Category Archives: Engineering

Why random experimental design is better

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In a previous post I claimed that, to do good research, you want to arrange experiments so there is no pre-hypothesis of how the results will turn out. As the post was long, I said nothing direct on how such experiments should be organized, but only alluded to my preference: experiments should be organized at randomly chosen conditions within the area of interest. The alternative, shown below is that experiments should be done at the cardinal points in the space, or at corner extremes: the Wilson Box and Taguchi design of experiments (DoE), respectively. Doing experiments at these points implies a sort of expectation of the outcome; generally that results will be linearly, orthogonal related to causes; in such cases, the extreme values are the most telling. Sorry to say, this usually isn’t how experimental data will fall out. First experimental test points according to a Wilson Box, a Taguchi, and a random experimental design. The Wilson box and Taguchi are OK choices if you know or suspect that there are no significant non-linear interactions, and where experiments can be done at these extreme points. Random is the way nature works; and I suspect that's best -- it's certainly easiest.

First experimental test points according to a Wilson Box, a Taguchi, and a random experimental design. The Wilson box and Taguchi are OK choices if you know or suspect that there are no significant non-linear interactions, and where experiments can be done at these extreme points. Random is the way nature works; and I suspect that’s best — it’s certainly easiest.

The first test-points for experiments according to the Wilson Box method and Taguchi method of experimental designs are shown on the left and center of the figure above, along with a randomly chosen set of experimental conditions on the right. Taguchi experiments are the most popular choice nowadays, especially in Japan, but as Taguchi himself points out, this approach works best if there are “few interactions between variables, and if only a few variables contribute significantly.” Wilson Box experimental choices help if there is a parabolic effect from at least one parameter, but are fairly unsuited to cases with strong cross-interactions.

Perhaps the main problems with doing experiments at extreme or cardinal points is that these experiments are usually harder than at random points, and that the results from these difficult tests generally tell you nothing you didn’t know or suspect from the start. The minimum concentration is usually zero, and the minimum temperature is usually one where reactions are too slow to matter. When you test at the minimum-minimum point, you expect to find nothing, and generally that’s what you find. In the data sets shown above, it will not be uncommon that the two minimum W-B data points, and the 3 minimum Taguchi data points, will show no measurable result at all.

Randomly selected experimental conditions are the experimental equivalent of Monte Carlo simulation, and is the method evolution uses. Set out the space of possible compositions, morphologies and test conditions as with the other method, and perhaps plot them on graph paper. Now, toss darts at the paper to pick a few compositions and sets of conditions to test; and do a few experiments. Because nature is rarely linear, you are likely to find better results and more interesting phenomena than at any of those at the extremes. After the first few experiments, when you think you understand how things work, you can pick experimental points that target an optimum extreme point, or that visit a more-interesting or representative survey of the possibilities. In any case, you’ll quickly get a sense of how things work, and how successful the experimental program will be. If nothing works at all, you may want to cancel the program early, if things work really well you’ll want to expand it. With random experimental points you do fewer worthless experiments, and you can easily increase or decrease the number of experiments in the program as funding and time allows.

Consider the simple case of choosing a composition for gunpowder. The composition itself involves only 3 or 4 components, but there is also morphology to consider including the gross structure and fine structure (degree of grinding). Instead of picking experiments at the maximum compositions: 100% salt-peter, 0% salt-peter, grinding to sub-micron size, etc., as with Taguchi, a random methodology is to pick random, easily do-able conditions: 20% S and 40% salt-peter, say. These compositions will be easier to ignite, and the results are likely to be more relevant to the project goals.

The advantages of random testing get bigger the more variables and levels you need to test. Testing 9 variables at 3 levels each takes 27 Taguchi points, but only 16 or so if the experimental points are randomly chosen. To test if the behavior is linear, you can use the results from your first 7 or 8 randomly chosen experiments, derive the vector that gives the steepest improvement in n-dimensional space (a weighted sum of all the improvement vectors), and then do another experimental point that’s as far along in the direction of that vector as you think reasonable. If your result at this point is better than at any point you’ve visited, you’re well on your way to determining the conditions of optimal operation. That’s a lot faster than by starting with 27 hard-to-do experiments. What’s more, if you don’t find an optimum; congratulate yourself, you’ve just discovered an non-linear behavior; something that would be easy to overlook with Taguchi or Wilson Box methodologies.

The basic idea is one Sherlock Holmes pointed out (Study in Scarlet): “It is a capital mistake to theorize before one has data. Insensibly one begins to twist facts to suit theories, instead of theories to suit facts.” (Case of Identity). “Life is infinitely stranger than anything which the mind of man could invent.“

Robert E. Buxbaum, September 11, 2013. A nice description of the Wilson Box method is presented in Perry’s Handbook (6th ed). SInce I had trouble finding a free, on-line description, I linked to a paper by someone using it to test ingredient choices in baked bread. Here’s a link for more info about random experimental choice, from the University of Michigan, Chemical Engineering dept. Here’s a joke on the misuse of statistics, and a link regarding the Taguchi Methodology. Finally, here’s a pointless joke on irrational numbers, that I posted for pi-day.

Surrealists art joke

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How many surrealists does it take to screw in a lightbulb.

 

The fish.

 

Surrealism aims to show the reality that exceeds realism; the dream-like absurd that is beyond the rational, common-sensical and practical. Beyond control engineering.

And you know “How many engineers would it take to screw in a lightbulb?” —- “Minimally two, and it would have to be a very large lightbulb.”

Even if the insights of surrealism are common-place, for example, that the eye is a false mirror of the world, I like is that they become real (if the surrealist is talented.)

False Mirror by Magritte; The idea, I suppose is that the eye is a false mirror of the world, seeing what's already within it.

False Mirror by Magritte; the idea, I suppose is that we see what’s already within us.

“The greatest obstacle to discovering the shape of the earth, the continents, and the oceans was not ignorance but the illusion of knowledge.” What I particularly like is the falseness of the mirror is shown as both false and true. The world is rarely this or that. Another insight / joke.

We all have masks, especially with those we love.

We all have masks, especially with those we love.

I imagine most I could make second-rate surrealistic works. The way to know your work is second rate it’s beautiful and insightful, but not funny.

Creation of Man-the-militant in the style of Michelangelo

Creation of Man-the-militant. Kuksi. It’s well done, and interesting (a retake on Michelangelo), but it’s not funny. See my cartoon in mechanical v civil engineers joke.

And then there is bad modern art. You could argue that this isn’t surreal, but some sort of other modern art, or post modern art. But that’s all false: it’s just bad art.

Bad modern art: little skill, little meaning, no humor. If you have to ask: "is it art?" It usually isn't.

Bad modern art: little skill, little meaning, no humor. If you have to ask: “is it art?” It usually isn’t.

If you buy something like this, and put it in your corporate headquarters lobby, the joke’s on you, and the artist is laughing his or her way to the bank.  Here is a link to why surrealism should be funny, And why architecture should not be (someone’s got to live in that joke).

R. E. Buxbaum, August 5, 2013

Mechanical Engineer v Civil Engineer Joke

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What’s the difference between a mechanical engineer and a civil engineer?

 

 

Mechanical engineers make weapons; civil engineers make targets.

 

 

Is funny because ….. it’s sort of true. Much of engineering is war-related, and always was. In earlier times, an engineer was someone who made engines of war: catapults, battering rams, and the like. Nowadays, mechanical engineers are the main designers for tanks, cannons, and ships. A civil engineer is one whose projects have civilian applications. But as these projects have military uses (roads, ports, offices, and bridges, for example), civilian projects are major targets for an opposing army.

An observation about war and peace: if you are really at making peacetime products, you’re a hero in your country and outside; if you design weapons, you are vilified by the enemy and likely to become a prisoner in your own land. Consider the designers of the atom bomb in the US, Russia, Israel, India, or Iran. They can’t go abroad, and are likely suspect at home. The leaders have to worry that these scientists will give the same weapons to their enemies (it’s happened) or that they will not be dedicated enough to make the next iteration of the weapon (ditto).

My advice: specialize items for peacetime or civilian use if you can. Those who make better cars, music, art or architecture are welcome everywhere; advances in death usually rebound on the inventor. Here’s a joke comparing chemists and chemical engineers, a piece on a favorite car engine advance, on perfect tuning of musical instruments, an architecture joke, and a control engineer joke. People like civil engineers.

What sort of guy does a king keep locked in the castle dungeon — not the common thief.  #wordstothewise.

R. E. Buxbaum, August 1, 2013. I’m a chemical engineer, who makes hydrogen stuff and consults, mostly for peace-time use.

Global warming takes a 15 year rest

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I have long thought that global climate change was chaotic, rather than steadily warming. Global temperatures show self-similar (fractal) variation with time and long-term cycles; they also show strange attractors generally states including ice ages and El Niño events. These are sudden rests of the global temperature pattern, classic symptoms of chaos. The standard models of global warming is does not predict El Niño and other chaotic events, and thus are fundamentally wrong. The models assume that a steady amount of sun heat reaches the earth, while a decreasing amount leaves, held in by increasing amounts of man-produced CO2 (carbon dioxide) in the atmosphere. These models are “tweaked” to match the observed temperature to the CO2 content of the atmosphere from 1930 to about 2004. In the movie “An Inconvenient Truth” Al Gore uses these models to predict massive arctic melting leading to a 20 foot rise in sea levels by 2100. To the embarrassment of Al Gore, and the relief of everyone else, though CO2 concentrations continue to rise, global warming took a 15 year break starting shortly before the movie came out, and the sea level is, more-or-less where it was except for temporary changes during periodic El Niño cycles.

Global temperature variation Fifteen years and four El Niño cycles, with little obvious change. Most models predict .25°C/decade.

Fifteen years of global temperature variation to June 2013; 4 El Niños but no sign of a long-term change.

Hans von Storch, a German expert on global warming, told the German newspaper, der Spiegel: “We’re facing a puzzle. Recent CO2 emissions have actually risen even more steeply than we feared. As a result, according to most climate models, we should have seen temperatures rise by around 0.25 degrees Celsius (0.45 degrees Fahrenheit) over the past 10 years. That hasn’t happened. [Further], according to the models, the Mediterranean region will grow drier all year round. At the moment, however, there is actually more rain there in the fall months than there used to be. We will need to observe further developments closely in the coming years.”

Aside from the lack of warming for the last 15 years, von Storch mentions that there has been no increase in severe weather. You might find that surprising given the news reports; still it’s so. Storms are caused by temperature and humidity differences, and these have not changed. (Click here to see why tornadoes lift stuff up).

At this point, I should mention that the majority of global warming experts do not see a problem with the 15 year pause. Global temperatures have been rising unsteadily since 1900, and even von Storch expects this trend to continue — sooner or later. I do see a problem, though, highlighted by the various chaotic changes that are left out of the models. A source of the chaos, and a fundamental problem with the models could be with how they treat the effects of water vapor. When uncondensed, water vapor acts as a very strong thermal blanket; it allows the sun’s light in, but prevents the heat energy from radiating out. CO2 behaves the same way, but weaker (there’s less of it).

More water vapor enters the air as the planet warms, and this should amplify the CO2 -caused run-away heating except for one thing. Every now and again, the water vapor condenses into clouds, and then (sometimes) falls as rain or show. Clouds and snow reflect the incoming sunlight, and this leads to global cooling. Rain and snow drive water vapor from the air, and this leads to accelerated global cooling. To the extent that clouds are chaotic, and out of man’s control, the global climate should be chaotic too. So far, no one has a very good global model for cloud formation, or for rain and snowfall, but it’s well accepted that these phenomena are chaotic and self-similar (each part of a cloud looks like the whole). Clouds may also admit “the butterfly effect” where a butterfly in China can cause a hurricane in New Jersey if it flaps at the right time.

For those wishing to examine the longer-range view, here’s a thermal history of central England since 1659, Oliver Cromwell’s time. At this scale, each peak is an El Niño. There is a lot of chaotic noise, but you can also notice either a 280 year periodicity (lat peak around 1720), or a 100 year temperature rise beginning about 1900.

Global warming; Central England Since 1659; From http://www.climate4you.com

It is not clear that the cycle is human-caused,but my hope is that it is. My sense is that the last 100 years of global warming has been a good thing; for agriculture and trade it’s far better than an ice age. If we caused it with our  CO2, we could continue to use CO2 to just balance the natural tendency toward another ice age. If it’s chaotic, as I suspect, such optimism is probably misplaced. It is very hard to get a chaotic system out of its behavior. The evidence that we’ve never moved an El Niño out of its normal period of every 3 to 7 years (expect another this year or next). If so, we should expect another ice age within the next few centuries.

Global temperatures measured from the antarctic ice showing stable, cyclic chaos and self-similarity.

Global temperatures measured from the antarctic ice showing 4 Ice ages.

Just as clouds cool the earth, you can cool your building too by painting the roof white. If you are interested in more weather-related posts, here’s why the sky is blue on earth, and why the sky on Mars is yellow.

Robert E. Buxbaum July 27, 2013 (mostly my business makes hydrogen generators and I consult on hydrogen).

Escher Architecture – joke?

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Caption will say where this is from.

Robert  Leighton, from the New Yorker,

Is funny because …. there’s an Escher-like impossible structure and a dirty word (ass, tee hee). Besides that, this joke highlights a fundamental conflict between the architect and the client (customer): what is good architecture?

Typically the customer whats a home or office that “looks nice”, “doesn’t cost too much”, and “works,” perhaps as an advertisement for the company. Often the architect wants to make a statement for him/herself, or wants to produce a work of art. Left to their own, architects can produce expensive monuments that no one can live in.

A wonderful (horrible) case concerns The Cooper Union, my alma mater, and more-or-less the only free college in America. The Cooper Union was founded by an inventive mechanic, Peter Cooper, see my biography, who invented jello, and rolled steel, laid the transatlantic cable, founded AT&T, and managed to give free education to a century and a half of students. The trustees of the school tore down the old, serviceable building, sold the land, and built a $270,000,000 dollar monstrosity. Hailed by the New York Times as great architecture, it bankrupted the school, and is unusable for the sort of hands-on education that Peter Cooper devised.

In hopes of attracting a rich donor, Cooper Union borrowed $175 million to erect this grotesque building for its engineering department. No donor materialized, and, as a result, the school’s 155-year-old policy of free tuition has vaporized.

In hopes of attracting a rich donor, Cooper Union sold its engineering building and borrowed $175 million to erect this replacement. No donor materialized, and, with it, a 155-year-old policy of free tuition.

Here’s a surrealist joke, an engineer joke, and a thought on control engineering. Here too is a  sculpture I put on top of my building; the eyes follow you.

R.E. Buxbaum, July 8, 2013; I do consulting on hydrogen, and my company makes hydrogen products.

Control engineer joke

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What made the control engineer go crazy?

 

He got positive feedback.

Is funny because …… it’s a double entente, where both meanings are true: (1) control engineers very rarely get compliments (positive feedback); the aim of control is perfection, something that’s unachievable for a dynamic system (and generally similar to near perfection: the slope at a maximum is zero). Also (2) systems go unstable if the control feedback is positive. This can happen if the controller was set backwards, but more usually happens when the response is too fast or too extreme. Positive feedback pushes a system further to error and the process either blows up, or (more commonly) goes wildly chaotic, oscillating between two or more “strange attractor” states.

It seems to me that hypnosis, control-freak love, and cult behaviors are the result of intentionally produced positive feedback. Palsies, economic cycles, and global warming are more likely the result of unintentional positive feedback. In each case, the behavior is oscillatory chaotic.

The normal state of Engineering is lack of feedback. Perhaps this is good because messed up feedback leads to worse results. From xykd.

Our brains give little reliable feedback on how well they work, but that may be better than strong, immediate feedback, as that could lead to bipolar instability. From xkcd. For more on this idea, see Science and Sanity, by Alfred Korzbski (mini youtube)

Control engineers tend to be male (85%), married (80%), happy people (at least they claim to be happy). Perhaps they know that near-perfection is close enough for a complex system in a dynamic world, or that one is about as happy as believes ones-self to be. It also helps that control engineer salaries are about $95,000/ year with excellent benefits and low employment turnover.

Here’s a chemical engineer joke I made up, and an older engineering joke. If you like, I’ll be happy to consult with you on the behavior of your processes.

By Dr. Robert E. Buxbaum, July 4, 2013

Chemist v Chemical Engineer joke

What’s the difference between a chemist and a chemical engineer?

 

How much they make.

 

I made up this joke up as there were no other chemical engineer jokes I knew. It’s an OK double entente that’s pretty true — both in terms of product produced and the amount of salary (there’s probably a cause-and-effect relation here). Typical of these puns, this joke ignores the internal differences in methodologies and background (see my post, How is Chemical engineering?). If you like, here’s another engineering joke,  a chemistry joke, and a dwarf joke.

R.E. Buxbaum –  June 28, 2013.

What’s Holding Gilroy on the Roof

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We recently put a sculpture on our roof: Gilroy, or “Mr Hydrogen.” It’s a larger version of a creepy face sculpture I’d made some moths ago. Like it, and my saber-toothed tiger, the eyes follow you. A worry about this version: is there enough keeping it from blowing down on the cars? Anyone who puts up a large structure must address this worry, but I’m a professional engineer with a PhD from Princeton, so my answer is a bit different from most.

Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you.

Gilroy (Mr Hydrogen) sculpture on roof of REB Research & Consulting. The eyes follow you. Aim is that it should withstand 50 mph winds.

The main force on most any structure is the wind (the pyramids are classic exceptions). Wind force is generally proportional to the exposed area and to the wind-speed squared: something called form-drag or quadratic drag. Since force is related to wind-speed, I start with some good statistics for wind speed, shown in the figure below for Detroit where we are.

The highest Detroit wind speeds are typically only 16 mph, but every few years the winds are seen to reach 23 mph. These are low relative to many locations: Detroit has does not get hurricanes and rarely gets tornadoes. Despite this, I’ve decided to brace the sculpture to withstand winds of 50 mph, or 22.3 m/s. On the unlikely chance there is a tornado, I figure there would be so much other flotsam that I would not have to answer about losing my head. (For why Detroit does not get hurricanes or tornadoes, see here. If you want to know why tornadoes lift things, see here).

The maximum area Gilroy presents is 1.5 m2. The wind force is calculated by multiplying this area by the kinetic energy loss per second 1/2ρv2, times a form factor.  F= (Area)*ƒ* 1/2ρv2, where ρ is the density of air, 1.29Kg/m3, and v is velocity, 22.3 m/s. The form factor, ƒ, is about 1.25 for this shape: ƒ is found to be 1.15 for a flat plane, and 1.1 to 1.3 a rough sphere or ski-jumper. F = 1.5*1.25* (1/2 *1.29*22.32) = 603 Nt = 134 lb.; pressure is this divided by area. Since the weight is only about 40 lbs, I find I have to tie down the sculpture. I’ve done that with a 150 lb rope, tying it to a steel vent pipe.

Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

Wind speed for Detroit month by month. Used to calculate the force. From http://weatherspark.com/averages/30042/Detroit-Michigan-United-States

It is possible that there’s a viscous lift force too, but it is likely to be small given the blunt shape and the flow Reynolds number: 3190. There is also the worry that Gilroy might fall apart from vibration. Gilroy is made of 3/4″ plywood, treated for outdoor use and then painted, but the plywood is held together with 25 steel screws 4″ long x 1/4″ OD. Screws like this will easily hold 134 lbs of steady wind force, but a vibrating wind will cause fatigue in the metal (bend a wire often enough and it falls apart). I figure I can leave Gilroy up for a year or so without worry, but will then go up to replace the screws and check if I have to bring him/ it down.

In the meantime, I’ll want to add a sign under the sculpture: “REB Research, home of Mr Hydrogen” I want to keep things surreal, but want to be safe and make sales.

by Robert E. Buxbaum, June 21, 2013

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.