Designation: C1340/C1340M − 10 (Reapproved 2015)Standard Practice forEstimation of Heat Gain or Loss Through Ceilings UnderAttics Containing Radiant Barriers by Use of a ComputerProgram1This standard is issued under the fixed designation C1340/C1340M; the number immediately following the designation indicates theyear of original adoption or, in the case of revision, the year of last revision. A number in parentheses indicates the year of lastreapproval. A superscript epsilon (´) indicates an editorial change since the last revision or reapproval.1. Scope1.1 This practice covers the estimation of heat gain or lossthrough ceilings under attics containing radiant barriers by useof a computer program. The computer program included as anadjunct to this practice provides a calculational procedure forestimating the heat loss or gain through the ceiling under anattic containing a truss or rafter mounted radiant barrier. Theprogram also is applicable to the estimation of heat loss or gainthrough ceilings under an attic without a radiant barrier. Thisprocedure utilizes hour-by-hour weather data to estimate thehour-by-hour ceiling heat flows. The interior of the housebelow the ceiling is assumed to be maintained at a constanttemperature. At present, the procedure is applicable to sloped-roof attics with rectangular floor plans having an unshadedgabled roof and a horizontal ceiling. It is not applicable tostructures with flat roofs, vaulted ceilings, or cathedral ceilings.The calculational accuracy also is limited by the quality ofphysical property data for the construction materials, princi-pally the insulation and the radiant barrier, and by the qualityof the weather data.1.2 Under some circumstances, interactions between radiantbarriers and HVAC ducts in attics can have a significant effecton the thermal performance of a building. Ducts are included inan extension of the computer model given in the appendix.1.3 The values stated in either SI units or inch-pound unitsare to be regarded separately as standard. The values stated ineach system may not be exact equivalents; therefore, eachsystem shall be used independently of the other. Combiningvalues from the two systems may result in non-conformancewith the standard.2. Referenced Documents2.1 ASTM Standards:2C168 Terminology Relating to Thermal Insulation2.2 ANSI Standards:X3.5 Flow Chart Symbols and Their Usage in InformationProcessing3X3.9 Standard for Fortran Programming Language32.3 ASTM Adjuncts:Computer Program for Estimation of Heat Gain or Lossthrough Ceilings Under Attics Containing Radiant Barri-ers43. Terminology3.1 Definitions—For definitions of terms used in thispractice, refer to Terminology C168.3.2 Symbols—Symbols will be introduced and defined in thedetailed description of the development.4. Summary of Practice4.1 The procedures used in this practice are based on thethermal response factor method for calculating dynamic heatconduction through multilayer slabs (1, 2),5along with a modelfor convective and radiative heat exchanges inside and outsidethe attic.4.2 The operation of the computer program involves thefollowing steps:1This practice is under the jurisdiction of ASTM Committee C16 on ThermalInsulation and is the direct responsibility of Subcommittee C16.21 on ReflectiveInsulation.Current edition approved Sept. 1, 2015. Published September 2015. Originallyapproved in 1999. Last previous edition approved in 2010 as C1340 –10. DOI:10.1520/C1340_C1340M-10R152For referenced ASTM standards, visit the ASTM website, www.astm.org, orcontact ASTM Customer Service at

[email protected] For Annual Book of ASTMStandards volume information, refer to the standard’s Document Summary page onthe ASTM website.3Available from American National Standards Institute (ANSI), 25 W. 43rd St.,4th Floor, New York, NY 10036, http://www.ansi.org.4Available from ASTM International Headquarters. Order Adjunct No.ADJC1340.5The boldface numbers in parentheses refer to the list of references at the end ofthis standard.Copyright © ASTM International, 100 Barr Harbor Drive, PO Box C700, West Conshohocken, PA 19428-2959. United States14.2.1 Response Factors—A separate computer programmust be used to calculate the thermal response factors of thesolid materials surrounding the attic. Input to this programwould consist of the thermal conductivity, specific heat,density, and thickness of each layer, or the thermal resistance ofthe layer if it has negligible density, and the fraction of thecross-sectional area occupied by the framing. Output of such aprogram would be a set of response factors for use as input tothe main program. The adjunct to this practice contains datafiles with response factors for several typical attic construc-tions.4.2.2 Data Input to the Main Program—This input includesthe response factors, total hemispherical emittances of theinside and outside surfaces of the attic envelope, solar absorp-tances of the outside surfaces of the attic envelope, length andwidth of the attic, slopes of the two roof sections, distancebetween attic floor and roof at edge of attic, orientation ofhouse, vent areas and type of vents, water vapor permeances ofattic surfaces, area of exposed wood inside attic, mass of woodin attic, initial moisture content of wood in attic, rate ofexfiltration of air from house into attic space, latitude andlongitude, time zone indicator, solar reflectance of the ground,indoor temperature, and indoor humidity.4.2.3 Analysis—Using hourly weather data consisting ofoutdoor temperature and humidity ratio, atmospheric pressure,total horizontal and direct solar radiation, wind speed anddirection, cloud amount, cloud type, and atmospheric clearnessnumber, the computer program calculates the inside andoutside temperatures of the attic envelope and the temperatureof the air inside the attic. Using these temperatures, theprogram calculates the heat flux through the ceiling.4.2.4 Output—The hourly heat flux through the ceiling iswritten to a file which can be used for further processing, suchas seasonal or annual heat gains or losses.5. Significance and Use5.1 Manufacturers of radiant barriers express the perfor-mance of their products in terms of the total hemisphericalemittance. The purpose of a radiant barrier is to decrease theradiation heat transfer across the attic air space, and hence, todecrease the heat loss or gain through the ceiling below theattic. The amount of decrease in heat flow will depend upon anumber of factors, such as weather conditions, amount of massor reflective insulation in the attic, solar absorptance of theroof, geometry of the attic and roof, and amount and type ofattic ventilation. Because of the infinite combinations of thesefactors, it is not practical to publish data for each possible case.5.2 The calculation of heat loss or gain of a systemcontaining radiant barriers is mathematically complex, andbecause of the iterative nature of the method, it is best handledby computers.5.3 Computers are now widely available to most producersand consumers of radiant barriers to permit the use of thispractice.5.4 The user of this practice may wish to modify the datainput to represent accurately the structure. The computerprogram also may be modified to meet individual needs. Also,additional calculations may be desired, for example, to sum thehourly heat flows in some fashion to obtain estimates ofseasonal or annual energy usages. This might be done using thehourly data as inputs to a whole-house model, and by choosinghouse balance points to use as cutoff points in the summations.6. Method of Calculation6.1 Approach:6.1.1 This calculation of heat loss or gain requires that thefollowing be known:6.1.1.1 The thermal conductivity, specific heat, and densityof the construction materials (that is, insulation, plywood,roofing materials, sheathing, gypsum board);6.1.1.2 The total hemispherical emittance of all materialsfacing the attic air space;6.1.1.3 The solar absorptance of the exterior surfaces of theattic (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 ofthe materials facing the attic space; and6.1.1.6 The weather conditions.6.1.2 The solution is a computer procedure that estimatestemperatures of both sides of the components of the atticenvelope and the temperature of the air in the attic space, usesthese estimates of temperatures to refine estimates of convec-tion and radiation heat transfer coefficients, reestimates thetemperatures using the new heat transfer coefficients, continuesiterating on the temperatures and heat transfer coefficients untilconvergence is reached, and uses the last estimates of tempera-tures to calculate the heat gain or loss through the ceiling. Thisprocedure is repeated for each hour of the simulation period(typically a full year).6.2 Development of Equations—The model that is the basisfor this practice is based on the model developed by B. Peavy(3), which was later extended by Wilkes (4-6). The sketch of anattic given in Fig. 1 shows the various heat transfer mecha-nisms that occur within an attic. Although the sketch showsFIG. 1 Schematic of Residential Attic Showing Heat Transfer Phe-nomenaC1340/C1340M − 10 (2015)2ventilation occurring at soffit and ridge vents, the location ofthe vents may be at other locations, such as at the gables. Themodel treats all of these phenomena through a system of heatbalance equations at the interior and exterior surfaces of theceiling, roof sections, and gables, as well as a heat balance onthe air mass within the attic. To handle the case of raisedtrusses, short vertical walls at the eaves also are included. Eachof the surfaces is assumed to be isothermal; thus, for an atticconsisting of a ceiling, two roof sections, two gables, twovertical eave sections, and one air space, a total of 15 heatbalance equations is used.6.3 Equations—Conduction:6.3.1 The model developed here utilizes the thermal re-sponse factor method to analyze conduction through buildingenvelope sections. The thermal response factor method wasdeveloped by Mitalas and Arseneault (1) and was extended byKusuda (2). The method is based on an exact analyticalsolution of the heat conduction equation for one-dimensionalheat flow through a multilayer slab having temperature-independent thermal properties. The only approximation is thatthe surface temperatures are taken to vary linearly with timebetween time steps. For analysis of buildings, the time step isnormally taken to be 1 h. The response factor equations relatethe heat fluxes at the surfaces of the slab to the present andprevious temperatures at the two surfaces. The equations are:QI 5(j50`Z ~j!~TIS~j! 2 TR! 2(j50`Y ~j!~TOS~j! 2 TR! (1)QO 5(j50`Y ~j!~TIS~j! 2 TR! 2(j50`X ~j!~TOS~j! 2 TR! (2)where:QI = heat flux at inside surface at presenttime (note that the positive heat flowdirection is from the inside to theoutside), W/m2[Btu/h·ft2],QO = heat flux at outside surface at presenttime, W/m2[Btu⁄h·ft2],TIS(j) = temperature at inside surface j hoursprevious to present time, °C [°F],TOS(j) = temperature at outside surface j hoursprevious to present time, °C [°F]X (j), Y (j), Z (j) = response factors, W/m2· K [Btu/h·ft2·°F], andTR = reference temperature, °C [°F].6.3.2 The response factors are determined from a sequenceof calculations that involve the thermal diffusivity, thermalconductivity, specific heat, density, and thickness of each of thelayers in the multilayer slab.An efficient computer program forcalculating the response factors has been developed by GeorgeWalton of the National Institute of Standards and Technology(NIST) (7).6.3.2.1 The efficiency of the response factor calculationscan be increased by making use of the fact that after a sufficientnumber of terms, the ratio of two consecutive response factorsbecomes constant. This is expressed by:X ~j11!X ~j!5Y ~j11!Y ~j!5Z ~j11!Z ~j!5 CR for j$N (3)CR = the common ratio, andN = a sufficiently large number.6.3.2.2 The common ratio is used to define a new set offunctions, called the first order conduction transfer functions orsimply 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)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 heatfluxes and surface temperatures are related by:QI 5(j50NZ~j!~TIS~j! 2 TR! (13)2(j50NY~j!~TOS~j! 2 TR!1CR QI QO 5(j50NY~j!~TIS~j! 2 TR! (14)2(j50NX~j!~TOS~j! 2 TR!1CR QO where:QI = heat flux at inside surface at previous time step,QO = heat flux at outside surface at previous time step, andN = number of significant conduction transfer functions.6.3.2.4 When parallel heat flow paths occur in an envelopecomponent, separate response factors for each path may beneeded. If the boundary temperatures of the two paths may beassumed to be equal, however, then the response factors maybe added together as:X 5 A1X 11A2X 2(15)where:A1,A2= area fractions forpaths 1 and 2, and(X 1,X 2), (Y 1,Y 2) and (Z 1,Z 2) = the response factorsfor paths 1 and 2.Parallel conduction transfer functions may be calculatedfrom these parallel response factors, provided that the commonratio for the path with the largest number of significant terms isused.6.3.2.5 The original derivation of the response factor tech-nique relied upon the assumption of temperature-independentC1340/C1340M − 10 (2015)3thermal properties. An approximate method has been devel-oped to account for the temperature dependence of the thermalproperties (5). The thermal transmission coefficient of thecomponent is taken to vary linearly with temperature as:U 5

[email protected]~T 2 TR!# (16)The conduction transfer function equations then become:QI 5(j50NZ~j!~TIS~j! 2 TR! (17)2(j50NY~j!~TOS~j! 2 TR!1CR QI 1b/2(j50NZ~j!~TIS~j! 2 TR!22 b/2(j50NY~j!~TOS~j! 2 TR!2QO 5(j50NY~j!~TIS~j! 2 TR! (18)2(j50NX~j!~TOS~j! 2 TR!1CR QO 1b/2(j50NY~j!~TIS~j! 2 TR!22 b/2(j50NX~j!~TOS~j! 2 TR!2These equations are used in the system of heat balanceequations.6.4 Equations—Convection:6.4.1 Convection heat transfer from the interior and exteriorsurfaces of the envelope components is calculated usingcorrelations from the literature (8). The coefficients are basedon correlations that have been developed for isolated isother-mal flat plates. The correlations are in the form of a Nusseltnumber, Nu, as a function of a Rayleigh number, Ra, Grashofnumber, Gr, or a Reynolds number, Re, where:Nu 5 hL/k (19)Ra 5gβpCρ∆TL3vk(20)Gr 5 Ra/Pr (21)Pr 5 v/α (22)Re 5 VL/v (23)and:h = convection heat transfer coefficient, W/ m2· K [Btu/h·ft2·°F],L = characteristic length of plate, m [ft],k = thermal conductivity of air, W/m·K[Btu/h·ft·°F],g = acceleration of gravity, m/s2[ft/h2],β = volume coefficient of expansion of air, K−1[°R−1],ρ = density of air, kg/m3[lb/ft3],Cp= specific hea