# Insulation is really easy

Why do people get so hung up about thermal insulation, it really is not difficult.

The main thing to remember is that the power, in watts (W, J.s^{-1}) that passes though a material is approximately proportional to three things, conductivity (*k, λ*), temperature difference (∆T) in kelvin (K), and thickness in metres (m).

In arithmetic terms, the thermal conductivity of a material is written as W.m^{-1}.K^{-1}, or W/m.K (as I have never found a way to write a superscript negative sign on my Android phone).

Different materials have different thermal conductivity. Taking extreme ends of the spectrum, natural diamond conducts at a rate if 2200 W.m^{-1}.K^{-1} and a pure vacuum, for these purposes, is 0 W.m^{-1}.K^{-1}.

Now we don't, in the real world, work at the extremes.

So let us stick to some more common building materials.

Ordinary brick, *k* = 0.72 W.m^{-1}.K^{-1}. Concrete *k* = 1.28 W.m^{-1}.K^{-1}. Timber *k* = 0.14 W.m^{-1}.K^{-1}. Mineral wool insulation *k* = 0.038 W.m^{-1}.K^{-1}. Expanded polystyrene *k* = 0.04 W.m^{-1}.K^{-1}. Polyurethane foam *k* = 0.03 W.m^{-1}.K^{-1}. Orientated Stand Board (OSB), 6% adhesive *k* = 0.16 W.m^{-1}.K^{-1}. Plasterboard *k* = 0.19 W.m^{-1}.K^{-1}.

There is, obviously a lot more materials and it is down to whoever is calculating to find and check figures. An example of this is granite, there are a lot of different types and the *k*-Value can range from 1.73 to 3.98 W.m^{-1}.K^{-1}. So do your research.

It is not normal to fit a metre thickness of any insulating material, If we did, none of this write up would be necessary.

Because we use fractional dimensions i.e. 0.2m when planning the insulation levels of a building, the more common thermal resistance (R = K.m^{-2}.W^{-1}) is used. There are two advantages of using the R-Value, it takes the thickness of the material, and the area of the material, into account. Converting from the *k*-Value to the R-Value is really easy, just divide the thickness by the *k*-Value.

R = *l* / *k*

R-Value is often quoted and one thing to be careful of is that imperial units are often quoted.

You may have noticed that R-Value has somehow introduced a m^{-}^{2} unit, this comes about from dimensional analysis of all the International System of Units (SI) units W, which is kg.m^{2}.s^{-3} and some arithmetic rearranging when combined with the other units, m and K. This is a useful as we do not have building elements that only have thickness, they also have area.

The most useful thing about R-Values is that they can be added together to give a total thermal resistance (ΣR).

Taking a simple wall build up of:)

Outer: Brick, *k* = 0.72, 0.1m thickness.

Full Fill Mineral Wool, *k* = 0.038, 0.2m thickness.

Inner: Brick, *k* = 0.72, 0.1m thickness.

Plasterboard, *k* = 0.19, 0.012m thickness.

The overall thickness is 0.412m (dimensions may vary, so check).

Using the sum (Σ) of *l* / *k* for every component makes for a long equation, and it is usual to use a spreadsheet.

ΣR = 0.1 / 0.72_{(outer brick)} + 0.2 / 0.038_{(mineral wool)} + 0.1 / 0.72_{(inner brick)} + 0.012 / 0.19_{(plaster board)}

ΣR = 0.139_{(outer brick)} + 5.263_{(mineral wool)} + 0.139_{(inner brick)} + 0.063_{(plaster board)}

ΣR = 5.604 K.m^{-2}.W^{-1}. Note here that the effects of the mineral wool are dominant and that large R-Values are better.

It is not normal to talk about a house, or wall, having an R-Value, but a U-Value (anyone know if R should be proceeded with 'a' or 'an', sounds like it should be 'an', but U sounds better with 'a').

Changing to U-Value, which is W.m^{-2}.K^{-1} is simply a matter of taking the inverse of the R-Value K.m^{-2}.W^{-1}

U-Value = 1 / R

So in this example:

U = 1 / 5.604

U = 0.178 W.m^{-2}.K^{-1}.

If the wall, ceiling, roof or floor is of timber construction, the technique is just the same, just that the appropriate areas also have to be included in the final solution, so you work out the U-Value for all the studs and noggins, plus the OSB thickness, and then the U-Value for all the insulation and the OSB thickness.

There is one other thing when looking at heat losses, and that is the air film surrounding them. Air has a very good *k*-Value of 0.026 W.m^{-1}.K^{-1} and is really the component that is doing the majority of the work in insulation, the material i.e. mineral wool or polyurethane foam is just there to stop the air conducting by trapping it in place. Because of this, some allowance has to be made for any air voids in the wall build up i.e. a service gap.

To simplify this, it has been decided that two standard values are used, one for walls and one for roofs, with no regard to thickness. The wall R-Value is 0.18 K.m^{-2}.W^{-1}, roof R-Value 0.04 K.m^{-2}.W^{-1}.

So taking the above example, and extra 0.18 K.m^{-2}.W^{-1} must be added to the sum of the R-Value.

ΣR = 5.604 + 0.18

ΣR = 5.784

Convert to U-Value

U = 1 / 5.784

U= 0.173 W.m^{-2}.K^{-1}.

It only makes a small difference, and at the third decimal place, but is still worth including because when the numbers are rounded, it may be the difference between the desired value or not.

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