# magnetic separation of air

As some of you will know, oxygen is paramagnetic, attracted slightly by a magnet. Oxygen’s paramagnetism is due to the two unpaired electrons in every O2 molecule. Oxygen has a triple-bond structure as discussed here (much of the chemistry you were taught is wrong). Virtually every other common gas is diamagnetic, repelled by a magnet. These include nitrogen, water, CO2, and argon — all diamagnetic. As a result, you can do a reasonable job of extracting oxygen from air by the use of a magnet. This is awfully cool, and could make for a good science fair project, if anyone is of a mind.

But first some math, or physics, if you like. To a good approximation the magnetization of a material, M = CH/T where M is magnetization, H is magnetic field strength, C is the Curie constant for the material, and T is absolute temperature.

Ignoring for now, the difference between entropy and internal energy, but thinking only in terms of work derived by lowering a magnet towards a volume of gas, we can say that the work extracted, and thus the decrease in energy of the magnetic gas is ∫∫HdM  = MH/2. At constant temperature and pressure, we can say ∆G = -CH2/2T.

With a neodymium magnet, you should be able to get about 50 Tesla, or 40,000 ampere meters At 20°C, the per-mol, magnetic susceptibility of oxygen is 1.34×10−6  This suggests that the Curie constant is 1.34 x293 = 3.93 ×10−4  At 20°C, this energy difference is 1072 J/mole. = RT ln ß where ß is the concentration ratio between the O2 content of the magnetized and un-magnetized gas.

From the above, we find that, at room temperature, 298K ß = 1.6, and thus that the maximum oxygen concentration you’re likely to get is about 1.6 x 21% = 33%. It’s slightly more than this due to nitrogen’s diamagnetism, but this effect is too small the matter. What does matter is that 33% O2 is a good amount for a variety of medical uses.

I show below my simple design for a magnetic O2 concentrator. The dotted line is a permeable membrane of no selectivity – with a little O2 permeability the design will work better. All you need is a blower or pump. A coffee filter could serve as a membrane.

This design is as simple as the standard membrane-based O2 concentrator – those based on semi-permeable membranes, but this design should require less pressure differential — just enough to overcome the magnet. Less pressure means the blower should be smaller, and less noisy, with less energy use.  I figure this could be really convenient for people who need portable oxygen. With several stages and low temperature operation, this design could have commercial use.

On the theoretical end, an interesting thing I find concerns the effect on the entropy of the magnetic oxygen. (Please ignore this paragraph if you have not learned statistical thermodynamics.) While you might imagine that magnetization decreases entropy, other-things being equal because the molecules are somewhat aligned with the field, temperature and pressure being fixed, I’ve come to realize that entropy is likely higher. A sea of semi-aligned molecules will have a slightly higher heat capacity than nonaligned molecules because the vibrational Cp is higher, other things being equal. Thus, unless I’m wrong, the temperature of the gas will be slightly lower in the magnetic area than in the non-magnetic field area. Temperature and pressure are not the same within the separator as out, by the way; the blower is something of a compressor, though a much less-energy intense one than used for most air separators. Because of the blower, both the magnetic and the non magnetic air will be slightly warmer than in the surround (blower Work = ∆T/Cp). This heat will be mostly lost when the gas leaves the system, that is when it flows to lower pressure, both gas streams will be, essentially at room temperature. Again, this is not the case with the classic membrane-based oxygen concentrators — there the nitrogen-rich stream is notably warm.

Robert E. Buxbaum, October 11, 2017. I find thermodynamics wonderful, both as science and as an analog for society.

# Activated sludge sewage treatment bioreactors

I ran for water commissioner of Oakland county in 2016, a county with 1.3 million people and eight sewage treatment plants. One of these plants uses the rotating disk contractor, described previously, but the others process sewage by bubbling air through it in a large tank — the so-called, activated sludge process. A description is found here in Wikipedia, but with no math, and thus, far less satisfying than it could be. I thought I might describe this process relevant mathematics, for my understanding and those interested: what happens to your stuff after you flush the toilet or turn on the garbage disposal.

Simplified sewage plant: a bubbling, plug-flow bio-reactor with 90% solids recycle and a settler used to extract floc solids and bio-catalyst material.

In most of the USA, sanitary sewage, the stuff from your toilet, sink, etc. flows separately from storm water to a treatment plant. At the plant, the sewage is first screened (rough filtered) and given a quick settle to remove grit etc. then sent to a bubbling flow, plug-flow bioreactor like the one shown at right. Not all cities use this type of sludge processes, but virtually every plant I’ve seen does, and I’ve come to believe this is the main technology in use today.

The sewage flows by gravity, typically, a choice that provides reliability and saves on operating costs, but necessitates that the sewage plant is located at the lowest point in the town, typically on a river. The liquid effluent of the sewage, after bio-treatment is typically dumped in the river, a flow that is so great more than, during dry season, more than half the flow of several rivers is this liquid effluent of our plants – an interesting factoid. For pollution reasons, it is mandated that the liquid effluent leaves the plant with less than 2 ppm organics; that is, it leaves the plant purer than normal river water. After settling and screening, the incoming flow to the bio-reactor typically contains about 400 ppm of biomaterial (0.04%), half of it soluble, and half as suspended colloidal stuff (turd bits, vegetable matter, toilet paper, etc). Between the activated sludge bio-reactor and the settler following it manage to reduce this concentration to 2 ppm or less. Soluble organics, about 200 ppm, are removed by this cellular oxidation (metabolism), while the colloidal material, the other 200 ppm, is removed by adsorption on the sticky flocular material in the tank (the plug-flow tank is called an oxidation ditch, BTW). The sticky floc is a product of the cells. The rate of oxidation and of absorption processes are proportional to floc concentration, F and to organic concentration, C. Mathematically we can say that

dC/dt = -kFC

where C and F are the concentration of organic material and floc respectively; t is time, and k is a reaction constant. It’s not totally a constant, since it is proportional to oxygen concentration and somewhat temperature dependent, but I’ll consider it constant for now.

As shown in the figure above, the process relies on a high recycle of floc (solids) to increase the concentration of cells, and speed the process. Because of this high recycle, we can consider the floc concentration F to be a constant, independent of position along the reactor length.

The volume of the reactor-ditch, V, is fixed -it’s a concrete ditch — but the flow rate into the ditch, Q, is not fixed. Q is high in the morning when folks take showers, and low at night. It’s also higher — typically about twice as high — during rain storms, the result of leakage and illegal connections. For any flow rate, Q, there is a residence time in the tank, τ where τ = V/Q. We can now solve the above equation assuming an incoming concentration C° = 400 ppm and an outgoing concentration Co of 2 ppm:

ln (C°/Co) = kFτ

Where τ equals the residence time in the tank. Since τ = V/Q,

ln (C°/Co) = kFV/Q.

The required volume of reactor, V, is related to the flow rate, Q, as follows for typical feed and exit concentrations:

V = Q/kF ln( 400/2) = 5.3 Q/kF.

The volume is seen to be dependent on F. In Oakland county, thank volume V is chosen to be one or two times the maximum expected value of Q. To keep the output organic content to less than 2 ppm, F is maintained so that kF≥ 5.3 per day. Thus, in Oakland county, a 2 million gallon per day sewage plant is built with a 2-4 million gallon oxidation ditch. The extra space allows for growth of the populations and for heavy rains, and insures that most of the time, the effluent contains less than 2 ppm organics.

Bob Martin chief engineer the South Lyon, MI, Activated Sludge plant, 2016. His innovation was to control the air bubblers according to measurements of the oxygen content. The O2 sensor is at bottom; the controller is at right. When I was there, some bubblers were acting up.

As you may guess, the activated sludge process requires a lot of operator control, far more than the rotating disk contractor we described. There is a need for constant monitoring and tweaking. The operator deals with some of the variations in Q by adjusting the recycle amount, with other problems by adjusting the air flow, or through the use of retention tanks upstream or downstream of the reactor, or by adding components — sticky polymer, FeCl3, etc. Finally, in have rains, the settler-bottom fraction itself is adjusted (increased). Because of all the complexity. sewer treatment engineer is a high-pay, in demand, skilled trade. If you are interested, contact me or the county. You’ll do yourself and the county a service.

I’d mentioned that the effluent water goes to the rivers in Oakland county. In some counties it goes to the fields, a good idea, I think. As for the solids, in Oakland county, the solid floc is concentrated to a goo containing about 5% solids. (The goo is called unconsolidated sludge) It is shipped free to farmer fields, or sometimes concentrated to more than 5% (consolidated sludge), and provided with additional treatment, anaerobic digestion to improve the quality and extract some energy. If you’d like to start a company to do more with our solids, that would be very welcome. In Detroit the solids are burned, a very wasteful, energy-consuming process, IMHO. In Wisconsin, the consolidated sludge is dried, pelletized, and sold as a popular fertilizer, Milorganite.

Dr. Robert Buxbaum, August 1, 2017. A colleague of mine owned (owns?) a company that consulted on sewage-treatment and manufactured a popular belt-filter. The name of his company: Consolidated Sludge. Here are some sewer jokes and my campaign song.

# Another Quantum Joke, and Schrödinger’s waves derived

Quantum mechanics joke. from xkcd.

Is funny because … it’s is a double entente on the words grain (as in grainy) and waves, as in Schrödinger waves or “amber waves of grain” in the song America (Oh Beautiful). In Schrödinger’s view of the quantum world everything seems to exist or move as a wave until you observe it, and then it always becomes a particle. The math to solve for the energy of things is simple, and thus the equation is useful, but it’s hard to understand why,  e.g. when you solve for the behavior of a particle (atom) in a double slit experiment you have to imagine that the particle behaves as an insubstantial wave traveling though both slits until it’s observed. And only then behaves as a completely solid particle.

Math equations can always be rewritten, though, and science works in the language of math. The different forms appear to have different meaning but they don’t since they have the same practical predictions. Because of this freedom of meaning (and some other things) science is the opposite of religion. Other mathematical formalisms for quantum mechanics may be more comforting, or less, but most avoid the wave-particle duality.

The first formalism was Heisenberg’s uncertainty. At the end of this post, I show that it is identical mathematically to Schrödinger’s wave view. Heisenberg’s version showed up in two quantum jokes that I explained (beat into the ground), one about a lightbulb  and one about Heisenberg in a car (also explains why water is wet or why hydrogen diffuses through metals so quickly).

Yet another quantum formalism involves Feynman’s little diagrams. One assumes that matter follows every possible path (the multiple universe view) and that time should go backwards. As a result, we expect that antimatter apples should fall up. Experiments are underway at CERN to test if they do fall up, and by next year we should finally know if they do. Even if anti-apples don’t fall up, that won’t mean this formalism is wrong, BTW: all identical math forms are identical, and we don’t understand gravity well in any of them.

Yet another identical formalism (my favorite) involves imagining that matter has a real and an imaginary part. In this formalism, the components move independently by diffusion, and as a result look like waves: exp (-it) = cost t + i sin t. You can’t observe the two parts independently though, only the following product of the real and imaginary part: (the real + imaginary part) x (the real – imaginary part). Slightly different math, same results, different ways of thinking of things.

Because of quantum mechanics, hydrogen diffuses very quickly in metals: in some metals quicker than most anything in water. This is the basis of REB Research metal membrane hydrogen purifiers and also causes hydrogen embrittlement (explained, perhaps in some later post). All other elements go through metals much slower than hydrogen allowing us to make hydrogen purifiers that are effectively 100% selective. Our membranes also separate different hydrogen isotopes from each other by quantum effects (big things tunnel slower). Among the uses for our hydrogen filters is for gas chromatography, dynamo cooling, and to reduce the likelihood of nuclear accidents.

Dr. Robert E. Buxbaum, June 18, 2013.

To see Schrödinger’s wave equation derived from Heisenberg for non-changing (time independent) items, go here and note that, for a standing wave there is a vibration in time, though no net change. Start with a version of Heisenberg uncertainty: h =  λp where the uncertainty in length = wavelength = λ and the uncertainty in momentum = momentum = p. The kinetic energy, KE = 1/2 p2/m, and KE+U(x) =E where E is the total energy of the particle or atom, and U(x) is the potential energy, some function of position only. Thus, p = √2m(E-PE). Assume that the particle can be described by a standing wave with a physical description, ψ, and an imaginary vibration you can’t ever see, exp(-iωt). And assume this time and space are completely separable — an OK assumption if you ignore gravity and if your potential fields move slowly relative to the speed of light. Now read the section, follow the derivation, and go through the worked problems. Most useful applications of QM can be derived using this time-independent version of Schrödinger’s wave equation.