HKDSE • Elective 3: Energy and Use of Energy • Buildings & air conditioning

Building Heat Gain, U-value & Air-Conditioner COP

Heat leaks into a cooled room through its envelope at a rate $Q/t = U A\,\Delta T$, and sunlight adds more through the window. An air conditioner must pump all of it back out, using electrical power $P = (Q/t)/\text{COP}$. Change the wall, window, shading and air conditioner and watch the energy bill respond.

Parameters

Hong Kong's recommended summer setting is 25.5 °C.

Wall material ($k$ in W m⁻¹ K⁻¹)

Fraction of solar radiation transmitted; sunshades, blinds and tinted glass lower $SC$.

$\text{COP} = Q_{\text{removed}} / W_{\text{electrical}}$. A Grade 1 energy-label unit has a higher COP.

Presets

Live readings

Wall $U$-value 8.50 W/m²K
Conduction: walls + roof 6.26 kW
Conduction: window 186 W
Solar gain: $SC \cdot I \cdot A_{\text{win}}$ 1.28 kW
Total heat gain 7.72 kW
OTTV (gain per m² of envelope) 80 W/m²
AC electrical power $= Q/\text{COP}$ 2.57 kW
Heat rejected outdoors $= Q + W$ 10.3 kW
8 h of cooling (≈ HK$1.3/kWh) 20.6 kWh · HK$27

HKDSE formulae

$$ \frac{Q}{t} = U A\,\Delta T, \qquad U = \frac{k}{d} $$
$$ \text{COP} = \frac{Q_{\text{removed}}}{W} $$
$U$: thermal transmittance (W m⁻² K⁻¹)
$k$: thermal conductivity of the material (W m⁻¹ K⁻¹)
OTTV: overall thermal transfer value = total envelope heat gain ÷ envelope area

Layers in series add their thermal resistances: $\frac{1}{U} = \frac{d_1}{k_1} + \frac{d_2}{k_2}$ (surface air films are ignored here).

3D room 6 m × 5 m × 3 m, window faces south
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Heat conduction Solar gain Cooled air Electrical input

Heat gain breakdown and air-conditioner input

Physics note

A thin layer of foam ($k$ = 0.035) adds far more thermal resistance $d/k$ than a thick concrete wall, so the $U$-value — and the conduction gain — collapses. Solar gain through glass does not depend on $\Delta T$ at all: only shading, tinting or a smaller window reduces it. The air conditioner is a heat pump; by energy conservation it dumps $Q + W$ outdoors, and a higher COP means less electrical work $W$ for the same cooling. Hong Kong's Building (Energy Efficiency) Regulation limits the OTTV of commercial building envelopes.