Building Insulation

How the model works

Heat flows through each layer of a wall in turn, so their thermal resistances add up. A layer of thickness d and conductivity k has R = d / k (m²K/W); the surfaces add small film resistances (0.13 inside, 0.04 outside). The U-value is the inverse of the total:

U = 1 / (R_si + Σ dᵢ/kᵢ + R_se) Q = U A ΔT

The temperature falls through each layer in proportion to its share of the resistance, so most of the drop happens across the insulation. Air leakage adds Q_vent ≈ 0.33 × n × V × ΔT watts, with n air changes per hour and V the house volume in m³.

Annual heating energy is estimated with heating degree-days (HDD): the sum over the year of how far the daily mean temperature falls below the base temperature. Energy = total heat-loss coefficient × HDD × 24 h.

The house is 10 m × 8 m with two storeys (5 m tall): 180 m² of external wall including windows, an 80 m² roof and an 80 m² ground floor (fixed U = 0.45), and 400 m³ of air. The bare wall is 13 mm plaster on 220 mm brick (k = 0.77).

Idealized steady state: no thermal bridges, solar or internal heat gains, thermal mass, moisture effects or heating-system losses. Real savings are usually smaller than this model suggests.

Guided experiments

1. Old house. Solid brick walls, single glazing, no loft insulation, one air change per hour; 0 °C outside. Prediction: wall U ≈ 2.08 and roof U ≈ 2.56 W/m²K. The house loses about 17.3 kW, most through the walls (6.7 kW) and roof (4.3 kW). Old house
2. Insulate the loft with 270 mm of mineral wool. Prediction: roof U falls to 0.14, roof loss from 4.3 kW to 0.24 kW, and total loss to about 13.3 kW: the cheapest big win. Loft insulation
3. Diminishing returns. Add 100 mm, then 200 mm of wall insulation. Prediction: 100 mm cuts the wall U from 2.08 to 0.34 and saves 5.6 kW. Doubling to 200 mm saves only another 0.5 kW, because U = 1/R flattens as R grows. 100 mm 200 mm
4. Modern house. 150 mm PIR walls, 300 mm loft, triple glazing, 0.3 air changes per hour. Prediction: total loss about 2.7 kW, one-sixth of the old house. Air leakage and the uninsulated floor are now the largest losses. Modern

Challenge

A wall has U = 0.30 W/m²K. You add a further 50 mm of PIR (k = 0.022). What is the new U-value, and by how much does the heat loss through 100 m² of this wall fall on a day with a 20 K temperature difference?

References

• ISO 6946:2017, Building components and building elements: thermal resistance and thermal transmittance, calculation methods.
• CIBSE, Guide A: Environmental Design (2015), sections 3 and 4.
• Szokolay, S. V., Introduction to Architectural Science, 3rd ed. (2014), ch. 1.