A 40 ft run of rigid PVC drain pipe sitting in an unvented attic is 2.13 inches longer on an August afternoon than it was on a February morning. Not two millimeters. Two and an eighth inches, on a run short enough to fit in most houses, from nothing more exotic than the pipe going from 40 to 140 degrees Fahrenheit.
Solvent weld that run end to end, hard-clip it at both ends, and it has nowhere to put those two inches. It does not stop growing because the fittings are tight. It bows between the brackets, and it keeps bowing every summer until something with a joint in it gives up. That is the first bill.
What a temperature change actually charges you for
The same swing does three separate things, and they are usually handled by three different people who never compare notes. It changes the size of everything it touches. It demands energy from whoever wants the temperature put back. And while the difference exists at all, it leaks heat continuously through every surface between the two sides.
Three quantities, three formulas, and each one is short enough to write on a business card.
| The bill | The formula | What sets the size of it | A real example |
|---|---|---|---|
| Things change size | dL = L x alpha x dT | The material, and how long the run is | 2.13 in on 40 ft of PVC |
| Energy to put it back | Q = m x c x dT | The mass, and what it is made of | 7.95 kWh for a 50 gallon tank |
| Heat leaking while it lasts | Q = A x dT / R | The R-value, and the area | 1,175.6 W through one brick wall |
Notice what the three have in common. The temperature difference appears in all of them, linearly. Double the swing and every one of the three doubles. Notice what they do not have in common: the property that decides the answer is different every time, and knowing one of them tells you nothing whatsoever about the other two. Copper is superb at conducting, mediocre at storing and middling at expanding.
Bill one: the pipe that grew two inches
PVC moves about 80 millionths of its own length for every kelvin it warms. That is nearly seven times what carbon steel does, and it is the entire reason plastic drainage is assembled with slip couplings while steel is welded solid without anyone worrying much.
Run the same attic swing across the materials that might be up there and the spread is the story. On that identical 40 ft, carbon steel moves 0.32 in and copper 0.44 in. The PVC moves 2.13 in, which is six and a half times the steel. Same attic, same hundred degrees, same length. Only the material changed.
| Run length of rigid PVC | Growth from 40 to 140 deg F | In millimeters | What has to absorb it |
|---|---|---|---|
| 10 ft | 0.53 in | 13.55 mm | One slip joint, or a fitting with room to move |
| 20 ft | 1.07 in | 27.09 mm | More than most hub fittings will give you |
| 25 ft | 1.33 in | 33.87 mm | An expansion coupling, not a guess |
| 40 ft | 2.13 in | 54.19 mm | The run in the photograph above |
| 60 ft | 3.20 in | 81.28 mm | Two joints, or one very generous one |
| 100 ft | 5.33 in | 135.47 mm | Nearly half a foot, on one straight run |
The column on the left is a straight line and that is worth saying out loud, because it is the only one of today's three quantities that behaves so obediently. Twice the run, twice the movement. There is no threshold below which it stops mattering and no length at which it suddenly gets worse. There is only the point where the gap you left runs out, and the pipe finds out about it before you do.
One detail decides where that gap has to sit: the temperature on the day of the install. Fit on a cold morning and almost the whole allowance has to be on the hot side. Fit at the top of the range and it goes entirely the other way. Fit in the middle and the movement splits, which is why an expansion joint set on a mild day is doing half the work of one set in a heatwave.
Bill two: the energy nobody argues about until it arrives
Water is the awkward substance in every building. Its specific heat is 4,186 J/(kg K), more than four times aluminum and nearly eleven times copper, which is exactly why it is used to move heat around and exactly why heating it is expensive.
A 50 US gallon tank holds 417.3 lb of it. Bringing that from a mains temperature of 55 to a stored 120 degrees Fahrenheit takes 7.95 kWh, which is 27,119 BTU. On a 4.5 kW element that is 1.77 hours of continuous running, and that is with a perfect transfer and nothing lost to the room.
Turn the same equation around and it explains something people argue about endlessly. One kilowatt-hour, one unit on the bill, heats about 21.5 kg of water through 40 K. That is roughly a bucketful taken from the cold faucet to bath temperature. Every claim you have ever read about the cost of a shower, a dishwasher cycle or a hot tank reduces to that single line, and most of them can be checked against it in about ten seconds.
The same arithmetic settles a much commoner puzzle. The air in a small room, about 60 kg of it, takes only 422.1 kJ to go up 7 K, which a 2 kW heater delivers in under four minutes. Yet no 2 kW heater warms a cold room in four minutes. The reason is the floor: 500 kg of concrete slab going up 40 K needs 4.89 kWh, more than forty times the air, and until the slab, the plaster and the furniture have warmed, they pull the air straight back down. Thermal mass is not a myth. It is just answering a different question from the one on the thermostat.
Bill three: the one that never stops running
The first two bills arrive once per swing. The third one runs continuously for as long as the two sides are at different temperatures, and it is the one that quietly dwarfs the others over a season.
A wall of solid brick, 38 cm thick and 30 m² in area, with 21 degrees C inside and -5 outside, loses 1,175.6 W. Continuously. That is a fan heater running day and night in one wall of one room, and the wall is not thin: nearly half a meter of masonry, which most people would call substantial. Its total resistance is R-3.8, and R-1.0 of that is nothing but the films of still air clinging to its two faces.
Put 15 cm of mineral wool on the outside and the same wall loses 169.2 W. That is 1,006.4 W less, a cut of 85.6 percent, and 24.15 kWh saved on every day the weather holds like that. The wool alone is worth R-22.4, eight times what the entire brick wall was contributing, from a layer a little over a third of its thickness.
Now the part that decides how much insulation is worth buying. Take a 100 sq ft ceiling with 70 degrees F inside and 10 outside, and add fiberglass batt an inch at a time.
| Fiberglass batt | Total R | Heat crossing 100 sq ft | In BTU/h | Cut against the row above |
|---|---|---|---|---|
| 2 in | R-7.5 | 233.8 W | 797.8 | - |
| 4 in | R-14.1 | 124.9 W | 426.2 | 108.9 W |
| 6 in | R-20.6 | 85.2 W | 290.8 | 39.7 W |
| 8 in | R-27.2 | 64.7 W | 220.7 | 20.5 W |
| 10 in | R-33.7 | 52.1 W | 177.8 | 12.6 W |
| 12 in | R-40.3 | 43.6 W | 148.9 | 8.5 W |
Read the last column downward. The step from two inches to four kills 108.9 W. The step from ten to twelve kills 8.5 W, about a thirteenth as much for the same material and the same labor. Nothing about the insulation got worse. Resistances add in a straight line, but the heat flow is one divided by their sum, and one over a growing number flattens fast. That single fact is why the argument is always about whether to insulate at all, and almost never about the last two inches.
It also explains the worst surface in most houses. A single pane of glass, 4 mm thick and 2 m², at 20 against -5 degrees C, loses 287.4 W and grades out at R-1.0 in total. Of that resistance, 97.7 percent is the air films. The glass itself contributes essentially nothing, which is why a window insulates through the gas between its panes and never through the glass.
Five mistakes that cost real money
Mistake one: judging the swing by the weather forecast. The attic in the opening reaches 140 degrees F while the air outside is nowhere near it, and a dark south-facing surface goes further still. The forecast describes the shade. Your pipe, your cladding and your roof deck live somewhere hotter.
Mistake two: assuming a small percentage is a small number. That 40 ft PVC run grows by 0.44 percent of its length. Trivial as a percentage, and it is still two inches of steel-hard plastic pushing against a fitting that was designed to move by nothing at all.
Mistake three: weighing the container with the contents. A pan and the water in it are two calculations, not one, because iron carries 490 J/(kg K) and water carries 4,186. Averaging them gives an answer that is wrong in both directions at once.
Mistake four: comparing an R-value of a material against a code figure for an assembly. Codes quote numbers that include the two surface air films, worth about R-1. On a well-insulated wall that gap is small. On a single pane it is the entire answer, which is how a product with almost no resistance of its own ends up quoted at R-1.
Mistake five: expecting thermal mass to do the job of insulation. The slab in the middle of this article stores a great deal and still leaks whatever its R-value allows. Mass changes when the heat arrives. Resistance changes how much of it leaves. Over a long cold spell, only the second one is on your side.
Five rules that follow from the numbers
One. Size every gap, joint and clearance for the widest span the material will ever see, then fit at a middling temperature so the allowance splits both ways.
Two. Energy scales with mass and with the span, and with nothing else. If a heating job feels expensive, one of those two is larger than you assumed, and it is usually the mass.
Three. The first inch of insulation is worth more than the last four. Spend on getting something onto every uninsulated surface before spending on making an insulated one thicker.
Four. Resistances add, conductances do not. Two layers at R-10 give you R-20. Two U-values of 0.1 do not give you 0.05, and adding U-values is the commonest arithmetic error in this whole subject.
Five. Check which of the three bills you are actually paying before optimizing anything. Expansion is a one-off you design around, energy is a one-off you pay for, and leakage is a standing order that runs every hour the weather holds.
Tools discussed in this article
Thermal Expansion Calculator - how far a length, an area or a volume moves for a given temperature change, across fifteen solids and four liquids, in Celsius or Fahrenheit and with the answer echoed in millimeters and inches. Ready-made for a 20 ft PVC run, 25 ft, 30 ft, 40 ft, 50 ft and 100 ft.
Specific Heat Calculator - the heat a job needs, the mass a given amount of energy will cover, the temperature rise it buys, or the coefficient itself from a measurement, across eighteen substances and six energy units including BTU and kilowatt-hours.
Thermal Conductivity Calculator - heat loss through a wall in watts and BTU per hour, with the R-value in both unit systems and an optional second layer that prints the before, the after and the percentage saved. Ready-made for 3 inches of fiberglass, 5 inches, 6 inches, 8 inches, 10 inches and 12 inches.
More Physics tools
Mechanical Power Calculator - the same watts this article counts as heat loss, arriving from a motor instead of a heating element.
Work Calculator - force multiplied by distance, and the reason a pipe held against its own expansion is storing energy rather than releasing it.
Potential Energy Calculator - what a raised mass is holding, in the same joules the heat calculations here are counted in.
Kinetic Energy Calculator - the third place the same joule shows up, and a useful check on how little energy motion really carries.
Force Calculator - what a restrained expansion turns into when the material is not allowed to move.
Momentum Calculator - mass times velocity, and the quantity a collision cannot destroy.
Torque Calculator - the pull a fastener demands from a handle of a given length, in newton-meters and pound-feet.
Kinetic Friction Calculator - what a surface takes back, with the static and the kinetic coefficient kept apart.
Pendulum Period Calculator - the exact period of a swing at any amplitude, not just the small-angle approximation.