Here are 26 pages of notes in advance for a workshop at the first
Pennsylvania Renewable Energy Festival this Saturday morning
from 9-12, September 24, 2005 near Allentown, PA. See
http://www.paenergyfest.com/workshop-info.shtml .

Comments welcome...

Nick

Notes for a Pennsylvania Renewable Energy Festival Workshop on Solar House Heating and Natural Cooling Techniques

Kempton, PA September 24, 2005

Written by Nick Pine, with Drew Gillett and Rich Komp

Debatable Conclusions

1. Heat flows like electricity. 2. Solar heat can be 100 times cheaper than solar electricity. 3. Superinsulated houses have to be very small or very large. 4. Direct gain houses can be improved. 5. Indirect gain can be more efficient. 6. We might store heat in the ceiling. 7. We might have a separate cloudy-day heat store. 8. Low temp heat storage and distribution are difficult. 9. Shurcliff's lung might be a good air-air heat exchanger. 10. Greywater heat exchangers, Big Fins and solar ponds can help. 11. We might also gather heat from PVs. 12. Smart ventilation can be helpful. 13. Swamp coolers can be improved.

0.0 Introduction

The US has 5% of the world's population and consumes 26% of the world's energy. House heating and cooling accounts for about one third of that. In 1980, "envelope house" inventor Tom Smith said:

It's a snap to save energy in the US. As soon as more people become involved in the basic math of heat transfer and get a gut-level, as well as intellectual, grasp on how a house works, solution after solution will appear.

This workshop aims at improving that grasp, which we can control better than our US cheap energy policy... If we paid related costs of healthcare and air pollution and Gulf wars at the pump, gasoline would be a lot more expensive. Drew says this writeup needs exercises for the reader. OK:

Exercise 0.1: The US consumed 21 million 42 gallon $41 barrels of oil per day in 2004. What goes into the real cost per gallon? (Debatable answers appear at the end of these notes :-)

Most people think "electricity" when they hear "energy," even though most houses need more heating energy than electrical energy (the ratio is 1:1 in Hawaii and 5:1 in Vermont.) It's easy to shrink the small electrical slice of the home energy pie with compact fluorescent (CF) lights and more efficient appliances. Solar heat can be very inexpensive compared to solar electricity. PV panels at $3 per peak watt cost 150X more than polycarbonate glazing at $1/50W = 0.02/Pw. And sunspaces add floorspace to a house.

A square foot of "solar collector" only collects about $1/year at $1/gallon so anything (except PVs??? :-) that costs more than $10/ft^2 (half labor) and only collects energy with no other purpose seems economically-doomed...

Exercise 0.2: Should we a) replace a 60 W bulb with a 14 watt CF or b) buy 60-14 = 46 additional watts of PV power?

Most of us "know" how to design passive solar houses with well-established rules of thumb, but let's relax and take a fresh look from a standpoint of basic physics...

Berlin is a nice town and there were many opportunities for a student to spend his time in an agreeable manner, for instance with the nice girls. But instead of that we had to perform big and awful calculations.

Konrad Zuse, inventor of the 1936 Z1 computer

Overview

This is a workshop on "Ohm's law for heatflow" with applications to solar water and house heating and natural cooling. We'll discuss a solar pond and a simple greywater heat exchanger, some inexpensive plastic pipe coiled inside a 55-gallon drum. With hot water bursts of 13 gallons or less, it could be 97% efficient. If it is, why bother with solar hot water?

We'll provide arithmetic tools and data and strategies needed to site-build effective house heating and cooling systems using inexpensive materials and skills. Participants will need some familiarity with high school algebra.

We'll discuss power, energy, heatflow, and overnight and cloudy day heat storage at a high-school math and physics level, with insulation values and heat capacities of materials, simple equations involving time constants, evaporative and night ventilation cooling, passive and low-energy solar heating, climate data, and schemes for houses that are 100% solar-heated and naturally cooled, by design. We'll provide a calculator (Steve Baer says "Throw away your calculator." :-) and a CD-ROM. Promising techniques include solar closets, trickle collectors, "pancake houses," soap bubble foam insulation, and solar attics, including systems to collect heat and electricity from water-cooled standard PV panels.

Rich Komp is president of the Maine Solar Energy Association and a PV author with a PChem PhD, Drew Gillett is a Professional Engineer with civil engineering and architectural degrees, and I'm an EE by training.

Disclaimer

Some of the techniques we describe are experimental. Some have never been tried. We do not accept responsibility for their safety or functionality.

1. Power and energy

Energy is the stuff we pay for, measured in Joules or watt-hours or kilowatt-hours (kWh) or Calories or "British thermal units" (Btu), no longer used in Britain :-) The British now use joules or kWh. A Btu is a quantity of heat, about the same as the energy in a kitchen match or a mouse-hour. One Btu can heat one pound (16 ounces) of water one degree F. One Calorie (capital C, 4.19 kilojoules) can heat one kilogram of water 1 C.

Exercise 1.1: How many Btu [joules] are needed to heat 8 ounces [0.25 kg] of water from 50 to 212 F [10 to 100 C] to make a cup of tea? [the brackets describe

Power is the rate of energy flow over time. A mere number, vs the stuff we pay for. Energy is power times time. One watt-hour of energy is equivalent to 3.41 Btu. If energy were miles traveled, power would be miles per hour. If energy were a paycheck, power would be an hourly rate of pay.

Exercise 1.2: How long would it take to heat the tea water with a 300 W immersion heater?

We might check this with an immersion heater and a watch and a $100 Raytek IR thermometer. Or a HOBO from Onset Computer Corp (1-800-LOGGERS.) Their $119 battery-powered U12-013 HOBO is about the size of a matchbox. It can record 43,000 12-bit samples at 1 second to 18 hour intervals of its own temperature and relative humidity (RH), with jacks for 2 more temperature probes or other devices on cables, and upload them to a PC spreadsheet via a USB port.

People often confuse power and energy, as in "My house uses lots of power" (vs energy) or "My furnace capacity is 50,000 Btu," (vs Btu/h.) Power is measured in watts or kilowatts (kW.) Unlike energy, it can't be used or consumed.

People confuse heat and temperature, too. A bathtub full of hot water contains a lot of useful house heat, compared to a candle, but the candle is much hotter. A lower minimum usable temperature increases useful heat. Temperature is a measure of heat intensity. A 12-volt 100 amp-hour 50 pound automobile battery stores 267 times more energy (12Vx100Ah = 1200 Wh) than a 9-volt 500 milliamp-hour (9Vx0.5Ah = 4.5 Wh) 2 ounce transistor radio battery, at a lower voltage (ie "electrical temperature.") The $40 battery can store about 200 kWh over its lifetime, at 20 cents/kWh. A $1 cubic foot of water cooling from 130 to 80 F stores (130F-80F)64Btu/F = 3200 Btu, ie about 1 kWh, with a much longer lifetime and simpler I/O.

2. Rich Komp, Ohm, and Newton

Rich Komp (who is still alive) says heat moves by conduction (a hot frying pan handle), convection (including air movement), radiation (the sun brings about 1000 watts per square meter or 300 Btu per hour per square foot on a clear day at noon in the Sahara), and phase change (144 Btu melts a pound of ice and 1000 Btu evaporates a pound of water.)

About 300 years ago, Isaac Newton said the amount of heat that flows through a wall is proportional to its area and the temperature difference from one side to the other and its thermal conductance. About 100 years later, Georg Ohm said the same about electricity: V = IR, ie a current I in amps times a resistance R in ohms produces a voltage difference V. Electrical conductance is measured on "mhos," which is ohms spelled backwards (or "siemens" which is snemeis spelled backwards :-) If 6 amps flows through a 2 ohm resistor, we'll see V = IR = 6Ax2ohms = 12 volts across it. Another example: 120 volts across 48 ohms makes I = V/R = 120V/48ohms = 2.5 amps flow, with electrical power P = IV = 2.5Ax120V = 300 watts. (These units are capitalized only when abbreviated, A vs amperes, V vs volts, and so on.)

Exercise 2.1: If 24 A flows through a 12 V headlamp, what's the resistance?

3. Thermal ohms?

Ohm's law for heatflow (aka Newton's law of cooling) uses a temperature difference instead of a voltage difference. Heatflow is measured in units of power, in watts or Btu per hour. There's no such thing as a thermal "ohm." The closest thing is the US "R-value" stamped on foamboards and batts in hardware stores.

Beadboard (expanded white polystyrene coffee cup material) has a US R-value of 4 (ft^2-F-h/Btu) per inch (in these notes, all R-values are US unless otherwise specified.) Blue, pink or green Styrofoam board has US R5/inch. So does air, for downward heatflow. The slow-moving indoor and windier outdoor air films near a single layer of glass have a combined US R-value of about 1. A smooth square foot of surface in slow-moving air loses about 1.5 Btu/h-F, with a U1.5 or an R2/3 airfilm resistance. A rough square foot of surface in V mph air loses about 2+V/2 Btu/h-F. Larger objects have lower airfilm conductances, since air warms or cools more as it passes over them, which reduces the temperature difference between the air and the surface.

Tiny cold soap bubbles (1/16" at 50 F) have about R3 per inch. Bill Sturm filled the space between two polyethylene film covers of his Calgary greenhouse with air during the day and soap bubble foam insulation at night and measured an 82% propane energy savings with and without the foam on -20 F nights, without much solar energy storage. In one system, air from a shop vac makes bubbles from a long PVC pipe full of holes in a trough near the ground containing a 10% detergent solution. When the bubbles reach a return air vent at the top of the greenhouse at night, they push on a piece of window screen and a microswitch turns off the shop vac until the bubbles collapse a bit in a few minutes, and the shop vac runs for a few more seconds as needed. Bill thinks about heating poly film refugee shelters this way in cold climates.

Fiberglass has R3.5/inch, or half that, if it contains 2% moisture (which it might, in a cold climate, with some holes in a vapor barrier and condensation), or even less, if air circulates within it, in very cold places. But R-values are not quite thermal ohms. To find the thermal resistance of a wall, divide the R-value by the area. An 8'x10' US R20 wall has a resistance of R20/(8'x10') = 0.25 F-h/Btu. We might call this "0.25 fhubs" or "4 buhfs." We can add buhfs for different kinds of house walls and windows like resistors in parallel and add fhubs for series layers of wall insulation.

Exercise 3.1: If 6 Btu/h [2 watts] of heat power flows through a wall with a 12 F [7 C] temp diff, what's the wall's thermal resistance in fhubs [C/W]?

Exercise 3.2: What's the conductance of an 8'x16' [2mx5m] R10 [metric R2] wall?

If it's 70 F indoors and 30 F outdoors, (70F-30F)/5F-h/Btu= 8 Btu/h [2.3 watts] of heat power flows through 1 ft^2 of R5 wall. But stud walls have lower stud resistances (R1/inch "thermal bridging") in parallel with the insulation resistance, and drywall and sheathing on each side of the wall act as stud fins for heat collection and distribution, which lowers the overall R-value.

Ignoring the fins, we can look at a 10.67 ft^2 16"x8' wall section with R13 insulation framed with R1 per inch 2x4s on all edges as a "stud conductor" with (16/12+8)1.5/12 = 1.17 ft^2 of stud surface in parallel with a 10.67-1.17 = 9.5 ft^2 "insulation conductor." The studs have a 1/(R1x3.5") = US U0.286 U-value and the insulation has 1/R13 = U0.0769. The combined conductance is 1.17x0.286+9.5x0.0769 = 1.07 Btu/h-F, for an effective US R-value of 10.67ft^2/1.07 buhf = R10.0, vs R13, without the studs.

Structural Insulated Panels (SIPs, glued plywood-foam-plywood sandwiches) have less thermal bridging and fewer air leaks. So do walls with plywood I-joists used as studs and loose-fill insulation (as used by ME Al Eggen, snipped-for-privacy@mindspring.org) and walls with a layer of foamboard insulation over the studs. An 8'x24' R24 6" SIP wall with a 40 F temperature difference would allow (70F-30F)8'x24'/R24 = 320 Btu/h of heatflow.

Metric (European, Canadian, Australian...) U-values are 5.68 times bigger than US U-values. A metric U1 window has 1 W/m^2C of thermal conductance. It's like a low-loss US R5.68 window. A 2 m^2 metric U1 window with 20 C air on one side and 5 C air on the other would pass (20C-5C)2m^2x1W/m^2C = 30 watts of heat power.

Exercise 3.3: How much heat flows through a 2mx6m [8'x16'] metric U0.5 [US U0.088] wall with a 20 C [36 F] temperature difference?

Exercise 3.4: How much heat would flow through a 3'x4' US U0.25 window with 70 F air inside and 30 F air outside?

Exercise 3.5: How much heat flows through a 24'x32' [7mx10m] R64 [metric U0.089] ceiling with a 40 F [22 C] temperature difference?

4. A three-dog house?

Now we can talk about superinsulated houses. They need to be small, like sleeping bags. Can a person heat a house with the help of a few dogs? An ASHRAE- (American Society of Heating, Refrigeration and AC Engineers) standard 154 pound 19.6 ft^2 human at rest makes about 350 Btu/h, like a 100 W light bulb...

Exercise 4.1 How big can an L' US R10 cube be, if it's 70 F inside and 30 F outside, with six outdoor faces, and it's heated by one person?

With R20 walls, L = sqrt(350/12) = 5.4'. Better, but cramped...

Exercise 4.2: How big can it be with the help of a 100 Btu/h dog? With two dogs? With three dogs? With three dogs and R50 strawbale walls?

If internal heat gain grows with volume, superinsulated apartment houses have minimum sizes. If each family of 8 1.65 Btu/h ASHRAE-standard 0.046 lb mice in a 1 foot cubical room makes 8x1.65 = 13.2 Btu/h-ft^3, what's the smallest cubical "mouse motel" with R5 external walls that can stay 70 F on a 30 F day? If 13.2L^3 = (70-30)6L^2/R5, L = 48/13.2 = 3.6', say 4’, with 512 mice in 64 rooms.

Exercise 4.3: What’s the smallest size for a mouse motel with R10 walls?

Germans build "Passive Houses" for people like this, with a "Passive House Institute" that certifies them. Jackson Lab in Bar Harbor, Maine is heated by 972,000 mice, even on -11 F cloudy days :-) A 6.61 pound ASHRAE-standard cat makes 68.02 Btu/h. A 50.0 pound normally-active ASHRAE dog makes 354.9 Btu/h. A 1,000 pound Jersey calf makes 4,000 Btu/h. Larger creatures with lower surface-to-volume ratios produce less heat per pound. In "The Return of the Solar Cat" (Patty Paw Press, 2003) PE Jim Augustyn mentions DOE's secret $1.6 trillion 100 megaton solar cat demonstration project "designed to mimic a solar cat's basic method for capturing energy on a grand scale."

A frugal 100 kWh per month of indoor electrical use (vs the US average of 833 kWh/mo or the UK average of 383 or the Indian average of 55) would add 100kWhx3412/(30dx24h) = 474 Btu/h to a person's 300 Btu/h, making an internal heat gain of 474+300 Btu/h = (70F-30F)6L^2/R20, which makes L = sqrt(774/12) = 8.03 feet for an R20 cube. With R50 strawbales, 474+300 Btu/h = (70-30)6L^2/R50 makes L = 12.7 feet max.

Exercise 4.4: If N families each use 774 Btu/h in 1024 ft^2 of floorspace with an 8' ceiling inside an L' R20 cube, what's Lmin?

But some people define a "solar house" as "one with no other form of heat," not even electrical usage or creatures-in-residence...

5. The sun, and weather

Suppose the R20 8' cube has an A ft^2 R2 window that admits 200 Btu/h-ft^2 of sun. If 200A = (70F-30F)(A/2+(6x8'x8'-A)/R20) = 18A+768, we can make it 70 F indoors on a 30 F day with an A = 4 ft^2 window, but the temperature would instantly drop to 30 F when night falls, unless the cube contained some thermal mass. Of course the sun doesn't put a constant 200 Btu/h-ft^2 into the window all day. This is a crude but useful full-sun approximation.

Exercise 5.1: What's the steady-state 24-hour temperature of an 8' [2m] R20 [metric R4] cube containing a very large thermal mass with little temp change over a day and a 4 ft^2 [0.4 m^2] US R2 [metric R0.4] window with 80% solar transmission, if 1000 Btu/ft^2/day [3.15 kWh/m^2/day] falls on the window over a long string of average 30 F [-1 C] days?

Keeping it warm at night requires a larger south window. The NREL (National Renewable Energy Lab) Solar Radiation Data Manual for Buildings (the "blue book" at http://rredc.nrel.gov ) implies that January is the worst-case month for solar house heating in Boulder, CO, when 1370 Btu/ft^2 of solar heat falls on a south wall on an average 29.7 F day, and 1370/(65-29.7) = 38.8. December is the worst-case month for solar house heating in Portland, OR, where solar heating is harder, with 470/(65-40.2) = 19.0. January is the worst-case month in Albuquerque, where solar heating is much easier, with 1640/(65-34.2) = 53.2.

PE Norman Saunders says the "worst-case month" is the one with the least solar heat per degree day, on average. This is usually December or January. Aerospace engineers often use "worst-case" specs instead of saying "I know you are going to be very happy with this airplane" or "It will go VERY far and VERY fast and carry a LOT of weight." :-)

People often wishfully overestimate passive solar performance, saying things like "we only burn a few cords of wood per year." How vague. What's "a few"? What kind of wood? A 4'x4'x8' cord of oak with 20% moisture content burned at 60% efficiency delivers 158 therms of heat (158x10^5 Btu), about the same as 158 gallons of oil. White cedar can only deliver 72 therms. And wood is ongoing work, compared to solar heat. Was it a cold year or a warm year or an average year?

What's the solar heating fraction of this house?

Every day is sunny for George and Charlotte Britton of Lafayette Hill. Their 2,900-square-foot house is blessed with energy bills 20 percent lower than one of comparable size... The design incorporates "passive" solar principles. There are large double pane windows and sliding glass doors on the south side. Inside, tile floors and a Trombe wall absorb the sun's heat during the day and radiate it at night... A fireplace on the south wall of the living area provides additional heat during colder months... Britton says "We have a fire every day of the winter."

Solar houses with no other form of heat are 100% solar-heated, with no doubt. Then it's just a matter of comfort, or temperature swing. You might buy an inexpensive 100% solar-heated tent with a -20 to 120 F temperature swing :-) A house that can keep itself warm in the worst-case month should do fine in other months. Should America's Cup boats have outboard motors? Should solar architects pay clients backup heating fuel bills, if any?

NREL's "blue book" contains long-term monthly average solar weather data for 239 US locations. NREL's Typical Meteorological Year (TMY2) hourly data files are useful for simple solar heating simulations with equations from high-school physics. My 12 Dec 2003 "BASIC TMY2 simulations" posting in the postlog file on the workshop CD has simple programs for direct gain, sunspace, ceiling mass, and solar closet houses with night setbacks.

PE Howard Reichmuth says a house that can pass an hourly TMY2 simulation with no backup fuel requirement can also survive a 30-year hourly simulation for the same location. But a calculator can be enough, with simple worst-case monthly average weather data. Can we connect these basic insights and understandings with LEED criteria which seem to have more to do with feng-shui and bicycle racks than fossil fuels? People cared more about heating bills in 1975...

I wish we could circulate hot air between a house and sunspace and stop the airflow at night with an Energy-10 simulation, but it looks like that's not part of the present package (available from www.sbicouncil.org) or future Energy-10 improvements. I still think of SBIC (aka PSIC) as crooked masonry salesmen out to raise the price and cripple performance of solar houses by loading up sunspaces with thermal mass, but people can change :-)

An 8' US R20 cube might be 70 F (eg 75 at dusk and 65 at dawn) over an average Boulder, CO December day with an A ft^2 R2 window with 80% transmission that admits 0.8x1370 = 1096 Btu/ft^2. If 1096A/24h = (70-30)(A/2+(6x64-A)/R20) = 18A+768, A = 28 ft^2. Thermal mass and a 4'x8' window would do, with a cube conductance of 32ft^2/R2+(6x64-32)/R20 = 34 Btu/h-F, about half window and half walls, like an early Los Alamos Labs passive solar test cell.

Exercise 5.2: What's A in Allentown, PA, where 800 Btu/ft^2 of sun falls on a south wall on an average 31.8 F December day?

6. Thermal mass and direct gain, aka "direct loss"

Water is cheap and heavy, but easily moved. When moving, it has a low thermal resistance. A square foot of 1" drywall (a "C1" board foot) stores 1 Btu/F, like a pound of water. A 95 lb cubic foot of dry sand with a 0.191 Btu/F-lb specific heat can store 0.191x95 = 18 Btu/F. Adding 40% water by volume would increase this to about 43 Btu/F-ft^3 and let it support something above.

Moving water can greatly lower its stagnant R0.4 per inch resistance. A cubic foot of concrete stores about 25 Btu/F, like wet sand, vs 64 for a cubic foot of water or steel, but concrete's R0.2/inch resistance makes it hard to move heat in and out. A 32 pound 8"x8"x16" hollow concrete block with 5.8 ft^2 of naked surface has a specific heat of 0.16 Btu/F-lb and stores 0.16x32 = 5 Btu/F.

An R fhub cube outside C Btu/F of thermal capacitance has a "time constant" RC = C/G in hours (dimensionally, (F-h/Btu)/(Btu/F) = h.) In RC hours, the indoor-outdoor temp difference decreases to about 1/3 (e^-1) of its initial value. If RC = 24 hours and T(0) = 75 F, T(h) = 30+(75-30)e^(-h/24) after h 30 F hours and T(d) = 30+(75-30)e^(-d) after d days, where e^x is the inverse of the natural log ln(x) function on a Casio fx-260 calculator. T starts at 75 F when e^(-0) = 1 and ends up at 30 F (the outdoor temp) much later, as e^(-t) approaches 0. Between those times, the exponential factor gradually squashes the initial temp diff of (75-30) = 45 degrees.

Exercise 6.1: What's T(d) after 1 30 F cloudy day?

A liter of water weighs 2.2 pounds, so it stores 2.2 Btu/F [4.2 kJ/kg-C or 1.2 Wh/kg-C.] N 2-liter bottles inside an 8' US R20 cube with an 8'x8' R2 south window and G = 64ft^2/R2+5x64ft^2/R20 = 48 Btu/h-F would make RC = 4.4N/48/24h = 0.00382N days.

Starting at 75 F, T(d) = 30+(75-30)e^(-d/0.00382N) = 30+(75-30)e^(-262d/N), d days later. When 65 = 30+(75-30)e^(-262d/N), ln((65-30)/(75-30)) = -262d/N, so N = -262d/ln((65-30)/(75-30)) = 1042d.

So storing heat for 1 day (d=1) requires 1042 2-liter bottles, 2 days takes 2084, and so on. If cloudy days are like coin flips, storing heat for 1 day makes the cube's solar heating fraction 50% max, with 75% for 2 days, 88% for 3, 94% for 4, and 97% for 5 cloudy days in a row, like 5 tails in a row, with a 2^-5 = 0.03 probability. We might flip a coin to simulate sun, with a constant outdoor temp and heads for an average day and tails for a cloudy day (no sun.) Real weather has more persistence, like a 3-state (cloudy, average, clear) Markov chain. We might roll a die with 1-2 for cloudy day, 3-4 for an average day with an average amount of sun, and 5-6 for a clear day with twice the average amount of sun.

But 5x1042 = 5210 is 5 pickup trucks full of bottles, and 23,000 pounds of water is a good ballast foundation :-) And PET bottles leak water vapor. They might need topping up once a year. If 9 4" diameter by 12" tall bottles fit in a cubic foot, they would occupy 5210/9 = 579 ft^3 of the cube's 8^3 = 512 ft^3 volume, rather intrusively :-) This fraction shrinks with larger cubes with smaller heat-losing surface to heat-storage volume ratios, so larger houses are easier to solar heat.

Exercise 6.2: What's the time constant of an empty windowless 8' cube with a 1/2" layer of drywall inside R20 foamboard insulation? If it's 70 F indoors and 30 F outdoors when we turn off the furnace at 10 PM, what will the temp be at 7 AM? What would it be if we add 1000 concrete blocks?

Exercise 6.3: What would it be if we filled it with water?

7. Indirect gain and night insulation

How often do we look out of black windows at night? They are often covered with curtains. With an R20 wall between the cube and a "low-thermal-mass sunspace" containing lots of south windows, we can circulate warm air between the sunspace and living space during the day and stop the air circulation at night, so we can have the best of both worlds, the daytime gain and views and of the window without nighttime and cloudy-day loss, and the window and thermal mass can be smaller, compared to direct gain, but it's hard to store solar heat from warm air, compared to mass in direct sun, because of the high airfilm resistance at the mass surface.

Exercise 7.1: Can we use less than the 5210 2-liter bottles in section 6, with a low-thermal-mass sunspace, ie indirect gain? How many are needed to keep the cube at least 65 F after 5 cloudy days, if it starts at 75 F?

We can stop air circulation at night with a one-way passive light plastic film damper hung over a vent hole in the R20 wall, with a mesh that only allows the film to swing open in one direction. I think Doug Kelbaugh (now Dean of the U. Mich architecture school) invented this "7-cent solution" in Princeton in 1973. Drew thinks he described it in the proceedings of the First Passive Solar Conference in Albuquerque. PE Norman Saunders says they need inspecting every two weeks for folded or torn or hung-up films, but automatic systems are nice, in general. Over the years, people usually stop moving movable insulation that requires twice-a-day people power, even the famous outdoor shutter over the drumwall of Steve Baer's house.

with plastic film dampers, described at http://www.BuildItSolar.com , which is on our workshop CD.

that might replace a plastic film damper. It would allow a downgoing cool indoor airstream near a window to cross an upgoing warm airstream behind an absorber mesh.

For room temperature control, the damper might be in series with an automatic foundation vent like Leslie-Locke's $12 8"x16" AFV-1B. Its louvers open as air temperature rises, but the bimetallic coil spring that opens them can be reversed to close the louvers as air temp rises and adjusted to change its soft threshold temperature by turning the spring mounting screw. NASA satellites use "deep-space coolers" that open to radiate heat as needed.

A motorized damper and thermostat could control a room air temp more accurately. Honeywell's $50 6161B1000 damper motor uses 2 watts when moving and 0 watts when stopped. If it runs 1 minute per day, that's 0.03 Wh. It might move with low-power electronics and a rechargeable battery with a 120 V charge pump or a 10 milliwatt PV cell. Drew says Tamarack Tech and Thermal Technology (John Schnebley) made a PV-powered shutter like this. Rich Komp would like to help make them in Nicaragua.

Fans use more power, but can work with accurate room temp thermostats, and they can double the collection efficiency of a low-temp sunspace, compared to natural thermosyphoning air. And fans can work through smaller holes than passive dampers, with less conduction loss to the sunspace at night.

8. Airflow and heatflow

One Btu can raise the temperature of 55 cubic feet of 70 F air 1 F, with a specific heat of 0.24 Btu/lb-F [1 kJ/kg-C or 0.28 Wh/kg-C], so 1 cubic foot per minute (cfm) of airflow with a 1 F temp diff moves about 1 (60/55) Btu per hour of heat. A 1 liter/second airstream with a 1 C temp diff moves about 1.2 W. ASHRAE says a person needs 15 cfm [7.1 L/s] of outdoor air to stay healthy. If this just leaks through a house, it adds about 15 Btu/h-F to its thermal conductance, but the house may leak less on a warm still day. An air-air heat exchanger can avoid this by preheating incoming cold air with outgoing warm air.

Bill Shurcliff proposed attaching a "lung" (picture a giant bellows) to the outside of a house, with a fan that periodically inflates and deflates the lung with house air, thus turning all the cracks and crevices in the house envelope into bidirectional heat exchangers with latent heat recovery, as in a camel's nose. An "infinite virtual lung" might divide a house into 2 partitions with a fan between them that periodically reverses. This could be efficient, done slowly, with lots of heat exchange area. (An exhaust fan or stack-effect chimney with Scandinavian-style "breathing walls" seems less efficient, with no heat recovered from exhaust air.)

If every 10'x10' wall section has a 4x10'x6"deepx1/32"wide perimeter crack with 40/32/12 = 0.104 ft^2 of cross-sectional area, a 4096 ft^2 envelope would have 4096/100x0.104 = 4.27 ft^2. If 30 cfm flows through it, the air speed would be 2x30/4.27 = 14 fpm. If each section has 40x0.5x2 = 40 ft^2 of heat exchange surface, Cmin = 30 and NTU = AU/Cmin = 4096/100x40x1.5/30 = 82 and E = 1-exp(-82). Very close to 100% efficiency :-)

LBNL tested a finite lung in the 80s, but the lung volume was small compared to the stud cavity volume, so there was little fresh air exchange. They aren't able to test it now, since their funding was cut by a factor of four. Someone else might, with a humidistat and Lasko's $50 2155 16" reversible fan (61 watts at 1360 cfm on high speed) and their 2A179 $88.15 programmable cycle timer and its $4.37 5X852 octal socket.

What is the mean size and standard deviation of house envelope cracks? A few big holes or cracks could make this work poorly. Are envelope cracks small in volume all the way through, or do they admit air to a large stud cavity? What about freezing and condensation?

Most US houses leak a lot more than 15 cfm/occupant. An old house might leak 2 air changes per hour (ACH), eg 2x2400x8/60 = 640 cfm for a 2400 ft^2 one- story house. A new US house might leak 1 house volume per hour. An "Energy Star" house might leak 0.5 ACH. Nisson and Dutt's Superinsulated House book suggests a 0.2 ACH target. Jeff Christian at Oak Ridge has built full-size 0.04 ACH test houses. We measure leaks by pressurizing and depressurizing a house to 50 Pascals (0.00725 psi or 0.2" H20) with a blower door and often divide airflow in cfm by 20 to estimate natural air leakage in wintertime.

Exercise 8.1: What's the conductance of a 40'60’x8' house with US R30 walls and an R60 ceiling and 8% of the floorspace as R4 windows and 0.5 ACH?

http://www.odpm.gov.uk/stellent/groups/odpm_buildreg/documents/page/odpm_ breg_600356.pdf (with a line-wrap) is a 1999 UK report. Page 2 says a 1981 Canadian housing development holds the world record for low air leakage. Page 5 mentions the world's tightest voluntary standard, Canada's IDEAS (post R2000) 0.15 m^3/h per m^2 of envelope at 50 Pa, for a natural air leakage of 2.5 cfm, or 0.008 ACH for a 2400 ft^2 1-story house :-)

The blower door as we know and love it today springs from technology first used in Sweden in 1977, where it was actually a blower window. The idea migrated to the United States with Ake Blomsterberg, who came to Princeton University to do research in 1979...

The Princeton researchers decided to mount the fan in a door because door sizes are more uniform than windows... the researchers found that hidden leaks accounted for a greater proportion of air leakage in a home than the more obvious culprits, such as windows, doors, and electrical outlets, a giant leap forward in our understanding of how a house operates...

A monster blower door is being used to test large residential and commercial buildings in Canada. The Super Sucker is a whopping 55,000 CFM fan that is 40 ft long and 5 ft in diameter. It is transported to the site on a flatbed trailer. It takes a team of five people [with safety belts?] to hook it up to a pair of double doors and perform the test.

From: http://hem.dis.anl.gov/eehem/95/951109.html

One empirical formula says an H foot chimney with A ft^2 vents at the top and bottom and an average temp Ti (F) inside and To outside has Q = 16.6Asqrt(HdT) cfm of airflow, where dT = Ti-To. The heatflow is QdT = 16.6Asqrt(H)dT^1.5 Btu/h, approximately.

A square foot of R1 vertical sunspace glazing with 90% transmission would gain 0.9x800 = 720 Btu over 6 hours on an average December day in Allentown, or more, with reflective ground to the south. With an atn(490/800) = 31 degree tilt to the south, it would gain 0.9sqrt(490^2+800^2) = 844 Btu/day. It could be one layer of clear flat GE Lexan polycarbonate plastic, which comes in 0.020"x49"x50' rolls and costs about $1.50 per square foot, or one layer of corrugated Dynaglas "solar siding," a greenhouse roofing material which costs about $1/ft^2. Both last at least 10 years. We might make an 8' long x 8' radius quarter-cylindrical sunspace with 3 $5 bows on 4' centers by bending 2 12' 1x3s into an 8' radius, with 1x3 spacer blocks and screws every 2'. A freestanding greenhouse might have 4-year greenhouse polyethylene film glazing (which costs about 5 cents/ft^2 in wide rolls) stretched over slightly-curved bows to avoid wind fatigue, in a 12' equilateral triangle, hinged at the top.

This could be a Food and Heat Producing Greenhouse like Bill Yanda's or Tom Lawand's. PE Howard Reichmuth's Ecotope greenhouse near Seattle has a steep transparent south wall and a reflective parabolic north wall that concentrates sun into a water trench. Greenhouses are more humid than houses, and they need to be warmer at night to avoid freezing plants, and it's hard to insulate them at night and also provide light for plants. Filling the space between two plastic film covers with air during the day and soap bubble foam at night is useful. Engineer Bob Quist in Toronto has developed a standard "replacement foam insulation" system for Venlo glass greenhouses (the most popular brand in the Netherlands.)

With A ft^2 of US R1 sunspace glazing and lots of thermal mass, we can keep an 8' R20 cube 70 F over an average 24h 31.8 F December day with a 39.2 max in Allentown if 720A = 6h(70-31.8)A/R1 (the daytime sunspace loss) + 18h(70-31.8)A/R20 (the nighttime sunspace loss) + 24h(70-31.8)(6x64-A)/R20 (the 24-hour loss from the rest of the cube), which makes A = 35 ft^2, eg a 5'x7'window, with a 6x64ft^2/R20 = 19.2 Btu/h-F cloudy-day conductance.

To size the sunspace vents, we might figure that 0.9x250 = 225 Btu/h-ft^2 of peak sun enters a square foot of R1 sunspace glazing and 70 F air near the glazing (on the south side of a dark screen north of the glazing, with warmer air north of the screen) loses (70-35)1ft^2/R1 = 35 Btu/h, for a net gain of 190 Btu/h-ft^2, or 6.7K Btu/h [1.1 kW] for 35 ft^2. If 70 F room air enters the sunspace through a lower vent and exits into the house at 120 F through a vent 8' above, the average temp inside the sunspace is 97.5 F, and 6.7K Btu/h = 16.6Asqrt(8')(97.5-70)^1.5 makes A = 0.98 ft^2.

9. Store the heat with less mass in the ceiling and a larger temp swing?

We might warm ceiling mass (eg soil "pugging" in Scotland) with hot air from a sunspace. Ignoring sunspace vent and ceiling airfilm resistances, on an average December day in Allentown, an 8' R20 cube with a ceiling temp T (F) and 8'x8' of R1 sunspace glazing might collect 0.9x800x64 = 46.1K Btu and lose about 6h(T-32)64ft^2/R1 from glazing during the day + 18h(65-32)64ft^2/R20 from the glazing at night + 24h(T-32)64ft^2/R20 up through the ceiling + 24h(65-32)3x64ft^2/R20 from the other 3 walls at night, which makes T = 46322/460.8 = 101 F, if gain equals loss on an average day.

If the ceiling conductance to slow-moving air below is 1.5x64 = 96 Btu/h-F and the cube needs (70-32)19.2 = 730 Btu/h of peak heat on a cloudy day, the ceiling must be at least 70+730/96 = 78 F to provide it. If the average temp is (101+78)/2 = 89.5 F for 5 cloudy days and the cube loses about 5x24h((65-32)4x64ft^2/R20 + (89.5-32)64ft^2/R20) = 72768 Btu, we need 72768/(101-78) = 3164 Btu/F of ceiling mass, eg 3164/64ft^2 = 49 pounds (9.5") of water per square foot of ceiling.

Or slightly less, using a more accurate differential equation. At ceiling temp T, I = (T-32)64ft^2/R20 + (65-32)4x64ft^2/R20 = 3.2T + 320 Btu/h flows from the ceiling mass C, so dT/dt = -I/C = aT + b = -3.2T/C - 320/C, and -a/b = -100, so T(t) = -a/b +(T(0)+a/b)e^-at = -100 + 201e^(-3.2t/C). After 5 days, T(120) = 78 = -100 + 201e^(-384/C) makes ln(178/201) = - 384/C, so C = 3160 Btu/F.

Exercise 9.1: How many inches of water are needed in December in Portland, OR when 470 Btu/ft^2 falls on a south wall on an average 40.2 F day, for an 8' R32 cube with an 8'x8' R2 sunspace window with 80% solar transmission?

Keeping the heat in the ceiling allows the cube to be cooler when vacant, so stored solar heat can last longer, especially if we reduce the ceiling's downward heatflow by radiation with a low-emissivity ceiling surface.

Exercise 9.2: How might this change if the cube only needs 8h/day of heat?

Solar pioneer Harold Hay built houses with roofponds and movable insulation. In the Barra system, hot sunspace air warms a hollow spancrete ceiling. We might collect and distribute heat with ceiling fin tubes, pumping cool water up during the day and thermosyphoning at night, with the help of a slow ceiling fan.

10. Radiation

Gustav Robert Kirchoff (1824-1887) said all radiation hitting a surface must be transmitted, absorbed, or reflected. Radiation obeyed. T+A+R=1. Kirchoff's identity says emissivity equals absorptivity for opaque black bodies or gray surfaces with a constant emissivity that doesn't depend on wavelength.

A surface emits se(T^4) of heat flux by radiation, where T is an absolute temperature in Rankine (F+460) or Kelvin (C+273) degrees, and s is the Stefan-Boltzman constant, 0.1714x10^-8 Btu/ft^2-R^4 or 5.660x10^-8 W/m^2-K^4. The surface's emissivity E varies from 0 (shiny) to 1 (dull.) Most natural surfaces are close to 1, but heat mirrors have emissivities close to 0.

The net heatflow from a surface at T1 degrees facing a T2 degree surface is se(T1^4-T2^4). For example, a square foot of 80 F Trombe wall might lose 0.1714x10^-8((80+460)^4-(30+460)^4) = 47 Btu/h by radiation on a 30 F night. In still air, it might lose 1.5(80-30) = 75 Btu/h by radiation and convection combined. A "selective surface" can absorb well at short solar wavelengths (<3 microns) and radiate poorly at longer heat wavelengths (10 microns at 80 F.) A 100% solar-heated house might have thick SIPs and radiant floors and a big enough well-insulated hot water tank to store heat for 5 cloudy days in a row and lots of $12/ft^2 vertical unglazed stainless steel ES solar roof collectors with durable selective surfaces from www.energie-solaire.com. With collector and ambient temps Tm0 and Ta=0 C and insolation I = 800 W/m^2 (full sun), x = (Tm-Ta)/I = 0.375 makes E = 0.959- 8.91x-0.047x^2 = 0.62, ie 62% in their efficiency formula.

Exercise 10.1: What's the temperature of a surface with e = 0.2 in 300 Btu/h-ft^2 sun on a 20 F day?

The most serious mistake was making the outer container of the receiver of plywood. We thought that the plywood would be sufficiently insulated from the copper panel which was the receiver proper, that it would not get too hot. The copper panel was separated from the plywood by 4" of fiberglass insulation. Nevertheless, the plywood caught fire and the unit was completely destroyed. We suppose this is a success, of sorts.

The copper panel which was plated with chrome black to provide a selective surface originally had a copper tube fastened to the back by a high melting point soft solder. When we first attempted to operate the unit, the soft solder melted, and the tubing became detatched from the panel. We attempted to repair this failure by silver soldering copper bars to the copper tube and screwing the bars to the plated copper sheet. This worked, after a fashion...

The glass window was originally tempered glass. This shattered due to thermal shock. We replaced it with ordinary window glass. This cracked due to thermal shock, but we were able to hold it in place well enough to make some measurements... The measurements we were able to make before the fire generally confirmed our thinking concerning the design...

from "A solar collector with no convection losses," (a downward-facing receiver over a 4:1 concentrating parabolic mirror), by H. Hinterberger and J. O'Meara of Fermilab, in "Sharing the Sun," A joint ASES/SESC conference, August 15th-20th, Winnepeg, Volume 2, pp 138-145.

If Tb is their approximate average temperature, the "linearized radiation conductance" between 2 surfaces Gr = 4seTb^3. A 96 F ceiling in a 70 F room with Tb = 543 R has Gr= 1.097 Btu/h-F-ft^2, roughly R0.9. If our cube needs (70F-30F)19.2Btu/h-F = 768 Btu/h, e(96-70)64x1.097 = 768 makes e = 0.42. We might make about 60% of the ceiling a low-e (0.05) paint or foil surface and the rest an ordinary surface and let radiation warm the room on an average day and use a ceiling fan and a thermostat to bring warm air down on cloudy days. A 72 F ceiling could supply 0.42(72-65)64ft^2x1.097 = 206 Btu/h of radiant heat to a 65 F room. The other 672-206 = 466 Btu/h might come from Q cfm of airflow. View figure 1 below in a fixed font:

1/96 1/Q (96+Q)/(96Q) 72 F ---www---www---65 F equivalent to 72------www------65 -----------> -----------> 466 Btu/h 466 Btu/h

where (72-65)96Q/(96+Q) = 466, so Q = 217 cfm. Grainger's $120 48" 315 rpm 86 W 21K cfm 4C853 ceiling fan might move 217 cfm at 217/21Kx315 = 3.3 rpm with 86(217/21K)^3 = 100 microwatts, according to the fan laws :-) Large slow fans can be very efficient and quiet. The Gossamer ceiling fan develope by Danny Parker at the Florida Solar Energy Center can move 1907 cfm with 9.1 watts. It might move 217 cfm with 1 average watt, running 11% of the time.

11. Even less ceiling mass, with a separate cloudy-day store?

If we only want to store 13.8K Btu of overnight heat on an average day in C Btu/F in a ceiling, with T(6) = 120+(72-120)e^(-6x96/C) = 120-48e^(-576/C) and (T(6)-72)C = 18,432, so C = 384/(1-e^(-576/C)). C = 384 on the right makes C = 494 on the left, and plugging that in on the right again makes C = 558, 596, 620, 635, 644, 649, 653, 655, 657, and 657 (whew!), so it looks like we can store overnight heat with 657/64ft^2 = 10.3 psf (about 2") of water in the ceiling, with T(6) = 120-48e^(-576/657) = 100 F.

If the 8' cube needs 4x24(70-30)19.2 = 73,728 Btu for 4 more cloudy days, this might come from a "solar closet" (see the CD paper) inside a sunspace. The heat lost from the closet air heater glazing efficiently heats sunspace air that heats the cube :-) with about 73,728/(120-70) = 1475 lb or 184 gal. or 23 ft^3 of water cooling from 120 F to 70 F over 4 days.

Kallwall's "solar battery" was an early solar closet. Tom Hopper built an insulated box containing fiberglass water cylinders with double-glazed south wall and an insulating curtain (5 layers of aluminized Mylar) that automatically rolled down over the glazing at night. Bill Shurcliff (in his 1980 Brick House Thermal Shutters and Shades book) said this remarkable Mylar curtain (with spacers that unfurled when it deployed) was invented by the Insulating Shade Co. of Branford, CN, and demonstrated at a 9/9/77 exhibition in Hartford, CN, and featured in the January 1979 issue of Popular Science Monthly.

With 1792 pounds of water in 56 10"x10"x13" 4-gallon ROPAK plastic tubs, a solar closet can supply 73,728 Btu as it cools from 120 to 120-73728/1792 = 78.9 F, with the tubs stacked 7-high and 4-wide and 2-deep in a 2'x4'x8' tall closet completely surrounded by insulation, with an air heater with its own closet vs sunspace glazing over the closet's insulated south wall and one-way dampers in that wall. With 56x4x10x13/144 = 202 ft^2 of tub surface and 300 Btu/h-F (buhfs) of water-air thermal conductance, we have fig 1 again with a 78.9-70 = 8.9 F temp diff and a (300+Q)/(300Q) resistor, so Q = 121 cfm.

With an 8' height difference and 121 cfm = 16.6Asqrt(8'(78.9-70)), we need 2 vents with A = 0.86 ft^2 for natural airflow into the cube.

As an alternative, the cloudy-day heat might come from 3072 pounds of water inside a 2'x4'x8' tall "shelfbox" with a 2'x4'x2' tall water tank below 18 2'x4' wire shelves on 4" centers, with 2" of water inside a $20 continuous piece of poly film duct folded to lay flat on each shelf and a small pump to circulate tank water up through the duct as needed. The tank might have a plastic pipe coil inside to make hot water for showers, with the help of an efficient external greywater heat exchanger, eg 300' of 1" plastic pipe pushed into two coils inside a 35"x23.5"" ID 55 gallon drum, if possible. Pushing the pipe into the drum without kinking is awkward but doable.

12. Greywater heat exchange, Big Fins, and solar ponds

About 15 years ago, Eric Olsen at Earthstar manufactured a heat exchanger with a copper coil in a 55 gallon drum to recover heat in kidney dialysis centers. Drew says professor Jane Davidson at U Minnesota is an expert on plastic heat exchangers. She says "The efficiency you calculate is actually effectiveness which is very different from efficiency. A high effectiveness does not mean a good heat exchanger!" I wonder what she means by that.

Gary and I are working on inexpensive "Big Fins" for water heating in sunspaces and a solar pond water heater and what might become an efficient counterflow greywater heat exchanger with heat storage. See http://BuildItSolar.com . We are using a new $35 55 gallon lined steel drum with a strong removable lid (because the drum might end up under 2' of greywater head, with the inlet and outlet above the lid) and bolt ring and a 3/4" bung and a 2" bung with a 3/4" threaded knockout, with 100 psi/ 73.4 F pipe from PT Industries at (800) 44 ENDOT. Their PBJ10041010001 $60 1"x300'100psi NSF-certified pipe is tested to 500 psi.

Lowes sells the rest of the hardware. It's all installed through the lid, so the drum itself has no holes. Here's a tentative parts list:

sales total # qty price description

25775 1 $5.73 24' of 1.25" sump pump hose (for greywater I/O) 105473 1 1.28 2 SS 1.75" hose clamps (for greywater hose) 54129 2 3.24 1.25" female adapter (greywater barb inlet and outlet) 23859 2 2.36 1.25x1.5" reducing male adapter (bulkhead fittings) 75912 1 0.51 2 1.25" conduit locknuts (bulkhead fittings) 28299 1 1.53 2 1.25" reducing conduit washers (") 22716 1 1.36 1.5" PVC street elbow (horizontal greywater inlet) 23830 1 2.98 10' 1.5" PVC pipe (for 3' greywater outlet dip tube)

The parts above are the greywater plumbing ($18.99.)

23766 4 1.28 3/4" CPVC male adapter (for 1" pipe barbs) 23766 2 0.64 3/4" CPVC male adapter (fresh water I/O) 42000 2 3.84 3/4" FIP to 3/4" male hose adapter 23813 1 1.39 10' 3/4" CPVC pipe (for 1"x3/4" fresh water outlet) 23760 2 0.96 3/4" CPVC T (fresh water I/O) 22643 2 0.86 3/4" CPVC street elbow (fresh water I/O) 4 - 1" 3/4" CPVC pipes (fresh water I/0) 1 - 3' 3/4" CPVC pipe (fresh water inlet) 22667 2 2.56 2 SS 1.125" hose clamps (fresh water I/O) 219980 1 4.87 10.1 oz DAP silicone ultra caulk (bulkhead fittings) 150887 1 3.94 4 oz primer and 4 oz PVC cement

The parts above are fresh water plumbing. Subtotal $39.33.

26371 1 6.83 1500 W electric water heater element 22230 1 2.31 1" galvanized T ("nut" for heating element) 61294 1 11.76 single element thermostat with safety 136343 1 0.56 5 10-24x3/4" machine screws (mount thermostat with 3) 33368 1 0.37 5 #10 SS flat washers (mount thermostat with 3) 198806 1 1.38 10 #0 rubber faucet washers (mount thermostat with 3) 8763 1 0.67 5 10-24 SS nuts (mount thermostat with 3)

The above would make a standalone water heater, if needed. Grand total: $63.21.

For 4 10 min showers per day and 20 minutes of dishwashing at 1.25 gpm we might heat 75 gallons of 55 F water to 110 with 8x75(110-55) = 33K Btu with about 10 kWh worth about $1/day at 10 cents/kWh. If the "heat capacity flow rate" Cmin = Cmax = 75gx8/24h = 25 Btu/h-F and the coil has A = 300Pi/12 = 78.5 ft^2 of surface with U = 10 Btu/h-F-ft^2 (for an HDPE pipe wall with slow-moving warm dirty water outside and 8x300Pi(1/2/12)^2 = 13 gallons of water inside), the "Number of heat Transfer Units" for this counterflow heat exchanger NTU = AU/Cmin = 78.5ft^2x10Btu/h-F-ft^2/25Btu/h-F = 31.4, so the "efficiency" E = NTU/(NTU+1) = 97% for hot water usage in bursts of less than 13 gallons. This works best with equal greywater and cold water flows, with either a 110 F water heater setting (preferable), or the heat exchanger output feeding the cold water shower inlet as well as the water heater.

The Hazen-Williams equation says L' of d" smooth pipe with G gpm flow has a 0.0004227LG^1.852d^-4.871 psi loss. At 1.25 gpm, the pressure drop for 2 150' coils of 1" pipe is 0.0004227x150x(1.25/2)^1.852x1^-4.871 = 0.03 psi.

If greywater leaves a shower drain at 100 F and fresh water enters at 50 F, the fresh water should leave at 50+0.97(100-50) = 98.5 F. Warming it further to 110 F would take 8x75(110-98.5) = 6.9K Btu/day with 2 kWh worth about 20 cents/at 10 cents/kWh, for a yearly savings of about ($1-0.20)365 = $292, or more, with a tighter shower enclosure and higher drain temperature. The 1500 W heater might operate 2kWh/1.5kW = 1.3 hours per day.

So far, we've discovered the steel drum wall needs insulation inside to preserve stratification because it conducts heat from warm greywater above to cooler greywater below. We might put foil foamboard with kerfs (knife cuts partially through the board) inside the drum, with a plastic film liner to keep it dry. We'd also like to improve incoming greywater stratification, to let it find its thermal level via the holey dip tube...

Zomeworks makes expensive "Big Fins" aluminum extrusions that snap onto copper pipes to heat water in sunspaces. The pressurized water inside the pipes thermosypohons through an insulated tank above, with no pumps, eat exchangers, controls, or antifreeze. We might use the wrapping jig at http://www.redrok.com/misc1.htm#wrappingjig to make less expensive Big Fins like this: . . . . . . aluminum flashing rivet . . 1/2" . . .........................|... . copper . . .........................|... . pipe . . rivet . . . . . . . . --- . . .

Comments welcome...

Nick

Notes for a Pennsylvania Renewable Energy Festival Workshop on Solar House Heating and Natural Cooling Techniques

Kempton, PA September 24, 2005

Written by Nick Pine, with Drew Gillett and Rich Komp

Debatable Conclusions

1. Heat flows like electricity. 2. Solar heat can be 100 times cheaper than solar electricity. 3. Superinsulated houses have to be very small or very large. 4. Direct gain houses can be improved. 5. Indirect gain can be more efficient. 6. We might store heat in the ceiling. 7. We might have a separate cloudy-day heat store. 8. Low temp heat storage and distribution are difficult. 9. Shurcliff's lung might be a good air-air heat exchanger. 10. Greywater heat exchangers, Big Fins and solar ponds can help. 11. We might also gather heat from PVs. 12. Smart ventilation can be helpful. 13. Swamp coolers can be improved.

0.0 Introduction

The US has 5% of the world's population and consumes 26% of the world's energy. House heating and cooling accounts for about one third of that. In 1980, "envelope house" inventor Tom Smith said:

It's a snap to save energy in the US. As soon as more people become involved in the basic math of heat transfer and get a gut-level, as well as intellectual, grasp on how a house works, solution after solution will appear.

This workshop aims at improving that grasp, which we can control better than our US cheap energy policy... If we paid related costs of healthcare and air pollution and Gulf wars at the pump, gasoline would be a lot more expensive. Drew says this writeup needs exercises for the reader. OK:

Exercise 0.1: The US consumed 21 million 42 gallon $41 barrels of oil per day in 2004. What goes into the real cost per gallon? (Debatable answers appear at the end of these notes :-)

Most people think "electricity" when they hear "energy," even though most houses need more heating energy than electrical energy (the ratio is 1:1 in Hawaii and 5:1 in Vermont.) It's easy to shrink the small electrical slice of the home energy pie with compact fluorescent (CF) lights and more efficient appliances. Solar heat can be very inexpensive compared to solar electricity. PV panels at $3 per peak watt cost 150X more than polycarbonate glazing at $1/50W = 0.02/Pw. And sunspaces add floorspace to a house.

A square foot of "solar collector" only collects about $1/year at $1/gallon so anything (except PVs??? :-) that costs more than $10/ft^2 (half labor) and only collects energy with no other purpose seems economically-doomed...

Exercise 0.2: Should we a) replace a 60 W bulb with a 14 watt CF or b) buy 60-14 = 46 additional watts of PV power?

Most of us "know" how to design passive solar houses with well-established rules of thumb, but let's relax and take a fresh look from a standpoint of basic physics...

Berlin is a nice town and there were many opportunities for a student to spend his time in an agreeable manner, for instance with the nice girls. But instead of that we had to perform big and awful calculations.

Konrad Zuse, inventor of the 1936 Z1 computer

Overview

This is a workshop on "Ohm's law for heatflow" with applications to solar water and house heating and natural cooling. We'll discuss a solar pond and a simple greywater heat exchanger, some inexpensive plastic pipe coiled inside a 55-gallon drum. With hot water bursts of 13 gallons or less, it could be 97% efficient. If it is, why bother with solar hot water?

We'll provide arithmetic tools and data and strategies needed to site-build effective house heating and cooling systems using inexpensive materials and skills. Participants will need some familiarity with high school algebra.

We'll discuss power, energy, heatflow, and overnight and cloudy day heat storage at a high-school math and physics level, with insulation values and heat capacities of materials, simple equations involving time constants, evaporative and night ventilation cooling, passive and low-energy solar heating, climate data, and schemes for houses that are 100% solar-heated and naturally cooled, by design. We'll provide a calculator (Steve Baer says "Throw away your calculator." :-) and a CD-ROM. Promising techniques include solar closets, trickle collectors, "pancake houses," soap bubble foam insulation, and solar attics, including systems to collect heat and electricity from water-cooled standard PV panels.

Rich Komp is president of the Maine Solar Energy Association and a PV author with a PChem PhD, Drew Gillett is a Professional Engineer with civil engineering and architectural degrees, and I'm an EE by training.

Disclaimer

Some of the techniques we describe are experimental. Some have never been tried. We do not accept responsibility for their safety or functionality.

1. Power and energy

Energy is the stuff we pay for, measured in Joules or watt-hours or kilowatt-hours (kWh) or Calories or "British thermal units" (Btu), no longer used in Britain :-) The British now use joules or kWh. A Btu is a quantity of heat, about the same as the energy in a kitchen match or a mouse-hour. One Btu can heat one pound (16 ounces) of water one degree F. One Calorie (capital C, 4.19 kilojoules) can heat one kilogram of water 1 C.

Exercise 1.1: How many Btu [joules] are needed to heat 8 ounces [0.25 kg] of water from 50 to 212 F [10 to 100 C] to make a cup of tea? [the brackets describe

___comparable but not identical___metric exercises.]Power is the rate of energy flow over time. A mere number, vs the stuff we pay for. Energy is power times time. One watt-hour of energy is equivalent to 3.41 Btu. If energy were miles traveled, power would be miles per hour. If energy were a paycheck, power would be an hourly rate of pay.

Exercise 1.2: How long would it take to heat the tea water with a 300 W immersion heater?

We might check this with an immersion heater and a watch and a $100 Raytek IR thermometer. Or a HOBO from Onset Computer Corp (1-800-LOGGERS.) Their $119 battery-powered U12-013 HOBO is about the size of a matchbox. It can record 43,000 12-bit samples at 1 second to 18 hour intervals of its own temperature and relative humidity (RH), with jacks for 2 more temperature probes or other devices on cables, and upload them to a PC spreadsheet via a USB port.

People often confuse power and energy, as in "My house uses lots of power" (vs energy) or "My furnace capacity is 50,000 Btu," (vs Btu/h.) Power is measured in watts or kilowatts (kW.) Unlike energy, it can't be used or consumed.

People confuse heat and temperature, too. A bathtub full of hot water contains a lot of useful house heat, compared to a candle, but the candle is much hotter. A lower minimum usable temperature increases useful heat. Temperature is a measure of heat intensity. A 12-volt 100 amp-hour 50 pound automobile battery stores 267 times more energy (12Vx100Ah = 1200 Wh) than a 9-volt 500 milliamp-hour (9Vx0.5Ah = 4.5 Wh) 2 ounce transistor radio battery, at a lower voltage (ie "electrical temperature.") The $40 battery can store about 200 kWh over its lifetime, at 20 cents/kWh. A $1 cubic foot of water cooling from 130 to 80 F stores (130F-80F)64Btu/F = 3200 Btu, ie about 1 kWh, with a much longer lifetime and simpler I/O.

2. Rich Komp, Ohm, and Newton

Rich Komp (who is still alive) says heat moves by conduction (a hot frying pan handle), convection (including air movement), radiation (the sun brings about 1000 watts per square meter or 300 Btu per hour per square foot on a clear day at noon in the Sahara), and phase change (144 Btu melts a pound of ice and 1000 Btu evaporates a pound of water.)

About 300 years ago, Isaac Newton said the amount of heat that flows through a wall is proportional to its area and the temperature difference from one side to the other and its thermal conductance. About 100 years later, Georg Ohm said the same about electricity: V = IR, ie a current I in amps times a resistance R in ohms produces a voltage difference V. Electrical conductance is measured on "mhos," which is ohms spelled backwards (or "siemens" which is snemeis spelled backwards :-) If 6 amps flows through a 2 ohm resistor, we'll see V = IR = 6Ax2ohms = 12 volts across it. Another example: 120 volts across 48 ohms makes I = V/R = 120V/48ohms = 2.5 amps flow, with electrical power P = IV = 2.5Ax120V = 300 watts. (These units are capitalized only when abbreviated, A vs amperes, V vs volts, and so on.)

Exercise 2.1: If 24 A flows through a 12 V headlamp, what's the resistance?

3. Thermal ohms?

Ohm's law for heatflow (aka Newton's law of cooling) uses a temperature difference instead of a voltage difference. Heatflow is measured in units of power, in watts or Btu per hour. There's no such thing as a thermal "ohm." The closest thing is the US "R-value" stamped on foamboards and batts in hardware stores.

Beadboard (expanded white polystyrene coffee cup material) has a US R-value of 4 (ft^2-F-h/Btu) per inch (in these notes, all R-values are US unless otherwise specified.) Blue, pink or green Styrofoam board has US R5/inch. So does air, for downward heatflow. The slow-moving indoor and windier outdoor air films near a single layer of glass have a combined US R-value of about 1. A smooth square foot of surface in slow-moving air loses about 1.5 Btu/h-F, with a U1.5 or an R2/3 airfilm resistance. A rough square foot of surface in V mph air loses about 2+V/2 Btu/h-F. Larger objects have lower airfilm conductances, since air warms or cools more as it passes over them, which reduces the temperature difference between the air and the surface.

Tiny cold soap bubbles (1/16" at 50 F) have about R3 per inch. Bill Sturm filled the space between two polyethylene film covers of his Calgary greenhouse with air during the day and soap bubble foam insulation at night and measured an 82% propane energy savings with and without the foam on -20 F nights, without much solar energy storage. In one system, air from a shop vac makes bubbles from a long PVC pipe full of holes in a trough near the ground containing a 10% detergent solution. When the bubbles reach a return air vent at the top of the greenhouse at night, they push on a piece of window screen and a microswitch turns off the shop vac until the bubbles collapse a bit in a few minutes, and the shop vac runs for a few more seconds as needed. Bill thinks about heating poly film refugee shelters this way in cold climates.

Fiberglass has R3.5/inch, or half that, if it contains 2% moisture (which it might, in a cold climate, with some holes in a vapor barrier and condensation), or even less, if air circulates within it, in very cold places. But R-values are not quite thermal ohms. To find the thermal resistance of a wall, divide the R-value by the area. An 8'x10' US R20 wall has a resistance of R20/(8'x10') = 0.25 F-h/Btu. We might call this "0.25 fhubs" or "4 buhfs." We can add buhfs for different kinds of house walls and windows like resistors in parallel and add fhubs for series layers of wall insulation.

Exercise 3.1: If 6 Btu/h [2 watts] of heat power flows through a wall with a 12 F [7 C] temp diff, what's the wall's thermal resistance in fhubs [C/W]?

Exercise 3.2: What's the conductance of an 8'x16' [2mx5m] R10 [metric R2] wall?

If it's 70 F indoors and 30 F outdoors, (70F-30F)/5F-h/Btu= 8 Btu/h [2.3 watts] of heat power flows through 1 ft^2 of R5 wall. But stud walls have lower stud resistances (R1/inch "thermal bridging") in parallel with the insulation resistance, and drywall and sheathing on each side of the wall act as stud fins for heat collection and distribution, which lowers the overall R-value.

Ignoring the fins, we can look at a 10.67 ft^2 16"x8' wall section with R13 insulation framed with R1 per inch 2x4s on all edges as a "stud conductor" with (16/12+8)1.5/12 = 1.17 ft^2 of stud surface in parallel with a 10.67-1.17 = 9.5 ft^2 "insulation conductor." The studs have a 1/(R1x3.5") = US U0.286 U-value and the insulation has 1/R13 = U0.0769. The combined conductance is 1.17x0.286+9.5x0.0769 = 1.07 Btu/h-F, for an effective US R-value of 10.67ft^2/1.07 buhf = R10.0, vs R13, without the studs.

Structural Insulated Panels (SIPs, glued plywood-foam-plywood sandwiches) have less thermal bridging and fewer air leaks. So do walls with plywood I-joists used as studs and loose-fill insulation (as used by ME Al Eggen, snipped-for-privacy@mindspring.org) and walls with a layer of foamboard insulation over the studs. An 8'x24' R24 6" SIP wall with a 40 F temperature difference would allow (70F-30F)8'x24'/R24 = 320 Btu/h of heatflow.

Metric (European, Canadian, Australian...) U-values are 5.68 times bigger than US U-values. A metric U1 window has 1 W/m^2C of thermal conductance. It's like a low-loss US R5.68 window. A 2 m^2 metric U1 window with 20 C air on one side and 5 C air on the other would pass (20C-5C)2m^2x1W/m^2C = 30 watts of heat power.

Exercise 3.3: How much heat flows through a 2mx6m [8'x16'] metric U0.5 [US U0.088] wall with a 20 C [36 F] temperature difference?

Exercise 3.4: How much heat would flow through a 3'x4' US U0.25 window with 70 F air inside and 30 F air outside?

Exercise 3.5: How much heat flows through a 24'x32' [7mx10m] R64 [metric U0.089] ceiling with a 40 F [22 C] temperature difference?

4. A three-dog house?

Now we can talk about superinsulated houses. They need to be small, like sleeping bags. Can a person heat a house with the help of a few dogs? An ASHRAE- (American Society of Heating, Refrigeration and AC Engineers) standard 154 pound 19.6 ft^2 human at rest makes about 350 Btu/h, like a 100 W light bulb...

Exercise 4.1 How big can an L' US R10 cube be, if it's 70 F inside and 30 F outside, with six outdoor faces, and it's heated by one person?

With R20 walls, L = sqrt(350/12) = 5.4'. Better, but cramped...

Exercise 4.2: How big can it be with the help of a 100 Btu/h dog? With two dogs? With three dogs? With three dogs and R50 strawbale walls?

If internal heat gain grows with volume, superinsulated apartment houses have minimum sizes. If each family of 8 1.65 Btu/h ASHRAE-standard 0.046 lb mice in a 1 foot cubical room makes 8x1.65 = 13.2 Btu/h-ft^3, what's the smallest cubical "mouse motel" with R5 external walls that can stay 70 F on a 30 F day? If 13.2L^3 = (70-30)6L^2/R5, L = 48/13.2 = 3.6', say 4’, with 512 mice in 64 rooms.

Exercise 4.3: What’s the smallest size for a mouse motel with R10 walls?

Germans build "Passive Houses" for people like this, with a "Passive House Institute" that certifies them. Jackson Lab in Bar Harbor, Maine is heated by 972,000 mice, even on -11 F cloudy days :-) A 6.61 pound ASHRAE-standard cat makes 68.02 Btu/h. A 50.0 pound normally-active ASHRAE dog makes 354.9 Btu/h. A 1,000 pound Jersey calf makes 4,000 Btu/h. Larger creatures with lower surface-to-volume ratios produce less heat per pound. In "The Return of the Solar Cat" (Patty Paw Press, 2003) PE Jim Augustyn mentions DOE's secret $1.6 trillion 100 megaton solar cat demonstration project "designed to mimic a solar cat's basic method for capturing energy on a grand scale."

A frugal 100 kWh per month of indoor electrical use (vs the US average of 833 kWh/mo or the UK average of 383 or the Indian average of 55) would add 100kWhx3412/(30dx24h) = 474 Btu/h to a person's 300 Btu/h, making an internal heat gain of 474+300 Btu/h = (70F-30F)6L^2/R20, which makes L = sqrt(774/12) = 8.03 feet for an R20 cube. With R50 strawbales, 474+300 Btu/h = (70-30)6L^2/R50 makes L = 12.7 feet max.

Exercise 4.4: If N families each use 774 Btu/h in 1024 ft^2 of floorspace with an 8' ceiling inside an L' R20 cube, what's Lmin?

But some people define a "solar house" as "one with no other form of heat," not even electrical usage or creatures-in-residence...

5. The sun, and weather

Suppose the R20 8' cube has an A ft^2 R2 window that admits 200 Btu/h-ft^2 of sun. If 200A = (70F-30F)(A/2+(6x8'x8'-A)/R20) = 18A+768, we can make it 70 F indoors on a 30 F day with an A = 4 ft^2 window, but the temperature would instantly drop to 30 F when night falls, unless the cube contained some thermal mass. Of course the sun doesn't put a constant 200 Btu/h-ft^2 into the window all day. This is a crude but useful full-sun approximation.

Exercise 5.1: What's the steady-state 24-hour temperature of an 8' [2m] R20 [metric R4] cube containing a very large thermal mass with little temp change over a day and a 4 ft^2 [0.4 m^2] US R2 [metric R0.4] window with 80% solar transmission, if 1000 Btu/ft^2/day [3.15 kWh/m^2/day] falls on the window over a long string of average 30 F [-1 C] days?

Keeping it warm at night requires a larger south window. The NREL (National Renewable Energy Lab) Solar Radiation Data Manual for Buildings (the "blue book" at http://rredc.nrel.gov ) implies that January is the worst-case month for solar house heating in Boulder, CO, when 1370 Btu/ft^2 of solar heat falls on a south wall on an average 29.7 F day, and 1370/(65-29.7) = 38.8. December is the worst-case month for solar house heating in Portland, OR, where solar heating is harder, with 470/(65-40.2) = 19.0. January is the worst-case month in Albuquerque, where solar heating is much easier, with 1640/(65-34.2) = 53.2.

PE Norman Saunders says the "worst-case month" is the one with the least solar heat per degree day, on average. This is usually December or January. Aerospace engineers often use "worst-case" specs instead of saying "I know you are going to be very happy with this airplane" or "It will go VERY far and VERY fast and carry a LOT of weight." :-)

People often wishfully overestimate passive solar performance, saying things like "we only burn a few cords of wood per year." How vague. What's "a few"? What kind of wood? A 4'x4'x8' cord of oak with 20% moisture content burned at 60% efficiency delivers 158 therms of heat (158x10^5 Btu), about the same as 158 gallons of oil. White cedar can only deliver 72 therms. And wood is ongoing work, compared to solar heat. Was it a cold year or a warm year or an average year?

What's the solar heating fraction of this house?

Every day is sunny for George and Charlotte Britton of Lafayette Hill. Their 2,900-square-foot house is blessed with energy bills 20 percent lower than one of comparable size... The design incorporates "passive" solar principles. There are large double pane windows and sliding glass doors on the south side. Inside, tile floors and a Trombe wall absorb the sun's heat during the day and radiate it at night... A fireplace on the south wall of the living area provides additional heat during colder months... Britton says "We have a fire every day of the winter."

Solar houses with no other form of heat are 100% solar-heated, with no doubt. Then it's just a matter of comfort, or temperature swing. You might buy an inexpensive 100% solar-heated tent with a -20 to 120 F temperature swing :-) A house that can keep itself warm in the worst-case month should do fine in other months. Should America's Cup boats have outboard motors? Should solar architects pay clients backup heating fuel bills, if any?

NREL's "blue book" contains long-term monthly average solar weather data for 239 US locations. NREL's Typical Meteorological Year (TMY2) hourly data files are useful for simple solar heating simulations with equations from high-school physics. My 12 Dec 2003 "BASIC TMY2 simulations" posting in the postlog file on the workshop CD has simple programs for direct gain, sunspace, ceiling mass, and solar closet houses with night setbacks.

PE Howard Reichmuth says a house that can pass an hourly TMY2 simulation with no backup fuel requirement can also survive a 30-year hourly simulation for the same location. But a calculator can be enough, with simple worst-case monthly average weather data. Can we connect these basic insights and understandings with LEED criteria which seem to have more to do with feng-shui and bicycle racks than fossil fuels? People cared more about heating bills in 1975...

I wish we could circulate hot air between a house and sunspace and stop the airflow at night with an Energy-10 simulation, but it looks like that's not part of the present package (available from www.sbicouncil.org) or future Energy-10 improvements. I still think of SBIC (aka PSIC) as crooked masonry salesmen out to raise the price and cripple performance of solar houses by loading up sunspaces with thermal mass, but people can change :-)

An 8' US R20 cube might be 70 F (eg 75 at dusk and 65 at dawn) over an average Boulder, CO December day with an A ft^2 R2 window with 80% transmission that admits 0.8x1370 = 1096 Btu/ft^2. If 1096A/24h = (70-30)(A/2+(6x64-A)/R20) = 18A+768, A = 28 ft^2. Thermal mass and a 4'x8' window would do, with a cube conductance of 32ft^2/R2+(6x64-32)/R20 = 34 Btu/h-F, about half window and half walls, like an early Los Alamos Labs passive solar test cell.

Exercise 5.2: What's A in Allentown, PA, where 800 Btu/ft^2 of sun falls on a south wall on an average 31.8 F December day?

6. Thermal mass and direct gain, aka "direct loss"

Water is cheap and heavy, but easily moved. When moving, it has a low thermal resistance. A square foot of 1" drywall (a "C1" board foot) stores 1 Btu/F, like a pound of water. A 95 lb cubic foot of dry sand with a 0.191 Btu/F-lb specific heat can store 0.191x95 = 18 Btu/F. Adding 40% water by volume would increase this to about 43 Btu/F-ft^3 and let it support something above.

Moving water can greatly lower its stagnant R0.4 per inch resistance. A cubic foot of concrete stores about 25 Btu/F, like wet sand, vs 64 for a cubic foot of water or steel, but concrete's R0.2/inch resistance makes it hard to move heat in and out. A 32 pound 8"x8"x16" hollow concrete block with 5.8 ft^2 of naked surface has a specific heat of 0.16 Btu/F-lb and stores 0.16x32 = 5 Btu/F.

An R fhub cube outside C Btu/F of thermal capacitance has a "time constant" RC = C/G in hours (dimensionally, (F-h/Btu)/(Btu/F) = h.) In RC hours, the indoor-outdoor temp difference decreases to about 1/3 (e^-1) of its initial value. If RC = 24 hours and T(0) = 75 F, T(h) = 30+(75-30)e^(-h/24) after h 30 F hours and T(d) = 30+(75-30)e^(-d) after d days, where e^x is the inverse of the natural log ln(x) function on a Casio fx-260 calculator. T starts at 75 F when e^(-0) = 1 and ends up at 30 F (the outdoor temp) much later, as e^(-t) approaches 0. Between those times, the exponential factor gradually squashes the initial temp diff of (75-30) = 45 degrees.

Exercise 6.1: What's T(d) after 1 30 F cloudy day?

A liter of water weighs 2.2 pounds, so it stores 2.2 Btu/F [4.2 kJ/kg-C or 1.2 Wh/kg-C.] N 2-liter bottles inside an 8' US R20 cube with an 8'x8' R2 south window and G = 64ft^2/R2+5x64ft^2/R20 = 48 Btu/h-F would make RC = 4.4N/48/24h = 0.00382N days.

Starting at 75 F, T(d) = 30+(75-30)e^(-d/0.00382N) = 30+(75-30)e^(-262d/N), d days later. When 65 = 30+(75-30)e^(-262d/N), ln((65-30)/(75-30)) = -262d/N, so N = -262d/ln((65-30)/(75-30)) = 1042d.

So storing heat for 1 day (d=1) requires 1042 2-liter bottles, 2 days takes 2084, and so on. If cloudy days are like coin flips, storing heat for 1 day makes the cube's solar heating fraction 50% max, with 75% for 2 days, 88% for 3, 94% for 4, and 97% for 5 cloudy days in a row, like 5 tails in a row, with a 2^-5 = 0.03 probability. We might flip a coin to simulate sun, with a constant outdoor temp and heads for an average day and tails for a cloudy day (no sun.) Real weather has more persistence, like a 3-state (cloudy, average, clear) Markov chain. We might roll a die with 1-2 for cloudy day, 3-4 for an average day with an average amount of sun, and 5-6 for a clear day with twice the average amount of sun.

But 5x1042 = 5210 is 5 pickup trucks full of bottles, and 23,000 pounds of water is a good ballast foundation :-) And PET bottles leak water vapor. They might need topping up once a year. If 9 4" diameter by 12" tall bottles fit in a cubic foot, they would occupy 5210/9 = 579 ft^3 of the cube's 8^3 = 512 ft^3 volume, rather intrusively :-) This fraction shrinks with larger cubes with smaller heat-losing surface to heat-storage volume ratios, so larger houses are easier to solar heat.

Exercise 6.2: What's the time constant of an empty windowless 8' cube with a 1/2" layer of drywall inside R20 foamboard insulation? If it's 70 F indoors and 30 F outdoors when we turn off the furnace at 10 PM, what will the temp be at 7 AM? What would it be if we add 1000 concrete blocks?

Exercise 6.3: What would it be if we filled it with water?

7. Indirect gain and night insulation

How often do we look out of black windows at night? They are often covered with curtains. With an R20 wall between the cube and a "low-thermal-mass sunspace" containing lots of south windows, we can circulate warm air between the sunspace and living space during the day and stop the air circulation at night, so we can have the best of both worlds, the daytime gain and views and of the window without nighttime and cloudy-day loss, and the window and thermal mass can be smaller, compared to direct gain, but it's hard to store solar heat from warm air, compared to mass in direct sun, because of the high airfilm resistance at the mass surface.

Exercise 7.1: Can we use less than the 5210 2-liter bottles in section 6, with a low-thermal-mass sunspace, ie indirect gain? How many are needed to keep the cube at least 65 F after 5 cloudy days, if it starts at 75 F?

We can stop air circulation at night with a one-way passive light plastic film damper hung over a vent hole in the R20 wall, with a mesh that only allows the film to swing open in one direction. I think Doug Kelbaugh (now Dean of the U. Mich architecture school) invented this "7-cent solution" in Princeton in 1973. Drew thinks he described it in the proceedings of the First Passive Solar Conference in Albuquerque. PE Norman Saunders says they need inspecting every two weeks for folded or torn or hung-up films, but automatic systems are nice, in general. Over the years, people usually stop moving movable insulation that requires twice-a-day people power, even the famous outdoor shutter over the drumwall of Steve Baer's house.

with plastic film dampers, described at http://www.BuildItSolar.com , which is on our workshop CD.

that might replace a plastic film damper. It would allow a downgoing cool indoor airstream near a window to cross an upgoing warm airstream behind an absorber mesh.

For room temperature control, the damper might be in series with an automatic foundation vent like Leslie-Locke's $12 8"x16" AFV-1B. Its louvers open as air temperature rises, but the bimetallic coil spring that opens them can be reversed to close the louvers as air temp rises and adjusted to change its soft threshold temperature by turning the spring mounting screw. NASA satellites use "deep-space coolers" that open to radiate heat as needed.

A motorized damper and thermostat could control a room air temp more accurately. Honeywell's $50 6161B1000 damper motor uses 2 watts when moving and 0 watts when stopped. If it runs 1 minute per day, that's 0.03 Wh. It might move with low-power electronics and a rechargeable battery with a 120 V charge pump or a 10 milliwatt PV cell. Drew says Tamarack Tech and Thermal Technology (John Schnebley) made a PV-powered shutter like this. Rich Komp would like to help make them in Nicaragua.

Fans use more power, but can work with accurate room temp thermostats, and they can double the collection efficiency of a low-temp sunspace, compared to natural thermosyphoning air. And fans can work through smaller holes than passive dampers, with less conduction loss to the sunspace at night.

8. Airflow and heatflow

One Btu can raise the temperature of 55 cubic feet of 70 F air 1 F, with a specific heat of 0.24 Btu/lb-F [1 kJ/kg-C or 0.28 Wh/kg-C], so 1 cubic foot per minute (cfm) of airflow with a 1 F temp diff moves about 1 (60/55) Btu per hour of heat. A 1 liter/second airstream with a 1 C temp diff moves about 1.2 W. ASHRAE says a person needs 15 cfm [7.1 L/s] of outdoor air to stay healthy. If this just leaks through a house, it adds about 15 Btu/h-F to its thermal conductance, but the house may leak less on a warm still day. An air-air heat exchanger can avoid this by preheating incoming cold air with outgoing warm air.

Bill Shurcliff proposed attaching a "lung" (picture a giant bellows) to the outside of a house, with a fan that periodically inflates and deflates the lung with house air, thus turning all the cracks and crevices in the house envelope into bidirectional heat exchangers with latent heat recovery, as in a camel's nose. An "infinite virtual lung" might divide a house into 2 partitions with a fan between them that periodically reverses. This could be efficient, done slowly, with lots of heat exchange area. (An exhaust fan or stack-effect chimney with Scandinavian-style "breathing walls" seems less efficient, with no heat recovered from exhaust air.)

If every 10'x10' wall section has a 4x10'x6"deepx1/32"wide perimeter crack with 40/32/12 = 0.104 ft^2 of cross-sectional area, a 4096 ft^2 envelope would have 4096/100x0.104 = 4.27 ft^2. If 30 cfm flows through it, the air speed would be 2x30/4.27 = 14 fpm. If each section has 40x0.5x2 = 40 ft^2 of heat exchange surface, Cmin = 30 and NTU = AU/Cmin = 4096/100x40x1.5/30 = 82 and E = 1-exp(-82). Very close to 100% efficiency :-)

LBNL tested a finite lung in the 80s, but the lung volume was small compared to the stud cavity volume, so there was little fresh air exchange. They aren't able to test it now, since their funding was cut by a factor of four. Someone else might, with a humidistat and Lasko's $50 2155 16" reversible fan (61 watts at 1360 cfm on high speed) and their 2A179 $88.15 programmable cycle timer and its $4.37 5X852 octal socket.

What is the mean size and standard deviation of house envelope cracks? A few big holes or cracks could make this work poorly. Are envelope cracks small in volume all the way through, or do they admit air to a large stud cavity? What about freezing and condensation?

Most US houses leak a lot more than 15 cfm/occupant. An old house might leak 2 air changes per hour (ACH), eg 2x2400x8/60 = 640 cfm for a 2400 ft^2 one- story house. A new US house might leak 1 house volume per hour. An "Energy Star" house might leak 0.5 ACH. Nisson and Dutt's Superinsulated House book suggests a 0.2 ACH target. Jeff Christian at Oak Ridge has built full-size 0.04 ACH test houses. We measure leaks by pressurizing and depressurizing a house to 50 Pascals (0.00725 psi or 0.2" H20) with a blower door and often divide airflow in cfm by 20 to estimate natural air leakage in wintertime.

Exercise 8.1: What's the conductance of a 40'60’x8' house with US R30 walls and an R60 ceiling and 8% of the floorspace as R4 windows and 0.5 ACH?

http://www.odpm.gov.uk/stellent/groups/odpm_buildreg/documents/page/odpm_ breg_600356.pdf (with a line-wrap) is a 1999 UK report. Page 2 says a 1981 Canadian housing development holds the world record for low air leakage. Page 5 mentions the world's tightest voluntary standard, Canada's IDEAS (post R2000) 0.15 m^3/h per m^2 of envelope at 50 Pa, for a natural air leakage of 2.5 cfm, or 0.008 ACH for a 2400 ft^2 1-story house :-)

The blower door as we know and love it today springs from technology first used in Sweden in 1977, where it was actually a blower window. The idea migrated to the United States with Ake Blomsterberg, who came to Princeton University to do research in 1979...

The Princeton researchers decided to mount the fan in a door because door sizes are more uniform than windows... the researchers found that hidden leaks accounted for a greater proportion of air leakage in a home than the more obvious culprits, such as windows, doors, and electrical outlets, a giant leap forward in our understanding of how a house operates...

A monster blower door is being used to test large residential and commercial buildings in Canada. The Super Sucker is a whopping 55,000 CFM fan that is 40 ft long and 5 ft in diameter. It is transported to the site on a flatbed trailer. It takes a team of five people [with safety belts?] to hook it up to a pair of double doors and perform the test.

From: http://hem.dis.anl.gov/eehem/95/951109.html

One empirical formula says an H foot chimney with A ft^2 vents at the top and bottom and an average temp Ti (F) inside and To outside has Q = 16.6Asqrt(HdT) cfm of airflow, where dT = Ti-To. The heatflow is QdT = 16.6Asqrt(H)dT^1.5 Btu/h, approximately.

A square foot of R1 vertical sunspace glazing with 90% transmission would gain 0.9x800 = 720 Btu over 6 hours on an average December day in Allentown, or more, with reflective ground to the south. With an atn(490/800) = 31 degree tilt to the south, it would gain 0.9sqrt(490^2+800^2) = 844 Btu/day. It could be one layer of clear flat GE Lexan polycarbonate plastic, which comes in 0.020"x49"x50' rolls and costs about $1.50 per square foot, or one layer of corrugated Dynaglas "solar siding," a greenhouse roofing material which costs about $1/ft^2. Both last at least 10 years. We might make an 8' long x 8' radius quarter-cylindrical sunspace with 3 $5 bows on 4' centers by bending 2 12' 1x3s into an 8' radius, with 1x3 spacer blocks and screws every 2'. A freestanding greenhouse might have 4-year greenhouse polyethylene film glazing (which costs about 5 cents/ft^2 in wide rolls) stretched over slightly-curved bows to avoid wind fatigue, in a 12' equilateral triangle, hinged at the top.

This could be a Food and Heat Producing Greenhouse like Bill Yanda's or Tom Lawand's. PE Howard Reichmuth's Ecotope greenhouse near Seattle has a steep transparent south wall and a reflective parabolic north wall that concentrates sun into a water trench. Greenhouses are more humid than houses, and they need to be warmer at night to avoid freezing plants, and it's hard to insulate them at night and also provide light for plants. Filling the space between two plastic film covers with air during the day and soap bubble foam at night is useful. Engineer Bob Quist in Toronto has developed a standard "replacement foam insulation" system for Venlo glass greenhouses (the most popular brand in the Netherlands.)

With A ft^2 of US R1 sunspace glazing and lots of thermal mass, we can keep an 8' R20 cube 70 F over an average 24h 31.8 F December day with a 39.2 max in Allentown if 720A = 6h(70-31.8)A/R1 (the daytime sunspace loss) + 18h(70-31.8)A/R20 (the nighttime sunspace loss) + 24h(70-31.8)(6x64-A)/R20 (the 24-hour loss from the rest of the cube), which makes A = 35 ft^2, eg a 5'x7'window, with a 6x64ft^2/R20 = 19.2 Btu/h-F cloudy-day conductance.

To size the sunspace vents, we might figure that 0.9x250 = 225 Btu/h-ft^2 of peak sun enters a square foot of R1 sunspace glazing and 70 F air near the glazing (on the south side of a dark screen north of the glazing, with warmer air north of the screen) loses (70-35)1ft^2/R1 = 35 Btu/h, for a net gain of 190 Btu/h-ft^2, or 6.7K Btu/h [1.1 kW] for 35 ft^2. If 70 F room air enters the sunspace through a lower vent and exits into the house at 120 F through a vent 8' above, the average temp inside the sunspace is 97.5 F, and 6.7K Btu/h = 16.6Asqrt(8')(97.5-70)^1.5 makes A = 0.98 ft^2.

9. Store the heat with less mass in the ceiling and a larger temp swing?

We might warm ceiling mass (eg soil "pugging" in Scotland) with hot air from a sunspace. Ignoring sunspace vent and ceiling airfilm resistances, on an average December day in Allentown, an 8' R20 cube with a ceiling temp T (F) and 8'x8' of R1 sunspace glazing might collect 0.9x800x64 = 46.1K Btu and lose about 6h(T-32)64ft^2/R1 from glazing during the day + 18h(65-32)64ft^2/R20 from the glazing at night + 24h(T-32)64ft^2/R20 up through the ceiling + 24h(65-32)3x64ft^2/R20 from the other 3 walls at night, which makes T = 46322/460.8 = 101 F, if gain equals loss on an average day.

If the ceiling conductance to slow-moving air below is 1.5x64 = 96 Btu/h-F and the cube needs (70-32)19.2 = 730 Btu/h of peak heat on a cloudy day, the ceiling must be at least 70+730/96 = 78 F to provide it. If the average temp is (101+78)/2 = 89.5 F for 5 cloudy days and the cube loses about 5x24h((65-32)4x64ft^2/R20 + (89.5-32)64ft^2/R20) = 72768 Btu, we need 72768/(101-78) = 3164 Btu/F of ceiling mass, eg 3164/64ft^2 = 49 pounds (9.5") of water per square foot of ceiling.

Or slightly less, using a more accurate differential equation. At ceiling temp T, I = (T-32)64ft^2/R20 + (65-32)4x64ft^2/R20 = 3.2T + 320 Btu/h flows from the ceiling mass C, so dT/dt = -I/C = aT + b = -3.2T/C - 320/C, and -a/b = -100, so T(t) = -a/b +(T(0)+a/b)e^-at = -100 + 201e^(-3.2t/C). After 5 days, T(120) = 78 = -100 + 201e^(-384/C) makes ln(178/201) = - 384/C, so C = 3160 Btu/F.

Exercise 9.1: How many inches of water are needed in December in Portland, OR when 470 Btu/ft^2 falls on a south wall on an average 40.2 F day, for an 8' R32 cube with an 8'x8' R2 sunspace window with 80% solar transmission?

Keeping the heat in the ceiling allows the cube to be cooler when vacant, so stored solar heat can last longer, especially if we reduce the ceiling's downward heatflow by radiation with a low-emissivity ceiling surface.

Exercise 9.2: How might this change if the cube only needs 8h/day of heat?

Solar pioneer Harold Hay built houses with roofponds and movable insulation. In the Barra system, hot sunspace air warms a hollow spancrete ceiling. We might collect and distribute heat with ceiling fin tubes, pumping cool water up during the day and thermosyphoning at night, with the help of a slow ceiling fan.

10. Radiation

Gustav Robert Kirchoff (1824-1887) said all radiation hitting a surface must be transmitted, absorbed, or reflected. Radiation obeyed. T+A+R=1. Kirchoff's identity says emissivity equals absorptivity for opaque black bodies or gray surfaces with a constant emissivity that doesn't depend on wavelength.

A surface emits se(T^4) of heat flux by radiation, where T is an absolute temperature in Rankine (F+460) or Kelvin (C+273) degrees, and s is the Stefan-Boltzman constant, 0.1714x10^-8 Btu/ft^2-R^4 or 5.660x10^-8 W/m^2-K^4. The surface's emissivity E varies from 0 (shiny) to 1 (dull.) Most natural surfaces are close to 1, but heat mirrors have emissivities close to 0.

The net heatflow from a surface at T1 degrees facing a T2 degree surface is se(T1^4-T2^4). For example, a square foot of 80 F Trombe wall might lose 0.1714x10^-8((80+460)^4-(30+460)^4) = 47 Btu/h by radiation on a 30 F night. In still air, it might lose 1.5(80-30) = 75 Btu/h by radiation and convection combined. A "selective surface" can absorb well at short solar wavelengths (<3 microns) and radiate poorly at longer heat wavelengths (10 microns at 80 F.) A 100% solar-heated house might have thick SIPs and radiant floors and a big enough well-insulated hot water tank to store heat for 5 cloudy days in a row and lots of $12/ft^2 vertical unglazed stainless steel ES solar roof collectors with durable selective surfaces from www.energie-solaire.com. With collector and ambient temps Tm0 and Ta=0 C and insolation I = 800 W/m^2 (full sun), x = (Tm-Ta)/I = 0.375 makes E = 0.959- 8.91x-0.047x^2 = 0.62, ie 62% in their efficiency formula.

Exercise 10.1: What's the temperature of a surface with e = 0.2 in 300 Btu/h-ft^2 sun on a 20 F day?

The most serious mistake was making the outer container of the receiver of plywood. We thought that the plywood would be sufficiently insulated from the copper panel which was the receiver proper, that it would not get too hot. The copper panel was separated from the plywood by 4" of fiberglass insulation. Nevertheless, the plywood caught fire and the unit was completely destroyed. We suppose this is a success, of sorts.

The copper panel which was plated with chrome black to provide a selective surface originally had a copper tube fastened to the back by a high melting point soft solder. When we first attempted to operate the unit, the soft solder melted, and the tubing became detatched from the panel. We attempted to repair this failure by silver soldering copper bars to the copper tube and screwing the bars to the plated copper sheet. This worked, after a fashion...

The glass window was originally tempered glass. This shattered due to thermal shock. We replaced it with ordinary window glass. This cracked due to thermal shock, but we were able to hold it in place well enough to make some measurements... The measurements we were able to make before the fire generally confirmed our thinking concerning the design...

from "A solar collector with no convection losses," (a downward-facing receiver over a 4:1 concentrating parabolic mirror), by H. Hinterberger and J. O'Meara of Fermilab, in "Sharing the Sun," A joint ASES/SESC conference, August 15th-20th, Winnepeg, Volume 2, pp 138-145.

If Tb is their approximate average temperature, the "linearized radiation conductance" between 2 surfaces Gr = 4seTb^3. A 96 F ceiling in a 70 F room with Tb = 543 R has Gr= 1.097 Btu/h-F-ft^2, roughly R0.9. If our cube needs (70F-30F)19.2Btu/h-F = 768 Btu/h, e(96-70)64x1.097 = 768 makes e = 0.42. We might make about 60% of the ceiling a low-e (0.05) paint or foil surface and the rest an ordinary surface and let radiation warm the room on an average day and use a ceiling fan and a thermostat to bring warm air down on cloudy days. A 72 F ceiling could supply 0.42(72-65)64ft^2x1.097 = 206 Btu/h of radiant heat to a 65 F room. The other 672-206 = 466 Btu/h might come from Q cfm of airflow. View figure 1 below in a fixed font:

1/96 1/Q (96+Q)/(96Q) 72 F ---www---www---65 F equivalent to 72------www------65 -----------> -----------> 466 Btu/h 466 Btu/h

where (72-65)96Q/(96+Q) = 466, so Q = 217 cfm. Grainger's $120 48" 315 rpm 86 W 21K cfm 4C853 ceiling fan might move 217 cfm at 217/21Kx315 = 3.3 rpm with 86(217/21K)^3 = 100 microwatts, according to the fan laws :-) Large slow fans can be very efficient and quiet. The Gossamer ceiling fan develope by Danny Parker at the Florida Solar Energy Center can move 1907 cfm with 9.1 watts. It might move 217 cfm with 1 average watt, running 11% of the time.

11. Even less ceiling mass, with a separate cloudy-day store?

If we only want to store 13.8K Btu of overnight heat on an average day in C Btu/F in a ceiling, with T(6) = 120+(72-120)e^(-6x96/C) = 120-48e^(-576/C) and (T(6)-72)C = 18,432, so C = 384/(1-e^(-576/C)). C = 384 on the right makes C = 494 on the left, and plugging that in on the right again makes C = 558, 596, 620, 635, 644, 649, 653, 655, 657, and 657 (whew!), so it looks like we can store overnight heat with 657/64ft^2 = 10.3 psf (about 2") of water in the ceiling, with T(6) = 120-48e^(-576/657) = 100 F.

If the 8' cube needs 4x24(70-30)19.2 = 73,728 Btu for 4 more cloudy days, this might come from a "solar closet" (see the CD paper) inside a sunspace. The heat lost from the closet air heater glazing efficiently heats sunspace air that heats the cube :-) with about 73,728/(120-70) = 1475 lb or 184 gal. or 23 ft^3 of water cooling from 120 F to 70 F over 4 days.

Kallwall's "solar battery" was an early solar closet. Tom Hopper built an insulated box containing fiberglass water cylinders with double-glazed south wall and an insulating curtain (5 layers of aluminized Mylar) that automatically rolled down over the glazing at night. Bill Shurcliff (in his 1980 Brick House Thermal Shutters and Shades book) said this remarkable Mylar curtain (with spacers that unfurled when it deployed) was invented by the Insulating Shade Co. of Branford, CN, and demonstrated at a 9/9/77 exhibition in Hartford, CN, and featured in the January 1979 issue of Popular Science Monthly.

With 1792 pounds of water in 56 10"x10"x13" 4-gallon ROPAK plastic tubs, a solar closet can supply 73,728 Btu as it cools from 120 to 120-73728/1792 = 78.9 F, with the tubs stacked 7-high and 4-wide and 2-deep in a 2'x4'x8' tall closet completely surrounded by insulation, with an air heater with its own closet vs sunspace glazing over the closet's insulated south wall and one-way dampers in that wall. With 56x4x10x13/144 = 202 ft^2 of tub surface and 300 Btu/h-F (buhfs) of water-air thermal conductance, we have fig 1 again with a 78.9-70 = 8.9 F temp diff and a (300+Q)/(300Q) resistor, so Q = 121 cfm.

With an 8' height difference and 121 cfm = 16.6Asqrt(8'(78.9-70)), we need 2 vents with A = 0.86 ft^2 for natural airflow into the cube.

As an alternative, the cloudy-day heat might come from 3072 pounds of water inside a 2'x4'x8' tall "shelfbox" with a 2'x4'x2' tall water tank below 18 2'x4' wire shelves on 4" centers, with 2" of water inside a $20 continuous piece of poly film duct folded to lay flat on each shelf and a small pump to circulate tank water up through the duct as needed. The tank might have a plastic pipe coil inside to make hot water for showers, with the help of an efficient external greywater heat exchanger, eg 300' of 1" plastic pipe pushed into two coils inside a 35"x23.5"" ID 55 gallon drum, if possible. Pushing the pipe into the drum without kinking is awkward but doable.

12. Greywater heat exchange, Big Fins, and solar ponds

About 15 years ago, Eric Olsen at Earthstar manufactured a heat exchanger with a copper coil in a 55 gallon drum to recover heat in kidney dialysis centers. Drew says professor Jane Davidson at U Minnesota is an expert on plastic heat exchangers. She says "The efficiency you calculate is actually effectiveness which is very different from efficiency. A high effectiveness does not mean a good heat exchanger!" I wonder what she means by that.

Gary and I are working on inexpensive "Big Fins" for water heating in sunspaces and a solar pond water heater and what might become an efficient counterflow greywater heat exchanger with heat storage. See http://BuildItSolar.com . We are using a new $35 55 gallon lined steel drum with a strong removable lid (because the drum might end up under 2' of greywater head, with the inlet and outlet above the lid) and bolt ring and a 3/4" bung and a 2" bung with a 3/4" threaded knockout, with 100 psi/ 73.4 F pipe from PT Industries at (800) 44 ENDOT. Their PBJ10041010001 $60 1"x300'100psi NSF-certified pipe is tested to 500 psi.

Lowes sells the rest of the hardware. It's all installed through the lid, so the drum itself has no holes. Here's a tentative parts list:

sales total # qty price description

25775 1 $5.73 24' of 1.25" sump pump hose (for greywater I/O) 105473 1 1.28 2 SS 1.75" hose clamps (for greywater hose) 54129 2 3.24 1.25" female adapter (greywater barb inlet and outlet) 23859 2 2.36 1.25x1.5" reducing male adapter (bulkhead fittings) 75912 1 0.51 2 1.25" conduit locknuts (bulkhead fittings) 28299 1 1.53 2 1.25" reducing conduit washers (") 22716 1 1.36 1.5" PVC street elbow (horizontal greywater inlet) 23830 1 2.98 10' 1.5" PVC pipe (for 3' greywater outlet dip tube)

The parts above are the greywater plumbing ($18.99.)

23766 4 1.28 3/4" CPVC male adapter (for 1" pipe barbs) 23766 2 0.64 3/4" CPVC male adapter (fresh water I/O) 42000 2 3.84 3/4" FIP to 3/4" male hose adapter 23813 1 1.39 10' 3/4" CPVC pipe (for 1"x3/4" fresh water outlet) 23760 2 0.96 3/4" CPVC T (fresh water I/O) 22643 2 0.86 3/4" CPVC street elbow (fresh water I/O) 4 - 1" 3/4" CPVC pipes (fresh water I/0) 1 - 3' 3/4" CPVC pipe (fresh water inlet) 22667 2 2.56 2 SS 1.125" hose clamps (fresh water I/O) 219980 1 4.87 10.1 oz DAP silicone ultra caulk (bulkhead fittings) 150887 1 3.94 4 oz primer and 4 oz PVC cement

The parts above are fresh water plumbing. Subtotal $39.33.

26371 1 6.83 1500 W electric water heater element 22230 1 2.31 1" galvanized T ("nut" for heating element) 61294 1 11.76 single element thermostat with safety 136343 1 0.56 5 10-24x3/4" machine screws (mount thermostat with 3) 33368 1 0.37 5 #10 SS flat washers (mount thermostat with 3) 198806 1 1.38 10 #0 rubber faucet washers (mount thermostat with 3) 8763 1 0.67 5 10-24 SS nuts (mount thermostat with 3)

The above would make a standalone water heater, if needed. Grand total: $63.21.

For 4 10 min showers per day and 20 minutes of dishwashing at 1.25 gpm we might heat 75 gallons of 55 F water to 110 with 8x75(110-55) = 33K Btu with about 10 kWh worth about $1/day at 10 cents/kWh. If the "heat capacity flow rate" Cmin = Cmax = 75gx8/24h = 25 Btu/h-F and the coil has A = 300Pi/12 = 78.5 ft^2 of surface with U = 10 Btu/h-F-ft^2 (for an HDPE pipe wall with slow-moving warm dirty water outside and 8x300Pi(1/2/12)^2 = 13 gallons of water inside), the "Number of heat Transfer Units" for this counterflow heat exchanger NTU = AU/Cmin = 78.5ft^2x10Btu/h-F-ft^2/25Btu/h-F = 31.4, so the "efficiency" E = NTU/(NTU+1) = 97% for hot water usage in bursts of less than 13 gallons. This works best with equal greywater and cold water flows, with either a 110 F water heater setting (preferable), or the heat exchanger output feeding the cold water shower inlet as well as the water heater.

The Hazen-Williams equation says L' of d" smooth pipe with G gpm flow has a 0.0004227LG^1.852d^-4.871 psi loss. At 1.25 gpm, the pressure drop for 2 150' coils of 1" pipe is 0.0004227x150x(1.25/2)^1.852x1^-4.871 = 0.03 psi.

If greywater leaves a shower drain at 100 F and fresh water enters at 50 F, the fresh water should leave at 50+0.97(100-50) = 98.5 F. Warming it further to 110 F would take 8x75(110-98.5) = 6.9K Btu/day with 2 kWh worth about 20 cents/at 10 cents/kWh, for a yearly savings of about ($1-0.20)365 = $292, or more, with a tighter shower enclosure and higher drain temperature. The 1500 W heater might operate 2kWh/1.5kW = 1.3 hours per day.

So far, we've discovered the steel drum wall needs insulation inside to preserve stratification because it conducts heat from warm greywater above to cooler greywater below. We might put foil foamboard with kerfs (knife cuts partially through the board) inside the drum, with a plastic film liner to keep it dry. We'd also like to improve incoming greywater stratification, to let it find its thermal level via the holey dip tube...

Zomeworks makes expensive "Big Fins" aluminum extrusions that snap onto copper pipes to heat water in sunspaces. The pressurized water inside the pipes thermosypohons through an insulated tank above, with no pumps, eat exchangers, controls, or antifreeze. We might use the wrapping jig at http://www.redrok.com/misc1.htm#wrappingjig to make less expensive Big Fins like this: . . . . . . aluminum flashing rivet . . 1/2" . . .........................|... . copper . . .........................|... . pipe . . rivet . . . . . . . . --- . . .