Single-pane window: film resistances and the overnight loss
SPH4C Grade 12 Physics (College) · Energy Transformations
For a home-energy project, a student instruments their own house for a night: one logging thermometer taped to the kitchen wall an arm's length from the back door, another hung outside under the eave, both set to record while the house sleeps. In the morning the two logs are averaged over the 8.00 h winter night: 20.0 °C in, −5.0 °C out — a difference a bare hand pressed to the glass could have guessed at. The back door's single-glazed pane measures 1.20 m by 0.80 m (0.960 m²) by the tape, and its glass is 4.0 mm thick with conductivity 0.96 W/(m·K). The glass, though, is only the middle of the path: a building-science table supplies what the glass alone cannot — the still-air film that clings to the inside face at RSI 0.12, and the thinner winter-wind film outside at RSI 0.030.
Every number in this problem is editable — change any value below and the whole chain recalculates.
- A = 0.96 m² — Area of the pane (1.20 m × 0.80 m)
- d = 4 mm — Glass thickness
- k = 0.96 W/(m·K) — Thermal conductivity of glass
- R_in = 0.12 m²·K/W — Inside still-air film resistance
- R_out = 0.03 m²·K/W — Outside wind film resistance
- ΔT = 25 C° — Inside-to-outside difference (20.0 − (−5.0))
- t = 8 h — Length of the night
- (a)the R-value of the glass layer itself
- (b)the total R-value of the path, air films included
- (c)the steady heat loss through the pane
- (d)the energy that walks out through this one window by morning
R = L/k for the glass alone: four thin millimetres of a decent conductor is RSI 0.0042 — next to nothing. Anyone who stops here and computes Q = kAΔT/d gets 5,760 W, a space heater's output times four, and the absurdity is the lesson: bare Fourier through the glass is not how a window loses heat.
Carried onward at full precision, not this rounded figure.
Heat must cross three layers in series — still air clinging inside, glass, moving air outside — and series resistances simply add: RSI 0.154 in total. Run the shares: the two invisible films carry 36 parts in 37 of the resistance, the glass one part. A single pane is mostly a frame for its own air films.
Carried onward at full precision, not this rounded figure.
Q = A·ΔT/R_tot: about 156 W flows through this one pane all night — a couple of old light bulbs' worth, continuously. The ΔT is 20.0 − (−5.0) = 25.0 K; mishandling the negative sign to get 15 K is the arithmetic slip this scenario is built to catch.
Carried onward at full precision, not this rounded figure.
Open the Heat Loss Through an Assembly (Q = A·ΔT/R) solver →
Energy is power held for a time: 155.7 W × 28 800 s ≈ 4.48 MJ by dawn, which the utility would bill as 1.25 kWh — from one window, in one night. The 8.00 h enters as seconds; hours fed raw would understate the loss 3,600-fold.
Carried onward at full precision, not this rounded figure.
Therefore the glass itself resists almost nothing (RSI 0.0042), the air films raise the path to RSI 0.154, the pane bleeds 156 W across the 25.0 K night, and by morning 4.48 MJ — 1.25 kWh — has left through one small window.
Why this order
The chain is ordered to spring a trap and then disarm it. Part (a) computes the honest R-value of the glass and invites the bare-conduction answer — 5,760 W through a pane, which would boil a kettle every four minutes — before part (b) adds the two air films and the estimate collapses by a factor of 37 to something a hand on cold glass actually feels. The physics point is that a series path is governed by its largest resistances, and here the largest resistances are layers no ruler can measure: the millimetres of still air the glass traps against itself indoors, and the thinner, wind-scrubbed film outside. That is why the outside film shrinks on a windy night (and the window loses faster), and why the R-value of any bare pane is almost independent of its glass. Steps (c) and (d) are then plain bookkeeping: rate from resistance, energy from rate and the night's length, each unit conversion done where the note says.
The engineering consequence is the entire glazing industry. If the films are the only real resistance, the way to a better window is more films — which is exactly what double glazing is: a sealed gap of still air (RSI ≈ 0.17 by itself) buys more than doubling the glass a hundred times over would. Argon fills, low-e coatings that suppress the radiative bypass, triple glazing — each is another still layer or a quieter parallel path, and each shows up directly in this same three-step arithmetic. The 1.25 kWh answer also scales soberingly: a house with fifteen such panes loses ~19 kWh a night through glass alone, which is why window upgrades headline every energy audit, and why the RSI/R-value language this chain practices is the one Canadian building code and contractor both speak.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.