Heat is lost from a house in two ways: through the walls, floor and roof; and by exchanging heated air for unheated air. The first of these is tackled with insulation, the second with draughtproofing.
Insulation can be carried to as great an extreme as we wish, but draughtproofing has an upper limit: if the house is sealed completely, the inhabitants will suffocate. By reducing draughts it is normally considered that a house will change its air three times an hour. Since a cubic metre of air has a heat capacity of approximately 630 joules per degree, that means that in a perfectly insulated house 1kW of heating per cubic metre produces a temperature rise of about 1900oC, or our 240 cubic metre house needs 126W per degree of temperature difference to overcome air circulation. Once the house is reasonably well draughtproofed, insulation is clearly the important factor.
Our house has a volume of 210 cubic metres, so air circulation removes 630 cubic metres an hour. That means a loss of 396,900 joules per hour, or 110W, for each degree of difference between inside and outside. So for our 13oC temperature difference, we need 1.43kW to heat the incoming air.
All modern heating systems consist of a source of heat and a thermostat to control that source of heat. Older heating systems are the same, but the thermostat consists of a human to throw another log on the fire when it gets cold. For this reason, when the desired temperature is reached the amount of fuel used is equal to the amount of heat lost.
So to calculate the amount of heat that will be needed, we need only calculate the amount of heat lost. The amount of heat lost is equal to the area of the walls, floor and roof, multiplied by the insulation values for these, multiplied by the temperature difference between inside and outside. The insulation value is often called the "U-value", and is measured either in imperial units (BTU per hour per square foot per degree Farenheit) or in metric units (watts per hour per square metre per degree centigrade). I will use metric units.
Although we have not yet bought a house, we have looked at a fair number, and most of the ones we have looked at have about the same dimensions and construction. Typically the building might be a series of rooms in a row, maybe 14 metres long and 5 metres wide. The walls might average 3 metres high, so the walls are 114 square metres. The floor is 70 square metres. The roof is pitched at 45o, so it is about 100 square metres.
The floor and walls are made of granite: for that thickness the U-value is between 6 and 10 watts per square metre per degree centigrade. The roof is made of slate, which might lose 40W per square metre per degree. Plugging these values in gives the following heating requirements:-| Part of house | Area | U-Value | Loss |
|---|---|---|---|
| Roof | 100 | 40 | 4000 |
| Walls | 114 | 6-10 | 684-1140 |
| Floor | 70 | 6-10 | 420-700 |
| TOTAL | 5104-5840 | ||
This simple calculation shows that for each degree of difference between inside and outside, the house loses somewhere between 5 and 6 kW. So, given an average outside temperature of 5oC, and an inside temperature of 18oC, that's still a heater of 78kW, and a daily usage of 1,872kWh!
However, the table also gives us the clue about what would be best to insulate: the roof. Polyurethane foam has a U-value of 0.03watt per metre per degree, so putting 75mm of polyurethane foam between the felt and the rafters is likely to reduce the roof loss to 0.39 Watts per square metre. Some insulation between the rafters, maybe only 150mm of glassfibre bats, will reduce this to the 0.35 figure required by British building regulations. This then reduces our total loss for 100 square metres of roof to just 38W per degree of temperature difference.
Now our table looks like this:-
| Part of house | Area | U-Value | Loss |
|---|---|---|---|
| Roof | 100 | 0.35 | 35 |
| Walls | 114 | 6-10 | 684-1140 |
| Floor | 70 | 6-10 | 420-700 |
| TOTAL | 1139-1875 | ||
Now for each degree of difference between inside and outside, the house loses somewhere between 1.4 and 1.8kW. So, given an average outside temperature of 5oC, and an inside temperature of 18oC, our heater only needs to produce 23.4kW, and our daily usage is 561.6kWh. That's a lot, but it's much less than it was.
The walls can also be insulated, and so can the floor. Removing the floor screed, and re-pouring it with polystyrene insulation (0.03 watts per metre per degree) of 75mm thickness (0.4 watts per square metre per degree) gives 0.38W per square metre per degree - well below the 0.5W per square metre per degree needed for British regulations. Alternatively a wooden (0.14) floor may be constructed: removing the screed and replacing it with a 25mm air gap and 75mm of polystyrene between 100mm x 50mm rafters on 400mm centres with 18mm of wood flooring on top gives rafters at 0.7 and insulation at 0.38 gives an overall 0.42 beneath the floor, which is better than the 0.5 required by the building regs. It also gives a 25mm gap between insulation and floor, suitable for fitting underfloor heating. A similar scheme on the walls creates a 25mm cavity and also adds 12mm plasterboard to the insualtion, giving the walls an insulation rating of 0.38, significantly better than the regulations again. A good door and double-glazed windows should keep the wall average below 0.5. This insulation reduces the wall loss to 57W per degree of difference; and the floors to 34W per degree of difference.
Once the walls are well insulated, the insulation value of windows and doors comes into play. To calculate the U-value of a wall with a window in it, the area of the window and the area of the wall are calculated. The U-value of the wall-with-window is the average of the wall and the window. So a 20 square metre wall with a U-value of 0.4 that has a 2 square metre window with a U-value of 1.4 has an average U-value of 2 x 1.4/20 + 18 x 0.4/20 = 0.5. Fortunately modern double glazed windows have similar U-values to walls, so for now I'm going to pretend the house has no windows. Alternatively, windows can be added to the table.
Here is our well-insulated house table:-
| Part of house | Area | U-Value | Loss |
|---|---|---|---|
| Roof | 100 | 0.35 | 35 |
| Walls | 114 | 0.5 | 52 |
| Floor | 70 | 0.42 | 29.4 |
| TOTAL | 116.4 | ||
Now the 13oC temperature difference needs a heater of 1.5kW, and the average daily bill is about 36kWh. Even when the temperature outside is a chilly -12oC, the heating will only need to find 3.5kW, and 84kWh per day.
An uninsulated house of the type we've described needs about 6kW to get a difference of one degree, or 78kW to heat it on an average winters' day. That adds up to a heating bill of 1800kWh per day, or 168,480kWh for a winter quarter. Clearly insulation is required to get a bare house warm.
An insulated house needs 110W per degree for insulation, and about the same for air movement. So a total heat of 3kW is all that is needed on an average winter's day, and that leads to daily usage of 72kWh or a winter quarter bill for 6480 units.
It can be seen that the savings from insulation have two effects: not only do they save on heating bills, but they make it possible to provide heating to obtain a reasonable inside temperature.
This page is some notes on Domestic Power from Renewable Sources, and is written and maintained by Simon. At this stage these pages are constantly under revision. Thoughts and comments are welcome.