Conduction occurs when a temperature differential causes heat flow within a material or between materials in thermal contact. The ability of materials to transfer heat as a result of a given temperature difference varies by factors of many thousands - from metals like silver and copper which are excellent conductors, to gases like argon which are very poor conductors.
The rate at which heat flows through a slab of homogenous material under steady-state conditions is given by:
- where
- Q = the resultant heat flow (Watts)
A = the surface area through which the heat flows (m²)
dT = the temperature difference between the warm and cold sides of the material (K), and
R = the thermal resistance per unit area of the piece of material (m²K/W).
The value of the thermal resistance of a piece of material can be thought of as the temperature difference across it required to produce one unit of heat flow per unit area.
Resistance vs Resistivity
Resistivity is a material property and refers to that material's ability to resist the flow of heat. Resistance on the other hand is an object property and depends on both the resistivity of the material and its overall thickness within that particular object.
Conductance vs Conductivity
Conductivity (k) is a material property and means its ability to conduct heat through its internal structure. Conductance on the other hand is an object property and depends on both its material and thickness.
Conductance equals conductivity multiplied by thickness, in units of W/m²K. As conductivity is the reciprocal of resistivity, the total resistance of a material can therefore be given as its total thickness divided by total conductivity.
As resistivity is the inverse of conductivity, and conductivity values are far more readily available for most building materials than resistivities, it is possible to calculate the material's resistance using conductivity as follows:
- where
- R = the thermal resistance per unit area of the piece of material (m²K/W),
t = represents the thickness of the material (m), and
k = represents the conductivity of the material(W/mK).
Total Thermal Resistance
A building structure is usually composed of a number of different materials which may be considered to act:
- In Series
When materials are place in series, their thermal resistances are added so that the same area will conduct less energy for a given temperature difference. An example of this is a cavity-brick wall, with two layers of brick, an air gap and 12mm of plasterboard - all in series. - In Parallel
When materials are placed in parallel, their thermal conductances are added and the total energy flow is increased for a given temperature difference. An example of this would be a cavity-brick wall with a window inserted within it.
The total resistance of an element includes all of the resistances of the individual materials that make it up as well as both the internal and external air-film resistance. It's units are the inverse of conductivity, i.e.: m²K/W.
Air Film Resistance
Air film resistance results from convection currents at the surface of a material. As the surface heats up or cools down, it affects the temperature of the air immediately adjacent. This then starts to rise or fall depending on whether it is hotter or colder. This has the same effect as increasing the resistance of the material to the flow of heat. The following table gives standard air-film resistance values.
| CONDITION | R (m²K/W) |
|---|---|
| Internal Air-Film Resistance (Rsi) | 8.13 |
| External Air-Film Resistance (Rso) | 18.18 |
Composite Building Materials
For a composite building element made up of a number of layers of different materials, its total resistance is given as:
And the resistance of the nth layer is:
- where
- Rt = the total overall resistance of the element (m²K/W),
Rn = the resistance of the nth material within a composite element (m²K/W),
tn = the thickness of the nth material in a composite element (m), and
kn = is the conductivity in of the nth material in a composite element.
Fortunately, enough is known about various materials to enable the calculation of an overall thermal character for most common fixed-dimension building systems so that an overall resistance (or conductance) can be derived.
Figures can be derived for single glazed and double glazed windows, concrete slab floors, suspended wooden floors, walls and so on. These characteristics are usually written as either an R-Value (for insulations) or a U-Vaue for other elements.
The R-Value
Resistance is usually given as an "R" value which is the resistance of one square metre of the material subject to a one degree temperature difference. Thus an R value of a typical fibreglass bat may be given as R = 2.4, with the implication that it has the units m²K/Watt. This means that if one takes the area of insulation in square metres multiplied by the temperature difference in degrees Kelvin and divided by 2.4, one gets the heat flow in Watts. For example, 100 square metres of R2.4 insulation, exposed to a 20°K difference, will pass about 833 Watts.
In fact, the heat loss would be expected to be slightly from this because there is an additional resistance in getting the energy from the inside air to the wall surface, and from the outside wall surface to the outside air. Moreover, the heat transfer on the outside surface may vary with wind speed.
The U-Value
The U-Value is an important concept in building design. It represents the air-to-air transmittance of an element. This refers to how well an element conducts heat from one side to the other, which makes it the reciprocal of its thermal resistance. Thus, if we calculate the thermal resistance of an element, we can simply invert it to obtain the U-Value (U = 1 / Rt):
The U-Value is a property of a material. Thus its units are Watts per metre squared Kelvin ('W/m² K'). This means that, if a wall material had a U-Value of 1 W/m² K, for every degree of temperature difference between the inside and outside surface, 1 Watt of heat energy would flow through each metre squared of its surface.
As an example, assume a wall with a U-Value of 4.5 W/m²K and a surface area of 10 m². If the outside temperature was 30°C and the inside was 25°C, we could calculate the total heat gain due to conduction through the wall as follows:
- where
- Q = the resultant heat flow (Watts)
A = the surface area through which the heat flows (m²)
dT = the temperature difference between the warm and cold sides of the material (K), and
R = the thermal resistance per unit area of the piece of material (m²K/W).
Cavities and Air Spaces
Heat is transferred across an air space by a combination of conduction, convection and radiation. Heat transfer by conduction is inversely proportional to depth of the air space. Convection is mainly dependant on the height of the air space and its depth. Heat transfer by radiation is relatively independent of both thickness and height, but is greatly dependent on the reflectivity of the internal surfaces. All three mechanisms are dependent on the temperatures of surface temperatures. When all three heat transfer processes occur at the same time, the overall thermal resistance of air spaces, both reflective and non-reflective, becomes virtually independent of gap depth when it is greater than around 25mm.
The resistance of a thick air space can be increased by subdividing it into several thin layers. The resistance of the whole space is then the sum of the resistances of the thin air spaces plus the resistances of the separators. This approach is most effective when the material used to subdivide the space has a low emissivity, such as an aluminium foil.
In this case the heat transfer by radiation is reduced to less than 10 per cent of what it would be if the separator were ordinary craft paper. It is important to remember, however, that foil by itself does not provide thermal resistance; foil facing an air space increases the resistance of the air space. The increase in resistance achieved by subdividing will be reduced if any air is allowed to move from one space to another or if the reflective surface becomes coated with dirt or condensation.
Thermal Bridges
The introduction of a section of good conductor in parallel with materials of high resistance is often referred to as "thermal bridging" because it provides a path for heat flow which by-passes the main insulation.
An example of this may be a stud wall with metal C-channels. Even if the spaces between are insulated, each stud provides a direct conduit for heat to flow from one side of the wall to the other. Some of the references below provide an excellent discussion on thermal bridges.
Related Links
- Heating and Cooling Requirements, CBD-105
- http://www.nrc.ca/irc/cbd/cbd105e.html
- Thermal Resistance of Building Insulation, CBD-149
- http://www.nrc.ca/irc/cbd/cbd149e.html
- Thermal Bridges in Buildings, CBD-44
- http://www.nrc.ca/irc/cbd/cbd009e.html