Category Archives: Physics

Heat conduction in insulating blankets, aerogels, space shuttle tiles, etc.

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A lot about heat conduction in insulating blankets can be explained by the ordinary motion of gas molecules. That’s because the thermal conductivity of air (or any likely gas) is much lower than that of glass, alumina, or any likely solid material used for the structure of the blanket. At any temperature, the average kinetic energy of an air molecule is 1/2kT in any direction, or 3/2kT altogether; where k is Boltzman’s constant, and T is absolute temperature, °K. Since kinetic energy equals 1/2 mv2, you find that the average velocity in the x direction must be v = √kT/m = √RT/M. Here m is the mass of the gas molecule in kg, M is the molecular weight also in kg (0.029 kg/mol for air), R is the gas constant 8.29J/mol°C, and v is the molecular velocity in the x direction, in meters/sec. From this equation, you will find that v is quite large under normal circumstances, about 290 m/s (650 mph) for air molecules at ordinary temperatures of 22°C or 295 K. That is, air molecules travel in any fixed direction at roughly the speed of sound, Mach 1 (the average speed including all directions is about √3 as fast, or about 1130 mph).

The distance a molecule will go before hitting another one is a function of the cross-sectional areas of the molecules and their densities in space. Dividing the volume of a mol of gas, 0.0224 m3/mol at “normal conditions” by the number of molecules in the mol (6.02 x10^23) gives an effective volume per molecule at this normal condition: .0224 m3/6.0210^23 = 3.72 x10^-26 m3/molecule at normal temperatures and pressures. Dividing this volume by the molecular cross-section area for collisions (about 1.6 x 10^-19 m2 for air based on an effective diameter of 4.5 Angstroms) gives a free-motion distance of about 0.23×10^-6 m or 0.23µ for air molecules at standard conditions. This distance is small, to be sure, but it is 1000 times the molecular diameter, more or less, and as a result air behaves nearly as an “ideal gas”, one composed of point masses under normal conditions (and most conditions you run into). The distance the molecule travels to or from a given surface will be smaller, 1/√3 of this on average, or about 1.35×10^-7m. This distance will be important when we come to estimate heat transfer rates at the end of this post.

 

Molecular motion of an air molecule (oxygen or nitrogen) as part of heat transfer process; this shows how some of the dimensions work.

Molecular motion of an air molecule (oxygen or nitrogen) as part of heat transfer process; this shows how some of the dimensions work.

The number of molecules hitting per square meter per second is most easily calculated from the transfer of momentum. The pressure at the surface equals the rate of change of momentum of the molecules bouncing off. At atmospheric pressure 103,000 Pa = 103,000 Newtons/m2, the number of molecules bouncing off per second is half this pressure divided by the mass of each molecule times the velocity in the surface direction. The contact rate is thus found to be (1/2) x 103,000 Pa x 6.02^23 molecule/mol /(290 m/s. x .029 kg/mol) = 36,900 x 10^23 molecules/m2sec.

The thermal conductivity is merely this number times the heat capacity transfer per molecule times the distance of the transfer. I will now calculate the heat capacity per molecule from statistical mechanics because I’m used to doing things this way; other people might look up the heat capacity per mol and divide by 6.02 x10^23: For any gas, the heat capacity that derives from kinetic energy is k/2 per molecule in each direction, as mentioned above. Combining the three directions, that’s 3k/2. Air molecules look like dumbbells, though, so they have two rotations that contribute another k/2 of heat capacity each, and they have a vibration that contributes k. We begin with an approximate value for k = 2 cal/mol of molecules per °C; it’s actually 1.987 but I round up to include some electronic effects. Based on this, we calculate the heat capacity of air to be 7 cal/mol°C at constant volume or 1.16 x10^-23 cal/molecule°C. The amount of energy that can transfer to the hot (or cold) wall is this heat capacity times the temperature difference that molecules carry between the wall and their first collision with other gases. The temperature difference carried by air molecules at standard conditions is only 1.35 x10-7 times the temperature difference per meter because the molecules only go that far before colliding with another molecule (remember, I said this number would be important). The thermal conductivity for stagnant air per meter is thus calculated by multiplying the number of molecules times that hit per m2 per second, the distance the molecule travels in meters, and the effective heat capacity per molecule. This would be 36,900 x 10^23  molecules/m2sec x 1.35 x10-7m x 1.16 x10^-23 cal/molecule°C = 0.00578 cal/ms°C or .0241 W/m°C. This value is (pretty exactly) the thermal conductivity of dry air that you find by experiment.

I did all that math, though I already knew the thermal conductivity of air from experiment for a few reasons: to show off the sort of stuff you can do with simple statistical mechanics; to build up skills in case I ever need to know the thermal conductivity of deuterium or iodine gas, or mixtures; and finally, to be able to understand the effects of pressure, temperature and (mainly insulator) geometry — something I might need to design a piece of equipment with, for example, lower thermal heat losses. I find, from my calculation that we should not expect much change in thermal conductivity with gas pressure at near normal conditions; to first order, changes in pressure will change the distance the molecule travels to exactly the same extent that it changes the number of molecules that hit the surface per second. At very low pressures or very small distances, lower pressures will translate to lower conductivity, but for normal-ish pressures and geometries, changes in gas pressure should not affect thermal conductivity — and does not.

I’d predict that temperature would have a larger effect on thermal conductivity, but still not an order-of magnitude large effect. Increasing the temperature increases the distance between collisions in proportion to the absolute temperature, but decreases the number of collisions by the square-root of T since the molecules move faster at high temperature. As a result, increasing T has a √T positive effect on thermal conductivity.

Because neither temperature nor pressure has much effect, you might expect that the thermal conductivity of all air-filed insulating blankets at all normal-ish conditions is more-or-less that of standing air (air without circulation). That is what you find, for the most part; the same 0.024 W/m°C thermal conductivity with standing air, with high-tech, NASA fiber blankets on the space shuttle and with the cheapest styrofoam cups. Wool felt has a thermal conductivity of 0.042 W/m°C, about twice that of air, a not-surprising result given that wool felt is about 1/2 wool and 1/2 air.

Now we can start to understand the most recent class of insulating blankets, those with very fine fibers, or thin layers of fiber (or aluminum or gold). When these are separated by less than 0.2µ you finally decrease the thermal conductivity at room temperature below that for air. These layers decrease the distance traveled between gas collisions, but still leave the same number of collisions with the hot or cold wall; as a result, the smaller the gap below .2µ the lower the thermal conductivity. This happens in aerogels and some space blankets that have very small silica fibers, less than .1µ apart (<100 nm). Aerogels can have much lower thermal conductivities than 0.024 W/m°C, even when filled with air at standard conditions.

In outer space you get lower thermal conductivity without high-tech aerogels because the free path is very long. At these pressures virtually every molecule hits a fiber before it hits another molecule; for even a rough blanket with distant fibers, the fibers bleak up the path of the molecules significantly. Thus, the fibers of the space shuttle (about 10 µ apart) provide far lower thermal conductivity in outer space than on earth. You can get the same benefit in the lab if you put a high vacuum of say 10^-7 atm between glass walls that are 9 mm apart. Without the walls, the air molecules could travel 1.3 µ/10^-7 = 13m before colliding with each other. Since the walls of a typical Dewar are about 0.009 m apart (9 mm) the heat conduction of the Dewar is thus 1/150 (0.7%) as high as for a normal air layer 9mm thick; there is no thermal conductivity of Dewar flasks and vacuum bottles as such, since the amount of heat conducted is independent of gap-distance. Pretty spiffy. I use this knowledge to help with the thermal insulation of some of our hydrogen generators and hydrogen purifiers.

There is another effect that I should mention: black body heat transfer. In many cases black body radiation dominates: it is the reason the tiles are white (or black) and not clear; it is the reason Dewar flasks are mirrored (a mirrored surface provides less black body heat transfer). This post is already too long to do black body radiation justice here, but treat it in more detail in another post.

RE. Buxbaum

Heisenberg joke and why water is wet

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I love hydrogen in large part because it is a quantum fluid. To explain what that means and how that leads to water being wet, let me begin with an old quantum physics joke.

Werner Heisenberg is speeding down a highway in his car when he’s stopped by a police officer. “Do you know how fast you were going?” asks the officer. “No idea” answers Heisenberg, “but I know exactly where I am.”

The joke relates to a phenomenon of quantum physics that states that the more precisely you can know the location of something, the less precisely you can infer the speed. Thus, the fact that Heisenberg knew precisely where he was implied that he could have no idea of the car’s speed. Of course, this uncertainty is mostly seen with small things like light and electrons –and a bit with hydrogen, but hardly at all with a car or with Dr. Heisenberg himself (and that’s why it’s funny).

This funky property is related to something you may have wondered about: why is water wet? That is, why does water cling to your hands or clothes while liquid teflon repels. Even further, you may have wondered why water is a liquid at normal conditions when H2S is a gas; H2S is a heavier analog, so if one of the two were a liquid, you’d think it was H2S.

Both phenomena are understood through hydrogen behaving as the quantum car above. Oxygen atoms are pretty small, and hydrogen atoms are light enough to start behaving in a quantum way. When a hydrogen atom attaches to an oxygen atom to form part of a water molecule, its location becomes fixed rather precisely. As a result, the hydrogen atom gains velocity (the hydrogen isn’t going anywhere with this velocity, and it’s sometimes called zero-point energy), but because of this velocity or energy, its bond to the oxygen becomes looser than it would be if you had heavier hydrogen. When the oxygen of another water molecule or of a cotton cellulose molecule comes close, the hydrogen starts to hop back and forth between the two oxygen atoms. This reduces the velocity of the hydrogen atom, and stabilizes the assemblage. There is now less kinetic energy (or zero-point energy) in the system, and this stability is seen as a bond that is caused not by electron sharing but by hydrogen sharing. We call the reasonably stable bond between molecules that share a hydrogen atom this way a “hydrogen bond.” (now you know).

The hydrogen bond is why water is a liquid and is the reason water is wet. The hydrogen atom jumping between water molecules stabilizes the liquid water more than it would stabilize liquid H2S. Since sulfur atoms are bigger than oxygen atoms, the advantage of hydrogen jumping is smaller. As a result, the heat of vaporization of water is higher than that of H2S, and water is a liquid at normal conditions while H2S is a gas.

Water sticks to cotton or your skin the same way, hydrogen atoms skip between the oxygen of water molecules and of these surfaces creating a bond. It is said to whet these surfaces, and the result is that water is found to be wet. Liquid teflon does not have hydrogen atoms that can jump so there is no band that could be made from that direction (there are some hydrogen atoms on the cotton that can jump to the teflon, but there is no advantage to bonding of this sort as there are only a few hydrogen atoms, and these already jump to other oxygens in the cotton. Thus, to jump to the teflon would mean breaking a bond with other oxygen atoms in the cotton — there would be no energy advantage. This then is just one of the reasons I love hydrogen: it’s a quantum-y material.

Why isn’t the sky green?

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Yesterday I blogged with a simple version of why the sky was blue and not green. Now I’d like to add mathematics to the treatment. The simple version said that the sky was blue because the sun color was a spectrum centered on yellow. I said that molecules of air scattered mostly the short wavelength, high frequency light colors, indigo and blue. This made the sky blue. I said that, the rest of the sunlight was not scattered, so that the sun looked yellow. I then said that the only way for the sky to be green would be if the sun were cooler, orange say, then the sky would be green. The answer is sort-of true, but only in a hand-waving way; so here’s the better treatment.

Light scatters off of dispersed small particles in proportion to wavelength to the inverse 4th power of the wavelength. That is to say, we expect air molecules will scatter more short wavelength, cool colors (purple and indigo) than warm colors (red and orange) but a real analysis must use the actual spectrum of sunlight, the light power (mW/m2.nm) at each wavelength.

intensity of sunlight as a function of wavelength (frequency)

intensity of sunlight as a function of wavelength

The first thing you’ll notice is that the light from our sun isn’t quite yellow, but is mostly green. Clearly plants understand this, otherwise chlorophyl would be yellow. There are fairly large components of blue and red too, but my first correction to the previous treatment is that the yellow color we see as the sun is a trick of the eye called additive color. Our eyes combine the green and red of the sun’s light, and sees it as yellow. There are some nice classroom experiment you can do to show this, the simplest being to make a Maxwell top with green and red sections, spin the top, and notice that you see the color as yellow.

In order to add some math to the analysis of sky color, I show a table below where I divided the solar spectrum into the 7 representative colors with their effective power. There is some subjectivity to this, but I took red as the wavelengths from 620 to 750nm so I claim on the table was 680 nm. The average power of the red was 500 mW/m2nm, so I calculate the power as .5 W/m2nm x 130 nm = 65W/m2. Similarly, I took orange to be the 30W/m2 centered on 640nm, etc. This division is presented in the first 3 columns of the following table. The first line of the table is an approximate of the Rayleigh-scatter factor for our atmosphere, with scatter presented as the percent of the incident light. That is % scattered = 9E11/wavelength^4.skyblue scatter

To use the Rayleigh factor, I calculate the 1/wavelength of each color to the 4th power; this is shown in the 4th column. The scatter % is now calculated and I apply this percent to the light intensities to calculate the amount of each color that I’d expect in the scattered and un-scattered light (the last two columns). Based on this, I find that the predominant wavelength in the color of the sky should be blue-cyan with significant components of green, indigo, and violet. When viewed through a spectroscope, I find that these are the colors I see (I have a pocket spectroscope and used it an hour ago to check). Viewed through the same spectroscope (with eye protection), I expect the sun should look like a combination of green and red, something our eyes see as yellow (I have not done this personally). At any rate, it appears that the sky looks blue because our eyes see the green+ cyan+ indigo + purple in the scattered light as sky blue.220px-RGB_illumination

At sunrise and sunset when the sun is on the horizon the scatter percents will be higher, so that all of the sun’s colors will be scattered except red and orange. The sun looks orange then, as expected, but the sky should look blue-green, as that’s the combination of all the other colors of sunlight when orange and red are removed. I’ve not checked this last yet. I’ll have to take my spectroscope to a fine sunset and see what I see when I look at the sky.

Why isn’t the sky green and the sun orange?

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Part of the reason the sky isn’t green has to do with the color of the sun. The sun’s color, and to a lesser extent, the sky color both are determined by the sun’s surface color, yellow. This surface color results from black body radiation: if you heat up a black object it will first glow red, then orange, yellow, green etc. Red is a relatively cool color because it’s a low frequency (long wavelength) and low frequencies are the lowest energy photons, and thus are the easiest for a black body to produce. As one increases the temperature of a black object, the total number of photons increases for all wavelengths, but the short wavelength (high frequency) colors increase faster than the of long wavelength colors. As a result, the object is seen to change color to orange, then yellow, or to any other color representative of objects at that particular temperature.

Our star is called a yellow sun because the center color of its radiation is yellow. The sun provides radiation in all colors and wavelengths, even colors invisible to the eye, infra red and ultra violet, but because of its temperature, most of the radiated energy appears as yellow. This being said, you may wonder why the sky isn’t yellow (the sky of Mars mostly is).

The reason the sky is blue, is that some small fraction of the light of the sun (about 10%) scatters off of the molecules of the air. This is called Rayleigh scatter — the scatter of large wavelegth waves off of small objects.  The math for this will be discussed in another post, but the most relevant aspect here is that the fraction that is scattered is proportional to the 4th power of the frequency. This is to say, that the high frequencies (blue, indigo, and violet) scatter a lot, about 20%, while the red hardly scatters at all. As a result the sky has a higher frequency color than the sun does. In our case, the sky looks blue, while the sun looks slightly redder from earth than it does from space — at least that’s the case for most of the day.

The sun looks orange-red at sundown because the sunlight has to go through more air. Because of this, a lot more of the yellow, green, and blue scatter away before we see it. Much more of the scatter goes off into space, with the result that the sky to looks dark, and somewhat more greenish at sundown. If the molecules were somewhat bigger, we’d still see a blue sky, maybe somewhat greener, with a lot more intensity. That’s the effect that carbon dioxide has — it causes more sunlight to scatter, making the sky brighter. If the sun were cooler (orange say), the sky would appear green. That’s because there would be less violet and blue in the sunlight, and the sky color would be shifted to the longer wavelengths. On planets where the sun is cooler than ours, the sky is likely green, but could be yellow or red.

Rayleigh scatter requires objects that are much smaller than the light wavelength. A typical molecule of air is about 1 nm in size (1E-9 of a meter), while the wavelength of yellow light is 580 nm. That’s much larger than the size of air molecules. Snow appears white because the size of the crystals are the size of the sun wavelengths, tor bigger, 500-2000 nm. Thus, the snow looks like all the colors of the sun together, and that’s white. White = the sum of all the colors: red + orange + blue + green + yellow + violet + indigo.

Robert Buxbaum  Jan. 27, 2013 (revised)

What causes the swirl of tornadoes and hurricanes

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Some weeks ago, I presented an explanation of why tornadoes and hurricanes pick up stuff based on an essay by A. Einstein that explained the phenomenon in terms of swirling fluids and Coriolis flows. I put in my own description that I thought was clearer since it avoided the word “Coriolis”, and attached a video so you could see how it all worked — or rather that is was as simple as all that. (Science teachers: I’ve found kids love it when I do this, and similar experiments with centrifugal force in the class-room as part of a weather demonstration).

I’d like to now answer a related question that I sometimes get: where does the swirl come from? hurricanes that answer follows, though I think you’ll find my it is worded differently from that in Wikipedia and kids’ science books since (as before) I don’t use the word Coriolis, nor any other concept beyond conservation of angular momentum plus that air flows from high pressure to low.

In Wikipedia and all the other web-sits I visited, it was claimed that the swirl came from “Coriolis force.” While this isn’t quite wrong, I find this explanation incomprehensible and useless. Virtually no-one has a good feel for Coriolis force as such, and those who do recognize that it doesn’t exist independently like gravity. So here is my explanation based on low and high pressure and on conservation of angular momentum.  I hope it will be clearer.

All hurricanes are associated with low pressure zones. This is not a coincidence as I understand it, but a cause-and-effect relationship. The low pressure center is what causes the hurricane to form and grow. It may also cause tornadoes but the relationship seems less clear. In the northern hemisphere, the lowest low pressure zones are found to form over the mid Atlantic or Pacific in the fall because the water there is warm and that makes the air wet and hot. Static air pressure is merely the weight of the air over a certain space, and as hot air has more volume and less density, it weighs less. Less weight = less pressure, all else being equal. Adding water (humidity) to air also reduces the air pressure as the density of water vapor is less than that of dry air in proportion to their molecular weights. The average molecular weight of dry air is 29 and the molecular weight of water is 18. As a result, every 9% increase in water content decreases the air pressure by 1% (7.6 mm or 0.3″ of mercury).

Air tends to flow from high pressure zones to low pressure zones. In the northern hemisphere, some of the highest high pressure zones form over northern Canada and Russia in the winter. High pressure zones form there by the late fall because these regions are cold and dry. Cold air is less voluminous than hot, and as a result additional hot air flows into these zones at high altitude. At sea level the air flows out from the high pressure zones to the low pressure zones and begins to swirl because of conservation of angular momentum.

All the air in the world is spinning with the earth. At the north pole the spin rate is 360 degrees every 24 hours, or 15 degrees per hour. The spin rate is slower further south, proportionally to the sine of the latitude, and it is zero at the equator. The spin of the earth at your location is observable with a Foucault pendulum (there is likely to be one found in your science museum). We normally don’t notice the spin of the air around us because the earth is spinning at the same rate, normally. However the air has angular momentum, and when air moves into into a central location the angular speed increases because the angular momentum must be conserved. As the gas moves in, the spin rate must increase in proportion; it eventually becomes noticeable relative to the earth’s spin. Thus, if the air starts out moving at 10 degrees per hour (that’s the spin rate in Detroit, MI 41.8° N), and moves from 800 miles away from a low pressure center to only 200 miles from the center, the angular momentum must increase four times, or to 40 degrees per hour. We would only see 30 degrees/hr of this because the earth is spinning, but the velocity this involves is significant: V= 200 miles * 2* pi *30/360 = 104 mph.

To give students a sense of angular momentum conservation, most science centers (and colleges) use an experiment involving bicycle wheels and a swivel chair. In the science centers there is usually no explanation of why, but in college they tend to explain it in terms of vectors and (perhaps) gauge theories of space-time (a gauge is basically a symmetry; angular momentum is conserved because space is symmetric in rotation). In a hurricane, the air at sea level always spins in the same direction of the earth: counter clockwise in the northern hemisphere, clockwise in the southern, but it does not spin this way forever.

The air that’s sucked into the hurricane become heated and saturated with water. As a result, it becomes less dense, expands, and rises, sucking fresh air in behind it. As the hot wet air rises it cools and much of the water rains down as rain. When the, now dry air reaches a high enough altitude its air pressure is higher than that above the cold regions of the north; the air now flows away north. Because this hot wet air travels north we typically get rain in Michigan when the Carolinas are just being hit by hurricanes. As the air flows away from the centers at high altitudes it begins to spin the opposite direction, by the way, so called counter-cyclonally because angular momentum has to be consevered. At high altitudes over high pressure centers I would expect to find cyclones too (spinning cyclonally) I have not found a reference for them, but suspect that airline pilots are aware of the effect. There is some of this spin at low altitudes, but less so most of the time.

Hurricanes tend to move to the US and north through the hurricane season because, as I understand it, the cold air that keeps coming to feed the hurricane comes mostly from the coastal US. As I understand it the hurricane is not moving as such, the air stays relatively stationary and the swirl that we call a hurricane moves to the US in the effective direction of the sea-level air flow.

For tornadoes, I’m sorry to say, this explanation does not work quite as well, and Wikipedia didn’t help clear things up for me either. The force of tornadoes is much stronger than of hurricanes (the swirl is more concentrated) and the spin direction is not always cyclonic. Also tornadoes form in some surprising areas like Kansas and Michigan where hurricanes never form. My suspicion is that most, but not all tornadoes form from the same low pressure as hurricanes, but by dry heat, not wet. Tornadoes form in Michigan, Texas, and Alabama in the early summer when the ground is dry and warmer than the surrounding lakes and seas. It is not difficult to imagine the air rising from the hot ground and that a cool wind would come in from the water and beginning to swirl. The cold, damp sea air would be more dense than the hot, dry land air, and the dry air would rise. I can imagine that some of these tornadoes would occur with rain, but that many the more intense?) would have little or none; perhaps rain-fall tends to dampen the intensity of the swirl (?)

Now we get to things that I don’t have good explanation for at all: why Kansas? Kansas isn’t particularly hot or cold; it isn’t located near lakes or seas, so why do they have so many tornadoes? I don’t know. Another issue that I don’t understand: why is it that some tornadoes rotate counter cyclonicly? Wikipedia says these tornadoes shed from other tornadoes, but this doesn’t quite seem like an explanation. My guess is that these tornadoes are caused by a relative high pressure source at ground level (a region of cold ground for example) coupled with a nearby low pressure zone (a warm lake?). My guess is that this produces an intense counter-cyclonic flow to the low pressure zone. As for why the pressure is very low in tornadoes, even these that I think are caused by high pressure, I suspect the intense low pressure is an epee-phenomenon caused by the concentration of spin — one I show in my video. That is, I suspect that the low pressure in the center of counter-cyclonic tornadoes is not the cause of the tornado but an artifact of the concentrated spin. Perhaps I’m wrong here, but that’s the explanation that seems to fit best with the info I’ve got. If you’ve got better explanations for these two issues, I’d love to hear them.

Engineering joke

An optimist says the cup is half full.

A pessimist says the cup is half empty.

An engineer says the cup is twice as big as it has to be.

(A quantum physicist might say that the water isn’t in the cup till he looks at it; then again, the quantum physicist isn’t there until someone looks at him. And that’s why I’m an engineer).

How hydrogen and/or water can improve automobile mileage (mpg)

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In case you’ve ever wondered why it was that performance cars got such poor milage, or why you got such bad milage in the city, the biggest single problem has to do with the vacuum drawn by the engine, some of the problem has to do with the speed of combustion, some has to do with rolling friction, and some with start/stop loss too. Only a very small fraction of the energy is lost on air friction until you reach highway speeds.

Lets consider vacuum loss first as it is likely the worst offender. A typical US car, e.g. a Chevy Malibu, has a 3.5 liter engine (a performance car has an engine that’s much larger). As you toodle down a street at 35 mph, your engine is going at about 2000 rpm, or 33 rps. Since the power required to move the car is far less than the 200 hp that the car could deliver, the air intake is throttled so that the engine is sucking a vacuum of about 75 kpa (10 psi for those using English units). To calculate the power loss this entails, multiply 33*3.5*80; this is about 8662 Watts, or 12 hp. To find the energy use per mile, divide by your average speed, 25 mph (it would be 35 mph, but you sometimes stop for lights). 8 kW/25 mph = .21 kW-hr/mile. One finds, as I’ll show that the car expends more energy sucking this vacuum than pushing the car itself. This is where the majority of the city mpg goes in a normal car, but it’s worse in a high performance car is worse since the engine is bigger. In city driving, the performance mpg will be lower than for a Malibu even if the performance car is lighter, if it has better aerodynamics (it does), and if you never stop at lights.

The two other big places were city mileage goes is overcoming rolling friction and the need to stop and go at lights, stop signs, etc. The energy used for rolling friction is the force it would take to push your car on level ground when in neutral times the distance. For a typical car, the push force is about 70 lbs or 32 kgs or 315 Nt; it’s roughly proportional to the car’s weight. At 35 mph, or 15.5 m/s, the amount of power this absorbs is calculated as the product of force and speed: 15.5*315 = 4882 W, or about 6.5 hp. The energy use is 4.9 kW/35 mph =.14 kWhr/mile. The energy loss from stop lights is similar to this, about .16 kWhr/mile, something you can tell by getting the car up to speed and seeing how far it goes before it stops. It’ll go about 2-3 blocks, a little less distance than you are likely to go without having to stop. Air resistance adds a very small amount at these speeds, about 2000 W, 2.7 hp, or .05 kWhr/mile; it’s far more relevant at 65 mph, but still isn’t that large.

If you add all this together, you find the average car uses about .56 kWhr/mile. For an average density gasoline of 5.6 lb/gal, and average energy-dense gasoline, 18,000 BTU/lb, and an average car engine efficiency of 11000 BTU/kWhr, you can now predict an average city gas mileage of 16.9 mpg, about what you find experimentally. Applying the same methods to highway traffic at 65 mph, you predict .38 kWhr/mile, or 25 mpg. Your rpms are the same on the highway as in the city, but the throttle is open so you get more power and loose less to vacuum.

Now, how do you increase a car’s mpg. If you’re a Detroit automaker you could reduce the weight of the car, or you the customer can clean the junk out of your trunk. Every 35 lbs or so increases the rolling friction by about 1%. These is another way to reduce rolling friction and that’s to get low resistance tires, or keep the tires you’ve got full of air. Still, what you’d really like to do is reduce the loss to vacuum energy, since vacuum loss is a bigger drain on mpg.

The first, simple way to reduce vacuum energy loss is to run lean: that is, to add more air than necessary for combustion. Sorry to say, that’s illegal now, but in the olden days before pollution control you could boost your mpg by adjusting your carburator to add about 10% excess of air. This reduced your passing power and the air pollution folks made it illegal (and difficult) after they noticed that it excess air increased NOx emissions. The oxygen sensor on most cars keeps you from playing with the carburator these days.

Another approach is to use a much smaller engine. The Japanese and Koreans used to do this, and they got good milage as a result. The problem here is that you now had to have a very light car or you’d get very low power and low acceleration — and no American likes that. A recent approach to make up for some of the loss of acceleration is by adding a battery and an electric motor; you’re now making a hybrid car. But the batteries add significant cost, weight and complexity to these cars, and not everyone feels this is worth it. So now on to my main topic: adding steam or hydrogen.

There is plenty of excess heat on the car manifold. A simile use of this heat is to warm some water to the point where the vapor pressure is, for example, 50 kPa. The pressure from this water adds to the power of your engine by allowing a reduction in the vacuum to 50 kPa or less. This cuts the vacuum loss at low speeds. At high speed and power the car automatically increases the air pressure and the water stops evaporating, so there is no loss of power. I’m currently testing this modification on my own automobile partly for the fun of it, and partly as a preface to my next step: using the car engine heat to run the reaction CH3OH + H2O –> CO2 + H2. I’ll talk more about our efforts adding hydrogen elsewhere, but thought you might be interested in these fundamentals.

http://www.rebresearch.com

How and why membrane reactors work

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What follows is a shorter, easier version of my old essay on how membrane reactors work to get you past the normal limits of thermodynamics. Also, I’m happy to say, our membrane reactors designs have gotten simpler, and that deserves an update.

At left, is our current, high pressure membrane reactor design, available in one-tube to  72 tube reactor assembly; high pressure, or larger, I suppose. Typically, the area around the shell is used for heat transfer. One needs to add heat to promote endothermic reactions like methanol reforming CH3OH + H2O –> 3H2 + CO2, or ammonia cracking 2NH3 –> N2 + 3H2. You need to take away heat from exothermic reactions, like the water gas shift reaction, CO + H2O –> CO2 + H2. Generally you want to have some heat transfer to help drive the reaction.

The reactor is a shell containing metallic tubes that filter gas. Normally the idea is for hydrogen to be formed in the shell area, and leave by diffusion through the tube walls and down the tubes, leaving as pure hydrogen (only hydrogen can go through metals). We typically suggest to have the reactor sit this way, with the tubes pointed up, and the body half-filled or more with catalyst. According to normal thermodynamics, the extent of a reaction like ammonia cracking will be negatively affected by overall pressure, and the extent of the WGS reaction is only affected by operating temperature. The membrane reactor allows you to remove hydrogen while the reaction progresses, and allows you to get good conversion at higher pressures too. That’s because hydrogen removal shifts the equilibrium so the reaction goes further. The effect is particularly significant at higher pressures. By combining the steps of reaction with separation, we can operate a higher pressures, delivering ultra high purity, avoiding the normal limitations of thermodynamics.

The water-gas reaction, CO + H2O –> CO2 + H2, deserves special mention since it’s common and exothermic. In a normal reactor, your only option to drive the reaction to near completion is by operating at low temperature where the catalyst barely works, 200 °C, or lower. You also have to remove the heat of reaction. In a membrane reactor, you can operate at a much higher temperature, especially if you work at higher pressure. At higher temperature the catalysts work better, and you don’t have to work so hard to remove heat.

At our company, REB Research, we sell membrane reactors; and catalysts, membrane tubes, and consulting.

True (magnetic) north

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Much of my wife’s family is Canadian, so I keep an uncommon interest in Canada — for an American. This is to say, I think about it once a month or so, more often during hockey season. So here is a semi-interesting factoid:

The magnetic north pole, the “true north” has been moving northwest for some time, but the rate has increased over the last few decades as the picture shows. It has now left the northern Canadian islands, so Canada is no longer “The true north, strong and free.” (It seems to be strong and free). True north  is now moving northwest, toward Siberia. true magnetic north heading to Russia