What If? Lab
What if a house had a tiny heater?
In winter, a heater keeps a house warm. What if the heater were tiny? Would the house stay cozy? Let's find out with Olly!
A warm house is always leaking heat into the cold outside. A heater keeps it warm only if it can put heat back as fast as it leaks out. In our model, a 10-kilowatt heater keeps a leaky house at 20 °C on a freezing day. Swap it for a tiny 3-kilowatt heater, and the house slowly cools until it finds a new balance.
Indoor temperature settles where heater power equals heat loss, Q = U·A·ΔT. In the model, a house with U = 1.5 W/m²·K and 300 m² of shell loses 9.00 kW at 20 °C inside and 0 °C outside. A 10 kW heater holds 20 °C running 90% of the time; a 3 kW heater can only balance the loss at 6.7 °C, which the house approaches over two days.
The big question
It is a freezing day. Inside, the house is warm and cozy.
A big heater keeps it warm. What if the heater were tiny?
On a freezing winter day, a heater hums away and the house stays comfortable. It seems simple: the heater makes heat, so the house is warm.
But heat is always escaping through the walls, roof, windows and floor. So what if the heater were much smaller? Would the house just take longer to warm up, or would something else happen? In this lab, you swap a big heater for a tiny one and watch the thermometer over two whole days.
Heating a building is a balance problem. Heat flows out through the shell at a rate set by insulation, area and the inside-outside temperature difference, and the heater must supply heat at the same rate to hold a steady temperature.
This lab uses a simple one-temperature house model with a thermostat set to 20 °C, 0 °C outdoors, and a fairly leaky shell. It compares a 10 kW heater with a 3 kW heater and asks where the house ends up when supply cannot match loss.
Try the experiment
Make a prediction
Olly swaps the big heater for a tiny one. What will the house do?
The 10-kilowatt heater is swapped for a 3-kilowatt one, on a 0 °C day. What happens to the house over two days?
A house losing 9.00 kW at 20 °C (0 °C outside) gets a 3 kW heater instead of 10 kW. What does the model's indoor temperature do?
- It stays at 20 °C, just warming up more slowly
- It cools down to the outside temperature of 0 °C
- It cools until it settles at a chilly but steady temperature
| Control | In our world | What if… | Range |
|---|---|---|---|
| Heater size · The most heat the heater can make each second, in kilowatts. This lab compares 10 with a tiny 3. | 10 kW | 3 kW | 0–20 kW |
| Outside temperature · How cold it is outdoors. A bigger difference pushes heat out faster. | 0 °C | — | -20–25 °C |
| How leaky the house is · How easily heat leaks through each square meter of wall, roof and floor for each degree of difference. Insulation makes this number smaller. | 1.5 W/m²·K | — | 0.1–3 W/m²·K |
| Thermostat setting · The temperature the thermostat tries to keep inside. | 20 °C | — | 10–25 °C |
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Established fact
Heat leaks out faster when it is much colder outside.
Heat flows out of a house faster when the difference between inside and outside is bigger, and when the walls are thinner.
Conductive heat flow is proportional to area and temperature difference and inversely proportional to thickness: Q/t = kA(T₂ − T₁)/d (OpenStax).
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What the model shows
In the model, a tiny heater can only keep the house at about 7 °C.
In the model, a 3-kilowatt heater runs nonstop but the house cools to about 9.5 °C in a day, heading for 6.7 °C.
U = 1.5 W/m²·K, 300 m², 0 °C outside: a 3 kW heater balances the loss at 6.7 °C; the 48-hour run reaches 9.5 °C at 24 h and 7.3 °C at 48 h.
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What the model shows
In the model, better insulation lets the tiny heater keep the house warm.
In the model, insulating the house so its heat-loss rate falls to 0.5 lets the 3-kilowatt heater just about hold 20 °C.
At U = 0.5 W/m²·K the loss at 20 °C is 3.00 kW, which the 3 kW heater matches; at U = 1.5 the minimum heater is 9 kW.
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Established fact
Lots of a home's energy goes into heating and cooling it.
In 2020, heating and air conditioning made up 52% of an average U.S. household's yearly energy use.
Space heating and air conditioning accounted for 52% of average U.S. household energy consumption in 2020 (U.S. Energy Information Administration).
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An idea scientists discuss
On a milder day, a tiny heater might be enough.
A small heater might keep up on milder days but not in a cold snap, so how cold it gets matters as much as the heater.
In the model the 3 kW heater holds 20 °C whenever it is 14 °C or warmer outside; a real house's balance point would also depend on drafts, sunshine and occupants.
A big enough heater
The house leaks heat all the time. The big heater puts it back.
It does not even need to run all the time. The house stays warm.
In the model, keeping this house at 20 °C while it's 0 °C outside means replacing 9 kilowatts of heat that leak out every moment. A 10-kilowatt heater can do that easily. Its thermostat switches it on and off, so it runs about 90% of the time.
Over a day, it puts in about 216 kilowatt-hours of heat. And if it were turned up full, it could warm the house to about 22 °C before the leaks caught up.
With U = 1.5 W/m²·K, A = 300 m² and ΔT = 20 K, the steady loss is Q = U·A·ΔT = 9.00 kW. A 10 kW heater exceeds it, so the thermostat cycles the heater on about 90% of the time and the house holds 20 °C, using about 216 kWh of heat a day.
The heater's ceiling is the temperature where its full output equals the loss: T = T_out + P/(U·A) ≈ 22.2 °C. Any setting below that is reachable.
Swap in a tiny heater
Now Olly puts in a tiny heater. It runs all the time.
But the house still gets colder and colder. After two days, it is a chilly 7 °C inside.
Now replace the heater with a 3-kilowatt one. It runs all the time, but the house loses 9 kilowatts at 20 °C, three times what the heater can make. So the house cools.
As it cools, the difference between inside and outside shrinks, so the heat leaks out more slowly. After about 6 hours the model's house is near 15.6 °C, after a day about 9.5 °C, and after two days about 7.3 °C. It's heading for 6.7 °C, the temperature where the tiny heater's 3 kilowatts finally match the leaks.
With P = 3 kW the heater runs continuously (100%) but cannot match 9.00 kW. The house cools exponentially toward the balance point T = 0 + 3,000/(1.5 × 300) ≈ 6.7 °C, with the model's time constant τ = C/(U·A) ≈ 15.4 h. The 48-hour run passes about 15.6 °C at 6 h, 12.8 °C at 12 h, 9.5 °C at 24 h and 7.3 °C at 48 h.
Note what does not change: the leak rate at any given indoor temperature. The tiny heater simply supplies less, so the balance moves to a smaller temperature difference.
Why the house finds a new balance
Heat leaks out faster when it is very warm inside and cold outside.
As the house cools, the leak slows down. When the leak is as small as the heater, the cooling stops.
Heat flows from warm to cold, and the bigger the temperature difference, the faster it flows. A warm house on a freezing day leaks heat fast; a chilly house leaks more slowly.
So when the heater is too small, the house cools until its leaks shrink to match the heater exactly. Then it stays there. Our model keeps things simple: the whole house is one temperature, the outside never changes, and it leaves out drafts, sunshine through windows and the warmth from people and appliances.
OpenStax gives the conduction rate as Q/t = kA(T₂ − T₁)/d: proportional to area and temperature difference, inversely proportional to thickness. The model lumps every surface into one U-value, so the loss is U·A·(T − T_out), and C·dT/dt = P − U·A·(T − T_out). The steady state is where P = U·A·ΔT, giving ΔT = P/(U·A).
The model's own limitations matter here: it ignores air leakage and ventilation, solar gains and internal heat from people and appliances, treats the house as one well-mixed heat store of 25 MJ/K, and holds the outdoor temperature fixed for 48 hours.
Bigger heater, or smaller leaks?
There are two ways to help a tiny heater. Make the heater bigger. Or stop the house leaking so much!
Thick, fluffy insulation helps keep the heat inside.
In the model, the house needs at least a 9-kilowatt heater to stay at 20 °C on a freezing day. But there's another fix: make the leaks smaller. Insulate the house so its heat-loss rate drops from 1.5 to 0.5, and the tiny 3-kilowatt heater can just about hold 20 °C, running nearly all the time.
This matters in real homes. The U.S. Energy Information Administration reports that in 2020, space heating and air conditioning together made up 52% of an average U.S. household's yearly energy use, and ENERGY STAR says sealing air leaks and adding insulation can save up to 10% on yearly energy bills.
In the model, the smallest heater that holds 20 °C at 0 °C outdoors is 9 kW; alternatively, cutting U from 1.5 to 0.5 W/m²·K drops the loss to 3.00 kW, which the 3 kW heater exactly matches. Insulation works by raising the R factor, thickness divided by conductivity; OpenStax cites R-11 for 3.5-inch fiberglass batts and R-19 for 6.5-inch batts.
The U.S. Energy Information Administration reports that space heating and air conditioning accounted for 52% of average household energy consumption in 2020, with a much smaller share in apartments, which are often partly insulated by neighboring units. ENERGY STAR states that sealing and insulating can provide up to a 10% saving on annual energy bills.
Amazing facts
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Thick fluffy insulation works because it is full of tiny air pockets.
Good insulators like fiberglass, wool and goose down all trap many small pockets of air, which conducts heat very poorly.
OpenStax notes that Styrofoam, fiberglass, wool and goose down all incorporate small pockets of air, exploiting air's poor thermal conductivity.
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Thicker insulation keeps more heat in.
Insulation is rated by its R-value: 3.5-inch fiberglass batts are about R-11, and 6.5-inch batts about R-19. A bigger number means better insulation.
OpenStax cites R-11 for 3.5-inch fiberglass batts and R-19 for 6.5-inch batts; the R factor is thickness over conductivity, and heat flow is inversely proportional to it.
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Apartments often need less heating than big houses.
Apartments usually use a smaller share of their energy for heating and cooling, partly because the apartments around them help keep them warm.
The EIA notes that space heating and air conditioning take a much smaller share in apartments, which are generally smaller and often partly insulated by adjacent units.
Words to know
- heat loss
- the heat that flows out of a warm building into colder surroundings
- insulation
- material that slows down the flow of heat
- thermostat
- a switch that turns a heater on and off to keep a set temperature
- kilowatt
- a unit of power: 1,000 watts
- steady state
- a balance where things stop changing because what comes in equals what goes out
- U-value
- a number that tells how easily heat leaks through a wall or roof; smaller is better insulated
Discovery challenge
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When does a house stay warm?
What has to be true for a heater to keep a house at a steady temperature?
A house holds a steady temperature when…
- The heater puts heat back as fast as it leaks out
- The heater is switched on all day
- The windows are open
Show the answer
The heater puts heat back as fast as it leaks out
Heat in must match heat out.
The heater must replace heat as fast as it escapes.
…heater power equals the heat loss, P = U·A·ΔT.
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In Olly's lab, how big must the heater be to keep the leaky house at 20 °C on a 0 °C day?
In the model, what is the smallest heater that keeps the leaky house at 20 °C when it is 0 °C outside?
With U = 1.5 W/m²·K, 300 m² and 0 °C outside, what is the minimum heater power to hold 20 °C in the model?
- 9 kilowatts
- 3 kilowatts
- 20 kilowatts
Show the answer
9 kilowatts
9 kilowatts, the same as the heat leaking out.
9 kilowatts, because that's how fast heat leaks out at 20 °C.
9.00 kW: U·A·ΔT = 1.5 × 300 × 20 W.
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In Olly's lab, what helps the tiny heater keep the house warm?
In the model, what lets the 3-kilowatt heater hold 20 °C on a 0 °C day?
In the model, which change lets the 3 kW heater hold 20 °C at 0 °C outside?
- Insulating the house well
- Setting the thermostat higher
- Making the house bigger
Show the answer
Insulating the house well
Insulation! It keeps the heat inside.
Insulating it so the heat-loss rate drops to 0.5 cuts the leak to 3 kilowatts.
Cutting U to 0.5 W/m²·K lowers the loss to 3.00 kW; a bigger house or higher setting would raise it.
Experiments to try
- Watch Try it What If? Lab What if a house had no insulation? Heat always leaks out of a warm house into the cold outside. Insulation in the walls and roof slows that leak down. In our model, an insulated house on a freezing day needs its heater only about a third of the time. Take the insulation away and the same heater runs nonstop, and the house still cools to about 17 °C.
- Watch Try it What If? Lab What if your cocoa went in a flask? A mug of hot cocoa cools fast: in our model it drops from 80 °C to 60 °C in under 12 minutes. Pour the same cocoa into a vacuum flask and it takes about 100 minutes. Find out how a flask blocks the three ways heat escapes.
- Watch Try it What If? Lab What if a house’s heating switched off on a cold night? A warm house is always leaking heat into the cold night. If the heating stops, nothing replaces it. In our model, an insulated house at 20 °C on a 0 °C night cools to about 15 °C in 12 hours, and then more and more slowly as it gets closer to the outside temperature.
- Watch Try it What If? Lab What if winter were much colder? Heat leaks out of a house faster when it is colder outside. In our model, an insulated house kept at 20 °C needs about 86 kilowatt-hours of heat a day when it is 0 °C outside. Drop the outside to −20 °C and the gap doubles, so it needs twice as much: about 173 kilowatt-hours.
Explore next
- Watch Try it What If? Lab What if a house had no insulation? Heat always leaks out of a warm house into the cold outside. Insulation in the walls and roof slows that leak down. In our model, an insulated house on a freezing day needs its heater only about a third of the time. Take the insulation away and the same heater runs nonstop, and the house still cools to about 17 °C.
- Watch Try it What If? Lab What if your cocoa went in a flask? A mug of hot cocoa cools fast: in our model it drops from 80 °C to 60 °C in under 12 minutes. Pour the same cocoa into a vacuum flask and it takes about 100 minutes. Find out how a flask blocks the three ways heat escapes.
- Watch Try it What If? Lab What if a house’s heating switched off on a cold night? A warm house is always leaking heat into the cold night. If the heating stops, nothing replaces it. In our model, an insulated house at 20 °C on a 0 °C night cools to about 15 °C in 12 hours, and then more and more slowly as it gets closer to the outside temperature.
- Watch Try it What If? Lab What if winter were much colder? Heat leaks out of a house faster when it is colder outside. In our model, an insulated house kept at 20 °C needs about 86 kilowatt-hours of heat a day when it is 0 °C outside. Drop the outside to −20 °C and the gap doubles, so it needs twice as much: about 173 kilowatt-hours.
- Watch Try it What If? Lab What if a battery had more voltage? A battery's voltage is how hard it pushes electric current around a circuit. Swap a 3-volt battery for a 9-volt one in our model and the same two bulbs carry three times the current, and each one uses nine times the power. Find out why, and why a real bulb can burn out on too much voltage.
- Watch Try it What If? Lab What if a battery wasted energy as heat? A battery's label tells you how much energy it stores. But on the way from the battery to a device, some energy always turns into heat in the wires and electronics. In our model, a 60-watt-hour battery runs a 10-watt lamp for 6 hours when nothing is lost. If 30% is lost as heat, the lamp goes dark after 4 hours 12 minutes.
- Watch Try it What If? Lab What if a battery were old and tired? Every battery has a little resistance inside it. A fresh battery has hardly any, but as a battery gets worn out, that inside resistance grows. In our model, four bulbs side by side glow at full brightness on a fresh 3-volt battery. Make the battery old and tired, and watch what happens to the bulbs and to the battery itself.
- Watch What If? Lab What if a machine could run forever? For more than a thousand years, inventors have drawn wheels and gadgets that were meant to keep turning forever with nothing to power them. None has ever worked. Follow the energy in this thought experiment to see why: energy is never made from nothing, and every moving machine leaks a little of it away as heat.
For parents & teachers
What children will learn
- Heat flows from warm to cold, faster when the temperature difference is bigger.
- A heater holds a steady temperature only if it replaces heat as fast as it is lost.
- An undersized heater lets a house cool until heat loss falls to match it.
- Insulation reduces heat loss, so a smaller heater can do the job.
Recommended ages: 11 to 17 · About 15 minutes
Talk about it
- Why does a house cool quickly at first and then more slowly?
- Would you rather fix a cold house with a bigger heater or with insulation? What are the trade-offs?
- What does this model leave out that would matter in a real home?
Try at home or in class
- Fill two identical jars with warm tap water (not hot). Wrap one in a towel or wool sweater and leave the other bare. Check them with a thermometer every 10 minutes for an hour.
- On a cold day, with a grown-up, hold a strip of tissue paper near the edges of a closed window or door and watch for drafts.
Key vocabulary: heat loss, insulation, thermostat, kilowatt, steady state, U-value
Where these facts come from
- College Physics 2e, 14.5 Conduction , OpenStax, Rice University
- Use of energy explained: Energy use in homes , U.S. Energy Information Administration
- Seal and Insulate with ENERGY STAR , ENERGY STAR (U.S. EPA)
Checked, not yet human-reviewed. Last updated October 26, 2026. Spotted a mistake? Tell us.