ISO 10211:2007

INTERNATIONAL ISO
STANDARD 10211
First edition
2007-12-15
Thermal bridges in building
construction — Heat flows and surface
temperatures — Detailed calculations
Ponts thermiques dans les bâtiments — Flux thermiques et
températures superficielles — Calculs détaillés

Reference number
©
ISO 2007
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ii © ISO 2007 – All rights reserved

Contents Page
Foreword. v
Introduction . vi
1 Scope .1
2 Normative references .1
3 Terms, definitions, symbols, units and subscripts.2
3.1 Terms and definitions .2
3.2 Symbols and units.6
3.3 Subscripts .7
4 Principles.7
5 Modelling of the construction .7
5.1 Dimension systems .7
5.2 Rules for modelling .7
5.3 Conditions for simplifying the geometrical model.13
6 Input data.17
6.1 General.17
6.2 Thermal conductivities of materials .18
6.3 Surface resistances.18
6.4 Boundary temperatures .18
6.5 Thermal conductivity of quasi-homogeneous layers .18
6.6 Equivalent thermal conductivity of air cavities .18
6.7 Determining the temperature in an adjacent unheated room .19
7 Calculation method.19
7.1 Solution technique.19
7.2 Calculation rules.19
8 Determination of thermal coupling coefficients and heat flow rate from 3-D calculations .20
8.1 Two boundary temperatures, unpartitioned model.20
8.2 Two boundary temperatures, partitioned model.20
8.3 More than two boundary temperatures .21
9 Calculations using linear and point thermal transmittances from 3-D calculations .21
9.1 Calculation of thermal coupling coefficient.21
9.2 Calculation of linear and point thermal transmittances .22
10 Determination of thermal coupling coefficient, heat flow rate and linear thermal
transmittance from 2-D calculations.23
10.1 Two boundary temperatures .23
10.2 More than two boundary temperatures .23
10.3 Determination of the linear thermal transmittance .23
10.4 Determination of the linear thermal transmittance for wall/floor junctions.24
10.5 Determination of the external periodic heat transfer coefficient for ground floors .25
11 Determination of the temperature at the internal surface .26
11.1 Determination of the temperature at the internal surface from 3-D calculations .26
11.2 Determination of the temperature at the internal surface from 2-D calculations .27
12 Input and output data .28
12.1 Input data.28
12.2 Output data.28
Annex A (normative) Validation of calculation methods .30
Annex B (informative) Examples of the determination of the linear and
point thermal transmittances. 37
Annex C (informative) Determination of values of thermal coupling coefficient and temperature
weighting factor for more than two boundary temperatures . 40
Bibliography . 45

iv © ISO 2007 – All rights reserved

Foreword
ISO (the International Organization for Standardization) is a worldwide federation of national standards bodies
(ISO member bodies). The work of preparing International Standards is normally carried out through ISO
technical committees. Each member body interested in a subject for which a technical committee has been
established has the right to be represented on that committee. International organizations, governmental and
non-governmental, in liaison with ISO, also take part in the work. ISO collaborates closely with the
International Electrotechnical Commission (IEC) on all matters of electrotechnical standardization.
International Standards are drafted in accordance with the rules given in the ISO/IEC Directives, Part 2.
The main task of technical committees is to prepare International Standards. Draft International Standards
adopted by the technical committees are circulated to the member bodies for voting. Publication as an
International Standard requires approval by at least 75 % of the member bodies casting a vote.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. ISO shall not be held responsible for identifying any or all such patent rights.
ISO 10211 was prepared by Technical Committee ISO/TC 163, Thermal performance and energy use in the
built environment, Subcommittee SC 2, Calculation methods.
This first edition of ISO 10211 cancels and replaces ISO 10211-1:1995 and ISO 10211-2:2001, which have
been technically revised.
The principal changes are as follows:
⎯ this first edition of ISO 10211 merges the title and general contents of ISO 10211-1:1995 and
ISO 10211-2:2001 into a single document;
⎯ Clause 3 indicates that ISO 10211 now uses only temperature factor, and not temperature difference
ratio;
⎯ 5.2.2 specifies that cut-off planes are to be located at the larger of 1 m and three times the thickness of
the flanking element;
⎯ 5.2.4 contains a revised version of Table 1 to correct error for three-dimensional calculations and to clarify
intentions;
⎯ 5.2.7 specifies that acceptable criterion is either on heat flow or on surface temperature; the heat flow
criterion has been changed from 2 % to 1 %;
⎯ 6.3 specifies that surface resistance values are to be obtained from ISO 6946 for heat flow calculations
and from ISO 13788 for condensation calculations; the contents of Annexes E and G of
ISO 10211-1:1995 have been deleted in favour of references to ISO 13788;
⎯ 6.6 specifies that data for air cavities is obtained from ISO 6946, EN 673 or ISO 10077-2; the contents of
Annex B of ISO 10211-1:1995 have been deleted in favour of these references;
⎯ 10.4 contains text formerly in ISO 13370, revised to specify that linear thermal transmittance values for
wall/floor junctions are the difference between the numerical result and the result from using ISO 13370
(a more consistent definition);
⎯ Annex A contains corrections to results for case 3; the conformity criterion for case 3 has been changed
from 2 % of heat flow to 1 %; a new case 4 has been added;
⎯ Annex C contains a corrected procedure;
⎯ all remaining annexes from ISO 10211-1:1995 and ISO 10211-2:2001 have been deleted.
Introduction
Thermal bridges, which in general occur at any junction between building components or where the building
structure changes composition, have two consequences compared with those of the unbridged structure:
a) a change in heat flow rate, and
b) a change in internal surface temperature.
Although similar calculation procedures are used, the procedures are not identical for the calculation of heat
flows and of surface temperatures.
A thermal bridge usually gives rise to three-dimensional or two-dimensional heat flows, which can be precisely
determined using detailed numerical calculation methods as described in this International Standard.
In many applications, numerical calculations based on a two-dimensional representation of the heat flows
provide results of adequate accuracy, especially when the constructional element is uniform in one direction.
A discussion of other methods for assessing the effect of thermal bridges is provided in ISO 14683.
ISO 10211 was originally published in two parts, dealing with three-dimensional and two-dimensional
calculations separately.
vi © ISO 2007 – All rights reserved

INTERNATIONAL STANDARD ISO 10211:2007(E)

Thermal bridges in building construction — Heat flows and
surface temperatures — Detailed calculations
1 Scope
This International Standard sets out the specifications for a three-dimensional and a two-dimensional
geometrical model of a thermal bridge for the numerical calculation of:
⎯ heat flows, in order to assess the overall heat loss from a building or part of it;
⎯ minimum surface temperatures, in order to assess the risk of surface condensation.
These specifications include the geometrical boundaries and subdivisions of the model, the thermal boundary
conditions, and the thermal values and relationships to be used.
This International Standard is based upon the following assumptions:
⎯ all physical properties are independent of temperature;
⎯ there are no heat sources within the building element.
This International Standard can also be used for the derivation of linear and point thermal transmittances and
of surface temperature factors.
2 Normative references
The following referenced documents are indispensable for the application of this document. For dated
references, only the edition cited applies. For undated references, the latest edition of the referenced
document (including any amendments) applies.
ISO 6946, Building components and building elements — Thermal resistance and thermal transmittance —
Calculation method
ISO 7345, Thermal insulation — Physical quantities and definitions
ISO 13370:2007, Thermal performance of buildings — Heat transfer via the ground — Calculation methods
ISO 13788, Hygrothermal performance of building components and building elements — Internal surface
temperature to avoid critical surface humidity and interstitial condensation — Calculation methods
3 Terms, definitions, symbols, units and subscripts
3.1 Terms and definitions
For the purposes of this document, the terms and definitions given in ISO 7345 and the following apply.
3.1.1
thermal bridge
part of the building envelope where the otherwise uniform thermal resistance is significantly changed by full or
partial penetration of the building envelope by materials with a different thermal conductivity, and/or a change
in thickness of the fabric, and/or a difference between internal and external areas, such as occur at
wall/floor/ceiling junctions
3.1.2
linear thermal bridge
thermal bridge with a uniform cross-section along one of the three orthogonal axes
3.1.3
point thermal bridge
localized thermal bridge whose influence can be represented by a point thermal transmittance
3.1.4
three-dimensional geometrical model
3-D geometrical model
geometrical model, deduced from building plans, such that for each of the orthogonal axes the cross-section
perpendicular to that axis changes within the boundary of the model
See Figure 1.
3.1.5
three-dimensional flanking element
3-D flanking element
part of a 3-D geometrical model which, when considered in isolation, can be represented by a 2-D geometrical
model
See Figures 1 and 2.
3.1.6
three-dimensional central element
3-D central element
part of a 3-D geometrical model which is not a 3-D flanking element
See Figure 1.
NOTE A central element is represented by a 3-D geometrical model.
3.1.7
two-dimensional geometrical model
2-D geometrical model
geometrical model, deduced from building plans, such that for one of the orthogonal axes the cross-section
perpendicular to that axis does not change within the boundaries of the model
See Figure 2.
NOTE A 2-D geometrical model is used for two-dimensional calculations.
3.1.8
two-dimensional flanking element
2-D flanking element
part of a 2-D geometrical model which, when considered in isolation, consists of plane, parallel material layers
2 © ISO 2007 – All rights reserved

3.1.9
two-dimensional central element
2-D central element
part of a 2-D geometrical model which is not a 2-D flanking element
3.1.10
construction planes
planes in the 3-D or 2-D geometrical model which separate different materials, and/or the geometrical model
from the remainder of the construction, and/or the flanking elements from the central element
See Figure 3.
3.1.11
cut-off planes
construction planes that are boundaries to the 3-D or 2-D geometrical model by separating the model from the
remainder of the construction
See Figure 3.
3.1.12
auxiliary planes
planes which, in addition to the construction planes, divide the geometrical model into a number of cells
3.1.13
quasi-homogeneous layer
layer which consists of two or more materials with different thermal conductivities, but which can be
considered as a homogeneous layer with an effective thermal conductivity
See Figure 4.
3.1.14
temperature factor at the internal surface
difference between internal surface temperature and external temperature, divided by the difference between
internal temperature and external temperature, calculated with a surface resistance R at the internal surface
si
3.1.15
temperature weighting factor
weighting factor which states the respective influence of the temperatures of the different thermal
environments upon the surface temperature at the point under consideration
3.1.16
external boundary temperature
external air temperature, assuming that the air temperature and the radiant temperature seen by the surface
are equal
3.1.17
internal boundary temperature
operative temperature, taken for the purposes of this International Standard as the arithmetic mean value of
internal air temperature and mean radiant temperature of all surfaces surrounding the internal environment
3.1.18
thermal coupling coefficient
heat flow rate per temperature difference between two environments which are thermally connected by the
construction under consideration
3.1.19
linear thermal transmittance
heat flow rate in the steady state divided by length and by the temperature difference between the
environments on either side of a thermal bridge
NOTE The linear thermal transmittance is a quantity describing the influence of a linear thermal bridge on the total
heat flow.
3.1.20
point thermal transmittance
heat flow rate in the steady state divided by the temperature difference between the environments on either
side of a thermal bridge
NOTE The point thermal transmittance is a quantity describing the influence of a point thermal bridge on the total
heat flow.
Key
F1, F2, F3, F4, F5 3-D flanking elements C 3-D central element
NOTE 3-D Flanking elements have constant cross-sections perpendicular to at least one axis; the 3-D central
element is the remaining part.
Figure 1 — 3-D geometrical model with five 3-D flanking elements and one 3-D central element
4 © ISO 2007 – All rights reserved

Key
F2, F3, F4, F5 3-D flanking elements C 3-D central element
NOTE F2 to F5 refer to Figure 1.
Figure 2 — Cross-sections of the 3-D flanking elements in a 3-D geometrical model
treated as 2-D geometrical models

Key
C construction planes perpendicular to the x-axis
x
C construction planes perpendicular to the y-axis
y
C construction planes perpendicular to the z-axis
z
NOTE Cut-off planes are indicated with enlarged arrows; planes that separate flanking elements from central element
are encircled.
Figure 3 — Example of a 3-D geometrical model showing construction planes
Figure 4 — Example of a minor point thermal bridge giving rise to three-dimensional heat flow,
incorporated into a quasi-homogeneous layer
3.2 Symbols and units
Symbol Quantity Unit
A area m
B′ characteristic dimension of floor m
b width m
d thickness m
f temperature factor at the internal surface _
Rsi
g temperature weighting factor _
h height m
L thermal coupling coefficient from two-dimensional calculation W/(m·K)
2D
L thermal coupling coefficient from three-dimensional calculation W/K
3D
l length m
q density of heat flow rate W/m
R thermal resistance m ·K/W
R external surface resistance m ·K/W
se
R internal surface resistance m ·K/W
si
T thermodynamic temperature K
U thermal transmittance W/(m ·K)
V volume m
w wall thickness m
Φ heat flow rate W
λ thermal conductivity W/(m·K)
θ Celsius temperature °C
∆θ temperature difference K
χ point thermal transmittance W/K
Ψ linear thermal transmittance W/(m·K)

6 © ISO 2007 – All rights reserved

3.3 Subscripts
Subscript Definition
e external
i internal
min minimum
s surface
4 Principles
The temperature distribution within, and the heat flow through, a construction can be calculated if the
boundary conditions and constructional details are known. For this purpose, the geometrical model is divided
into a number of adjacent material cells, each with a homogeneous thermal conductivity. The criteria which
shall be met when constructing the model are given in Clause 5.
In Clause 6, instructions are given for the determination of the values of thermal conductivity and boundary
conditions.
The temperature distribution is determined either by means of an iterative calculation or by a direct solution
technique, after which the temperature distribution within the material cells is determined by interpolation. The
calculation rules and the method of determining the temperature distribution are described in Clause 7.
The results of the calculations can be used to determine linear thermal transmittances, point thermal
transmittances and internal surface temperatures. The equations for doing so are provided in Clauses 9, 10
and 11.
Specific procedures for window frames are given in ISO 10077-2.
5 Modelling of the construction
5.1 Dimension systems
Lengths may be measured using internal dimensions, overall internal dimensions or external dimensions,
provided that the same system is used consistently for all parts of a building.
NOTE For further information on dimension systems, see ISO 13789.
5.2 Rules for modelling
5.2.1 General
It is not usually feasible to model a complete building using a single geometrical model. In most cases, the
building may be partitioned into several parts (including the subsoil, where appropriate) by using cut-off
planes. This partitioning shall be performed in such a way that all differences are avoided in the results of
calculation between the partitioned building and the building when treated as a whole. This partitioning into
several geometrical models is achieved by choosing suitable cut-off planes.
5.2.2 Cut-off planes for a 3-D geometrical model for calculation of total heat flow and/or surface
temperatures
The geometrical model includes the central element(s), the flanking elements and, where appropriate, the
subsoil. The geometrical model is delimited by cut-off planes.
Cut-off planes shall be positioned as follows:
⎯ at a symmetry plane if this is less than d from the central element (see Figure 5);
min
⎯ at least d from the central element if there is no nearer symmetry plane (see Figure 6);
min
⎯ in the ground, in accordance with 5.2.4,
where d is the greater of 1 m and three times the thickness of the flanking element concerned.
min
A geometrical model can contain more than one thermal bridge. In such cases, cut-off planes need to be
situated at least d from each thermal bridge, or need to be at a symmetry plane (see Figure 6).
min
Dimensions in millimetres
a
Arrows indicate the symmetry planes.
Figure 5 — Symmetry planes which can be used as cut-off planes
8 © ISO 2007 – All rights reserved

Dimensions in millimetres
a) b)
Key
1 1 000 mm or at a symmetry plane
A thermal bridge at the corner of the internal room
B thermal bridge around the window in the external wall
NOTE Thermal bridge B does not fulfil the condition of being at least d (= 1 m) from a cut-off plane [Figure 6 a)].
min
This is corrected by extending the model in two directions [Figure 6 b)].
Figure 6 — 3-D geometrical model containing two thermal bridges
5.2.3 Cut-off planes for a 2-D geometrical model
The same rules as given in 5.2.2 apply to a 2-D geometrical model. Examples are shown in Figures 7 and 8.
In Figure 8, the left-hand drawing may be used if the thermal bridge is symmetrical.
Key
d minimum thickness
min
Figure 7 — Location of cut-off planes at least d from the central element in a 2-D geometrical model
min
Key
d minimum thickness
min
l fixed distance
W
Figure 8 — Example of a construction with linear thermal bridges at fixed distances, l , showing
W
symmetry planes which can be used as cut-off planes
5.2.4 Cut-off planes in the ground
Where the calculation involves heat transfer via the ground (foundations, ground floors, basements), the cut-
off planes in the ground shall be positioned as indicated in Table 1.
10 © ISO 2007 – All rights reserved

Table 1 — Location of cut-off planes in the ground
Distance to central element
Purpose of the calculation
Direction
Heat flow and surface
Surface temperatures only
a
temperatures
b
Horizontal distance to vertical plane, inside the building at least three times wall thickness 0,5 × floor dimension
c, d
Horizontal distance to vertical plane, outside the building at least three times wall thickness 2,5 × floor width
c
Vertical distance to horizontal plane below ground level at least 3 m 2,5 × floor width
Vertical distance to horizontal plane below floor level
c
(applies only if the level of the floor under consideration is at least 1 m 2,5 × floor width
more than 2 m below the ground level)
a
See Figures 9 and 10.
b
In a 3-D geometrical model, the floor dimensions (length and width) inside the building are to be considered separately in each
direction (see Figure 9).
c
In a 3-D geometrical model, the distance outside the building and below ground is to be based on the smaller dimension (width) of
the floor (see Figure 9).
d
If vertical symmetry planes are known, for example as a result of adjacent buildings, they can be used as cut-off planes.

For two-dimensional calculations, there is a vertical symmetry plane in the middle of the floor (so that one half
of the building is modelled). For three-dimensional calculations on a rectangular building, vertical adiabatic
boundaries are taken in the ground mid-way across the floor in each direction (so that one quarter of the
building is modelled). For non-rectangular buildings, it is necessary either to model the complete building
(together with the ground on all sides), or to convert the problem to a two-dimensional one using a building of
infinite length and of width equal to the characteristic dimension of the floor, B′ (see ISO 13370).
EXAMPLE For the floor illustrated in Figure 9, B′ = bc/(b + c).
All cut-off planes shall be adiabatic boundaries.
5.2.5 Periodic heat flows via the ground
Similar criteria to those in 5.2.4 apply to time-dependent numerical calculations for the determination of
periodic heat transfer coefficients (as defined in ISO 13370), except that adiabatic cut-off planes may be taken
at positions equal to twice the periodic penetration depth measured from the edge of the floor in any direction
(if these dimensions are less than those specified in 5.2.4). For further details, see 10.5.
5.2.6 Adjustments to dimensions
Adjustments to the dimensions of the geometrical model with respect to the actual geometry are allowed if
they have no significant influence on the result of the calculation; this can be assumed if the conditions in
5.3.2 are satisfied.
Key
b, c dimensions of floor
NOTE The floor dimensions are b × c, with c > b
Figure 9 — Illustration of cut-off planes for 3-D geometrical model which includes the ground

Key
b width of floor
Figure 10 — Illustration of cut-off planes for 2-D geometrical model which includes the ground
12 © ISO 2007 – All rights reserved

5.2.7 Auxiliary planes
The number of auxiliary planes in the model shall be such that at least one of the following criteria is met:
⎯ doubling the number of subdivisions does not change the calculated heat flow through by more than 1 %,
or
⎯ doubling the number of subdivisions does not change the temperature factor at the inside surface, f , by
Rsi
more than 0,005.
NOTE 1 Requirements for validation of calculation methods are given in A.2.
NOTE 2 A satisfactory sub-division of the geometrical model will usually be obtained by arranging for the sub-divisions
to be smallest within any central element, and gradually increasing in size to larger sub-divisions near cut-off planes.
5.2.8 Quasi-homogeneous layers and materials
In a geometrical model, materials with different thermal conductivities may be replaced by a material with a
single thermal conductivity if the conditions in 5.3.3 are satisfied.
NOTE Examples are joints in masonry, wall-ties in thermally insulated cavities, screws in wooden laths, roof tiles and
the associated air cavity and tile battens.
5.3 Conditions for simplifying the geometrical model
5.3.1 General
Calculation results obtained from a geometrical model with no simplifications shall have precedence over
those obtained from a geometrical model with simplifications.
NOTE This is important when the results of a calculation are close to any required value.
The adjustments described in 5.3.2 can be made.
5.3.2 Conditions for adjusting dimensions to simplify the geometrical model
Adjustment to the dimensions may be made only to materials with thermal conductivity less than 3 W/(m·K),
as described below.
a) Change in the location of the surface of a block of material adjacent to the internal or external surface of
the geometrical model (see Figure 11): for the location of surfaces which are not flat, the local adjustment
perpendicular to the mean location of the internal or external surface, d , shall not exceed
c
dR= λ (1)
cc
where
R is equal to 0,03 m ·K/W;
c
λ is the thermal conductivity of the material in question.
EXAMPLE Inclined surfaces, rounded edges and profiled surfaces such as roof tiles.
Key
1 wall socket
d local adjustment perpendicular to the mean location of the internal or external surface
c
Figure 11 — Change in the location of the internal or external surface
b) Change in the interface of two regions of different material:
⎯ the relocation of the interface shall take place in a direction perpendicular to the internal surface;
⎯ the relocation of the interface shall be such that the material with the lower thermal conductivity is
replaced by the material with the higher thermal conductivity (see Figure 12).
EXAMPLE Recesses for sealing strips, kit joints, adjusting blocks, wall sockets, inclined surfaces and other
connecting details.
Combination Simplifications
Material block Thermal conductivity a b c d
1 λ λ > λ λ > λ λ < λ λ < λ
1 1 2 1 3 1 3 1 2
2 λ
3 λ λ > λ λ > λ λ < λ
3 3 2 3 2 3 2
Figure 12 — Four possibilities for relocating the interface between three material blocks,
depending on the ratio of their thermal conductivities, λ
14 © ISO 2007 – All rights reserved

c) Neglecting thin layers:
⎯ non-metallic layers with a thickness of not more than 1 mm may be ignored;
⎯ thin metallic layers may be ignored if it is established that they have an negligible effect on the heat
transfer.
EXAMPLE Thin membranes which resist the passage of moisture, water vapour or wind-driven air.
d) Neglecting appendages attached to the outside surface: components of the building which have been
attached to the outside surface (i.e. attached at discrete points) may be neglected.
EXAMPLE Rainwater gutters and discharge pipes.
5.3.3 Conditions for using quasi-homogeneous material layers to simplify the geometrical model
5.3.3.1 All calculations
The following conditions for incorporating minor linear and point thermal bridges into a quasi-homogeneous
layer apply in all cases:
⎯ the layers of material in question are located in a part of the construction which, after simplification,
becomes a flanking element;
⎯ the thermal conductivity of the quasi-homogeneous layer after simplification is not more than 1,5 times
the lowest thermal conductivity of the materials present in the layer before simplification.
5.3.3.2 Calculations performed to obtain the thermal coupling coefficient L or L
3D 2D
The effective thermal conductivity of the quasi-homogeneous layer, λ′, shall be calculated in accordance with
Equation (2) or (3):
d
λ′= (2)
d
A j
−−RR −
si se
∑
L λ
3D j
d
λ′= (3)
d
l
j
tb
−−RR −
si se ∑
L λ
2D j
where
d is the thickness of the thermally inhomogeneous layer;
A is the area of the building component;
l is the length of a linear thermal bridge;
tb
L is the thermal coupling coefficient of the building component determined by a 3-D calculation;
3D
L is the thermal coupling coefficient of the building component determined by a 2-D calculation;
2D
d is the thickness of any homogeneous layer which is part of the building element;
j
λ are the thermal conductivities of these homogeneous layers.
j
NOTE The use of Equation (2) or (3) is appropriate if a number of identical minor thermal bridges are present
(wall-ties, joints in masonry, hollow blocks, etc.). The calculation of the thermal coupling coefficient can be restricted to a
basic area that is representative of the inhomogeneous layer. For instance, a cavity wall with four wall-ties per square
metre can be represented by a basic area of 0,25 m with one wall-tie.
5.3.3.3 Calculations performed to obtain the internal surface temperature or the linear thermal
transmittance, Ψ, or the point thermal transmittance, χ
See Clause 9 for calculations using linear and point thermal transmittances from 3-D calculations.
The effective thermal conductivity of the quasi-homogeneous layer, λ′, may be taken as
AAλ++. λ
()
11 nn
λ′= (4)
AA++.
()
1 n
where
λ , . λ are the thermal conductivities of the constituent materials;
1 n
A . A are the areas of the constituent materials measured in the plane of the layer,
1 n
provided that
⎯ the thermal bridges in the layer under consideration are at, or nearly at, right angles to the internal or
external surface of the building element and penetrate the layer over its entire thickness;
⎯ the thermal resistance (surface to surface) of the building element after simplification is at least
1,5 (m ·K)/W;
⎯ the conditions of at least one of the groups stated in Table 2 are met (see Figure 13).
Table 2 — Specific conditions for incorporating linear or point thermal bridges
into a quasi-homogeneous layer
b c e f g h
λ A R R λ d
tb tb o t,i i i
a
Group
2 2 2
W/(m·K) m m ·K/W m ·K/W W/(m·K) m
d
1 u 1,5 u 0,05 × l u 0,5 — — —
tb
−6
2 > 3 u 30 × 10 u 0,5 — — —
−6
3 > 3 u 30 × 10 > 0,5 W 0,5 — —
−6
4 > 3 u 30 × 10 > 0,5 < 0,5 W 0,5 W 0,1
NOTE 1 Group 1 includes linear thermal bridges. Examples are joints in masonry, wooden battens in air cavities or in insulated
cavities of minor thickness.
NOTE 2 Group 2 includes such items as wall-ties, insofar as they are fitted in masonry or concrete or are located in an air cavity, as
well as nails and screws in layers of material or strips with the indicated maximum thermal resistance.
NOTE 3 Groups 3 and 4 include such items as cavity ties, insofar as they penetrate an insulation layer which has a higher thermal
resistance than that indicated for group 2. The inner leaf therefore needs to have thermal properties that Iimit the influence of the
thermal bridge on the internal surface temperature, e.g. if the inner leaf has a sufficient thermal resistance (group 3) or the thermal
conductivity of the inner leaf is such that the heat flow through the cavity ties is adequately distributed over the internal surface; most
masonry or concrete inner leaves are examples of group 4.
a
See Figure 13.
b
λ is the thermal conductivity of the thermal bridge to be incorporated into the quasi-homogeneous layer.
tb
c
A is the area of the cross-section of the thermal bridge.
tb
d
l is the length of a linear thermal bridge.
tb
e
R is the thermal resistance of the layer without the presence of the point thermal bridge.
f
R is the total thermal resistance of the layers between the quasi-homogeneous layer considered and the internal surface.
t,i
g
λ is the thermal conductivity of the material layer between the quasi-homogeneous layer considered and the internal surface with
i
the highest value of λ ⋅ d .
ii
h
d is the thickness of the same layer.
i
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a)  Group 1 b)  Group 2
c)  Group 3 d)  Group 4
For key of symbols, see Table 2.
Figure 13 — Specific conditions for incorporating linear and point thermal bridges in
a quasi-homogeneous layer for the groups given in Table 2
6 Input data
6.1 General
Use values as described in this clause unless non-standard values are justified for a particular situation.
NOTE Non-standard values can be justified by local conditions (e.g. established temperature distributions in the
ground) or by specific material properties (e.g. the effect of a low emissivity coating on the surface resistance).
6.2 Thermal conductivities of materials
The design values of thermal conductivities of building materials and products should either be calculated in
accordance with ISO 10456, or taken from tabulated values such as in ISO 10456.
The thermal conductivity of soil can be taken as 2,0 W/(m·K).
NOTE Other values for the thermal conductivity of the soil can be used if information on the local soil condition is
available (see ISO 13370).
6.3 Surface resistances
For the calculation of heat flow rates, surface resistances shall be in accordance with ISO 6946, depending on
the direction of heat flow. However, the value of R corresponding to horizontal heat flow may be used for all
si
surfaces when
a) the direction of heat flow is uncertain or is liable to vary, or
b) a whole building is being modelled in a single calculation.
For the calculation of internal surface temperatures for the purposes of evaluating condensation risk, surface
resistances shall be in accordance with ISO 13788.
6.4 Boundary temperatures
Table 3 gives the boundary temperatures which shall be used.
Table 3 — Boundary temperatures
Location Boundary temperature
Internal Internal boundary temperature
Internal in unheated rooms See 6.7
External External boundary temperature
At the distance below ground level
Soil (horizontal cut-off plane) given in Table 1: adiabatic boundary
condition
6.5 Thermal conductivity of quasi-homogeneous layers
The thermal conductivity of quasi-homogeneous layers shall be calculated in accordance with Equations (2),
(3) and (4).
6.6 Equivalent thermal conductivity of air cavities
An air cavity shall be considered as a homogeneous conductive material with a thermal conductivity λ .
g
If the thermal resistance of an air layer or cavity is known, its equivalent thermal conductivity, λ , is obtained
g
from
d
g
λ = (5)
g
R
g
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where
d is the thickness of the air layer;
g
R is the thermal resistance in the main direction of heat flow.
g
Thermal resistances of air layers and cavities bounded by opaque materials shall be obtained by the
procedure in ISO 6946.
For the thermal resistance of air layers in multiple glazing, see EN 673. Information about the treatment of
cavities in window frames is given in ISO 10077-2.
Air cavities with dimensions of more than 0,5 m along each one of the orthogonal axis shall be treated as
rooms (see 6.7).
6.7 Determining the temperature in an adjacent unheated room
If sufficient information is available, the temperature in an adjacent unheated room may be calculated in
accordance with ISO 13789.
If the temperature in an adjacent unheated room is unknown and cannot be calculated in accordance with
ISO 13789 because the necessary information is not available, the heat flows and internal surface
temperatures cannot be calculated. However, all required coupling coefficients and temperature weighting
factors can be calculated and presented in accordance with Annex C.
7 Calculation method
7.1 Solution technique
The geometrical model is divided into a number of cells, each with a characteristic point (called a node). By
applying the laws of energy conservation (div q = 0) and Fourier (q = − λ grad θ) and taking into account the
boundary conditions, a system of equations is obtained which is a function of the temperatures at the nodes.
The solution of this system, either by a direct solution technique or by an iterative method, provides the node
temperatures from which the temperature field can be determined. From the temperature distribution, the heat
flows can be calculated by applying Fourier’s law.
Calculation methods shall be verified in accordance with the requirements of Annex A.
7.2 Calculation rules
7.2.1 Heat flows between material cells and adjacent environment
The density of heat flow rate, q, perpendicular to the interface between a material cell and the adjacent
environment shall satisfy
θ − θ
( )
s
q = (6)
R
s
where
θ is the internal or external reference temperature;
θ is the temperature at the internal or external surface;
s
R is the internal or external surface resistance.
s
7.2.2 Heat flows at cut-off planes
The cut-off planes shall be adiabatic (i.e. zero heat flow).
7.2.3 Solution of the equations
The equations shall be solved
...