occasional energy-saving or water saving ideas. Another good energy saver is curtains. And here are some ideas on water-saving, and on toilet paper.

Bitcoin risks, uses, and bubble

Bitcoin prices over the last 3 years

As I write this, the price of a single bitcoin stands are approximately \$11,100 yesterday, up some 2000% in the last 6 months suggesting it is a financial bubble. Or maybe it’s not: just a very risky investment suited for inclusion in a regularly balanced portfolio. These are two competing views of bitcoin, and there are two ways to distinguish between them. One is on the basis of technical analysis — does this fast rise look like a bubble (Yes!), and the other is to accept that bitcoin has a fundamental value, one I’ll calculate that below. In either case, the price rise is so fast that it is very difficult to conclude that the rise is not majorly driven by speculation: the belief that someone else will pay more later. The history of many bubbles suggests that all bubbles burst sooner or later, and that everyone holding the item loses when it does. The only winners are the last few who get out just before the burst. The speculator thinks that’s going to be him, while the investor uses rebalancing to get some of benefit and fun, without having to know exactly when to get out.

That bitcoin is a bubble may be seen by comparing the price three years ago. At that point it was \$380 and dropping. A year later, it was \$360 and rising. One can compare the price rise of the past 2-3 years with that for some famous bubbles and see that bitcoin has risen 30 times approximately, an increase that is on a path to beat them all except, perhaps, the tulip bubble of 1622.

A comparison between Bitcoin prices, and those of tulips, 1929 stocks, and other speculative bubbles; multiple of original price vs year from peak.

That its price looks like a bubble is not to deny that bitcoin has a fundamental value. Bitcoin is nearly un-counterfeit-able, and its ownership is nearly untraceable. These are interesting properties that make bitcoin valuable mostly for illegal activity. To calculate the fundamental value of a bitcoin, it is only necessary to know the total value of bitcoin business transactions and the “speed of money.” As a first guess, lets say that all the transactions are illegal and add up to the equivalent of the GDP of Michigan, \$400 billion/year. The value of a single bitcoin would be this number divided by the number of bitcoin in circulation, 12,000,000, and by the “speed of money,” the number of business transactions per year per coin. I’ll take this to be 3 per year. It turns out there are 5 bitcoin transactions total per year per coin, but 2/5 of that, I’ll assume, are investment transactions. Based on this, a single bitcoin should be worth about \$11,100, exactly its current valuation. The speed number, 3, includes those bitcoins that are held as investments and never traded, and those actively being used in smuggling, drug-deals, etc.

If you assume that the bitcoin trade will grow to \$600 billion year in a year or so, the price rise of a single coin will surpass that of Dutch tulip bulbs on fundamentals alone. If you assume it will reach \$1,600 billion/year, the GDP of Texas in the semi-near future, before the Feds jump in, the value of a coin could grow to \$44,000 or more. There are several problems for bitcoin owners who are betting on this. One is that the Feds are unlikely to tolerate so large an unregulated, illegal economy. Another is that bitcoin are not likely to go legal. It is very hard (near impossible) to connect a bitcoin to its owner. This is great for someone trying to deal in drugs or trying hide profits from the IRS (or his spouse), but a legal merchant will want the protection of courts of law. For this, he or she needs to demonstrate ownership of the item being traded, and that is not available with bitcoin. The lack of good legitimate business suggests to me that the FBI will likely sweep in sooner or later.

Yet another problem is the existence of other cryptocurrencies: Litecoin (LTC), Ethereum (ETH), and Zcash (ZEC) as examples. The existence of these coins increase the divisor I used when calculating bitcoin value above. And even with bitcoins, the total number is not capped at 12,000,000. There are another 12,000,000 coins to be found — or mined, as it were, and these are likely to move faster (assume an average velocity of 4). By my calculations, with 24,000,000 bitcoin and a total use of \$1,600 billion/year, the fundamental value of each coin is only \$16,000. This is not much higher than its current price. Let the buyer beware.

For an amusing, though not helpful read into the price: here are Bill Gates, Warren Buffet, Charlie Munger, and Noam Chomsky discussing Bitcoin.

Robert Buxbaum, December 3, 2017.