1、2010 ASHRAE 169ABSTRACTThe paper represents the second part of a comprehensivestudy on modeling heat transfer in metal building roof insula-tion assemblies containing fiberglass insulation. Part 1described the development and experimental validation of acomprehensive three-dimensional mathematical m
2、odel forheat transfer (and air flow where applicable) in metal buildingroof insulation assemblies. The model was used to calculateU-factors for several standing seam roof (SSR) and through-fastened roof (TFR) assemblies assuming the insulationdrapes between the purlins to be parabolic. The second pa
3、rt ofthe work used the results from the model described in part 1 todevelop a relatively simple approach to predict the overall heattransfer coefficient (U-factor) of insulated metal building roofinsulation assemblies. It is shown that the U-factors obtainedusing the numerical model may be correlate
4、d, to a fairly highdegree ( R2 = 0.998), to an overall or system thermal resistanceparameter, Rinsul-sysas shown below.1/U = 0.8672 Rinsul-sys + 1.132In the above equation, U is in Btu/ft2hF and R is inft2hF/Btu. 1 Btu/ft2hF = 5.6782 W/m2K and 1ft2hF/Btu = 0.17611 m2K/W.The parameter Rinsul-sysis ca
5、lculated by combining threethermal resistances operating in parallel. Two of these resis-tances of equal value correspond to the insulation drapes oneither side of the purlin and the one in the middle representsthe thermal resistance of insulation over the purlin. An analyt-ical expression for the R
6、-value of the parabolic insulationdrape is derived. The approach developed in this paper greatlyenhances our ability to predict U-factors for metal buildingroof assemblies.INTRODUCTIONDuring the late 1990s, the American Society of Heating,Refrigeration and Air-Conditioning Engineers (ASHRAE)Standing
7、 Standard Project Committee (SSPC) 90.1 EnvelopeSubcommittee incorporated specific maximum allowable U-factors (the overall heat transfer coefficients) for metal build-ing roofs and walls into ANSI/ASHRAE/IESNA Standard90.1-1999. Recently ASHRAE and other industry organiza-tions have recognized that
8、 Standard 90.1 should be revised andupdated to account for a more accurate understanding of theinstallation of insulation in metal building roof and wallassemblies. A comprehensive mathematical modeling basedstudy of heat transfer in metal building roof insulation assem-blies containing fiberglass i
9、nsulation was undertaken to facil-itate the revision of U-factors in the Standard 90.1. The firstpart of the study, summarized in a previous paper (Choudharyet al. 2010), involved the development and validation of athree-dimensional mathematical model for heat transfer (andair flow where applicable)
10、 in metal building roof insulationassemblies.In the second part of the study, reported here, the validatedmodel was used to calculate the U-factors for several standingseam roof (SSR) and through-fastened roof (TFR) insulationassemblies. In these modeling studies, the shape of the fiber-glass insula
11、tion drape between two consecutive purlins wasassumed to be parabolic. The assumption of the parabolicshape was based on measurements of the drape in prototypesof several metal building roof insulation assemblies (Chris-tianson 2010). For the SSR, 30 assemblies were modeled; 15each for a 1.375 in. (
12、0.0349 m) tall clip and a 1.75 in. (0.0445m) tall clip. The 15 cases corresponded to five different insu-lation types (R-10, R-11, R-13, R-16, and R-19), each at threeASHRAE Standard 90.1 Metal Building U-FactorsPart 2: A Systems Based Approach for Predicting the Thermal Performance of Single Layer
13、Fiberglass Batt Insulation AssembliesM.K. Choudhary, PhD, PE C.P. KasprzakAssociate Member ASHRAEM.K. Choudhary is a member of senior technical staff at Owens Corning Science four for R-19 and 1 each for R-16, R-13, R-11,and R-10. The U-factors for the 38 cases are summarized inthis paper.One may us
14、e the three-dimensional numerical heat transfermodel described (Choudhary et al. 2010) to calculate U-factorsfor each of the cases included in the Standard 90.1. This,however, will be time consuming. Also, modeling tools andcapabilities may not be readily available to many people andorganizations wi
15、th interest in understanding and specifyingthermal performance of the metal building insulation assem-blies. Therefore, and at the request of ASHRAEs 90.1 Enve-lope Subcommittee, an approach was developed to calculate theU-factor from an overall or system thermal resistance parame-ter, Rinsul-sys, t
16、hat combined thermal resistance of the insulationdrapes between the purlins with the thermal resistance of theinsulation above the purlin. The paper presents a derivation foran analytical expression for the thermal resistance (the R-value)of the parabolic insulation drape, a key part of Rinsul-sys.T
17、he model calculated 38 U-factors for the various roofinsulation assemblies mentioned earlier were found to corre-late highly (R2 = 0.998) with the overall or system thermalresistance parameter, Rinsul-sys. Subsequently, four moreassemblies were modeled to correspond to Rinsul-sysvaluesconsiderably o
18、utside the range of the earlier 38 cases. Thecorrelation equation was found to work very well in predictingthe U-factor with the worst discrepancy between the valuesfrom the model and the correlation equation being 15%.The U-factor estimation approach outlined here is quitesimple and greatly expands
19、 our ability to predict U-factors formetal building roof assemblies. The correlation derived here isfor cases where several design parameters (e.g., spacingbetween the purlins and the dimensions of the purlins) were keptinvariant. It would be relatively straightforward to extend thepresent approach
20、to include other design parameters of interest.In the following, we will first derive an expression for theoverall or system thermal resistance parameter, Rinsul-sys, thensummarize the modeling results on U-factors for several roofinsulation assemblies, and finally correlate the U-values calcu-lated
21、 by the numerical model to Rinsul-sys. The correlation basedapproach described below has been adopted by the 90.1 Enve-lope Subcommittee for calculating the U-factors for the singlelayer fiberglass insulation assemblies. We have followed therecommendation from the 90.1 Envelope Subcommittee(McBride
22、and Waite 2009) to include the important steps in thederivation of expressions for various thermal resistances pres-ent in the metal building insulation assemblies.DERIVATION OF EQUATIONS FOR THE THERMAL RESISTANCE PARAMETERSThe first step in the calculation of the overall or systemthermal resistanc
23、e, Rinsul-sysis to calculate the thermal resis-tance or the R-value of the insulation drape between thepurlins. The drape is assumed to be parabolic.R-Value of a Parabolic Drape of Fiberglass InsulationLet us consider the part of a parabolic fiberglass insula-tion drape starting at the end of the pu
24、rlin flange. The nomen-clature is explained in Figure 1.The equation for the parabola is given below.(1)Let us introduce some additional variables. ; (2a);(2b); (2c) = /ym(2d)We may now writeY = X(2 X)(3b)y = yo+ Y (3c)The thermal conductivity of fiberglass insulation is givenby the following equati
25、on (Wilkes 1979).K = A + B + (C/)(4)where, A, B, C are empirically determined coefficients and is the fiberglass density.At any location x, the insulation thickness, y and thedensity, of the insulation are related, by mass conservation,as shown below.y = ryr(5)where, ris the reference density of the
26、 insulation of thick-ness yr.Equation (4) may now be expressed in terms of the localthickness.Figure 1 Schematic of the insulation drape between thepurlin flange and the mid point of purlin spacing.yyoymyo-xl- 2xl-=Xxl-=Yyyoymyo-= ymyo= 2010, American Society of Heating, Refrigerating and Air-Condit
27、ioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions 2010, Vol. 116, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission. ASHRAE Transactions 171(6)Let us
28、consider an insulation strip of length dx (see Figure1). Assuming heat flow to be in the insulation thickness direc-tion (i.e., in the y-direction) only, heat flowing through thisstrip, dQ is given by;(7)where,T = Temperature difference between the top and the bottom of the stripW = Width of the roo
29、f panelThe total heat flowing through the insulation drape, Qmay be obtained by integrating Equation (7).(8)In writing Equation (8) we have assumed that the temper-ature difference across the insulation thickness, T is constant(i.e., it does not depend on x).Let Rinsul-1be the R-value of the parabol
30、ic insulationdrape.(9)From Equations (8) and (9) we may derive the followingexpression for Rinsul-1.(10)Using Equations (3b) and (6), the integrand K/y may beexpressed as follows.(11)Thus, Equation (10) may be rewritten as shown below.(12)We will now integrate each of the three terms in Equation(12)
31、. Let us first express the denominator, yo+Y in terms of Xby using Equation (3a).yo+ Y = y0+ 2X X2 (13)Now, let us consider the integral . We note that denominator is of the form (a + bX + cX2),where a, b, and c are defined as follows. a = y0; b = 2; c = (14)Let us note that 4ac b2 0. Under these co
32、nditions(Tuma 1979, p. 259),(15)Using the definitions of a, b, and c from Equation (14) weget;b2 4ac = 4(y0+ ) = 4ym(16a)b + 2c = 0 (16b)On substituting Equations (16a) and (16b) into Equation(15), and using Equation (2d) we get;(17)Let us now put the integration limits of 0 and 1 for X toderive the
33、 following expression.(18)We will next consider the second integral in Equation (12),namely or .We will use the following relationship in evaluating thisintegral (Tuma 1979, p. 259).(19)It should be noted that the integral on the right hand sideof Equation (19) has already been evaluated and express
34、ed asEquation (17). On taking the integration limits of 0 and 1, thefirst term on the right hand side is simplified as shown below.KABryry-Cyryr-+=dQ KTy- Wdx=QTWKy- xd0l=QTWlRinsul-1-=1Rinsul-1-1l-Ky- xd0lKy- Xd01=Ky-AyoY+-BryryoY+()2-Cryr-+=1Rinsul-1-AyoY+-BryryoY+()2-Cryr-+Xd01=1yo2X X2+-Xd011abX
35、cX2+- Xd1b24ac-=b24ac b 2cX+()b24ac b 2cX+()+-ln1abXcX2+- Xd12 ym-ym 1 X()ym 1 X()+-ln=12ym-1 1 X()1 1 X+()+-ln=1abXcX2+- Xd0112ym-1 +1 -ln=1y02X X2+2-Xd011abXcX2+()2- Xd011abXcX2+()2- Xdb 2cX+b24ac()abXcX2+()-=2cb24ac()-1abXcX2+- Xd 2010, American Society of Heating, Refrigerating and Air-Condition
36、ing Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions 2010, Vol. 116, Part 1. For personal use only. Additional reproduction, distribution, or transmission in either print or digital form is not permitted without ASHRAEs prior written permission. 172 ASHRAE Transactions(20)The secon
37、d term in Equation (19) reduces to the following.(21)We are now in a position to express the R-value of theparabolic insulation drape.(22)Equation (22) may also be written as follows.(23)Let us check to see if Equation (23) reduces to the simpleform we expect when tends to zero (i.e., the insulation
38、 thick-ness is constant). It is seen that the last term on the right handside of Equation (23) becomes indeterminate when = 0. Letus evaluate the term as tends to zero.For ease in writing, we introduce the variable =. Ourtask now is to calculate the limiting value of the followingexpression.From Tum
39、a (1979, p. 108),Therefore,Thus for the special case of insulation of constant thick-ness, t (yo= ym= t) we get the simple expression given below.While our principal aim in this section was to derive anexpression for the Rinsul-1, it is also of interest to derive anexpression for k-value of the para
40、bolic drape.(24)Using the symbols defined earlier,(25)Substituting Equation (25) into Equation (24) and inte-grating gives the following expression.(26)R-Value of Insulation Over PurlinThis is relatively straight forward. We will assume that thespace above the purlin consists of constant thickness i
41、nsula-tion layers. Let us take the general case where the insulationabove the purlin consists of a foam layer of thermal conduc-tivity Kfoamand thickness tfoamon top of a fiberglass layer ofconductivity Kfgand thickness tfg.(27)Incorporation of Inside and Outside Air Film ResistancesLet us call the
42、inside and outside air film resistances to beRfilm,iand Rfilm,o. We may incorporate them into insulationresistances as follows.Rinsul-1+air = Rinsul,1+ Rfilm,i+ Rfilm,o(28a)Rinsul-2+air = Rinsul,2+ Rfilm,i+ Rfilm,o(28b)0b 2cX+b24ac()abXcX2+()-01b 2c+b24ac()abc+()-=+bb24ac()a-12yoym- =2cb24ac()-1abXc
43、X2+- Xd24ym-12ym-1 +1 -ln=14ym2-1 +1 -ln=1Rinsul-1- A12ym-1 +1 -ln=Bryr12y0ym-14ym2-1 +1 -ln+Cryr-+1Rinsul-1-Cryr-Bryr2y0ym-+=ABryr2ym-+12ym-1 +1 -ln+12()()1 +()1 ()ln12-1 +1 -ln 0lim1 +1 -ln 22i2i 1+-i=0=12-1 +1 - 1=ln 0lim1Rinsul-1-At-Bryrt2-Cryr-+Kt-=Kinsul-11l- Kxd0lKXd01=KABryryo2X X2+-Cyo2X X2+ryr-+=Kinsul-1ABryrym-12 -1 +1 -ln+=Cryr-2ymyo+3-+Rinsul-2tfoamKfoam-tfgKfg-+= 2010, American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.org). Published in ASHRAE Transactions 2010, Vol. 116, Part 1. For personal use only.
