Heat Loss Through a Wall: U-values

Set the insulation on a solid brick wall, how much of the wall is window and what glazing it has, and the temperatures inside and out. The page adds up the thermal resistance of every layer, works out the heat through the wall and through the glass, and the temperature of each surface you could touch.

Worked example — Heat Loss Through a Wall: U-values

Total resistance of the wall	0.509 m²K/W
U-value of the wall	1.966 W/m²K
U-value of the glazing	5.747 W/m²K
Window and wall areas	0.00 m² + 10.00 m²
Heat through the wall	491.4 W
Heat through the window	0.0 W
Total heat loss	491.4 W
Average U-value, window included	1.966 W/m²K

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Heat Loss Through a Wall: U-values

Heat flows through a flat layer at a rate proportional to the temperature difference across it and inversely to its thermal resistance, R = d/λ, the thickness over the material's thermal conductivity, in m²K/W. The still air against each face adds a surface resistance of its own. Layers the heat must cross one after another add like resistors in series, R_T = R_si + ΣR + R_se, and the U-value is its inverse, U = 1/R_T, the watts lost per square metre for each degree of difference. A window beside a wall is a second path in parallel: the heat flows Q = U·A·ΔT through each are added.

What each symbol means

λ: thermal conductivity of a material, W/mK. Brick 0.77, mineral wool 0.033, glass 1.0 on this page
R: thermal resistance of a layer, its thickness over λ, m²K/W. The surface films are resistances too: 0.13 inside, 0.04 outside
U: the U-value, 1/R_T, in W/m²K. The heat lost per square metre per degree of difference
ΔT: inside temperature less outside temperature, in K, which is the same number as in °C

Assumes: Steady conditions: the temperatures have held long enough for the wall to stop warming or cooling, which for 215 mm of brick takes many hours. Heat flows straight through, one-dimensionally, so nothing is lost at corners, round the window frame or through mortar joints and wall ties, and the insulation is continuous. The glazing is its centre-pane value; a real window's frame and edge usually lose more. The surface resistances are the standard values for heat flowing horizontally, BR 443's, which fix the wind outside and the air movement inside. No sun, and no air leaking through cracks.
Breaks when: Wet insulation or wet brick conducts much better than dry, and a solid wall insulated on the inside can become colder and damper in its outer part; this page has no moisture in it. Air leaking through gaps is a separate loss and a different calculation. A thermal bridge, a concrete lintel or a steel beam across the insulation, can carry more heat than the square metres around it, which is why U-value conventions such as BR 443 have to make allowances for them. And the heat a room needs is not only this: ventilation, floor and roof all lose heat by the same arithmetic with their own U-values.

The first 50 mm of insulation saves more than the next 250 mm together 🖖

Press Bare brick: 10 m² of solid brick, 20 °C inside and −5 °C out, loses 491.4 W. Press First 50 mm and 50 mm of mineral wool behind the plasterboard brings it down to 123.5 W, a saving of 367.9 W. Then 300 mm: six times the insulation, a loss of 26.0 W, so the other 250 mm saved 97.5 W between them. The heat flow is AΔT/R, and what insulation adds is resistance, the same 1.515 m²K/W for every 50 mm of this wool. Added to a bare wall of 0.509 it nearly quadruples R; added to a wall that already has 250 mm it raises R by less than a fifth. The table under the results prints the saving of each step, and each is smaller than the one before: 367.9, 52.9, 21.2, 11.4, 7.1 and 4.9 W. It is the rule for resistors in series, applied to heat, and it is why the 0.26 W/m²K that England's Building Regulations set for a new wall takes 110 mm of this wool and nobody fits a metre.

A window a fifth the size of the wall loses five times the heat 🖖

Press Single glazing. The wall has 100 mm of insulation and a 2 m² single-glazed window. The 8 m² of wall loses 56.5 W; the 2 m² of glass loses 287.4 W, 83.6% of the total. Heat takes both paths at once, so it is the flows that add, not the resistances, and the path of least resistance carries most of it, as the smaller resistor does in a parallel circuit. Now read the table: tripling the insulation to 300 mm takes the total from 343.9 W only to 308.2 W, and in the second panel the total can never fall below the window's flat line. Press Low-e double glazing instead. The same window at a centre-pane U-value of 1.000 W/m²K loses 50.0 W and the total falls to 106.5 W, a saving no thickness of insulation on the wall could reach.

Single glazing insulates with the still air on its two faces 🖖

In the results, single glazing has a U-value of 5.747 W/m²K, a resistance of 0.174 m²K/W, and 97.7% of that resistance is the two thin layers of still air on its faces: 0.13 on the room side and only 0.04 outside, where the wind keeps sweeping the layer away. The 4 mm of glass contributes 0.004, 2.3%. Glass conducts heat thirty times better than mineral wool, and there is very little of it; in the first panel it is a hairline between the two air films. So almost all of the 25 degrees are lost in air, most of them in the film against the room, which leaves the inside face of the glass at 1.32 °C. That is why a single-glazed window streams with water on a cold morning while the insulated wall beside it sits at 19.08 °C. The worked problem below finds how dry the room would have to be for the glass to stay clear.

Problem solved in full

Will the window mist up? 6 steps
A room is at 20 °C and 50% relative humidity, and it is −5 °C outside. Will its single-glazed window mist up, and would low-e double glazing? Press Single glazing to check the first two steps.

Open this problem in the tool
1. The glazing's resistance is the two air films plus the glass, and its U-value is the inverse:
  
  $U_{g} = \frac{1}{R _{si} + d / λ + R _{se}} = \frac{1}{0 . 13 + 0 . 004/1 . 0 + 0 . 04} = \frac{1}{0 . 174} = 5.747 W/ m^{2} K$
2. The inside face of the glass is colder than the room by the temperature drop across the inside film:
  
  $T_{s} = T_{i} - R_{si} U_{g} Δ T = 20 - 0.13 \times 5.747 \times 25 = 1.32^{\circ} C$
3. The Magnus formula gives the pressure of water vapour that saturates air at 20 °C. At 50% relative humidity the room carries half of it:
  
  $e_{s} (20) = 6.1094 exp (\frac{17 . 625 \times 20}{243 . 04 + 20}) = 23.33 hPa, e = 0.50 \times 23.33 = 11.67 hPa$
4. The dew point is the temperature at which that vapour would saturate the air:
  
  $γ = ln 0.50 + \frac{17 . 625 \times 20}{263 . 04} = 0.6470, T_{d} = \frac{243 . 04 γ}{17 . 625 - γ} = 9.26^{\circ} C$
5. The glass is well below the dew point, so water condenses on it. It would stay clear only while the room's humidity is below the ratio of the saturation pressures at the glass and in the room:
  
  $φ_{m a x} = \frac{e _{s} ( 1 . 32 )}{e _{s} ( 20 )} = \frac{6 . 72}{23 . 33} = 28 . 8%$
6. Low-e double glazing puts the inside face at 16.75 °C, and the same ratio is:
  
  $\frac{e _{s} ( 16 . 75 )}{e _{s} ( 20 )} = \frac{19 . 03}{23 . 33} = 81 . 6%$
Answer

The single-glazed window mists up. Its inside face, at 1.32 °C, is 7.94 degrees below the room's dew point of 9.26 °C, and it would stay clear only below 28.8% relative humidity. Low-e double glazing at 16.75 °C stays clear up to 81.6%. Both figures are for the centre of the pane; the edge of a sealed unit, near its spacer and frame, is colder, so condensation there starts at a lower humidity.

$T_{s} = 1.32^{\circ} C < T_{d} = 9.26^{\circ} C φ_{m a x} : 28 . 8% \to 81 . 6%$

References (3)

The UK convention for U-value calculations: the total resistance as Rsi + R1 + … + Rn + Rse, the surface resistances 0.13 and 0.04 m²K/W for a wall, and the default conductivities this page uses for brick (0.77 W/mK, an outer leaf) and standard plasterboard (0.21 W/mK): B. Anderson and L. Kosmina, Conventions for U-value calculations, BR 443, 2019 edition. BRE, 2019
Glass in the U-value calculation: under EN 673 the resistivity of soda-lime-silicate glass is taken as 1.0 m·K/W, and a double unit of two 4 mm panes with one low-emissivity coating and a 16 mm argon-filled gap reaches a centre-pane U-value of 1.0 W/m²K; triple glazing reaches 0.5: Saint-Gobain Building Glass UK, Solar & Thermal Technical Guidance ST-1E: Thermal insulation. 2018
The Magnus formula for the saturation vapour pressure of water used in the worked problem, e = 6.1094 exp(17.625 t / (243.04 + t)) hPa, fitted over −40 to 50 °C: O. A. Alduchov and R. E. Eskridge, Improved Magnus' form approximation of saturation vapor pressure. Report DOE/ER/61011-T6, National Climatic Data Center, NOAA, 1997. DOI 10.2172/548871. Also published in Journal of Applied Meteorology 35(4), 1996, pp. 601-609

By the 42Math team · Reviewed against published sources. Corrections welcome.