ASTM C1340/C1340M-10(2021)

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.  
1.4 This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.

General Information

• Status: Published
• Publication Date: 31-Aug-2021
• ICS: 91.120.10 - Thermal insulation of buildings
• Technical Committee: C16 - Thermal Insulation
• Drafting Committee: C16.21 - Reflective Insulation

Relations

• Refers: ASTM C168-24 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 15-Apr-2024
• Refers: ASTM C168-18 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 15-Apr-2018
• Refers: ASTM C168-17 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Jun-2017
• Refers: ASTM C168-15a - Standard Terminology Relating to Thermal Insulation
• Effective Date: 15-Oct-2015
• Refers: ASTM C168-15 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Jun-2015
• Refers: ASTM C168-13 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Apr-2013
• Refers: ASTM C168-10 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Jan-2010
• Refers: ASTM C168-08b - Standard Terminology Relating to Thermal Insulation
• Effective Date: 15-Dec-2008
• Refers: ASTM C168-08a - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Sep-2008
• Refers: ASTM C168-08 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Jun-2008
• Refers: ASTM C168-05a - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Nov-2005
• Refers: ASTM C168-05 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 01-Jun-2005
• Refers: ASTM C168-03 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 10-Apr-2003
• Refers: ASTM C168-02 - Standard Terminology Relating to Thermal Insulation
• Effective Date: 10-Jul-2002
• Refers: ASTM C168-00 - Standard Terminology Relating to Thermal Insulating Materials
• Effective Date: 10-May-2001

Overview

ASTM C1340/C1340M-10(2021): Standard Practice for Estimation of Heat Gain or Loss Through Ceilings Under Attics Containing Radiant Barriers by Use of a Computer Program provides a standardized methodology for estimating heat transfer through ceilings beneath attics. This practice is specifically designed for attics equipped with radiant barriers and employs a computer program to handle the complex, iterative calculations required. The standard supports both attics with and without radiant barriers and is particularly useful for sloped-roof attics with rectangular floor plans, an unshaded gabled roof, and a horizontal ceiling.

Heat transfer through attic ceilings is affected by multiple factors, including insulation, weather, attic geometry, attic ventilation, and the use of radiant barriers. Because of the large variety of variables and their combinations, the standard employs a flexible, computer-based approach, allowing users to input relevant construction and environmental data for accurate energy modeling.

Key Topics

• Radiant Barriers: Evaluation and impact of total hemispherical emittance in reducing radiation heat transfer across attic air space.
• Heat Gain and Loss Estimation: Use of hour-by-hour weather data and iterative mathematical models to determine heat flow through ceiling assemblies.
• Computer Modeling: Requirement for computer-based analysis due to the complexity and interdependence of variables such as material properties, weather, attic ventilation, and geometry.
• Customization: Flexibility to modify both input data and the computer program to match specific building structures or modeling requirements (e.g., seasonal or annual energy usage).
• Limitations: Scope limited to attics with sloped, gabled roofs and rectangular floor plans; not applicable to flat, vaulted, or cathedral ceilings. Calculational accuracy depends on the quality of physical property and weather data.
• Thermal Interaction: Consideration of HVAC duct effects within attics as included in model extensions.

Applications

ASTM C1340/C1340M-10(2021) is vital for:

• Building Envelope Analysis: Assessing and optimizing attic insulation strategies for improved thermal performance and reduced energy consumption.
• Radiant Barrier Evaluation: Quantifying the effectiveness of radiant barriers in diverse climatic conditions and buildings, supporting product selection and installation decisions.
• Energy Modeling: Supplying accurate, hour-by-hour heat flow data for use in whole-house energy simulations or estimating seasonal/annual energy requirements.
• Design and Retrofit Projects: Guiding architects, engineers, builders, and energy consultants in designing new buildings or retrofitting existing structures to enhance attic performance with radiant or traditional insulation.
• Regulatory Compliance and Product Development: Serving as a reference point for meeting building codes or developing new insulation materials and thermal management systems.

Related Standards

Key standards referenced or relevant to ASTM C1340/C1340M-10(2021) include:

• ASTM C168: Terminology Relating to Thermal Insulation
• ANSI X3.5: Flow Chart Symbols and Their Usage in Information Processing
• ANSI X3.9: Standard for Fortran Programming Language
• ASHRAE Handbook of Fundamentals: Guidelines for stack and wind pressure calculations and ventilation modeling

Practical Value

Adopting ASTM C1340/C1340M-10(2021) enables stakeholders to produce accurate, consistent estimations of attic-related heat gain or loss, facilitating informed decisions in energy efficiency improvement. By leveraging computer-based simulation, this standard ensures that diverse attic design configurations and site-specific variables are properly accounted for, ultimately leading to better indoor comfort, lower operational costs, and enhanced building sustainability.

Keywords: ASTM C1340, radiant barriers, attic heat gain, attic heat loss, computer modeling, ceiling insulation, energy modeling, building envelope, attic ventilation, thermal performance.

Standards Content (Sample)

This international standard was developed in accordance with internationally recognized principles on standardization established in the Decision on Principles for the
Development of International Standards, Guides and Recommendations issued by the World Trade Organization Technical Barriers to Trade (TBT) Committee.
Designation: C1340/C1340M − 10 (Reapproved 2021)
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 Development of International Standards, Guides and Recom-
mendations issued by the World Trade Organization Technical
1.1 This practice covers the estimation of heat gain or loss
Barriers to Trade (TBT) Committee.
through ceilings under attics containing radiant barriers by use
of a computer program. The computer program included as an
2. Referenced Documents
adjunct to this practice provides a calculational procedure for
2.1 ASTM Standards:
estimating the heat loss or gain through the ceiling under an
C168Terminology Relating to Thermal Insulation
attic containing a truss or rafter mounted radiant barrier. The
programalsoisapplicabletotheestimationofheatlossorgain
2.2 ANSI Standards:
through ceilings under an attic without a radiant barrier. This X3.5Flow Chart Symbols and Their Usage in Information
procedure utilizes hour-by-hour weather data to estimate the
Processing
hour-by-hour ceiling heat flows. The interior of the house X3.9Standard for Fortran Programming Language
below the ceiling is assumed to be maintained at a constant
2.3 ASTM Adjuncts:
temperature.At present, the procedure is applicable to sloped-
Computer Program for Estimation of Heat Gain or Loss
roof attics with rectangular floor plans having an unshaded
through Ceilings Under Attics Containing Radiant Barri-
gabled roof and a horizontal ceiling. It is not applicable to
ers
structureswithflatroofs,vaultedceilings,orcathedralceilings.
The calculational accuracy also is limited by the quality of 3. Terminology
physical property data for the construction materials, princi-
3.1 Definitions—For definitions of terms used in this
pally the insulation and the radiant barrier, and by the quality
practice, refer to Terminology C168.
of the weather data.
3.2 Symbols—Symbolswillbeintroducedanddefinedinthe
1.2 Undersomecircumstances,interactionsbetweenradiant
detailed description of the development.
barriers and HVAC ducts in attics can have a significant effect
onthethermalperformanceofabuilding.Ductsareincludedin
4. Summary of Practice
an extension of the computer model given in the appendix.
4.1 The procedures used in this practice are based on the
1.3 The values stated in either SI units or inch-pound units
thermal response factor method for calculating dynamic heat
are to be regarded separately as standard. The values stated in
conductionthroughmultilayerslabs (1, 2), alongwithamodel
each system may not be exact equivalents; therefore, each
for convective and radiative heat exchanges inside and outside
system shall be used independently of the other. Combining
the attic.
values from the two systems may result in non-conformance
4.2 The operation of the computer program involves the
with the standard.
following steps:
1.4 This international standard was developed in accor-
dance with internationally recognized principles on standard-
ization established in the Decision on Principles for the 2
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.
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, 2021. Published October 2021. Originally ADJC1340.
approved in 1999. Last previous edition approved in 2015 as C1340/C1340M–10 Theboldfacenumbersinparenthesesrefertothelistofreferencesattheendof
(2015). DOI: 10.1520/C1340_C1340M-10R21. this standard.
Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States
C1340/C1340M − 10 (2021)
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 (2021)
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 (2021)
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. Separate coefficients are calculated for
2 Y j TOS j 2 TR 1CRQI'
~ !~ ~ ! !
(
naturalandforcedflow,andamixedcoefficientiscalculatedby
j50
takingthethirdrootofthesumofthecubesofthetwoseparate
N N
coefficients (10).
2 2
1b/2 Z~j!~TIS~j! 2 TR! 2 b/2 Y~j!~TOS~j! 2 TR!
( (
j50 j50 6.4.2.2 The correlations used in the model are given in
N Table 1.They account for the effects of surface-to-air tempera-
QO 5 Y~j!~TIS~j! 2 TR! (18)
( ture difference, heat flow direction, film temperature, surface
j50
size, and surface orientation. The correlations are contained in
N
a subroutine called HCON. Values of surface temperature, air
2 X j TOS j 2 TR 1CRQO'
~ !~ ~ ! !
(
temperature, plate tilt angle, plate characteristic length, a flag
j50
to denote whether the plate faces up or down, and air speed
N N
past the plate are passed to the subroutine. The subroutine
2 2
1b/2 Y j TIS j 2 TR 2 b/2 X j TOS j 2 TR
~ !~ ~ ! ! ~ !~ ~ ! !
( (
j50 j50 returns the mixed convection coefficient. For exterior surfaces,
the air speed is taken to be the wind speed obtained from
These equations are used in the system of heat balance
meteorological data. For interior surfaces, a crude estimate of
equations.
air speed is obtained by dividing the attic ventilation volume
6.4 Equations—Convection:
flow rate by an average cross-sectional area for the attic. For
6.4.1 Convectionheattransferfromtheinteriorandexterior the interior surfaces, a crude estimate for air speed should
surfaces of the envelope components is calculated using
suffice, since natural convection should dominate over forced
correlations from the literature (8). The coefficients are based convection.
on correlations that have been developed for isolated isother-
6.5 Equations—Radiation:
mal flat plates. The correlations are in the form of a Nusselt
6.5.1 Radiation interchanges within the attic space are
number, Nu, as a function of a Rayleigh number, Ra, Grashof
handled using the enclosure method described by Sparrow and
number, Gr, or a Reynolds number, Re, where:
Cess (11).Withthismethod,eachofthesurfacesisassumedto
Nu 5 hL/k (19) beplane,opaque,gray,andisothermal,tobediffuselyemitting
3 and reflecting, and to have a uniform radiant flux over the
gβpC ∆TL
ρ
Ra 5 (20)
surface. The assumption of diffuse emission and reflection for
vk
radiant barrier surfaces may be questioned. Very flat radiant
Gr 5 Ra/Pr (21)
Pr 5 v/α (22)
TABLE 1 Correlations for Convection Coefficients
Re 5 VL/v (23)
I. Natural Convection:
A. Horizontal surface, heat flow up
and:
1/4 6
Nu=0.54Ra forRa<8×10
1/3 6
Nu=0.15Ra forRa>8×10
h = convection heat transfer coefficient, W/ m · K [Btu/
B. Horizontal surface, heat flow down
h·ft ·°F],
0.2
Nu =0.58 Ra
L = characteristic length of plate, m [ft],
C. Vertical surface
1/4 9
Nu=0.59Ra forRa<1×10
k = thermal conductivity of air, W/m·K [Btu/h·ft·°F],
1/3 9
2 2
Nu=0.10Ra forRa>1×10
g = acceleration of gravity, m/s [ft/h ],
D. Nearly horizontal surface (tilt angle less than 2°), heat flow down
−1 −1
β = volume coefficient of expansion of air, K [°R ],
0.2
Nu=0.58Ra
3 3
ρ = density of air, kg/m [lb/ft ],
E. Tilted surfaces (greater than 2° tilt), heat flow down
1/4
Nu = 0.56 (Ra sin(Φ)) Φ = tilt angle
C = specific heat of air, J/kg · k [Btu/lb·°F],
p
F. Tilted surface, heat flow up
∆T = temperature difference between surface and air, K
1/4
Nu = 0.56 (Ra sin(Φ)) for Ra/Pr < Gr
c
[°F], 1/3 1/3
Nu=0.14(Ra −(Gr Pr) )
c
2 2
1/4
v = kinematic viscosity of air, m /s [ft /h], +0.56(Gr Pr sin(Φ)) for Ra/Pr > Gr
c c
Gr =1×10 for Φ < 15°
Pr = Prandtl number for air, dimensionless, c
(Φ/(1.1870 + 0.0870×Φ))
2 2 Gr =10 for 15° < Φ <75°
c
α = thermal diffusivity of air, m /s [ft /h], and
Gr =5×10 for Φ > 75°
c
V = velocity of air stream, m/s (ft/h).
II. Forced Convection:
1/3 1/2 5
Nu = 0.664 Pr Re forRe<5×10
6.4.2 The model accounts for the temperature dependence 1/3 0.8 5
Nu=Pr (0.037 Re − 850) for Re>5×10
of the properties of air by evaluating them at the film
C1340/C1340M − 10 (2021)
barrier surfaces would be expected to exhibit specular proper- from the surrounding surfaces. This is expressed by the
ties.Realradiantbarriersurfacesusuallyarenotflat,andhence following first order ordinary differential equation:
the diffuse assumption is probably appropriate.
M
dT
m˙C 5 A j HC j TIS j 2 T (27)
~ ! ~ !~ ~ ! !
6.5.1.1 The net radiant heat flux away from surface i is p
(
dx
j51
given by:
m˙ = mass flow rate of ventilation air, kg/s [lb/h],
N
4 4
C = specific heat of air, J/kg ·K [Btu/lb·°F],
Q 5 G σ~T 2 T ! (24)
p
ri ( ij i j
j51
T = air temperature at position, °C [°F],
x = normalized position (0–1) along flow path,
with the following relations:
A(j) = area of surface j in contact with ventilation air, m
ε
i
[ft ],
G 5 ψ (25)
ij ij
1 2ε
i
HC(j) = convection heat transfer coefficient at surface j,
2 2
W/m · K [Btu/h·ft ·°F],
ψ 5 inverseofthematrixχ
ij ij
TIS(j) = temperature of surface j, °C [°F], and,
M = number of surfaces in contact with ventilation air.
δ 2 1 2ε F
~ !
ij i ij
χ 5
ij
ε
i 6.6.2 In the absence of a detailed picture of the ventilation
airflowpatternwithintheattic,thesimplifyingassumptionhas
δ 5 0 forifij; δ 5 1 fori 5 j
ij ij
been made that equal proportions of area of each surface are
where:
contacted as the air flows along a differential length of its flow
path. Integrating this equation yields the following expression
F = radiation view factor from surface i to surface j,
ij
for the temperature as a function of flow path position:
ε = emittance of surface i,
i
-8 2 4
σ = Stefan-Boltzmann constant, 5.6696×10 W/m ·K
T 5 T exp 2ax 1b@1 2 exp 2ax # (28)
~ ! ~ !
o
−9 2 4
[1.714 × 10 Btu/h·ft ·R ], and
where:
N = number of surfaces in enclosure.
T = temperature at x=0,
o
6.5.1.2 The heat flux equation may be cast into a form that
a =
M
is linear in the temperatures by factoring as:
A j HC j /m˙C , and
~ ! ~ !
p
(
j51
N
2 2
Q 5 G σ T 1T T 1T T 2 T (26)
~ !~ !~ !
ri ( ij i j i j i j
j51
b =
M
A j HC j TIS j
~ ! ~ ! ~ !
N
(
j51
5 HR T 2 T .
~ !
ij i j M
(
j51
A j HC j
~ ! ~ !
(
j51
6.5.1.3 The quantities appearing in these equations are
calculated in several subprograms: subroutine VIEW2, and 6.6.3 An average air temperature is defined by:
functions FMN, FP, and HRAD. Function FMN calculates the
TA 5 Tdx 5 b 11 exp 2a 2 1 /a 2 T /a exp 2a 2 1
* @ ~ ~ ! ! # @ ~ ! #
o
view factor between two rectangular surfaces that meet at an
angle and share a common edge, while function FP calculates
(29)
the view factor between two parallel surfaces that have the
and the temperature of the air exiting from the attic at x=1
same dimensions. Expressions for the two view factors are
is:
given by Sparrow and Cess (11). The functions themselves are
based on those given by Peavy, with some necessary modifi- TE 5 b 1 2 exp 2a 1T exp 2a (30)
@ ~ !# ~ !
o
cations (3). The functions have been verified by comparison
6.6.4 The flow rate of ventilation air is determined from a
with the calculations of Feingold (12). Subroutine VIEW2
combination of stack and wind pressure effects. For the stack
calculates the view factors among the surfaces that face the
effectcalculation,theairinsidetheatticspaceisassumedtobe
attic space, using view factor algebra and calls to functions
well-mixed, and relations from Chapter 22 of the 1985
FMN and FP. The G matrix is also calculated within VIEW2.
ij
ASHRAE Handbook of Fundamentals are used (13). The
The HR matrix is calculated by function HRAD using G , T,
ij ij i
volume flow rate due to the stack effect is given by:
and T as inputs.
j
˙ 1/2
6.5.1.4 Thealgorithmforradiationinterchangesallowseach
V 5 C A@2 gh ~TA 2 T !/TA# forTA.T (31)
S D o o
of the surfaces to have different emittances. The algorithm
1/2
˙
V 5 C A 2 gh T 2 TA /T forT .TA (32)
@ ~ ! #
S D o o o
accounts for all interreflections within the enclosure and
properly accounts for the T (Stefan-Boltzmann) law for
where:
3 3
radiation exchange.
˙
V = volume flow rate due to stack effect, m /s [ft h],
S
C = discharge coefficient taken to be 0.65,
6.6 Equations—Ventilation: D
A = the lesser of the net free areas of the vent inlet and
6.6.1 Heat transfer to the ventilation air is treated by an
2 2
outlet, m [ft ],
extension of the method used by Peavy (3). With this method,
h = height of the neutral pressure level from the lower
the temperature of the air is assumed to increase (or decrease)
opening, m [ft],
as it moves along a flow path and picks up heat by convection
C1340/C1340M − 10 (2021)
Additional experimental data appear to be needed to develop
T = absolute temperature of inlet air, K [°R], and
o
better algorithms for ventilation.
TA = average absolute temperature of air in attic, K [°R].
6.6.5 The height of the neutral pressure level is given by:
6.7 Equations—Moisture:
6.7.1 Approximations have been built into the models to
H
h 5 forTA.T (33)
o
account for the latent heat effects due to sorption and desorp-
AI TA
S D
tion of moisture at the wood surfaces that face the attic space.
AO T
o
These generally follow the suggestions given by Burch et al
H
h 5 forT .TA (34)
2 o (17) and Cleary (18). In this model, the wood surface is
AI T
o
S D assumed to be in moisture equilibrium with a thin layer of air
AO TA
adjacent to the surface. The humidity ratio of this layer of air
where:
is given by (18).
H = difference in elevation between inlet and outlet vents,
2 3
ω 5 ~b1c·u1d·u 1e·u !exp~T/a! (39)
s
m [ft],
2 2
AI = net free area of inlet vents, m [ft ], and
where:
2 2
AO = net free area of outlet vents, m [ft ].
ω = humidity ratio of air near wood surface, kg of
s
water/kg of dry air [lb of water/lb of dry air],
6.6.6 Thevolumeflowrateduetowindpressureisgivenby:
T = temperature of wood surface, °C [°F],
˙
V 5 C AWS (35)
u = moisture content of wood (dimensionless
W F
fraction),
where:
a, b, c, d = constants from fit to equilibrium moisture con-
C = discharge coefficient,
F
tent data.
A = the lesser of the net free areas of the vent inlet and
2 2
outlet, m [ft ], and
6.7.2 Therateoftransferofmoisturefromtheairintheattic
WS = wind speed, m/s [ft/h].
space to the surface is given by:
6.6.7 From the data given by Burch and Treado (14), the
m˙ 5 h ω 2ω (40)
~ !
w w a s
dischargecoefficientsforthewindpressureeffectareestimated
where:
tobe0.38forsoffitandridgeventsand0.54forsoffitandgable
m˙ = mass flow rate of moisture per unit area of exposed
vents. For soffit vents only, C is estimated from
F
2 2
wood surface, kg/s·m [lb/h·ft ],
C 5 0.08910.132 sin D (36)
F
ω = humidity ratio of air in attic space, kg of water/kg of
a
where: dry air [lb of water/lb of dry air], and
2 2
h = mass transfer coefficient, kg/s·m [lb/h·ft ].
w
D = winddirection,measuredfromadirectionparalleltothe
ridge, radians.
6.7.3 The mass transfer coefficient is obtained from the
analogy between heat and mass transfer as:
6.6.8 Massflowratesforthestackandwindpressureeffects
are obtained by multiplying the volume flow rates by the 2/3
h α
c
5 '1 (41)
S D
density of the air. A total mass flow rate is given by:
h C D
w P
2 2 1/2
m˙ 5 ~m˙ 1m˙ ! (37)
S W
where:
6.6.9 Finally, the adjustment recommended byASHRAE is h = convection heat transfer coefficient, W/m ·K [Btu/
c
made to account for unequal inlet and outlet vent areas: h·ft ·°F],
C = specific heat of air, J/kg·K [Btu/lb·°F],
1.5 2 2 1/2
P
m˙ 5 110.4077 1 2 A /A m˙ 1m˙ (38)
@ ~ ~ ! # ~ !
min max S W 2 2
α = thermal diffusivity of air, m /s [ft /h], and
where the quantity in parentheses is a fit to the graph in the D = coefficient for diffusion of water vapor through air,
2 2
m /s [ft /h].
1985ASHRAE Handbook of Fundamentals (13). The air flow
due to exfiltration of air from the house into the attic space is
6.7.4 The humidity ratio of the attic air is obtained by
added to the flow due to the stack and wind pressure effects.
performing a steady-state moisture balance on the attic space,
The algorithm for estimating ventilation rates is embodied in a
including diffusion of moisture through the boundary surfaces,
subroutine called VENT.
convection of moisture into the attic space from the outside air
6.6.10 Although some other results in the literature are in
and from exfiltration from the house, convection of moisture
qualitative agreement (15,16), there appears to be a good deal
out of the attic space by the ventilation air, and moisture
of uncertainty in this ventilation algorithm. D. M. Burch, in a
transfer to or from the wood surfaces. The attic moisture
private conversation with K. E. Wilkes in 1987, has suggested
balance is given by:
that the ventilation rates he and Treado measured may be in
M
errorbecauseofproblemsofmixingtracergasesinanattic.D.
A j Perm j P j 2 P 1m˙ ω 2ω 1m˙ ω 2ω (42)
~ ! ~ !~ ~ ! ! ~ ! ~ !
a v o a E i a
(
Ober, in a private conversation with K. E. Wilkes in 1988, has
j51
suggested that a better stack ventilation rate might be obtained
M
by using the roof temperature rather than the average air
1 h ~j! A'~j!~ω ~j! 2ω ! 5 0
( w s a
temperature as the driving force during the daytime hours. j51
C1340/C1340M − 10 (2021)
M
where:
1 HR i,k TIS i,0 2 TIS k,0 2 m˙ i h 5 0
~ !~ ~ ! ~ !! ~ !
( w v
Perm(j) = water vapor permeance of surface j, k51 kfii
m˙ = mass flow rate of ventilation air,
v
6.8.1.1 In Eq 44, the index i refers to the surface for which
m˙ = mass flow rate of exfiltration air,
E
the heat balance is being written. Index k refers to the other
ω (j) = humidity ratio at surface j,
s
surfaces that face the attic space, and j is the index for the
ω = humidity ratio of air in attic space,
a
conduction transfer function time sequence, with j = 0 repre-
ω = humidity ratio of outside air,
o
senting the current time. An equation of this form is obtained
ω = humidity ratio of indoor air,
i
for each of the surfaces facing the attic space.
P(j) = partialpressureofwatervaporinaironoutsideof
6.8.2 Theheatbalanceattheexteriorsurfacerelatestheheat
surface j,
conducted through the envelope surface to the heat convected
P = partial pressure of water vapor in attic air,
a
to the outdoor air, the heat radiated by the surface to the
h (j) = water vapor mass transfer coefficient at surface j,
w
surroundings, and the solar radiation absorbed by the surface.
A(j) = area of surface j, and
A'(j) = exposed wood area at surface j.
Since the exterior surface of the bottom of the attic is the
ceiling of the house, the balance is modified to allow convec-
6.7.4.1 Whenthehumidityratiooftheatticairandthemass
tion to the indoor air, radiation to the surfaces of the room
transfer rates have been calculated, the heat transferred to the
belowtheattic,andnodirectabsorptionofsolarradiation.The
surface by latent heat effects, Q , is given by
lat
exterior heat balances have the form:
Q 5 m˙ h (43)
lat w v
N N
Y i,j TIS i,j 2 TR 2 X i,j TOS i,j 2 TR (45)
~ !~ ~ ! ! ~ !~ ~ ! !
where: ( (
j50 j50
h = latent heat of vaporization of water. For this model, h
v v
N
has been taken to have a constant value of 2 466 kJ/kg
1b i /2 Y i,j TIS i,j 2 TR
~ ! ~ !~ ~ ! !
(
[1060 Btu/lb]. j50
N
6.7.4.2 The mass transfer to a surface is used to estimate a
2b i /2 X i,j TOS i,j 2 TR
~ ! ~ !~ ~ ! !
new surface moisture content, assuming that only a thin layer
(
j50
of wood participates in the moisture exchanges. It should be
noted that the intent of this treatment of moisture is only to 1CR i QO' i 1HC i T 2 TOS i,0
~ ! ~ ! ~ !~ ~ !!
o
estimatetheeffectoflatentheatsontheheatflowratesandnot
1HR i T 2 TOS i,0 1α i Q i 5 0
~ !~ ~ !! ~ ! ~ !
s s
todeterminetheaccumulationofmoistureitself.Anestimation
ofmoistureaccumulationratesoverlongperiodsoftimewould
where:
require a more detailed treatment than is used here.
T = temperature of outside air,
o
6.7.5 The algorithms for calculating moisture effects are
T = temperature of surroundings,
s
embodied in several subprograms. The humidity ratio at the
α(i) = solar absorptance of surface i, and
wood surface is calculated by function WDHUM. Subroutine
Q (i) = solar radiation incident on surface i.
s
PSY is used to calculate humidity ratios and partial pressures
6.8.2.1 The heat balance on the attic air mass accounts for
of water vapor from known values of air temperature and
the heat convected to the air from each of the surfaces facing
relative humidity. The mass transfer coefficients are calculated
the attic space, convection of outdoor air into the space, and
in subroutine HMASS, and the moisture balance is performed
convection of attic air out of the space. With the model given
in subroutine MOIST.
aboveforthetemperatureriseoftheventilationair,theatticair
6.8 Equations—Heat Balances:
heat balance is given by:
6.8.1 The heat balance equations combine the heat flows by
M
C
various mechanisms at the inside and outside surfaces of the
A~i! HC~i! TIS~i,0! 2 TA (46)
(
11C C
i51
2 3
componentsoftheatticenvelopeandontheatticairmass.The
heat balance at an interior surface, that is, one facing the attic
C
5m˙ C T
space, is obtained by summing the contributions due to v p o
11C C
2 3
conductionthroughthecomponent,radiationinterchangeswith
where:
each of the other surfaces that it sees, convection exchanges
M
C =
with the attic air mass, and latent heat loads due to moisture
A(i) HC(i),
(
i51
sorption/desorption. The interior heat balances are:
m˙ C
C =
v p
, and
N N C
C = exp(−1/C)−1.
3 2
Z~i,j!~TIS~i,j! 2 TR! 2 Y~i,j!~TOS~i,j! 2 TR! (44)
( (
j50 j50
6.8.2.2 The system of heat balance equations is arranged in
N N
a matrix form with the interior and exterior surface tempera-
1b i /2 Z i,j TIS i,j 2 TR 2 b i /2 Y i,j TOS i,j
~ ! ~ !~ ~ ! ! ~ ! ~ !~ ~ !
( ( turesandtheairtemperaturesatthecurrenttimestepbeingthe
j50 j50
unknown quantities. Values for these temperatures at previous
2TR) 1CR i QI' i 1HC i TIS i,0 2 TA time steps are known. The matrix equation may be written as:
~ ! ~ ! ~ !~ ~ ! !
C1340/C1340M − 10 (2021)
AA T 5 BB (47) 7.1.2 The program consists of a main program and several
~ !~ ! ~ !
subroutines. Other subroutines may be added to make the
where:
programmoreapplicabletothespecificproblemsofindividual
AA = a square matrix of coefficients,
users.
T = a vector of unknown temperatures, and
7.1.3 A flow chart for the program is given in Fig. 2. The
BB = a known factor.
program starts by initializing all temperatures to 75°F and all
Detailed expressions for the elements of the matrices are
heatfluxestozerothroughtheuseofDATAstatements.DATA
lengthy.They may be found by inspecting the Fortran code for
statements also are used to set various other quantities to zero
the computer program and will not be repeated here.
such as the elements of the (AA) matrix. Many of these
elements will remain at zero while others are recalculated later
6.8.3 This system of equations is solved by Gauss-Jordan
in the program.
elimination, using a subroutine named SOLVP, which was
7.1.3.1 Next, material property and geometrical input data
developedbyPeavy(giveninKusuda (19)).Sincemanyofthe
arereadin.Thisconsistsofconductiontransferfunctions,solar
coefficientsinthesquarematrixandtheconstantvectordepend
absorptances, total hemispherical emittances, the length and
upon the unknown temperatures, the system of equations is
widthoftheattic,roofpitches,heightofeavewalls,orientation
solved iteratively. When the temperatures have been
ofthehouse,ventinletandoutletareasandtypeofvents,water
determined, the heat flows are calculated using the conduction
vapor permeances, wood surface areas and participating
transfer function equations.
masses, and wood moisture contents. The house exfiltration
6.8.4 To avoid a numerical instability associated with the
rate and the latent heat of vaporization also are read as inputs.
latent heat term in the heat balance on interior surfaces, the
By setting the latent heat to zero, moisture effects may be
surface humidity ratio is expanded in aTaylor series such that:
ignored. The input read in also includes a series of flags that
m˙ i 5 h i A' i ω 2ω i (48)
~ ! ~ ! ~ !~ ~ !! specifywhethersurfacetemperaturesaretobeforcedtoknown
w w a s
values or are to be calculated from weather conditions.
dω ~i!
s
7.1.3.2 TheprogramthencallssubroutineVIEW2.Theattic
5h i A' i ω 2 ω ' i 2 TIS i 2 TIS' i
~ ! ~ ! FS ~ ! ~ ~ ! ~ !!G
w a s
dT
length,width,roofpitches,heightofeavewallsandemittances
of the surfaces facing the attic space are passed into VIEW2.
where:
ThesubroutineutilizesfunctionFMNtocalculateviewfactors
ω' (i) = is the surface humidity ratio evaluated at a previ-
s
between two rectangular surfaces that share an edge and
ously estimated surface temperature TIS' (i).
function FP to calculate view factors between two equal
The coefficients of TIS(i) are included in the (AA) matrix
parallel rectangles. The subroutine then uses view factor
while the other terms are included in the (BB) vector.With this
algebra to calculate the view factors among the surfaces that
scheme, the solutions have been found to converge.
face the attic space and the overall view factor matrix G ,
ij
6.8.5 The system of equations is set up to allow the model which is passed back out of the subroutine.
to be driven by weather conditions with all the surface and air
7.1.3.3 Next, the program calculates characteristic lengths
temperatures being unknown. For comparison of the model
and areas of surfaces. For the ceiling, the characteristic length
with experiments, often it is helpful to use measured values of is taken to be the average of the length and width of the attic.
exterior roof and ceiling temperatures as boundary conditions.
For a roof surface, it is taken to be the distance from the eave
Themodelissetuptoallowanyofthesurfacetemperaturesto totheridge.Forgables,itistakentobetheaverageheight,and
be forced to its measured value. This is accomplished using a for the eave walls, it is taken to be the height. These
characteristic lengths are the ones normally chosen for use in
methodsuggestedbyD.Ober,inaprivateconversationwithK.
correlations for convection heat transfer coefficients.
E. Wilkes in 1988, wherein the diagonal element of (AA) that
corresponds to the forced temperature is multiplied by a very
7.1.3.4 In the next step, the program reads a line of hourly
large number and the corresponding element of (BB) is set to weather data. This reading includes the outdoor temperature,
the product of the new element of (AA) and the known incident solar radiation, wind speed, wind direction, and
outdoor humidity ratio.At this step, the program also is set up
temperature.
to read in measured values for surface temperatures and
ventilation rate. If the appropriate flags are set, the program
7. Computer Program
will use the indicated measured surface temperatures or ven-
7.1 General:
tilation rate in the calculations. If this option is elected, a
7.1.1 A computer program that embodies the model de-
separate input file is required.
scribed in this practice is available from ASTM as an ajunct.
7.1.3.5 If the flag is set so that the measured ventilation rate
This program is only an example. Users of the practice may
is not to be used, then the program makes a call to subroutine
develop their own program if they wish. The computer
VENT. Quantities passed into the subroutine are outdoor air
program in the adjunct is written in Fortran 77.
temperature,averageairtemperaturewithintheairspace,wind
speed and direction, difference in elevation between inlet and
NOTE 1—Identical versions of this computer program have been
outlet vents, inlet and outlet vent areas, and a flag to indicate
compiled successfully and run on mainframe and personal computers.
the type of vent. The subroutine uses the algorithms described
Only minor modifications necessary for conformance to the resident
operating system were required for operation. above for the stack and wind pressure effects to calculate
C1340/C1340M − 10 (2021)
FIG. 2 Flow Chart for Attic Model
ventilation volume and mass flow rates, and the product of the set to use the measured ventilation volume flow rate, VENT is
mass flow rate and the specific heat of air. These last three bypassed and the measured volume flow rate is used to
quantities are passed back out of the subroutine. If the flag is calculate the mass flow rate and the product of the mass flow
C1340/C1340M − 10 (2021)
rate and the specific heat. Next, a crude estimate for the flow
velocity is calculated by dividing the volume flow rate by an
average cross sectional area for the attic.
7.1.3.6 The program then makes several calls to subroutine
HCONtocalculateconvectionheattransfercoefficientsforthe
interiorandexteriorsurfaces.QuantitiespassedintoHCONare
the previous estimates of the surface temperature and the
adjacent average air temperature, surface tilt angle, character-
istic length of surface, a flag to indicate whether the surface
faces up or down, and the air speed past the surface. For
interiorsurfaces,thecrudeestimateofventilationflowvelocity
is used. For exterior surfaces, the wind speed is used (except
FIG. 3 Geometry of Attic for Example Problem
for the exterior of the ceiling, where the air speed is set to
zero). HCON utilizes the correlations described above to
calculate natural, forced, and mixed convection heat transfer 7.1.3.11 When all of the new estimates for surface and air
coefficients. The mixed coefficient is passed back out of temperatures agree with old estimates to within 0.0005°C
[0.001°F], or when the limit of 15 iterations has been reached,
HCON.
heat fluxes are calculated using the conduction transfer func-
7.1.3.7 Radiation heat transfer coefficients are calculated
tion equations, moisture contents are updated, and results are
using the HRAD function. The function uses the previous
written out for the current hour. The WRITE and FORMAT
estimates of the two bounding temperatures and either the
statements may need to be changed to suit the user’s needs.
appropriateelementoftheG matrix(foraninteriorsurface)or
ij
7.1.3.12 The time indices on the temperatures and heat
the emittance (for an exterior surface).
fluxes are shifted, the program goes back to read another line
7.1.3.8 Ifthelatentheatisnotsetclosetozero,theprogram
of weather data and the calculations proceed as before for the
calls subroutine MOIST, which calculates moisture sorption/
next hour.
desorption rates for the surfaces facing each of the air space.
ThequantitiespassedintoMOISTarethepreviousestimatesof
8. Example Calculation
the surface temperatures, the outdoor and indoor air
8.1 The gabled attic shown in Fig. 3 was modeled using the
temperatures,thepreviousestimateoftheaverageairtempera-
computer program. The construction and properties of the
ture within the space, the convection heat transfer coefficients,
various surfaces of the attic are given in Tables 2 and 3. The
surface areas, wood surface areas and masses, wood moisture
ridge of the attic is oriented in the east-west direction, the attic
contents, indoor and outdoor relative humidities, water vapor
2 2
has soffit vents with an area of 0.64m [6.84 ft ] and a ridge
permeances, attic ventilation mass flow rate, and house exfil-
2 2
vent with an area of 0.32m [3.42 ft ]. The interior of the
tration mass flow rate. The subroutine calculates wood surface
house is maintained at 23°C [74°F] and 50% relative humid-
humidityratiosfromthesurfacemoisturecontentandtempera-
ity. No air from the house exfiltrates into the attic space. The
ture using function WDHUM. Subroutine PSY is used to
house is located in Phoenix, AZ, and the reflectance of the
calculate humidity ratios and water vapor partial pressures for
ground is 0.2.
the indoor and outdoor air using the given temperatures and
relative humidities. Mass transfer coefficients are calculated 8.2 An input file for the computer program is given in Fig.
using subroutine HMASS, which uses the analogy between
4, and a key to this file is given in Table 4. A file containing
heat and mass transfer. weather data for input to the program is given in Fig. 5, and a
7.1.3.9 The humidity ratio of the air in the air space then is
calculated from a moisture balance on the space. This calcu-
TABLE 2 Construction and Properties of Attic Surfaces for
lation is done iteratively since the moisture balance involves Example Problem
both the humidity ratio and the partial pressure. With the
Thermal Specific
Thickness,
Material Conductivity, Heat, Btu/ Density, lb/ft
humidity ratio of the air established, the moisture flow rates
in.
Btu/(h·ft·°F) lb·°F
from Eq 48 are calculated and pass back out of the subroutine.
Ceiling:
Ifthelatentheatissetclosetozero,allofthesecalculationsare
Gypsum board 0.5 0.0926 0.26 50
bypassed.
R-19 insulation 6.25 0.02741 0.19 0.6
A
Joists 0.2917 0.06833 0.39 28
7.1.3.10 At this point, all of the quantities for the matrices
Roof (both sides):
(AA)and(BB)areavailable.Thematricesaresetupinthemain
Shingles 0.25 0.04734 0.30 70
Felt 0.085 0.04734 0.36 70
program and are solved using subroutine SOLVP.The result of
Plywood 0.5 0.0667 0.29 34
this solution is a new set of estimates for the surface and air
A
Rafters 3.5 0.06833 0.39 28
space temperatures. The new estimates are compared with the
Gables (both ends):
Hardboard siding 0.4375 0.1242 0.28 40
old estimates. If any differences are greater than 0.0005°C
Studs 1.5 0.06833 0.39 28
[0.001°F], then the progra
...