Category Archives: Energy savings

Paint your factory roof white

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Standing on the flat roof of my lab / factory building, I notice that virtually all of my neighbors’ roofs are black, covered by tar or bitumen. My roof was black too until three weeks ago; the roof was too hot to touch when I’d gone up to patch a leak. That’s not quite egg-frying hot, but I came to believe my repair would last longer if the roof stayed cooler. So, after sealing the leak with tar and bitumen, we added an aluminized over-layer from Ace hardware. The roof is cooler now than before, and I notice a major drop in air conditioner load and use.

My analysis of our roof coating follows; it’s for Detroit, but you can modify it for your location. Sunlight hits the earth carrying 1300 W/m2. Some 300W/m2 scatters as blue light (for why so much scatters, and why the sky is blue, see here). The rest, 1000 W/m2 or 308 Btu/ft2hr, comes through or reflects off clouds on a cloudy day and hits buildings at an angle determined by latitude, time of day, and season of the year.

Detroit is at 42° North latitude so my roof shows an angle of 42° to the sun at noon in mid spring. In summer, the angle is 20°, and in winter about 63°. The sun sinks lower on the horizon through the day, e.g. at two hours before or after noon in mid spring the angle is 51°. On a clear day, with a perfectly black roof, the heating is 308 Btu/ft2hr times the cosine of the angle.

To calculate our average roof heating, I integrated this heat over the full day’s angles using Euler’s method, and included the scatter from clouds plus an absorption factor for the blackness of the roof. The figure below shows the cloud cover for Detroit.

Average cloud cover for Detroit, month by month.

Average cloud cover for Detroit, month by month; the black line is the median cloud cover. On January 1, it is strongly overcast 60% of the time, and hardly ever clear; the median is about 98%. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Based on this and an assumed light absorption factor of σ = .9 for tar and σ = .2 after aluminum. I calculate an average of 105 Btu/ft2hr heating during the summer for the original black roof, and 23 Btu/ft2hr after aluminizing. Our roof is still warm, but it’s no longer hot. While most of the absorbed heat leaves the roof by black body radiation or convection, enough enters my lab through 6″ of insulation to cause me to use a lot of air conditioning. I calculate the heat entering this way from the roof temperature. In the summer, an aluminum coat is a clear winner.

Detroit High and Low Temperatures Over the ear

High and Low Temperatures For Detroit, Month by Month. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Detroit has a cold winter too, and these are months where I’d benefit from solar heat. I find it’s so cloudy in winter that, even with a black roof, I got less than 5 Btu/ft2hr. Aluminizing reduced this heat to 1.2 Btu/ft2hr, but it also reduces the black-body radiation leaving at night. I should find that I use less heat in winter, but perhaps more in late spring and early fall. I won’t know the details till next year, but that’s the calculation.

The REB Research laboratory is located at 12851 Capital St., Oak Park, MI 48237. We specialize in hydrogen separations and membrane reactors. By Dr. Robert Buxbaum, June 16, 2013

What’s the quality of your home insulation

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By Dr. Robert E. Buxbaum, June 3, 2013

It’s common to have companies call during dinner offering to blow extra insulation into the walls and attic of your home. Those who’ve added this insulation find a small decrease in their heating and cooling bills, but generally wonder if they got their money’s worth, or perhaps if they need yet-more insulation to get the full benefit. Here’s a simple approach to comparing your home heat bill to the ideal your home can reasonably reach.

The rate of heat transfer through a wall, Qw, is proportional to the temperature difference, ∆T, to the area, A, and to the average thermal conductivity of the wall, k; it is inversely proportional to the wall thickness, ∂;

Qw = ∆T A k /∂.

For home insulation, we re-write this as Qw = ∆T A/Rw where Rw is the thermal resistance of the wall, measured (in the US) as °F/BTU/hr-ft2. Rw = ∂/k.

Lets assume that your home’s outer wall thickness is nominally 6″ thick (0.5 foot). With the best available insulation, perfectly applied, the heat loss will be somewhat higher than if the space was filled with still air, k=.024 BTU/ft. hr°F, a result based on molecular dynamics. For a 6″ wall, the R value, will always be less than .5/.024 = 20.8 °F/BTU/hr-ft2.. It will be much less if there are holes or air infiltration, but for practical construction with joists and sills, an Rw value of 15 or 16 is probably about as good as you’ll get with 6″ walls.

To show you how to evaluate your home, I’ll now calculate the R value of my walls based on the size of my ranch-style home (in Michigan) and our heat bills. I’ll first do this in a simplified calculation, ignoring windows, and will then repeat the calculation including the windows. Windows are found to be very important. I strongly suggest window curtains to save heat and air conditioning,

The outer wall of my home is 190 feet long, and extends about 11 feet above ground to the roof. Multiplying these dimensions gives an outer wall area of 2090 ft2. I could now add the roof area, 1750 ft2 (it’s the same as the area of the house), but since the roof is more heavily insulated than the walls, I’ll estimate that it behaves like 1410 ft2 of normal wall. I calculate there are 3500 ft2 of effective above-ground area for heat loss. This is the area that companies keep offering to insulate.

Between December 2011 and February 2012, our home was about 72°F inside, and the outside temperature was about 28°F. Thus, the average temperature difference between the inside and outside was about 45°F; I estimate the rate of heat loss from the above-ground part of my house, Qu = 3500 * 45/R = 157,500/Rw.

Our house has a basement too, something that no one has yet offered to insulate. While the below-ground temperature gradient is smaller, it’s less-well insulated. Our basement walls are cinderblock covered with 2″ of styrofoam plus wall-board. Our basement floor is even less well insulated: it’s just cement poured on pea-gravel. I estimate the below-ground R value is no more than 1/2 of whatever the above ground value is; thus, for calculating QB, I’ll assume a resistance of Rw/2.

The below-ground area equals the square footage of our house, 1750 ft2 but the walls extend down only about 5 feet below ground. The basement walls are thus 950 ft2 in area (5 x 190 = 950). Adding the 1750 ft2 floor area, we find a total below-ground area of 2700 ft2.

The temperature difference between the basement and the wet dirt is only about 25°F in the winter. Assuming the thermal resistance is Rw/2, I estimate the rate of heat loss from the basement, QB = 2700*25*(2/Rw) = 135,000/Rw. It appears that nearly as much heat leaves through the basement as above ground!

Between December and February 2012, our home used an average of 597 cubic feet of gas per day or 25497 BTU/hour (heat value = 1025 BTU/ ft3). QU+ QB = 292,500/Rw. Ignoring windows, I estimate Rw of my home = 292,500/25497 = 11.47.

We now add the windows. Our house has 230 ft2 of windows, most covered by curtains and/or plastic. Because of the curtains and plastic, they would have an R value of 3 except that black-body radiation tends to be very significant. I estimate our windows have an R value of 1.5; the heat loss through the windows is thus QW= 230*45/1.5 = 6900 BTU/hr, about 27% of the total. The R value for our walls is now re-estimated to be 292,500/(25497-6900) = 15.7; this is about as good as I can expect given the fixed thickness of our walls and the fact that I can not easily get an insulation conductivity lower than still air. I thus find that there will be little or no benefit to adding more above-ground wall insulation to my house.

To save heat energy, I might want to coat our windows in partially reflective plastic or draw the curtains to follow the sun. Also, since nearly half the heat left from the basement, I may want to lay a thicker carpet, or lay a reflective under-layer (a space blanket) beneath the carpet.

To improve on the above estimate, I could consider our furnace efficiency; it is perhaps only 85-90% efficient, with still-warm air leaving up the chimney. There is also some heat lost through the door being opened, and through hot water being poured down the drain. As a first guess, these heat losses are balanced by the heat added by electric usage, by the body-heat of people in the house, and by solar radiation that entered through the windows (not much for Michigan in winter). I still see no reason to add more above-ground insulation. Now that I’ve analyzed my home, it’s time for you to analyze yours.

My steam-operated, high pressure pump

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Here’s a miniature version of a duplex pump that we made 2-3 years ago at REB Research as a way to pump fuel into hydrogen generators for use with fuel cells. The design is from the 1800s. It was used on tank locomotives and steamboats to pump water into the boiler using only the pressure in the boiler itself. This seems like magic, but isn’t. There is no rotation, but linear motion in a steam piston of larger diameter pushes a liquid pump piston with a smaller diameter. Each piston travels the same distance, but there is more volume in the steam cylinder. The work from the steam piston is greater: W = ∫PdV; energy is conserved, and the liquid is pumped to higher pressure than the driving steam (neat!).

The following is a still photo. Click on the YouTube link to see the steam pump in action. It has over 4000 views!

Mini duplex pump. Provides high pressure water from steam power. Amini version of a classic of the 1800s Coffee cup and pen shown for scale.

Mini duplex pump. Provides high pressure water from steam power. A mini version of a classic of the 1800s Coffee cup and pen shown for scale.

You can get the bronze casting and the plans for this pump from Stanley co (England). Any talented machinist should be able to do the rest. I hired an Amish craftsman in Ohio. Maurice Perlman did the final fit work in our shop.

Our standard line of hydrogen generators still use electricity to pump the methanol-water. Even our latest generators are meant for nom-mobile applications where electricity is awfully convenient and cheap. This pump was intended for a future customer who would need to generate hydrogen to make electricity for remote and mobile applications. Even our non-mobile hydrogen is a better way to power cars than batteries, but making it mobile has advantages. Another advance would be to heat the reactors by burning the waste gas (I’ve been working on that too, and have filed a patent). Sometimes you have to build things ahead of finding a customer — and this pump was awfully cool.

Camless valves and the Fiat-500

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One of my favorite automobile engine ideas is the use of camless, electronic valves. It’s an idea whose advantages have been known for 100 years or more, and it’s finally going to be used on a mainstream, commercial car — on this year’s Fiat 500s. Fiat is not going entirely camless, but the plan is to replace the cams on the air intake valves with solenoids. A normal car engine uses cams and lifters to operate the poppet valves used to control the air intake and exhaust. Replacing these cams and lifters saves some weight, and allows the Fiat-500 to operate more efficiently at low power by allowing the engine to use less combustion energy to suck vacuum. The Fiat 500 semi-camless technology is called Multiair: it’s licensed from Valeo (France), and appeared as an option on the 2010 Alfa Romeo.

How this saves mpg is as follows: at low power (idling etc.), the air intake of a normal car engine is restricted creating a fairly high vacuum. The vacuum restriction requires energy to draw and reduces the efficiency of the engine by decreasing the effective compression ratio. It’s needed to insure that the car does not produce too much NOx when idling. In a previous post, I showed that the rate of energy wasted by drawing this vacuum was the vacuum pressure times the engine volume and the rpm rate; I also mentioned some classic ways to reduce this loss (exhaust recycle and adding water).

Valeo’s/Fiat’s semi-camless design does nothing to increase the effective compression ratio at low power, but it reduces the amount of power lost to vacuum by allowing the intake air pressure to be higher, even at low power demand. A computer reduces the amount of air entering the engine by reducing the amount of time that the intake valve is open. The higher air pressure means there is less vacuum penalty, both when the valve is open even when the valve is closed. On the Alfa Romeo, the 1.4 liter Multiair engine option got 8% better gas mileage (39 mpg vs 36 mpg) and 10% more power (168 hp vs 153 hp) than the 1.4 liter cam-driven engine.

David Bowes shows off his latest camless engines at NAMES, April 2013.

David Bowes shows off his latest camless engines at NAMES, April 2013.

Fiat used a similar technology in the 1970s with variable valve timing (VVT), but that involved heavy cams and levers, and proved to be unreliable. In the US, some fine engineers had been working on solenoids, e.g. David Bowes, pictured above with one of his solenoidal engines (he’s a sometime manufacturer for REB Research). Dave has built engines with many cycles that would be impractical without solenoids, and has done particularly nice work reducing the electric use of the solenoid.

Durability may be a problem here too, as there is no other obvious reason that Fiat has not gone completely camless, and has not put a solenoid-controlled valve on the exhaust too. One likely reason Fiat didn’t do this is that solenoidal valves tend to be unreliable at the higher temperatures found in exhaust. If so, perhaps they are unreliable on the intake too. A car operated at 1000-4000 rpm will see on the order of 100,000,000 cycles in 25,000 miles. No solenoid we’ve used has lasted that many cycles, even at low temperatures, but most customers expect their cars to go more than 25,000 miles without needing major engine service.

We use solenoidal pumps in our hydrogen generators too, but increase the operating live by operating the solenoid at only 50 cycles/minute — maximum, rather than 1000- 4000. This should allow our products to work for 10 years at least without needing major service. Performance car customers may be willing to stand for more-frequent service, but the company can’t expect ordinary customers to go back to the days where Fiat stood for “Fix It Again Tony.”

Most Heat Loss Is Black-Body Radiation

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In a previous post I used statistical mechanics to show how you’d calculate the thermal conductivity of any gas and showed why the insulating power of the best normal insulating materials was usually identical to ambient air. That analysis only considered the motion of molecules and not of photons (black-body radiation) and thus under-predicted heat transfer in most circumstances. Though black body radiation is often ignored in chemical engineering calculations, it is often the major heat transfer mechanism, even at modest temperatures.

One can show from quantum mechanics that the radiative heat transfer between two surfaces of temperature T and To is proportional to the difference of the fourth power of the two temperatures in absolute (Kelvin) scale.

Heat transfer rate = P = A ε σ( T^4 – To^4).

Here, A is the area of the surfaces, σ is the Stefan–Boltzmann constant, ε is the surface emissivity, a number that is 1 for most non-metals and .3 for stainless steel.  For A measured in m2, σ = 5.67×10−8 W m−2 K−4.

Infrared picture of a fellow wearing a black plastic bag on his arm. The bag is nearly transparent to heat radiation, while his eyeglasses are opaque. His hair provides some insulation.

Unlike with conduction, heat transfer does not depend on the distances between the surfaces but only on the temperature and the infra-red (IR) reflectivity. This is different from normal reflectivity as seen in the below infra-red photo of a lightly dressed person standing in a normal room. The fellow has a black plastic bag on his arm, but you can hardly see it here, as it hardly affects heat loss. His clothes, don’t do much either, but his hair and eyeglasses are reasonably effective blocks to radiative heat loss.

As an illustrative example, lets calculate the radiative and conductive heat transfer heat transfer rates of the person in the picture, assuming he has  2 m2 of surface area, an emissivity of 1, and a body and clothes temperature of about 86°F; that is, his skin/clothes temperature is 30°C or 303K in absolute. If this person stands in a room at 71.6°F, 295K, the radiative heat loss is calculated from the equation above: 2 *1* 5.67×10−8 * (8.43×109 -7.57×109) = 97.5 W. This is 23.36 cal/second or 84.1 Cal/hr or 2020 Cal/day; this is nearly the expected basal calorie use of a person this size.

The conductive heat loss is typically much smaller. As discussed previously in my analysis of curtains, the rate is inversely proportional to the heat transfer distance and proportional to the temperature difference. For the fellow in the picture, assuming he’s standing in relatively stagnant air, the heat boundary layer thickness will be about 2 cm (0.02m). Multiplying the thermal conductivity of air, 0.024 W/mK, by the surface area and the temperature difference and dividing by the boundary layer thickness, we find a Wattage of heat loss of 2*.024*(30-22)/.02 = 19.2 W. This is 16.56 Cal/hr, or 397 Cal/day: about 20% of the radiative heat loss, suggesting that some 5/6 of a sedentary person’s heat transfer may be from black body radiation.

We can expect that black-body radiation dominates conduction when looking at heat-shedding losses from hot chemical equipment because this equipment is typically much warmer than a human body. We’ve found, with our hydrogen purifiers for example, that it is critically important to choose a thermal insulation that is opaque or reflective to black body radiation. We use an infra-red opaque ceramic wrapped with aluminum foil to provide more insulation to a hot pipe than many inches of ceramic could. Aluminum has a far lower emissivity than the nonreflective surfaces of ceramic, and gold has an even lower emissivity at most temperatures.

Many popular insulation materials are not black-body opaque, and most hot surfaces are not reflectively coated. Because of this, you can find that the heat loss rate goes up as you add too much insulation. After a point, the extra insulation increases the surface area for radiation while barely reducing the surface temperature; it starts to act like a heat fin. While the space-shuttle tiles are fairly mediocre in terms of conduction, they are excellent in terms of black-body radiation.

There are applications where you want to increase heat transfer without having to resort to direct contact with corrosive chemicals or heat-transfer fluids. Often black body radiation can be used. As an example, heat transfers quite well from a cartridge heater or band heater to a piece of equipment even if they do not fit particularly tightly, especially if the outer surfaces are coated with black oxide. Black body radiation works well with stainless steel and most liquids, but most gases are nearly transparent to black body radiation. For heat transfer to most gases, it’s usually necessary to make use of turbulence or better yet, chaos.

Robert Buxbaum

Hydrogen versus Battery Power

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There are two major green energy choices that people are considering to power small-to-medium size, mobile applications like cars and next generation, drone airplanes: rechargeable, lithium-ion batteries and hydrogen /fuel cells. Neither choice is an energy source as such, but rather a clean energy carrier. That is, batteries and fuel cells are ways to store and concentrate energy from other sources, like solar or nuclear plants for use on the mobile platform.

Of these two, rechargeable batteries are the more familiar: they are used in computers, cell phones, automobiles, and the ill-fated, Boeing Dreamliner. Fuel cells are less familiar but not totally new: they are used to power most submarines and spy-planes, and find public use in the occasional, ‘educational’ toy. Fuel cells provided electricity for the last 30 years of space missions, and continue to power the international space station when the station is in the dark of night (about half the time). Batteries have low energy density (energy per mass or volume) but charging them is cheap and easy. Home electricity costs about 12¢/kWhr and is available in every home and shop. A cheap transformer and rectifier is all you needed to turn the alternating current electricity into DC to recharge a battery virtually anywhere. If not for the cost and weight of the batteries, the time to charge the battery (usually and hour or two), batteries would be the obvious option.

Two obvious problems with batteries are the low speed of charge and the annoyance of having to change the battery every 500 charges or so. If one runs an EV battery 3/4 of the way down and charges it every week, the battery will last 8 years. Further, battery charging takes 1-2 hours. These numbers are acceptable if you use the car only occasionally, but they get more annoying the more you use the car. By contrast, the tanks used to hold gasoline or hydrogen fill in a matter of minutes and last for decades or many thousands of fill-cycles.

Another problem with batteries is range. The weight-energy density of batteries is about 1/20 that of gasoline and about 1/10 that of hydrogen, and this affects range. While gasoline stores about 2.5 kWhr/kg including the weight of the gas tank, current Li-Ion batteries store far less than this, about 0.15 kWhr/kg. The energy density of hydrogen gas is nearly that of gasoline when the efficiency effect is included. A 100 kg of hydrogen tank at 10,000 psi will hold 8 kg of hydrogen, or enough to travel about 350 miles in a fuel-cell car. This is about as far as a gasoline car goes carrying 60 kg of tank + gasoline. This seems acceptable for long range and short-range travel, while the travel range with eVs is more limited, and will likely remain that way, see below.

The volumetric energy density of compressed hydrogen/ fuel cell systems is higher than for any battery scenario. And hydrogen tanks are far cheaper than batteries. From Battery University. http://batteryuniversity.com/learn/article/will_the_fuel_cell_have_a_second_life

The volumetric energy density of compressed hydrogen/ fuel cell systems is higher than for any battery scenario. And hydrogen tanks are far cheaper than batteries. From Battery University. http://batteryuniversity.com/learn/article/will_the_fuel_cell_have_a_second_life

Cost is perhaps the least understood problem with batteries. While electricity is cheap (cheaper than gasoline) battery power is expensive because of the high cost and limited life of batteries. Lithium-Ion batteries cost about $2000/kWhr, and give an effective 500 charge/discharge cycles; their physical life can be extended by not fully charging them, but it’s the same 500 cycles. The effective cost of the battery is thus $4/kWhr (The battery university site calculates $24/kWhr, but that seems overly pessimistic). Combined with the cost of electricity, and the losses in charging, the net cost of Li-Ion battery power is about $4.18/kWhr, several times the price of gasoline, even including the low efficiency of gasoline engines.

Hydrogen prices are much lower than battery prices, and nearly as low as gasoline, when you add in the effect of the high efficiency fuel cell engine. Hydrogen can be made on-site and compressed to 10,000 psi for less cost than gasoline, and certainly less cost than battery power. If one makes hydrogen by electrolysis of water, the cost is approximately 24¢/kWhr including the cost of the electrolysis unit.While the hydrogen tank is more expensive than a gasoline tank, it is much cheaper than a battery because the technology is simpler. Fuel cells are expensive though, and only about 50% efficient. As a result, the as-used cost of electrolysis hydrogen in a fuel cell car is about 48¢/kWhr. That’s far cheaper than battery power, but still not cheap enough to encourage the sale of FC vehicles with the current technology.

My company, REB Research provides another option for hydrogen generation: The use of a membrane reactor to make it from cheap, easy to transport liquids like methanol. Our technology can be used to make hydrogen either at the station or on-board the car. The cost of hydrogen made this way is far cheaper than from electrolysis because most of the energy comes from the methanol, and this energy is cheaper than electricity.

In our membrane reactors methanol-water (65-75% Methanol), is compressed to 350 psi, heated to 350°C, and reacted to produce hydrogen that is purified as it is made. CH3OH + H2O –> 3H2 + CO2, with the hydrogen extracted through a membrane within the reactor.

The hydrogen can be compressed to 10,000 psi and stored in a tank on board an automobile or airplane, or one can choose to run this process on-board the vehicle and generate it from liquid fuel as-needed. On-board generation provides a saving of weight, cost, and safety since you can carry methanol-water easily in a cheap tank at low pressure. The energy density of methanol-water is about 1/2 that of gasoline, but the fuel cell is about twice as efficient as a gasoline engine making the overall volumetric energy density about the same. Not including the fuel cell, the cost of energy made this way is somewhat lower than the cost of gasoline, about 25¢/kWhr; since methanol is cheaper than gasoline on a per-energy basis. Methanol is made from natural gas, coal, or trees — non-imported, low cost sources. And, best yet, trees are renewable.

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

A visit to the Buxbaum laboratory from Metromedia

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It’s a slow news day in Detroit, so the folks from Metromedia came to visit my laboratory at REB Research. You can visit too. We’re doing cool stuff most of the time, we’re working on a hydrogen-fueled plane that stays aloft for weeks (not that cool, actually, the Hindenberg did it in the 30s). On this particular day I’ve got a cool hat on, and a beige suit. I’m putting hydrogen in my car. Hydrogen increases the speed of combustion, and so it adds to milage — or it has when we’ve added it from electrolysis sources.buxbaum-003

The fun thing about science is that there are always surprises.

Adding hydrogen to a Malibu at REB Research

Adding hydrogen to a Malibu at REB Research

The joy of curtains

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By Dr. Robert E. Buxbaum January 18, 2013

In our northern climates most homes have double-paned windows; they cost a fortune, and are a lot better than plain glass, but they still lose a lot of heat: far more than the equivalent area of wall. The insulation value is poor mostly because the thickness is low: a typical double pane window is only ½” thick. The glass panes have hardly any insulation value, so the majority of the insulation is the 0.3″ air space between them. Our outer walls, by contrast, are typically 6” thick filled with glass –wool. The wall is 12 times as thick as the window, and it turns out that the R value is about 12 times as great. Since window area is about 1/10 the wall area, we can expect that about half your homes heat goes out through the windows (about half the air-conditioner cooling in the summer too). A good trick to improve your home’s insulation, then, is to add curtains as this provides a fairly thick layer of stagnant air inside the room, right next to your windows.

To see how much you can save by adding curtains, it’s nice (for me, and my mind-set mostly) to talk in terms of R values. In the northern USA, the “R” value of a typical, well-insulated outer wall is about 24. What that means is that it takes 24°F and one square foot of wall to remove 1 BTU per hour. That is, the resistance to heat loss is 24 °F.hr.ft2/BTU. The R value for a typical double pane window is about 2 in the same units, and is only 1 if you have single panes. The insulating quality of our windows is so poor that, for many homes, more heat is lost through the windows than through the rest of the wall space.

To figure out how much heat is lost through your windows take the area in square feet multiply by a typical temperature differential (50°F might be typical in Michigan), and divide by the R value of your paned windows (1 or 2) depending on whether it’s single or double paned. Since heat costs about $10/MMBTU ($10 per million BTU) for a gas heated house, you can figure out what a small, 10 ft2 window costs a typical Michigan householder as follows, assuming a single pane (R=1):

Q = Area* ∆T/R = 10 ft2 * 50°F/1 = 500 BTU/hr. Here Q is the heat lost per unit time, ∆T is the temperature difference between the window surface and the room, and A is the ara of the window surface.

Since there are 24 hours in a day, and 30.5 days in a month the dollar cost of that window is 500*24*31.5*10/1,000,000 = $3.78/month. After a few years, you’ll have paid $200 for that small window in lost heat and another $200 in air conditioning.

A cheap solution is to add curtains, shades, or plastic of some sort. These should not be placed too close to the window, or you won’t have a decent air gap, nor so far that the air will not be static in the gap. For small gaps between the window glass and your plastic or curtain, the heat transfer rate is proportional to the thermal conductivity of air, k, and inversely proportional to the air gap distance, ∂.

Q = ∆T A k /∂.

R  = ∂/k.

The thermal conductivity of air, k, is about .024 BTU/ft. hr°F. We thus confirm that the the R-value for an air gap of 9/16” or 1/20 foot is about 2 in these units. Though the typical air gap between the glass is less, about .3″ there is some more stagnant air outside the glass an that counts towards the 9/16″ of stagnant air. The k value of glass or plastic is much higher than of air, so the layers of glass or plastic add almost nothing to the total heat transfer resistance.

Because the R value of glass and plastic is so low, if you cover your window with a layer of plastic sheet that touches the window, the insulation effect is basically zero. To get insulation value you want to use a gap between about ½” and 1” in thickness. If you already have a 2 paned window of R value 2, you can expect to be able to raise your insulation value to 4 by adding a plastic sheet or single curtain at 9/16” from the glass.

Sorry to say, you can’t raise this insulation value much higher than 4 by use of a single air gap that’s more than 1″ thick. When a single gap exceeds this size, the insulating value drops dramatically as gas circulation in the gap (free convection) drives heat transfer. That’s why wall insulation has fiber-glass fill. For your home, you will want something more attractive than fiberglass between you and the window pane, and typical approaches  include cellular blinds or double layer drapes. These work on the same principle as the single sheet, but have extra layers that stop convection.

My favorite version of the double drapes is the federalist version, where the inner drape is near transparent, shim cloth hangs close to the window, with a heavier drape beyond that. The heavier curtain is closed at night and opened in daytime; where insulation is needed, the lighter cloth hangs day and night. This looks a lot better than a roll-type window shade, or bamboo screen. Besides, with a roll-shade or bamboo, you must put it close to the window where it will interfere with the convection flow, that is cold shedding from the shut window.

Another nice alternative is a “cell shade” These are folded lengths of two or more stiff cloths that are formed into honeycombs ½” to 2” apart. This empty thickness provides the insulating power of the shade. Placed at the right distance from the window, the cell shade will add 3 or more to the overall R value of the window (1/12 ft / .024 BTU/ft. hr°F = 3.5 ft2hr°F/BTU). As with a bamboo screen, all this R value goes away if the shade is set at more than about 1” from the window or an interior shade. At a greater thickness that this, the free convection flow of cold air between the window and the shade dominates, and you get a puddle of cold air on the floor. 

I would suggest a cellular shade that opens from the bottom only and is translucent. This provides light and privacy; a shade that is too dark will be left open. Behind this, my home has double-pane windows (when I was single the window was covered by a layer of plastic too). The see-through shade provides insulation while allowing one to see out the window (or let light in) when the shade is drawn. You want to be able to see out; that’s the reason you had a window in the first place. Very thick, insulating curtains and blinds seem like a waste to me – they are enough thicker to add any significant R-value, they block the light, and if they end up far from the window, the shedding heat loss will more than offset any small advantage from the thick cloth.

One last window insulation option that’s worth mentioning is a reflective coating on the glass (an e-coating). This is not as bad an idea as you might think, even in a cold climate as in Detroit. A surprising amount of heat tends to escape your windows in the form of radiation. That is, the heat leaves by way of invisible (infra –red) light that passes unimpeded through the double pane glass. In hot climates even more heat comes in this way, and a coating is even more useful to preserve air conditioning power. Reflective plastic coats are cheap enough and readily available, though they can be hard to apply, and are not always attractive.

You can expect to reduce the window heat loss by a factor of 3 or more using these treatments, reducing the heat loss through the small window to $1.00 or so per month, far enough that the main heat loss is through the walls. At that point, it may be worth putting your efforts elsewhere. Window treatments can save you money, make a previously uninhabitable room pleasant, and can help preserve this fair planet of ours. Enjoy.

Updated, Feb 9, 2022, REB.

What is the best hydrogen storage medium?

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Answering best questions is always tricky since best depends on situation, but I’ll cover some hydrogen storage options here, and I’ll try to explain where our product options (cylinder gas purifiers and methanol-water reformers) fit in.

The most common laboratory option for hydrogen storage is inside a tank; typically this tank is made of steel, but it can be made of aluminum, fiberglass or carbon fiber. Tanks are the most convenient source for small volume users since they are instantly ready for delivery at any pressure up to the storage pressure; typically that’s 2000 psi (135 atm) though 10,000 (1350 atm) is available by special order. The maximum practical density for this storage is about 50 g/liter, but this density ignores the weight of the tank. The tank adds a factor of 20 or to the weight, making tanks a less-favored option for mobile users. Tanks also add significantly to the cost. They also tend to add impurities to the gas, and there’s a safety issue too: tanks sometimes fall over, and compressed gas can explode. For small-volume, non-mobile users, one can address safety by chaining up ones tank and adding a metal membrane hydrogen purifier; This is one of our main products.

Another approach is liquid hydrogen; The density of liquid hydrogen is higher than of gas, about 68 g/liter, and you don’t need as a tank that’s a big or heavy. One problem is that you have to keep the liquid quite cold, about 25 K. There are evaporative losses too, and if the vent should freeze shut you will get a massive explosion. This is the storage method preferred by large users, like NASA.

Moving on to metal hydrides. These are heavy and rather expensive but they are safer than the two previous options. To extract hydrogen from a metal hydride bed the entire hydride bed has to be heated, and this adds complexity. To refill the bed, it generally has to be cooled, and this too adds complexity. Generally, you need a source of moderately high pressure, clean, dry hydrogen to recharge a bed. You can get this from either an electrolysis generator, with a metal membrane hydrogen purifier, or by generating the hydrogen from methanol using one of our membrane reactor hydrogen generators.

Borohydrides are similar to metal hydrides, but they can flow. Sorry to say, they are more expensive than normal metal hydrides and they can not be regenerated.They are ideal for some military use

And now finally, chemical materials: water, methanol, and ammonia. Chemical compounds are a lot cheaper than metal hydrides or metal borohydrides, and tend to be far more readily available and transportable being much lighter in weight. Water and/or methanol contains 110 gm of H2/liter;  ammonia contains 120 gms/liter, and the tanks are far lighter and cheaper too. Polyethylene jugs weighing a few ounces suffices to transport gallon quantities of water or methanol and, while not quite as light, relatively cheap metallic containers suffice to hold and transport ammonia.

The optimum choice of chemical storage varies with application and customer need. Water is the safest option, but it can freeze in the cold, and it does not contain its own chemical energy. The energy to split the water has to come externally, typically from electricity via electrolysis. This makes water impractical for mobile applications. Also, the hydrogen generated from water electrolysis tends to be impure, a problem for hydrogen that is intended for storage or chemical manufacture. Still, there is a big advantage to forming hydrogen from something that is completely non-toxic, non-flammable, and readily available, and water definitely has a place among the production options.

Methanol contains its own chemical energy, so hydrogen can be generated by heating alone (with a catalyst), but it is toxic to drink and it is flammable. I’ve found a  my unique way of making hydrogen from methanol-water using  a membrane reactor. Go to my site for sales and other essays.

Finally, ammonia provides it’s own chemical energy like methanol, and is flammable, like methanol; we can convert it to hydrogen with our membrane reactors like we can methanol, but ammonia is far more toxic than methanol, possessing the power to kill with both its vapors and in liquid form. We’ve made ammonia reformers, but prefer methanol.