Where Does the Heat in Your House Go?
Your heating is not warming your house. It is replacing, watt for watt, the heat that is leaving through the roof, the walls, the windows and the gaps. Here is the bill, itemised.
A bucket with holes in it
Junior level — plain language, no maths
Think of a warm house as a bucket with holes in the bottom. The heating is the tap. When the water level holds steady it does not mean the bucket is full and finished - it means the tap is pouring in exactly as fast as the holes are letting it out. Turn the tap off and the level drops. That is your house in winter, and every euro you spend on heating is buying water for a leaking bucket.
Heat always moves from warm to cold, and it never gets tired. On a January night your living room is twenty degrees warmer than the garden, so heat pours out through everything: the roof, the walls, the windows, the floor, and the gaps around the letterbox. The four surfaces leak by conduction - heat crawling through solid material - and the gaps leak by simply throwing your warm air outside and pulling cold air in.
Insulation does something cleverer than you might think. It does not block heat with sheer bulk. Still air is one of the best insulators there is, and every insulating material - mineral wool, foam, feathers in a duvet, the fur on an animal - is really a way of holding air still so it cannot carry heat away. Mineral wool is about 95% air. You are not installing a barrier; you are installing trapped air.
Where should the first hundred euros go? Almost always the roof, because heat rises and lofts are easy to reach - then the draughts, which cost almost nothing to seal. Windows feel cold and get blamed first, but a single pane loses roughly five times as much per square metre as an insulated wall, and there is far less window than wall. The simulation puts real numbers on all of it, so you can stop guessing and look.
Things worth knowing
- A duvet, a jumper and a loft full of mineral wool all work identically: they hold a thick layer of air still. The material barely matters - what matters is that the air inside it cannot circulate.
- The first ten centimetres of loft insulation cut heat loss through the roof by about 80%. The next ten cut it by another 8%. Insulation has sharply diminishing returns, which is why there is a sensible thickness and then a silly one.
- In a typical older house, a fifth to a third of the heat loss is simply air escaping through gaps. Sealing draughts is the cheapest energy saving available to almost anyone.
U-values, degree-days and where the watts actually go
Student level — the core equations
Every element of a building is described by a single number: its U-value, in watts per square metre per kelvin of temperature difference. Multiply by area and by the inside-outside difference and you have watts: \(Q = \sum U_i A_i \Delta T\). A solid brick wall is about 1.7 W/m²K, the same wall with 15 cm of mineral wool about 0.2, a single-glazed window 5.0, and modern triple glazing 0.8. Those numbers are the whole argument, and they are all measured the same way.
The U-value is the reciprocal of the total thermal resistance, \(U = 1/\left(R_{si}+\sum d_i/\lambda_i+R_{se}\right)\), where \(\lambda\) is conductivity and \(d\) is thickness. That form explains the most important practical fact about insulation: because thickness appears in the denominator, the returns are hyperbolic. Going from 0 to 10 cm on a wall takes it from 1.7 to about 0.29 W/m²K. Going from 10 to 20 cm takes it to 0.16. The first centimetres are transformative and the last ones are almost pointless.
Then there is the loss that has no U-value at all. Ventilation and infiltration carry heat out with the air itself: \(Q_v = 0.33\,n\,V\,\Delta T\), with \(n\) the air changes per hour and \(V\) the volume, the 0.33 being the heat capacity of air in convenient units. An old house at one and a half air changes an hour loses as much through its gaps as through its walls. You cannot seal a house completely - people need fresh air - which is why airtight buildings pair sealing with mechanical ventilation and heat recovery.
Annual energy comes from degree-days. Sum the temperature difference over the heating season - about 2500 kelvin-days for northern Italy or southern England - and \(E = \sum UA \times 24 \times \text{DD}/1000\) gives kilowatt-hours per year. The same sum also gives the peak load that sizes the boiler or heat pump, and that is the number that turns insulation into a bargain: halve the heat loss and you can install a heat pump half the size, which changes the capital cost as well as the running cost.
Key Formulas
| Fabric heat loss | \(Q = \sum U_i A_i \Delta T\) | watts |
|---|---|---|
| U-value | \(U = \dfrac{1}{R_{si}+\sum d_i/\lambda_i+R_{se}}\) | W/m²K |
| Insulation resistance | \(R = \dfrac{d}{\lambda}\) | λ ≈ 0.035 W/mK for wool |
| Ventilation loss | \(Q_v = 0.33\,n\,V\,\Delta T\) | n = air changes per hour |
| Total loss coefficient | \(H = \sum U_iA_i + 0.33nV\) | W/K |
| Annual energy | \(E = \dfrac{H \times 24 \times \mathrm{DD}}{1000}\) | kWh per year |
| Peak load | \(P = H\,\Delta T_{\text{design}}\) | sizes the heat pump |
| Typical values | \(U_{\text{brick}}\approx1.7,\ U_{3\text{-glaze}}\approx0.8\) | |
Things worth knowing
- Thermal bridges - balcony slabs, window reveals, steel lintels - can add 10-30% to a well-insulated building's heat loss, because they short-circuit the insulation. They are accounted for separately with linear ψ-values, in W/mK rather than W/m²K.
- Insulation changes where surfaces sit relative to the dew point. Insulate badly - inside the wall, without a vapour control layer - and you can move condensation into the structure, which is how well-meant retrofits grow mould.
- Mechanical ventilation with heat recovery reclaims 80-90% of the heat in outgoing air. It only makes sense once the building is airtight; in a leaky house the air simply goes around it.
Three transport mechanisms, the bridges nobody draws, and why the savings never quite arrive
Scholar level — full mathematical depth
01What a U-value hides
A U-value is a steady-state, one-dimensional lump that quietly contains all three heat transport mechanisms. Inside the insulation, conduction through the solid matrix and through the trapped air run in parallel with radiation between fibres and with any convection the pore structure permits; the measured \(\lambda\) of mineral wool, around 0.035 W/mK, is the sum of all of them and is why performance degrades if the material is compressed or wetted. The surface resistances \(R_{si} \approx 0.13\) and \(R_{se} \approx 0.04\) m²K/W are themselves combined convection and radiation coefficients, which is why an exposed windy wall performs slightly differently from a sheltered one, and why radiant barriers help in some assemblies and do nothing in others.
02Glazing, where the physics is most visible
A single pane is not a good insulator because glass conducts well and a single sheet has no cavity: \(U \approx 5.7\). Add a second pane and the gas gap does the work, with an optimum around 16 mm - narrower and conduction dominates, wider and convection cells start up inside the cavity. Fill it with argon instead of air to cut conduction, and coat one surface with a low-emissivity layer to cut the radiative term, which alone is worth roughly a third of the remaining loss. Triple glazing at \(U = 0.8\) is not three times better than double at 2.7 by being thicker; it is better because there are two cavities and two coatings. Meanwhile the frame, which nobody photographs, is often the worst part of the window.
03Thermal bridges and the gap between calculation and reality
One-dimensional U-values assume the wall is a wall everywhere. Real buildings have junctions, balconies, lintels and fixings where the insulation is interrupted, and heat takes the short cut. These are handled with linear transmittance \(\psi\) in W/mK and point transmittance \(\chi\), determined by two- and three-dimensional numerical modelling. In a poorly insulated building they are a rounding error; in a well insulated one they can be a fifth of the total, because as the plane elements improve, the junctions do not. This is the general shape of deep retrofit: every improvement promotes the next weakest term to being the dominant one.
04Air, and the awkward trade with health
Infiltration heat loss \(Q_v = \rho c_p \dot V \Delta T\), with the familiar 0.33 W/m³K constant folding in air's density and specific heat, is driven by wind pressure and stack effect, so it is neither constant nor controllable in a leaky building. Airtightness testing quotes \(q_{50}\) or \(n_{50}\) at 50 Pa, and modern standards demand values a leaky house would fail by a factor of ten. But a sealed house without deliberate ventilation accumulates moisture, CO₂ and volatile organics, so the standard answer is mechanical ventilation with heat recovery, recovering 80-90% of the outgoing heat. The order of operations matters: seal first, then ventilate deliberately - the reverse produces mould.
05Steady state is a convenient lie
All of the above assumes equilibrium. Real buildings are dynamic: thermal mass stores heat and shifts loads in time, described by admittance, decrement factor and time lag, which is why a heavy masonry building stays cool through a hot afternoon and a lightweight one does not. Solar gains through south-facing glass can supply a serious fraction of winter heating and a serious cooling problem in summer, so the same window is an asset in January and a liability in July - hence external shading, which works, versus internal blinds, which mostly do not. Dynamic simulation exists because the annual sum of a steady-state model can be right while every hour in it is wrong.
06Why measured savings are smaller than modelled ones
The energy performance gap is well documented and has several honest causes. Predicted savings assume the house was heated to 20 °C before the retrofit, and cold houses are frequently under-heated, so occupants take part of the improvement as comfort rather than as money - the rebound effect, typically 10-30% for heating. Workmanship gaps, thermal bridging not modelled, and airtightness worse than specified account for more. None of this argues against insulating; it argues for measuring, and for the one line that survives every version of this analysis: the cheapest kilowatt-hour is the one the building never needed.
Key Formulas
| Total resistance | \(R_{\text{tot}} = R_{si}+\sum\dfrac{d_i}{\lambda_i}+R_{se}\) | |
|---|---|---|
| U-value | \(U = \dfrac{1}{R_{\text{tot}}}\) | |
| With thermal bridges | \(H = \sum U_iA_i+\sum\psi_j\ell_j+\sum\chi_k\) | |
| Ventilation loss | \(Q_v = \rho c_p\dot V\Delta T \approx 0.33\,nV\Delta T\) | |
| With heat recovery | \(Q_v \to (1-\eta_{\text{hr}})\,0.33\,nV\Delta T\) | η ≈ 0.85 |
| Annual demand | \(E = \dfrac{24\,H\,\mathrm{DD}}{1000}-Q_{\text{gains}}\) | |
| Decrement factor | \(f = \dfrac{\Delta T_{\text{in}}}{\Delta T_{\text{out}}}\) | dynamic response of mass |
| Rebound | \(S_{\text{actual}} = (1-r)S_{\text{modelled}}\) | r ≈ 0.1–0.3 |
Things worth knowing
- The optimum cavity in double glazing is about 16 mm. Below that, conduction across the gas dominates; above it, a convection loop starts inside the cavity and the U-value gets worse again.
- Low-emissivity coatings are a few atoms thick and cut the radiative heat transfer across a glazing cavity by around two thirds. They are the single largest improvement in windows since the second pane.
- Degree-days let you compare buildings across climates: multiply the loss coefficient in W/K by 24 and by the local degree-day total to get annual kWh. It is crude, ignores solar and internal gains, and is still one of the most useful estimates in the business.