SIST EN ISO 13786:2008

SLOVENSKI STANDARD
SIST EN ISO 13786:2008
01-april-2008
1DGRPHãþD
SIST EN ISO 13786:2000
7RSORWQH]QDþLOQRVWLGHORYVWDYE'LQDPLþQHWRSORWQH]QDþLOQRVWL5DþXQVNH
PHWRGH,62
Thermal performance of building components -- Dynamic thermal characteristics --
Calculation methods (ISO 13786:2007)
Wärmetechnisches Verhalten von Bauteilen - Dynamisch-thermische Kenngrößen -
Berechnungsverfahren (ISO 13786:2007)
Performance thermique des composants de bâtiment - Caractéristiques thermiques
dynamiques - Méthodes de calcul (ISO 13786:2007)
Ta slovenski standard je istoveten z: EN ISO 13786:2007
ICS:
91.120.10 Toplotna izolacija stavb Thermal insulation
SIST EN ISO 13786:2008 en,fr
2003-01.Slovenski inštitut za standardizacijo. Razmnoževanje celote ali delov tega standarda ni dovoljeno.

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SIST EN ISO 13786:2008

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SIST EN ISO 13786:2008
EUROPEAN STANDARD
EN ISO 13786
NORME EUROPÉENNE
EUROPÄISCHE NORM
December 2007
ICS 91.060.01; 91.120.10 Supersedes EN ISO 13786:1999
English Version
Thermal performance of building components - Dynamic thermal
characteristics - Calculation methods (ISO 13786:2007)
Performance thermique des composants de bâtiment - Wärmetechnisches Verhalten von Bauteilen - Dynamisch-
Caractéristiques thermiques dynamiques - Méthodes de thermische Kenngrößen - Berechnungsverfahren (ISO
calcul (ISO 13786:2007) 13786:2007)
This European Standard was approved by CEN on 7 December 2007.
CEN members are bound to comply with the CEN/CENELEC Internal Regulations which stipulate the conditions for giving this European
Standard the status of a national standard without any alteration. Up-to-date lists and bibliographical references concerning such national
standards may be obtained on application to the CEN Management Centre or to any CEN member.
This European Standard exists in three official versions (English, French, German). A version in any other language made by translation
under the responsibility of a CEN member into its own language and notified to the CEN Management Centre has the same status as the
official versions.
CEN members are the national standards bodies of Austria, Belgium, Bulgaria, Cyprus, Czech Republic, Denmark, Estonia, Finland,
France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia, Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal,
Romania, Slovakia, Slovenia, Spain, Sweden, Switzerland and United Kingdom.
EUROPEAN COMMITTEE FOR STANDARDIZATION
COMITÉ EUROPÉEN DE NORMALISATION
EUROPÄISCHES KOMITEE FÜR NORMUNG
Management Centre: rue de Stassart, 36  B-1050 Brussels
© 2007 CEN All rights of exploitation in any form and by any means reserved Ref. No. EN ISO 13786:2007: E
worldwide for CEN national Members.

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SIST EN ISO 13786:2008
EN ISO 13786:2007 (E)
Contents Page
Foreword.3

2

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SIST EN ISO 13786:2008
EN ISO 13786:2007 (E)
Foreword
This document (EN ISO 13786:2007) has been prepared by Technical Committee ISO/TC 163 "Thermal
performance and energy use in the built environment" in collaboration with Technical Committee CEN/TC 89
"Thermal performance of buildings and building components", the secretariat of which is held by SIS.
This European Standard shall be given the status of a national standard, either by publication of an identical
text or by endorsement, at the latest by June 2008, and conflicting national standards shall be withdrawn at
the latest by June 2008.
Attention is drawn to the possibility that some of the elements of this document may be the subject of patent
rights. CEN [and/or CENELEC] shall not be held responsible for identifying any or all such patent rights.
This document supersedes EN ISO 13786:1999.
According to the CEN/CENELEC Internal Regulations, the national standards organizations of the following
countries are bound to implement this European Standard: Austria, Belgium, Bulgaria, Cyprus, Czech
Republic, Denmark, Estonia, Finland, France, Germany, Greece, Hungary, Iceland, Ireland, Italy, Latvia,
Lithuania, Luxembourg, Malta, Netherlands, Norway, Poland, Portugal, Romania, Slovakia, Slovenia, Spain,
Sweden, Switzerland and the United Kingdom.
Endorsement notice
The text of ISO 13786:2007 has been approved by CEN as a EN ISO 13786:2007 without any modification.

3

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SIST EN ISO 13786:2008

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SIST EN ISO 13786:2008

INTERNATIONAL ISO
STANDARD 13786
Second edition
2007-12-15

Thermal performance of building
components — Dynamic thermal
characteristics — Calculation methods
Performance thermique des composants de bâtiment —
Caractéristiques thermiques dynamiques — Méthodes de calcul




Reference number
ISO 13786:2007(E)
©
ISO 2007

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SIST EN ISO 13786:2008
ISO 13786:2007(E)
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©  ISO 2007
All rights reserved. Unless otherwise specified, no part of this publication may be reproduced or utilized in any form or by any means,
electronic or mechanical, including photocopying and microfilm, without permission in writing from either ISO at the address below or
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Published in Switzerland

ii © ISO 2007 – All rights reserved

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SIST EN ISO 13786:2008
ISO 13786:2007(E)
Contents Page
Foreword. iv
Introduction . v
1 Scope . 1
2 Normative references . 1
3 Terms, definitions, symbols and units . 2
3.1 Terms and definitions. 2
3.2 Symbols and units. 5
3.3 Subscripts . 6
3.4 Other symbols. 6
4 Period of the thermal variations. 6
5 Data required. 6
6 Heat transfer matrix of a multi-layer component. 7
6.1 General. 7
6.2 Procedure . 7
6.3 Heat transfer matrix of a homogeneous layer . 7
6.4 Heat transfer matrix of plane air cavities . 8
6.5 Heat transfer matrix of a building component. 8
7 Dynamic thermal characteristics . 8
7.1 Characteristics for any component . 8
7.2 Characteristics for components consisting of plane and homogeneous layers. 8
8 Report . 10
8.1 Calculation report . 10
8.2 Summary of results . 10
Annex A (normative) Simplified calculation of the heat capacity . 11
Annex B (informative) Principle of the method and examples of applications. 13
Annex C (informative) Further information for computer programming . 17
Annex D (informative) Examples . 19
Bibliography . 22

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SIST EN ISO 13786:2008
ISO 13786:2007(E)
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 13786 was prepared by Technical Committee ISO/TC 163, Thermal performance and energy use in the
built environment, Subcommittee SC 2, Calculation methods.
This second edition cancels and replaces the first edition (ISO 13786:1999), which has been technically
revised.
The following principal changes have been made to the first edition:
⎯ all equations in Clause 3 have been reviewed and corrected as appropriate; the definition of heat capacity
(3.1.1.5) has been modified;
⎯ all equations in 7.2.1 and 7.2.2 have been reviewed and corrected as appropriate;
⎯ 7.2.4 contains a new equation for periodic thermal transmittance, and a new note;
⎯ Equation (A.4) has been corrected;
⎯ B.2 has undergone minor revisions;
⎯ Table C.1 has been added;
⎯ Annex D contains amended examples to align with changes to the formulae in the main body of the text.
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
Introduction
This International Standard provides the means (in part) to assess the contribution that building products and
services make to energy conservation and to the overall energy performance of buildings.
The dynamic thermal characteristics of a building component describe the thermal behaviour of the
component when it is subject to variable boundary conditions, i.e. variable heat flow rate or variable
temperature on one or both of its boundaries. In this International Standard, only sinusoidal boundary
conditions are considered: boundaries are submitted to sinusoidal variations of temperature or heat flow rate.
The properties considered are thermal admittances and thermal dynamic transfer properties, relating cyclic
heat flow rate to cyclic temperature variations. Thermal admittance relates heat flow rate to temperature
variations on the same side of the component. Thermal dynamic transfer properties relate physical quantities
on one side of the component to those on the other side. From the aforementioned properties, it is possible to
define the heat capacity of a given component which quantifies the heat storage property of that component.
The dynamic thermal characteristics defined in this International Standard can be used in product
specifications of complete building components.
The dynamic thermal characteristics can also be used in the calculation of:
⎯ the internal temperature in a room;
⎯ the daily peak power and energy needs for heating or cooling;
⎯ the effects of intermittent heating or cooling, etc.

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SIST EN ISO 13786:2008

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SIST EN ISO 13786:2008
INTERNATIONAL STANDARD ISO 13786:2007(E)

Thermal performance of building components — Dynamic
thermal characteristics — Calculation methods
1 Scope
This International Standard specifies the characteristics related to the dynamic thermal behaviour of a
complete building component and provides methods for their calculation. It also specifies the information on
building materials required for the use of the building component. Since the characteristics depend on the way
materials are combined to form building components, this International Standard is not applicable to building
materials or to unfinished building components.
The definitions given in this International Standard are applicable to any building component. A simplified
calculation method is provided for plane components consisting of plane layers of substantially homogeneous
building materials.
Annex A specifies simpler methods for the estimation of the heat capacities in some limited cases. These
methods are suitable for the determination of dynamic thermal properties required for the estimation of energy
use. These approximations are not appropriate, however, for product characterization.
Annex B gives the basic principle and examples of applications of the dynamic thermal characteristics defined
in this International Standard.
Annex C provides information for programming the calculation method.
Annex D gives examples of calculation for a building component.
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 10211, Thermal bridges in building construction — Heat flows and surface temperatures — Detailed
calculations
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
3 Terms, definitions, symbols and units
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 Definitions valid for any component
3.1.1.1
component
part of a building, such as a wall, floor or roof, or a part of such an element
3.1.1.2
thermal zone of a building
part of a building throughout which the internal temperature is assumed to have negligible spatial variations
NOTE 1 A component separates two zones, designated in this International Standard by m and n.
NOTE 2 The external environment can also be considered a zone.
3.1.1.3
sinusoidal conditions
conditions in which the variations of the temperature and heat flows around their long term average values are
described by a sine function of time
NOTE Using complex numbers, the temperature in zone n can be described by Equation (1) and the heat flow by
Equation (2):
1
jjωωtt−
ˆˆˆ
⎡⎤
θθ()tt=+θ cos()ω+ψ=θ+ θe +θe (1)
nn n n +−n n
⎣⎦
2
1
jjωωtt−
ˆˆ⎡⎤ˆ
ΦΦ()tt=+Φ cosω+ϕ=Φ+ Φe +Φe (2)
()
nn n n +−n n
⎣⎦
2
where
θ and Φ are average values of temperature and heat flow;
n n
ˆ ˆ
θ and Φ are amplitudes of temperature and heat flow variations;
n n
ˆ ˆ
θ and Φ are complex amplitudes defined by:
±n ±n

±jψ ±jϕ
ˆ ˆˆ
θθ= e and ΦΦ= e (3)
±nn ±nn
ω is the angular frequency of the variations.
3.1.1.4
periodic thermal conductance
L
mn
complex number relating the periodic heat flow into a component to the periodic temperatures on either side of
it under sinusoidal conditions
Another representation of the concept:
ˆˆ
ˆ
Φθ=−LLθ (4)
mmmm mnn
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
NOTE 1 L relates the periodic heat flow on side m to the periodic temperature on side m when the temperature
mm
amplitude on side n is zero. L relates the periodic heat flow on side m to the periodic temperature on side n when the
mn
temperature amplitude on side m is zero.
NOTE 2 As a convention within this International Standard, the heat flow rate is defined as positive when it enters the
surface of the component.
3.1.1.5
heat capacity
modulus of the net periodic thermal conductance divided by the angular frequency
Another representation of the concept:
1
CL=−L (5)
mmm mn
ω
3.1.1.6
time shift
∆t
period of time between the maximum amplitude of a cause and the maximum amplitude of its effect
3.1.2 Definitions valid only for one dimensional heat flow
3.1.2.1
plane component
component for which the smallest curvature radius is at least five times its thickness
3.1.2.2
homogeneous material layer
layer of material in which the largest size of inhomogeneities does not exceed one fifth of the thickness of the
layer
3.1.2.3
thermal admittance
complex quantity defined as the complex amplitude of the density of heat flow rate through the surface of the
component adjacent to zone m, divided by the complex amplitude of the temperature in the same zone when
the temperature on the other side is held constant
Another representation of the concept:
qˆ
m
(6)
Y =
mm
ˆ
θ
m
3.1.2.4
periodic thermal transmittance
complex quantity defined as the complex amplitude of the density of heat flow rate through the surface of the
component adjacent to zone m, divided by the complex amplitude of the temperature in zone n when the
temperature in zone m is held constant
Another representation of the concept:
qˆ
m
Y =− (7)
mn
ˆ
θ
n
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
3.1.2.5
areal heat capacity
heat capacity divided by area of element
Another representation of the concept:
C
1
m
κ== YY− (8)
mmmmn
A ω
NOTE 1 Using Equation (8), the heat capacities are then:
CA= κ (9)
mm
NOTE 2 There are two thermal admittances and two heat capacities for a component separating two zones, all of
which depend on the period of the thermal variations.
3.1.2.6
decrement factor
ratio of the modulus of the periodic thermal transmittance to the steady-state thermal transmittance U
Another representation of the concept:
qYˆ
mmn
f== (10)
ˆ
U
θ U
n
where m ≠ n
3.1.2.7
periodic penetration depth
δ
depth at which the amplitude of the temperature variations are reduced by the factor “e” in a homogeneous
material of infinite thickness subjected to sinusoidal temperature variations on its surface
Another representation of the concept:
λ T
δ = (11)
πρ c
NOTE e is the base of natural logarithms; e = 2,718.
3.1.2.8
heat transfer matrix
Z
matrix relating the complex amplitudes of temperature and heat flow rate on one side of a component to the
complex amplitudes of temperature and heat flow rate on the other side
Another representation of the concept:
ˆˆ
⎛⎞ ⎛ ⎞
ZZ
θ⎛⎞ θ
11 12
21
Z== ⋅ (12)
⎜⎟ ⎜ ⎟
⎜⎟
ˆˆZZ
qq
21 22
21⎝⎠
⎝⎠ ⎝ ⎠
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
3.2 Symbols and units
Symbol Quantity Unit
2
A area m
C heat capacity J/K
L periodic thermal conductance W/K
mn
2
R thermal resistance m ⋅K/W
T period of the variations s
2
U thermal transmittance under steady state boundary conditions W/(m ⋅K)
2
Y thermal admittance W/(m ⋅K)
mm
2
Y periodic thermal transmittance W/(m ⋅K)
mn
Z heat transfer matrix environment to environment —
Z element of the heat transfer matrix —
mn
2
a thermal diffusivity m /s
c specific heat capacity J/(kg⋅K)
d thickness of a layer m
f decrement factor —
j unit on the imaginary axis for a complex number; j1= − —
2
q density of heat flow rate W/m
t time s or h
x distance through the component m
∆t time shift: time lead (if positive), or time lag (if negative) s or h
δ periodic penetration depth of a heat wave in a material m
Φ heat flow rate W
ξ ratio of the thickness of the layer to the penetration depth —
2
κ areal heat capacity J/(m ⋅K)
λ design thermal conductivity W/(m⋅K)
3
ρ density kg/m
θ temperature °C
2π
ω angular frequency; ω = rad/s
T
ϕ, ψ phase differences rad
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
3.3 Subscripts
Subscript Definition
a air layer
e external
i internal
m, n for the thermal zones
s related to surface
ee from environment to environment
3.4 Other symbols
Symbol Definition
^
complex amplitude
–
mean value
modulus of a complex number
⎪ ⎪
arg argument of a complex number
4 Period of the thermal variations
The definition of dynamic thermal characteristics and the formulae for their calculation are valid for any period
of thermal variations.
The values of dynamic thermal characteristics depend on the periods. If more than one period is considered,
an additional suffix shall be added to all quantities affected so as to distinguish between the values for
different periods.
Practical time periods are:
⎯ one hour (3 600 s), which corresponds to very short time variations, such as those resulting from
temperature control systems;
⎯ one day (86 400 s), corresponding to daily meteorological variations and temperature setback;
⎯ one week (604 800 s), corresponding to longer term averaging of the building;
⎯ one year (31 536 000 s), useful for treatment of heat transfer through the ground.
5 Data required
The data required to compute the dynamic thermal characteristics are:
a) the detailed drawings of the product, with dimensions;
b) for each material used in the product:
⎯ the thermal conductivity, λ;
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
⎯ the specific heat capacity, c;
⎯ the density, ρ.
These values shall be the design values of the materials used.
6 Heat transfer matrix of a multi-layer component
6.1 General
The procedure in 6.2 applies to building components consisting of plane homogeneous layers. Thermal
bridges usually present in such building components do not affect significantly the dynamic thermal
characteristics, and can hence be neglected.
The calculation of dynamic thermal characteristics of non-plane components and of components containing
very important thermal bridges shall be made by solving the equation of heat transfer under periodic boundary
conditions. For this purpose, the rules for modelling the component as given in ISO 10211 shall be used
together with numerical methods, such as finite difference and finite element techniques.
6.2 Procedure
The procedure is as follows:
a) identify the materials comprising the layers of the building component and the thickness of these layers,
and determine the thermal characteristics of the materials;
b) specify the period of the variations at the surfaces;
c) calculate the penetration depth for the material of each layer;
d) determine the elements of the heat transfer matrix for each layer;
e) multiply the layer heat transfer matrices, including those of the boundary layers, in the correct order, so as
to obtain the transfer matrix of the component.
6.3 Heat transfer matrix of a homogeneous layer
The periodic penetration depth for the material of the layer, δ, is calculated from its thermal properties and the
period T using Equation (11).
The ratio of the thickness of the layer to the penetration depth is then
d
ξ = (13)
δ
The matrix elements, Z , are calculated as follows:
mn
ZZ== coshξξcos + j sinhξ sinξ ;
( ) ( ) ( ) ( )
11 22
δ
Z =− sinhξξcos + coshξ sinξ + j ⎡coshξ sinξ − sinhξ cosξ ⎤ ;
() () ( ) () () ( ) ( ) ( )
{}
12
⎣ ⎦
2λ
λ
Z =− sinhξξcos − coshξ sinξ + j ⎡sinhξξcos + coshξ sinξ ⎤ . (14)
() () () ( ) ( ) ( ) () ( )
{}
21
⎣ ⎦
δ
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
6.4 Heat transfer matrix of plane air cavities
The specific heat capacity of such layers is neglected. Hence, if R is the thermal resistance of the air layer,
a
including convection, conduction and radiation, its heat transfer matrix is
1 −R
⎛⎞
a
Z = (15)
a⎜⎟
01
⎝⎠
The thermal resistance of the air layer shall be calculated in accordance with ISO 6946.
6.5 Heat transfer matrix of a building component
The heat transfer matrix of the building component from surface to surface is
⎛⎞ZZ
11 12
Z==Z Z . ZZZ (16)
⎜⎟ NN −1321
ZZ
⎝⎠21 22
where Z , Z , Z , …, Z , are the heat transfer matrices of the various layers of the building component,
1 2 3 N
beginning from layer 1. As a convention for building envelope components, layer 1 shall be the innermost
layer.
The heat transfer matrix from environment to environment through the building component is
ZZ= ZZ (17)
ee s2 s1
where Z and Z are the heat transfer matrices of the boundary layers, given by
s1 s2
⎛⎞1 −R
s
Z = (18)
s⎜⎟
01
⎝⎠
where R is the surface resistance of the boundary layer, including convection and radiation. Values of surface
s
resistance shall be in accordance with ISO 6946.
In most cases, the heat transfer matrix and the dynamic characteristics of a building component shall be
calculated using the surface resistance values appropriate to the intended orientation of the component. If the
orientation of the component is not known, the calculations shall be done for vertical orientation (heat flow
horizontal). For certain applications where boundary layers are taken into account separately, the periodic
heat capacity of the component should be calculated omitting the boundary layers.
7 Dynamic thermal characteristics
7.1 Characteristics for any component
The dynamic thermal characteristics of any component are four periodic thermal conductances, L , and two
mn
heat capacities, C , as given in 3.1.1.4 and 3.1.1.5.
m
7.2 Characteristics for components consisting of plane and homogeneous layers
7.2.1 Thermal admittances and periodic thermal conductances
The thermal admittances are
Z Z
11 22
Y =− and Y =− (19)
11 22
Z Z
12 12
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SIST EN ISO 13786:2008
ISO 13786:2007(E)
where Y is for the internal side of the component, while Y is for the external side.
11 22
The time shift of admittance, is:
T
∆=tYarg() (20)
Ymm
2π
with the argument evaluated in the range 0 to 2π.
7.2.2 Modified admittance for internal partitions
For internal partitions within a building, where the temperature variations are the same on either side of the
partition, the periodi
...