Living Room Heat Loss: Double Glazing vs Ceiling Insulation
Alex's living room is 5.0 m × 4.0 m × 2.4 m. On a winter day the inside is 20 °C and outside 10 °C. An energy audit estimates heat loss rates: windows 800 W, roof 1000 W, walls 600 W, floor 200 W. The room has a 2.0 kW heater. Alex can afford one upgrade: double glazing for $4000, which cuts window heat loss by 60%, or ceiling insulation for $1500, which cuts roof heat loss by 70%. Alex wants the room to stay at 20 °C and wants the best value for money. The heater has a thermostat and runs continuously on cold days. The family uses the room for 5 hours each evening. Electricity costs 30 cents per kWh.
1. Calculate the total heat loss from the room and decide whether the 2.0 kW heater can maintain 20 °C. Explain.
Read the worked answer
- Total heat loss = (800+1000+600+200=2600\text{ W}).
- Convert to kW: (2600\text{ W}=2.6\text{ kW}).
- The heater supplies 2.0 kW, which is less than the 2.6 kW loss. At a steady inside temperature of 20 °C the room loses more heat than the heater can replace.
- The room may stay warm for a while because of thermal mass, but eventually it will cool until the heat loss equals 2.0 kW, which would be at a lower inside temperature. The thermostat cannot fix this because the heater is already at maximum.
- Judgement: the 2.0 kW heater is not sufficient on its own to maintain 20 °C under the stated conditions.
2. Compare the two upgrades by heat saved and cost per watt saved. Which is better value, and which makes the heater sufficient?
Read the worked answer
- Double glazing saves (0.60\times800=480\text{ W}). New total loss = (2600-480=2120\text{ W}=2.12\text{ kW}), which is still greater than 2.0 kW, so the heater remains insufficient.
- Ceiling insulation saves (0.70\times1000=700\text{ W}). New total loss = (2600-700=1900\text{ W}=1.9\text{ kW}), which is less than 2.0 kW, so the heater becomes sufficient.
- Cost per watt saved:
- Double glazing: (4000/480=$8.33) per W saved.
- Ceiling insulation: (1500/700=$2.14) per W saved.
- Judgement: ceiling insulation saves more heat, costs less, and is the only single upgrade that makes the 2.0 kW heater adequate. It is clearly better value for money in this scenario.
3. Estimate the daily electricity cost saving from the insulation, and the payback period, using 5 hours per evening at 30 cents/kWh. State one assumption.
Read the worked answer
- Heat saved by insulation = 700 W = 0.70 kW.
- Energy saved per evening = (0.70\times5=3.5\text{ kWh}).
- Cost saving per evening = (3.5\times$0.30=$1.05).
- Payback = ($1500/$1.05\approx1429) evenings.
- If the room is heated every evening, that is about (1429/365\approx3.9) years. If heating is only used in winter, say 150 evenings per year, payback is about (1429/150\approx9.5) years.
- Assumption: the heater would otherwise have run long enough to supply the saved heat, electricity price stays at 30 cents/kWh, and the heat-loss rates are constant. If the family heats less often, the payback is longer.