ASTM C1340-99

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Designation: C 1340 – 99
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 C 1340; 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 (e) indicates an editorial change since the last revision or reapproval.
1. Scope 2.3 ASTM Adjuncts:
Computer Program for Estimation of Heat Gain or Loss
1.1 This practice covers the estimation of heat gain or loss
through Ceilings Under Attics Containing Radiant Barri-
through ceilings under attics containing radiant barriers by use
ers
of a computer program. The computer program included as an
adjunct to this practice provides a calculational procedure for
3. Terminology
estimating the heat loss or gain through the ceiling under an
3.1 Definitions—For definitions of terms used in this prac-
attic containing a truss or rafter mounted radiant barrier. The
tice, refer to Terminology C 168.
program also is applicable to the estimation of heat loss or gain
3.2 Symbols—Symbols will be introduced and defined in
through ceilings under an attic without a radiant barrier. This
the detailed description of the development.
procedure utilizes hour-by-hour weather data to estimate the
hour-by-hour ceiling heat flows. The interior of the house
4. Summary of Practice
below the ceiling is assumed to be maintained at a constant
4.1 The procedures used in this practice are based on the
temperature. At present, the procedure is applicable to sloped-
thermal response factor method for calculating dynamic heat
roof attics with rectangular floor plans having an unshaded
conduction through multilayer slabs (1,2), along with a model
gabled roof, a horizontal ceiling, and no HVAC ducts in the
for convective and radiative heat exchanges inside and outside
attic. It is not applicable to structures with flat roofs, vaulted
the attic.
ceilings, or cathedral ceilings. The calculational accuracy also
4.2 The operation of the computer program involves the
is limited by the quality of physical property data for the
following steps:
construction materials, principally the insulation and the radi-
4.2.1 Response Factors—A separate computer program
ant barrier, and by the quality of the weather data.
must be used to calculate the thermal response factors of the
1.2 Under some circumstances, interactions between radiant
solid materials surrounding the attic. Input to this program
barriers and HVAC ducts in attics can have a significant effect
would consist of the thermal conductivity, specific heat,
on the thermal performance of a building. When analysis of
density, and thickness of each layer, or the thermal resistance of
these interactions is completed they will be added to the
the layer if it has negligible density, and the fraction of the
computer program.
cross-sectional area occupied by the framing. Output of such a
program would be a set of response factors for use as input to
2. Referenced Documents
the main program. The adjunct to this practice contains data
2.1 ASTM Standards:
files with response factors for several typical attic construc-
C 168 Terminology Relating to Thermal Insulating Materi-
2 tions.
als
4.2.2 Data Input to the Main Program—This input includes
2.2 ANSI Standards:
the response factors, total hemispherical emittances of the
X3.5 Flow Chart Symbols and Their Usage in Information
3 inside and outside surfaces of the attic envelope, solar absorp-
Processing
tances of the outside surfaces of the attic envelope, length and
X3.9 Standard for Fortran Programming Language
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
This practice is under the jurisdiction of ASTM Committee C16 on Thermal
attic surfaces, area of exposed wood inside attic, mass of wood
Insulation and is the direct responsibility of Subcommittee C16.21 on Reflective
Insulation.
Current edition approved June 10, 1999. Published August 2001.
2 4
Annual Book of ASTM Standards, Vol 04.06. Available from ASTM Headquarters. Request Adjunct: ADJC1340.
3 5
Available from American National Standards Institute, 11 W. 42nd St., 13th The boldface numbers in parentheses refer to the list of references at the end of
Floor, New York, NY 10036. this standard.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959, United States.
C1340–99
in attic, initial moisture content of wood in attic, rate of 6.1.2 The solution is a computer procedure that estimates
exfiltration of air from house into attic space, latitude and temperatures of both sides of the components of the attic
longitude, time zone indicator, solar reflectance of the ground, envelope and the temperature of the air in the attic space, uses
indoor temperature, and indoor humidity. these estimates of temperatures to refine estimates of convec-
4.2.3 Analysis—Using hourly weather data consisting of tion and radiation heat transfer coefficients, reestimates the
outdoor temperature and humidity ratio, atmospheric pressure, temperatures using the new heat transfer coefficients, continues
total horizontal and direct solar radiation, wind speed and iterating on the temperatures and heat transfer coefficients until
direction, cloud amount, cloud type, and atmospheric clearness convergence is reached, and uses the last estimates of tempera-
number, the computer program calculates the inside and tures to calculate the heat gain or loss through the ceiling. This
outside temperatures of the attic envelope and the temperature procedure is repeated for each hour of the simulation period
of the air inside the attic. Using these temperatures, the (typically a full year).
program calculates the heat flux through the ceiling. 6.2 Development of Equations—The model that is the basis
4.2.4 Output—The hourly heat flux through the ceiling is for this practice is based on the model developed by B. Peavy
written to a file which can be used for further processing, such (3), which was later extended by Wilkes (4-6). The sketch of an
as seasonal or annual heat gains or losses. attic given in Fig. 1 shows the various heat transfer mecha-
nisms that occur within an attic. Although the sketch shows
5. Significance and Use
ventilation occurring at soffit and ridge vents, the location of
5.1 Manufacturers of radiant barriers express the perfor- the vents may be at other locations, such as at the gables. The
mance of their products in terms of the total hemispherical
model treats all of these phenomena through a system of heat
emittance. The purpose of a radiant barrier is to decrease the balance equations at the interior and exterior surfaces of the
radiation heat transfer across the attic air space, and hence, to
ceiling, roof sections, and gables, as well as a heat balance on
decrease the heat loss or gain through the ceiling below the the air mass within the attic. To handle the case of raised
attic. The amount of decrease in heat flow will depend upon a
trusses, short vertical walls at the eaves also are included. Each
number of factors, such as weather conditions, amount of mass
of the surfaces is assumed to be isothermal; thus, for an attic
or reflective insulation in the attic, solar absorptance of the
consisting of a ceiling, two roof sections, two gables, two
roof, geometry of the attic and roof, and amount and type of
vertical eave sections, and one air space, a total of 15 heat
attic ventilation. Because of the infinite combinations of these
balance equations is used.
factors, it is not practical to publish data for each possible case.
6.3 Equations—Conduction:
5.2 The calculation of heat loss or gain of a system
6.3.1 The model developed here utilizes the thermal re-
containing radiant barriers is mathematically complex, and
sponse factor method to analyze conduction through building
because of the iterative nature of the method, it is best handled
envelope sections. The thermal response factor method was
by computers.
developed by Mitalas and Arseneault (1) and was extended by
5.3 Computers are now widely available to most producers
Kusuda (2). The method is based on an exact analytical
and consumers of radiant barriers to permit the use of this
solution of the heat conduction equation for one-dimensional
practice. heat flow through a multilayer slab having temperature-
5.4 The user of this practice may wish to modify the data
independent thermal properties. The only approximation is that
input to represent accurately the structure. The computer the surface temperatures are taken to vary linearly with time
program also may be modified to meet individual needs. Also,
between time steps. For analysis of buildings, the time step is
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
FIG. 1 Schematic of Residential Attic Showing Heat Transfer
6.1.1.6 The weather conditions. Phenomena
C1340–99
normally taken to be 1 h. The response factor equations relate Y~j! 5 0 for j . N (11)
the heat fluxes at the surfaces of the slab to the present and
previous temperatures at the two surfaces. The equations are: Z~j! 5 0 for j . N (12)
‘ ‘
6.3.2.3 With the conduction transfer functions, the heat
QI 5 Z8 ~j! ~TIS~j! 2 TR! 2 Y8~j! ~TOS~j! 2 TR! (1)
( (
fluxes and surface temperatures are related by:
j50 j50
N
QI 5 Z~j! ~TIS~j! 2 TR!
‘ ‘
(
j50
QO 5 Y8 ~j! ~TIS~j! 2 TR! 2 X8~j! ~TOS~j! 2 TR! (2)
( (
N
j50 j50
2 Y~j! ~TOS~j! 2 TR! 1 CR QI8 (13)
(
j50
where:
QI = heat flux at inside surface at present
N
time (note that the positive heat flow
QO 5 Y j TIS j 2 TR
~ ! ~ ~ ! !
(
j50
direction is from the inside to the
N
outside), Btu/h·ft ,
2 X~j! ~TOS~j! 2 TR! 1 CR QO8 (14)
(
j50
QO = heat flux at outside surface at present
time, Btu/h·ft ,
TIS(j) = temperature at inside surface j hours
where:
previous to present time, °F,
QI8 = heat flux at inside surface at previous time step,
TOS(j) = temperature at outside surface j hours
QO8 = heat flux at outside surface at previous time step,
previous to present time, °F,
and
X8 (j), Y8 (j), Z8(j) = response factors, Btu/h·ft ·°F, and
N = number of significant conduction transfer functions.
TR = reference temperature, °F.
6.3.2.4 When parallel heat flow paths occur in an envelope
6.3.2 The response factors are determined from a sequence
component, separate response factors for each path may be
of calculations that involve the thermal diffusivity, thermal
needed. If the boundary temperatures of the two paths may be
conductivity, specific heat, density, and thickness of each of the
assumed to be equal, however, then the response factors may
layers in the multilayer slab. An efficient computer program for
be added together as:
calculating the response factors has been developed by George
X8 5 A X8 1 A X8 (15)
1 1 2 2
Walton of the National Institute of Standards and Technology
where:
(NIST) (7).
A ,A = area fractions for
1 2
6.3.2.1 The efficiency of the response factor calculations
paths 1 and 2, and
can be increased by making use of the fact that after a sufficient
(X8 ,X8 ), (Y8 ,Y8 ) and (Z8 ,Z8 ) = the response fac-
1 2 1 2 1 2
number of terms, the ratio of two consecutive response factors
tors for paths 1
becomes constant. This is expressed by:
and 2.
X8 j 1 1 Y8 j 1 1 Z8 j 1 1
~ ! ~ ! ~ !
Parallel conduction transfer functions may be calculated
5 5 5 CR for j $ N (3)
X8 ~j! Y8 ~j! Z8 ~j!
from these parallel response factors, provided that the common
ratio for the path with the largest number of significant terms is
used.
CR = the common ratio, and
6.3.2.5 The original derivation of the response factor tech-
N = a sufficiently large number.
nique relied upon the assumption of temperature-independent
6.3.2.2 The common ratio is used to define a new set of
thermal properties. An approximate method has been devel-
functions, called the first order conduction transfer functions or
oped to account for the temperature dependence of the thermal
simply the conduction transfer functions, X(j), Y(j), and Z(j),
properties (5). The thermal transmission coefficient of the
which are given by:
component is taken to vary linearly with temperature as:
X 0! 5 X8 0! (4)
~ ~
U 5 U @1 1 b~T 2 TR!# (16)
TR
The conduction transfer function equations then become:
Y~0! 5 Y8 ~0! (5)
N
QI 5 Z~j! ~TIS~j! 2 TR!
(
Z~0! 5 Z8 ~0! (6)
j50
N
2 Y j! TOS j! 2 TR! 1 CR QI8 (17)
~ ~ ~
(
j50
X~j! 5 X8~j! 2 CR X8 ~j 2 1! for j # N (7)
N N
2 2
1 b/2 Z~j! ~TIS~j! 2 TR! 2 b/2 Y~j! ~TOS~j! 2 TR!
( (
Y~j! 5 Y8~j! 2 CR Y8 ~j 2 1! for j # N (8) j50 j50
N
Z~j! 5 Z8~j! 2 CR Z8 ~j 2 1! for j # N (9)
QO 5 Y~j! ~TIS~j! 2 TR!
(
j50
N
X~j! 5 0 for j . N (10)
2 X~j! ~TOS~j! 2 TR! 1 CR QO8 (18)
(
j50
C1340–99
N N
TABLE 1 Correlations for Convection Coefficients
2 2
1 b/2 Y~j! ~TIS~j! 2 TR! 2 b/2 X~j! ~TOS~j! 2 TR!
( (
I. Natural Convection:
j50 j50
A. Horizontal surface, heat flow up
1/4 6
These equations are used in the system of heat balance Nu = 0.54 Ra for Ra < 8 3 10
1/3 6
Nu = 0.15 Ra for Ra > 8 3 10
equations.
B. Horizontal surface, heat flow down
0.2
6.4 Equations—Convection:
Nu =0.58 Ra
C. Vertical surface
6.4.1 Convection heat transfer from the interior and exterior
1/4 9
Nu = 0.59 Ra for Ra < 1 3 10
surfaces of the envelope components is calculated using 1/3 9
Nu = 0.10 Ra for Ra > 1 3 10
correlations from the literature (8). The coefficients are based D. Nearly horizontal surface (tilt angle less than 2°), heat flow down
0.2
Nu = 0.58 Ra
on correlations that have been developed for isolated isother-
E. Tilted surfaces (greater than 2° tilt), heat flow down
mal flat plates.
...