ASTM C1340/C1340M-10(2015)

Standard Practice for Estimation of Heat Gain or Loss Through Ceilings Under Attics Containing Radiant Barriers by Use of a Computer Program

SIGNIFICANCE AND USE
5.1 Manufacturers of radiant barriers express the performance of their products in terms of the total hemispherical emittance. The purpose of a radiant barrier is to decrease the radiation heat transfer across the attic air space, and hence, to decrease the heat loss or gain through the ceiling below the attic. The amount of decrease in heat flow will depend upon a number of factors, such as weather conditions, amount of mass or reflective insulation in the attic, solar absorptance of the roof, geometry of the attic and roof, and amount and type of attic ventilation. Because of the infinite combinations of these factors, it is not practical to publish data for each possible case.  
5.2 The calculation of heat loss or gain of a system containing radiant barriers is mathematically complex, and because of the iterative nature of the method, it is best handled by computers.  
5.3 Computers are now widely available to most producers and consumers of radiant barriers to permit the use of this practice.  
5.4 The user of this practice may wish to modify the data input to represent accurately the structure. The computer program also may be modified to meet individual needs. Also, additional calculations may be desired, for example, to sum the hourly heat flows in some fashion to obtain estimates of seasonal or annual energy usages. This might be done using the hourly data as inputs to a whole-house model, and by choosing house balance points to use as cutoff points in the summations.
SCOPE
1.1 This practice covers the estimation of heat gain or loss through ceilings under attics containing radiant barriers by use of a computer program. The computer program included as an adjunct to this practice provides a calculational procedure for estimating the heat loss or gain through the ceiling under an attic containing a truss or rafter mounted radiant barrier. The program also is applicable to the estimation of heat loss or gain through ceilings under an attic without a radiant barrier. This procedure utilizes hour-by-hour weather data to estimate the hour-by-hour ceiling heat flows. The interior of the house below the ceiling is assumed to be maintained at a constant temperature. At present, the procedure is applicable to sloped-roof attics with rectangular floor plans having an unshaded gabled roof and a horizontal ceiling. It is not applicable to structures with flat roofs, vaulted ceilings, or cathedral ceilings. The calculational accuracy also is limited by the quality of physical property data for the construction materials, principally the insulation and the radiant barrier, and by the quality of the weather data.  
1.2 Under some circumstances, interactions between radiant barriers and HVAC ducts in attics can have a significant effect on the thermal performance of a building. Ducts are included in an extension of the computer model given in the appendix.  
1.3 The values stated in either SI units or inch-pound units are to be regarded separately as standard. The values stated in each system may not be exact equivalents; therefore, each system shall be used independently of the other. Combining values from the two systems may result in non-conformance with the standard.

General Information

• Status: Historical
• Publication Date: 31-Aug-2015
• ICS: 91.120.10 - Thermal insulation of buildings
• Technical Committee: C16 - Thermal Insulation
• Drafting Committee: C16.21 - Reflective Insulation

Standards Content (Sample)

NOTICE: This standard has either been superseded and replaced by a new version or withdrawn.
Contact ASTM International (www.astm.org) for the latest information
Designation: C1340/C1340M − 10 (Reapproved 2015)
Standard Practice for
Estimation of Heat Gain or Loss Through Ceilings Under
Attics Containing Radiant Barriers by Use of a Computer
Program
This standard is issued under the fixed designation C1340/C1340M; the number immediately following the designation indicates the
year of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last
reapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope 2. Referenced Documents
1.1 This practice covers the estimation of heat gain or loss 2.1 ASTM Standards:
through ceilings under attics containing radiant barriers by use C168Terminology Relating to Thermal Insulation
of a computer program. The computer program included as an
2.2 ANSI Standards:
adjunct to this practice provides a calculational procedure for X3.5Flow Chart Symbols and Their Usage in Information
estimating the heat loss or gain through the ceiling under an
Processing
attic containing a truss or rafter mounted radiant barrier. The X3.9Standard for Fortran Programming Language
programalsoisapplicabletotheestimationofheatlossorgain
2.3 ASTM Adjuncts:
through ceilings under an attic without a radiant barrier. This
Computer Program for Estimation of Heat Gain or Loss
procedure utilizes hour-by-hour weather data to estimate the
through Ceilings Under Attics Containing Radiant Barri-
hour-by-hour ceiling heat flows. The interior of the house
ers
below the ceiling is assumed to be maintained at a constant
3. Terminology
temperature.At present, the procedure is applicable to sloped-
roof attics with rectangular floor plans having an unshaded
3.1 Definitions—For definitions of terms used in this
gabled roof and a horizontal ceiling. It is not applicable to
practice, refer to Terminology C168.
structureswithflatroofs,vaultedceilings,orcathedralceilings.
3.2 Symbols—Symbolswillbeintroducedanddefinedinthe
The calculational accuracy also is limited by the quality of
detailed description of the development.
physical property data for the construction materials, princi-
pally the insulation and the radiant barrier, and by the quality
4. Summary of Practice
of the weather data.
4.1 The procedures used in this practice are based on the
1.2 Undersomecircumstances,interactionsbetweenradiant
thermal response factor method for calculating dynamic heat
barriers and HVAC ducts in attics can have a significant effect
conductionthroughmultilayerslabs (1, 2), alongwithamodel
onthethermalperformanceofabuilding.Ductsareincludedin
for convective and radiative heat exchanges inside and outside
an extension of the computer model given in the appendix.
the attic.
1.3 The values stated in either SI units or inch-pound units
4.2 The operation of the computer program involves the
are to be regarded separately as standard. The values stated in
following steps:
each system may not be exact equivalents; therefore, each
system shall be used independently of the other. Combining
For referenced ASTM standards, visit the ASTM website, www.astm.org, or
values from the two systems may result in non-conformance
contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM
with the standard.
Standards volume information, refer to the standard’s Document Summary page on
the ASTM website.
1 3
This practice is under the jurisdiction of ASTM Committee C16 on Thermal Available fromAmerican National Standards Institute (ANSI), 25 W. 43rd St.,
Insulation and is the direct responsibility of Subcommittee C16.21 on Reflective 4th Floor, New York, NY 10036, http://www.ansi.org.
Insulation. Available from ASTM International Headquarters. Order Adjunct No.
Current edition approved Sept. 1, 2015. Published September 2015. Originally ADJC1340.
approved in 1999. Last previous edition approved in 2010 as C1340–10. DOI: Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
10.1520/C1340_C1340M-10R15 this standard.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
C1340/C1340M − 10 (2015)
4.2.1 Response Factors—A separate computer program additionalcalculationsmaybedesired,forexample,tosumthe
must be used to calculate the thermal response factors of the hourly heat flows in some fashion to obtain estimates of
solid materials surrounding the attic. Input to this program seasonalorannualenergyusages.Thismightbedoneusingthe
would consist of the thermal conductivity, specific heat, hourlydataasinputstoawhole-housemodel,andbychoosing
density,andthicknessofeachlayer,orthethermalresistanceof house balance points to use as cutoff points in the summations.
the layer if it has negligible density, and the fraction of the
6. Method of Calculation
cross-sectional area occupied by the framing. Output of such a
program would be a set of response factors for use as input to
6.1 Approach:
the main program. The adjunct to this practice contains data
6.1.1 This calculation of heat loss or gain requires that the
files with response factors for several typical attic construc-
following be known:
tions.
6.1.1.1 The thermal conductivity, specific heat, and density
4.2.2 Data Input to the Main Program—This input includes
of the construction materials (that is, insulation, plywood,
the response factors, total hemispherical emittances of the
roofing materials, sheathing, gypsum board);
inside and outside surfaces of the attic envelope, solar absorp-
6.1.1.2 The total hemispherical emittance of all materials
tances of the outside surfaces of the attic envelope, length and
facing the attic air space;
width of the attic, slopes of the two roof sections, distance
6.1.1.3 The solar absorptance of the exterior surfaces of the
between attic floor and roof at edge of attic, orientation of
attic (that is, the roof and gables);
house,ventareasandtypeofvents,watervaporpermeancesof
6.1.1.4 The geometry of the attic;
atticsurfaces,areaofexposedwoodinsideattic,massofwood
6.1.1.5 The moisture permeance and storage properties of
in attic, initial moisture content of wood in attic, rate of
the materials facing the attic space; and
exfiltration of air from house into attic space, latitude and
6.1.1.6 The weather conditions.
longitude, time zone indicator, solar reflectance of the ground,
6.1.2 The solution is a computer procedure that estimates
indoor temperature, and indoor humidity.
temperatures of both sides of the components of the attic
4.2.3 Analysis—Using hourly weather data consisting of envelope and the temperature of the air in the attic space, uses
outdoor temperature and humidity ratio, atmospheric pressure,
these estimates of temperatures to refine estimates of convec-
total horizontal and direct solar radiation, wind speed and
tion and radiation heat transfer coefficients, reestimates the
direction,cloudamount,cloudtype,andatmosphericclearness
temperaturesusingthenewheattransfercoefficients,continues
number, the computer program calculates the inside and
iteratingonthetemperaturesandheattransfercoefficientsuntil
outside temperatures of the attic envelope and the temperature
convergenceisreached,andusesthelastestimatesoftempera-
of the air inside the attic. Using these temperatures, the
tures to calculate the heat gain or loss through the ceiling.This
program calculates the heat flux through the ceiling. procedure is repeated for each hour of the simulation period
4.2.4 Output—The hourly heat flux through the ceiling is (typically a full year).
written to a file which can be used for further processing, such
6.2 Development of Equations—The model that is the basis
as seasonal or annual heat gains or losses.
for this practice is based on the model developed by B. Peavy
(3),whichwaslaterextendedbyWilkes (4-6).Thesketchofan
5. Significance and Use
attic given in Fig. 1 shows the various heat transfer mecha-
nisms that occur within an attic. Although the sketch shows
5.1 Manufacturers of radiant barriers express the perfor-
mance of their products in terms of the total hemispherical
emittance. The purpose of a radiant barrier is to decrease the
radiation heat transfer across the attic air space, and hence, to
decrease the heat loss or gain through the ceiling below the
attic. The amount of decrease in heat flow will depend upon a
numberoffactors,suchasweatherconditions,amountofmass
or reflective insulation in the attic, solar absorptance of the
roof, geometry of the attic and roof, and amount and type of
attic ventilation. Because of the infinite combinations of these
factors,itisnotpracticaltopublishdataforeachpossiblecase.
5.2 The calculation of heat loss or gain of a system
containing radiant barriers is mathematically complex, and
because of the iterative nature of the method, it is best handled
by computers.
5.3 Computers are now widely available to most producers
and consumers of radiant barriers to permit the use of this
practice.
5.4 The user of this practice may wish to modify the data
input to represent accurately the structure. The computer
FIG. 1 Schematic of Residential Attic Showing Heat Transfer Phe-
program also may be modified to meet individual needs.Also, nomena
C1340/C1340M − 10 (2015)
ventilation occurring at soffit and ridge vents, the location of
CR = the common ratio, and
the vents may be at other locations, such as at the gables. The
N = a sufficiently large number.
model treats all of these phenomena through a system of heat
6.3.2.2 The common ratio is used to define a new set of
balance equations at the interior and exterior surfaces of the
functions,calledthefirstorderconductiontransferfunctionsor
ceiling, roof sections, and gables, as well as a heat balance on
simply the conduction transfer functions, X(j), Y(j), and Z(j),
the air mass within the attic. To handle the case of raised
which are given by:
trusses,shortverticalwallsattheeavesalsoareincluded.Each
X~0! 5 X' ~0! (4)
of the surfaces is assumed to be isothermal; thus, for an attic
consisting of a ceiling, two roof sections, two gables, two
Y~0! 5 Y' ~0! (5)
vertical eave sections, and one air space, a total of 15 heat
Z 0 5 Z' 0 (6)
~ ! ~ !
balance equations is used.
X j 5 X' j 2 CRX' j 2 1 forj# N (7)
~ ! ~ ! ~ !
6.3 Equations—Conduction:
Y~j! 5 Y'~j! 2 CRY' ~j 2 1! forj# N (8)
6.3.1 The model developed here utilizes the thermal re-
sponse factor method to analyze conduction through building Z~j! 5 Z'~j! 2 CRZ' ~j 2 1! forj# N (9)
envelope sections. The thermal response factor method was
X j 5 0 forj.N (10)
~ !
developed by Mitalas andArseneault (1) and was extended by
Y j 5 0 forj.N (11)
~ !
Kusuda (2). The method is based on an exact analytical
Z~j! 5 0 forj.N (12)
solution of the heat conduction equation for one-dimensional
heat flow through a multilayer slab having temperature-
6.3.2.3 With the conduction transfer functions, the heat
independentthermalproperties.Theonlyapproximationisthat
fluxes and surface temperatures are related by:
the surface temperatures are taken to vary linearly with time
N
between time steps. For analysis of buildings, the time step is
QI 5 Z j TIS j 2 TR (13)
~ !~ ~ ! !
(
j50
normally taken to be 1 h. The response factor equations relate
the heat fluxes at the surfaces of the slab to the present and
N
previous temperatures at the two surfaces. The equations are:
2 Y j TOS j 2 TR 1CRQI'
~ !~ ~ ! !
(
j50
` `
QI 5 Z' j TIS j 2 TR 2 Y' j TOS j 2 TR (1)
~ !~ ~ ! ! ~ !~ ~ ! !
( (
j50 j50
` ` N
QO 5 Y' j TIS j 2 TR 2 X' j TOS j 2 TR (2) QO 5 Y j TIS j 2 TR (14)
~ !~ ~ ! ! ~ !~ ~ ! ! ~ !~ ~ ! !
( ( (
j50 j50 j50
N
where:
2 X~j!~TOS~j! 2 TR!1CRQO'
(
QI = heat flux at inside surface at present
j50
time (note that the positive heat flow
direction is from the inside to the
2 2
outside), W/m [Btu/h·ft ],
where:
QO = heat flux at outside surface at present
2 2 QI' = heat flux at inside surface at previous time step,
time, W/m [Btu⁄h·ft ],
QO' = heatfluxatoutsidesurfaceatprevioustimestep,and
TIS(j) = temperature at inside surface j hours
N = number of significant conduction transfer functions.
previous to present time, °C [°F],
TOS(j) = temperature at outside surface j hours
6.3.2.4 When parallel heat flow paths occur in an envelope
previous to present time, °C [°F]
component, separate response factors for each path may be
X' (j), Y' (j), Z'(j) = response factors, W/m · K [Btu/
needed. If the boundary temperatures of the two paths may be
h·ft ·°F], and
assumed to be equal, however, then the response factors may
TR = reference temperature, °C [°F].
be added together as:
6.3.2 The response factors are determined from a sequence
X' 5 A X' 1A X' (15)
1 1 2 2
of calculations that involve the thermal diffusivity, thermal
where:
conductivity,specificheat,density,andthicknessofeachofthe
A,A = area fractions for
layersinthemultilayerslab.Anefficientcomputerprogramfor
1 2
paths 1 and 2, and
calculating the response factors has been developed by George
(X' ,X' ), (Y',Y' ) and (Z',Z' ) = the response factors
Walton of the National Institute of Standards and Technology 1 2 1 2 1 2
for paths 1 and 2.
(NIST) (7).
6.3.2.1 The efficiency of the response factor calculations
Parallel conduction transfer functions may be calculated
canbeincreasedbymakinguseofthefactthatafterasufficient
fromtheseparallelresponsefactors,providedthatthecommon
number of terms, the ratio of two consecutive response factors
ratioforthepathwiththelargestnumberofsignificanttermsis
becomes constant. This is expressed by:
used.
X' ~j11! Y' ~j11! Z' ~j11! 6.3.2.5 The original derivation of the response factor tech-
5 5 5 CRforj$ N (3)
X' j Y' j Z' j nique relied upon the assumption of temperature-independent
~ ! ~ ! ~ !
C1340/C1340M − 10 (2015)
thermal properties. An approximate method has been devel- temperature, which is defined as the average of the tempera-
oped to account for the temperature dependence of the thermal tures of the surface and the air. Relationships for the tempera-
properties (5). The thermal transmission coefficient of the ture dependent properties were obtained from NBS Circular
component is taken to vary linearly with temperature as: 564 (9).
6.4.2.1 The model utilizes correlations that have been de-
U 5 U @11b T 2 TR # (16)
~ !
TR
veloped for various orientations of the plate with respect to
The conduction transfer function equations then become:
gravity and for the direction of heat flow (up versus down).
N
Correlations for both laminar and turbulent flow are used, with
QI 5 Z j TIS j 2 TR (17)
~ !~ ~ ! !
the choice depending upon the magnitude of the Rayleigh
(
j50
number for natural convection and of the Reynolds number for
N
forced convection. Sepa
...

This document is not an ASTM standard and is intended only to provide the user of an ASTM standard an indication of what changes have been made to the previous version. Because
it may not be technically possible to adequately depict all changes accurately, ASTM recommends that users consult prior editions as appropriate. In all cases only the current version
of the standard as published by ASTM is to be considered the official document.
Designation: C1340/C1340M − 10 C1340/C1340M − 10 (Reapproved 2015)
Standard Practice for
Estimation of Heat Gain or Loss Through Ceilings Under
Attics Containing Radiant Barriers by Use of a Computer
Program
This standard is issued under the fixed designation C1340/C1340M; the number immediately following the designation indicates the
year of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of last
reapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.
1. Scope
1.1 This practice covers the estimation of heat gain or loss through ceilings under attics containing radiant barriers by use of
a computer program. The computer program included as an adjunct to this practice provides a calculational procedure for
estimating the heat loss or gain through the ceiling under an attic containing a truss or rafter mounted radiant barrier. The program
also is applicable to the estimation of heat loss or gain through ceilings under an attic without a radiant barrier. This procedure
utilizes hour-by-hour weather data to estimate the hour-by-hour ceiling heat flows. The interior of the house below the ceiling is
assumed to be maintained at a constant temperature. At present, the procedure is applicable to sloped-roof attics with rectangular
floor plans having an unshaded gabled roof and a horizontal ceiling. It is not applicable to structures with flat roofs, vaulted
ceilings, or cathedral ceilings. The calculational accuracy also is limited by the quality of physical property data for the
construction materials, principally the insulation and the radiant barrier, and by the quality of the weather data.
1.2 Under some circumstances, interactions between radiant barriers and HVAC ducts in attics can have a significant effect on
the thermal performance of a building. Ducts are included in an extension of the computer model given in the appendix.
1.3 The values stated in either SI units or inch-pound units are to be regarded separately as standard. The values stated in each
system may not be exact equivalents; therefore, each system shall be used independently of the other. Combining values from the
two systems may result in non-conformance with the standard.
2. Referenced Documents
2.1 ASTM Standards:
C168 Terminology Relating to Thermal Insulation
2.2 ANSI Standards:
X3.5 Flow Chart Symbols and Their Usage in Information Processing
X3.9 Standard for Fortran Programming Language
2.3 ASTM Adjuncts:
Computer Program for Estimation of Heat Gain or Loss through Ceilings Under Attics Containing Radiant Barriers
3. Terminology
3.1 Definitions—For definitions of terms used in this practice, refer to Terminology C168.
3.2 Symbols—Symbols will be introduced and defined in the detailed description of the development.
4. Summary of Practice
4.1 The procedures used in this practice are based on the thermal response factor method for calculating dynamic heat
conduction through multilayer slabs (1, 2), along with a model for convective and radiative heat exchanges inside and outside the
attic.
This practice is under the jurisdiction of ASTM Committee C16 on Thermal Insulation and is the direct responsibility of Subcommittee C16.21 on Reflective Insulation.
Current edition approved June 1, 2010Sept. 1, 2015. Published July 2010September 2015. Originally approved in 1999. Last previous edition approved in 20042010 as
C1340 – 04.C1340 –10. DOI: 10.1520/C1340_C1340M-10.10.1520/C1340_C1340M-10R15
For referenced ASTM standards, visit the ASTM website, www.astm.org, or contact ASTM Customer Service at [e-mail]. For Annual Book of ASTM Standards
volume information, refer to the standard’s Document Summary page on the ASTM website.
Available from American National Standards Institute (ANSI), 25 W. 43rd St., 4th Floor, New York, NY 10036, http://www.ansi.org.
Available from ASTM International Headquarters. Order Adjunct No. ADJC1340.
The boldface numbers in parentheses refer to the list of references at the end of this standard.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
C1340/C1340M − 10 (2015)
4.2 The operation of the computer program involves the following steps:
4.2.1 Response Factors—A separate computer program must be used to calculate the thermal response factors of the solid
materials surrounding the attic. Input to this program would consist of the thermal conductivity, specific heat, density, and thickness
of each layer, or the thermal resistance of the layer if it has negligible density, and the fraction of the cross-sectional area occupied
by the framing. Output of such a program would be a set of response factors for use as input to the main program. The adjunct
to this practice contains data files with response factors for several typical attic constructions.
4.2.2 Data Input to the Main Program—This input includes the response factors, total hemispherical emittances of the inside
and outside surfaces of the attic envelope, solar absorptances of the outside surfaces of the attic envelope, length and width of the
attic, slopes of the two roof sections, distance between attic floor and roof at edge of attic, orientation of house, vent areas and type
of vents, water vapor permeances of attic surfaces, area of exposed wood inside attic, mass of wood in attic, initial moisture content
of wood in attic, rate of exfiltration of air from house into attic space, latitude and longitude, time zone indicator, solar reflectance
of the ground, indoor temperature, and indoor humidity.
4.2.3 Analysis—Using hourly weather data consisting of outdoor temperature and humidity ratio, atmospheric pressure, total
horizontal and direct solar radiation, wind speed and direction, cloud amount, cloud type, and atmospheric clearness number, the
computer program calculates the inside and outside temperatures of the attic envelope and the temperature of the air inside the attic.
Using these temperatures, the program calculates the heat flux through the ceiling.
4.2.4 Output—The hourly heat flux through the ceiling is written to a file which can be used for further processing, such as
seasonal or annual heat gains or losses.
5. Significance and Use
5.1 Manufacturers of radiant barriers express the performance of their products in terms of the total hemispherical emittance.
The purpose of a radiant barrier is to decrease the radiation heat transfer across the attic air space, and hence, to decrease the heat
loss or gain through the ceiling below the attic. The amount of decrease in heat flow will depend upon a number of factors, such
as weather conditions, amount of mass or reflective insulation in the attic, solar absorptance of the roof, geometry of the attic and
roof, and amount and type of attic ventilation. Because of the infinite combinations of these factors, it is not practical to publish
data for each possible case.
5.2 The calculation of heat loss or gain of a system containing radiant barriers is mathematically complex, and because of the
iterative nature of the method, it is best handled by computers.
5.3 Computers are now widely available to most producers and consumers of radiant barriers to permit the use of this practice.
5.4 The user of this practice may wish to modify the data input to represent accurately the structure. The computer program also
may be modified to meet individual needs. Also, additional calculations may be desired, for example, to sum the hourly heat flows
in some fashion to obtain estimates of seasonal or annual energy usages. This might be done using the hourly data as inputs to a
whole-house model, and by choosing house balance points to use as cutoff points in the summations.
6. Method of Calculation
6.1 Approach:
6.1.1 This calculation of heat loss or gain requires that the following be known:
6.1.1.1 The thermal conductivity, specific heat, and density of the construction materials (that is, insulation, plywood, roofing
materials, sheathing, gypsum board);
6.1.1.2 The total hemispherical emittance of all materials facing the attic air space;
6.1.1.3 The solar absorptance of the exterior surfaces of the attic (that is, the roof and gables);
6.1.1.4 The geometry of the attic;
6.1.1.5 The moisture permeance and storage properties of the materials facing the attic space; and
6.1.1.6 The weather conditions.
6.1.2 The solution is a computer procedure that estimates temperatures of both sides of the components of the attic envelope
and the temperature of the air in the attic space, uses these estimates of temperatures to refine estimates of convection and radiation
heat transfer coefficients, reestimates the temperatures using the new heat transfer coefficients, continues iterating on the
temperatures and heat transfer coefficients until convergence is reached, and uses the last estimates of temperatures to calculate
the heat gain or loss through the ceiling. This procedure is repeated for each hour of the simulation period (typically a full year).
6.2 Development of Equations—The model that is the basis for this practice is based on the model developed by B. Peavy (3),
which was later extended by Wilkes (4-6). The sketch of an attic given in Fig. 1 shows the various heat transfer mechanisms that
occur within an attic. Although the sketch shows ventilation occurring at soffit and ridge vents, the location of the vents may be
at other locations, such as at the gables. The model treats all of these phenomena through a system of heat balance equations at
the interior and exterior surfaces of the ceiling, roof sections, and gables, as well as a heat balance on the air mass within the attic.
To handle the case of raised trusses, short vertical walls at the eaves also are included. Each of the surfaces is assumed to be
isothermal; thus, for an attic consisting of a ceiling, two roof sections, two gables, two vertical eave sections, and one air space,
a total of 15 heat balance equations is used.
C1340/C1340M − 10 (2015)
FIG. 1 Schematic of Residential Attic Showing Heat Transfer Phenomena
6.3 Equations—Conduction:
6.3.1 The model developed here utilizes the thermal response factor method to analyze conduction through building envelope
sections. The thermal response factor method was developed by Mitalas and Arseneault (1) and was extended by Kusuda (2). The
method is based on an exact analytical solution of the heat conduction equation for one-dimensional heat flow through a multilayer
slab having temperature-independent thermal properties. The only approximation is that the surface temperatures are taken to vary
linearly with time between time steps. For analysis of buildings, the time step is normally taken to be 1 h. The response factor
equations relate the heat fluxes at the surfaces of the slab to the present and previous temperatures at the two surfaces. The
equations are:
` `
QI 5 Z' ~j! ~TIS~j! 2 TR! 2 Y'~j! ~TOS~j! 2 TR! (1)
( (
j50 j50
` `
QO 5 Y' ~j! ~TIS~j! 2 TR! 2 X'~j! ~TOS~j! 2 TR! (2)
( (
j50 j50
where:
QI = heat flux at inside surface at present time (note that the positive heat flow direction is from the inside to the
2 2
outside), W/m [Btu/h·ft ],
2 2
QO = heat flux at outside surface at present time, W/m [Btu ⁄h·ft ],
TIS(j) = temperature at inside surface j hours previous to present time, °C [°F],
TOS(j) = temperature at outside surface j hours previous to present time, °C [°F]
2 2
X' (j), Y' (j), Z'(j) = response factors, W/m · K [Btu/h·ft ·°F], and
TR = reference temperature, °C [°F].
6.3.2 The response factors are determined from a sequence of calculations that involve the thermal diffusivity, thermal
conductivity, specific heat, density, and thickness of each of the layers in the multilayer slab. An efficient computer program for
calculating the response factors has been developed by George Walton of the National Institute of Standards and Technology
(NIST) (7).
6.3.2.1 The efficiency of the response factor calculations can be increased by making use of the fact that after a sufficient number
of terms, the ratio of two consecutive response factors becomes constant. This is expressed by:
X' j11 Y' j11 Z' j11
~ ! ~ ! ~ !
5 5 5 CR for j $ N (3)
X' ~j! Y' ~j! Z' ~j!
CR = the common ratio, and
N = a sufficiently large number.
6.3.2.2 The common ratio is used to define a new set of functions, called the first order conduction transfer functions or simply
the conduction transfer functions, X(j), Y(j), and Z(j), which are given by:
X~0!5 X' ~0! (4)
Y 0 5 Y' 0 (5)
~ ! ~ !
Z 0 5 Z' 0 (6)
~ ! ~ !
C1340/C1340M − 10 (2015)
X j 5 X' j 2 CR X' j 2 1 for j # N (7)
~ ! ~ ! ~ !
Y j 5 Y' j 2 CR Y' j 2 1 for j # N (8)
~ ! ~ ! ~ !
Z~j! 5 Z'~j! 2 CR Z' ~j 2 1! for j # N (9)
X~j! 5 0 for j.N (10)
Y j 5 0 for j.N (11)
~ !
Z j 5 0 for j.N (12)
~ !
6.3.2.3 With the conduction transfer functions, the heat fluxes and surface temperatures are related by:
N
QI 5 Z~j! ~TIS~j! 2 TR! (13)
(
j50
N
2 Y~j! ~TOS~j! 2 TR!1CR QI'
(
j50
N
QO 5 Y j TIS j 2 TR (14)
~ ! ~ ~ ! !
(
j50
N
2 X~j! ~TOS~j! 2 TR!1CR QO'
(
j50
where:
QI' = heat flux at inside surface at previous time step,
QO' = heat flux at outside surface at previous time step, and
N = number of significant conduction transfer functions.
6.3.2.4 When parallel heat flow paths occur in an envelope component, separate response factors for each path may be needed.
If the boundary temperatures of the two paths may be assumed to be equal, however, then the response factors may be added
together as:
X'5 A X' 1A X' (15)
1 1 2 2
where:
A , A = area fractions for paths 1 and 2, and
1 2
(X' , X' ), (Y' , Y' ) and (Z' , Z' ) = the response factors for paths 1 and 2.
1 2 1 2 1 2
Parallel conduction transfer functions may be calculated from these parallel response factors, provided that the common ratio
for the path with the largest number of significant terms is used.
6.3.2.5 The original derivation of the response factor technique relied upon the assumption of temperature-independent thermal
properties. An approximate method has been developed to account for the temperature dependence of the thermal properties (5).
The thermal transmission coefficient of the component is taken to vary linearly with temperature as:
U 5 U @11b~T 2 TR!# (16)
TR
The conduction transfer function equations then become:
N
QI 5 Z j TIS j 2 TR (17)
...