1、2011 ASHRAE 859ABSTRACTHeat loss through tapered roof insulation is generallycomputed using an R-value based on the average thickness ofthe tapered section. However, this misrepresents the actualperformance of the tapered insulation, which is always lessefficient than an equal volume of untapered in
2、sulation. Forcommonly encountered slopes formed with one-way and four-way tapers, the true efficiency of tapered insulation (comparedwith an equal volume of untapered insulation) depends only onthe ratio of high- and low-point R-values, and ranges from100% to about 70% for R-value ratios between 1 a
3、nd 10. Theimpact of curved heat flow trajectories on the efficiency of thetapered forms is a function of taper angle or slope, andbecomes significant only at slopes much steeper than thosefound in typical tapered roof insulation applications. Equa-tions are derived, and tables are presented, for the
4、 efficiencyof tapered insulation considering volumetric forms typicallyencountered in practice. Examples illustrate how these toolscan be used to accurately calculate heat loss through a roofassembly with tapered insulation. INTRODUCTIONIt is commonly assumed that the average R-value oftapered insul
5、ation is equivalent to the R-value of its averagethickness (Graham 1995; PIMA). However, because heat lossthrough insulation is inversely proportional to insulationthickness, a unit increase or decrease of insulation thicknessdoes not result in a constant increase or decrease in heat loss:a unit cha
6、nge in thickness (assuming uniform R-value) from 4to 3 results in an increase in heat loss of 1/3 divided by 1/4, or133%; whereas a unit change in thickness from 3 to 2 resultsin an increase of 1/2 divided by 1/3, or 150%. As the materialgets thinner, “energy consumption,” or heat flux, gets biggera
7、t an increasing rate (Johnson 2009).For this reason, heat loss through tapered insulation is notequivalent to heat loss through the same quantity (volume, oraverage thickness) of constant-thickness insulation. Insulationhaving a simple taper will be less efficient than insulation withthe same volume
8、 configured with constant thickness, since theportion of tapered insulation that is thinner than average willlose more heat than the thicker part will save.Several tapered insulation forms are commonly used,including one-way slopes, two-way slopes, and four-way orpyramidal (along with inverse pyrami
9、dal) shapes. So-calledcrickets are often placed above a one-way or two-way slope todirect water to drains. Cricket geometry can be resolved intoone or more triangular solids having different thickness at eachvertex. While crickets may be manufactured as separate piecesof insulation placed on top of
10、tapered insulation with one- ortwo-way slopes, they will be analyzed as if they extendedvertically to an assumed horizontal plane (roof deck). Thesetapered roof geometries are illustrated in Figure 1.Certain geometries have the same underlying efficiencyand need not be separately analyzed: two-way s
11、lopes may beanalyzed as two one-way slopes; one-way slopes that convergeupward to a point have the same efficiency as pyramids (four-way slopes with external drainage); and one-way slopes thatconverge downward to a point have the same efficiency asinverted pyramids (four-way slopes with internal dra
12、inage).For the same reason that the efficiency of tapered panelsis not the same as that of flat panels with equal volume, onecannot simply add the equivalent average R-value, computedfor an isolated piece of tapered insulation, to R-valuescomputed for other elements of the roof assembly (e.g., roofD
13、etermining the Average R-Value of Tapered InsulationJonathan OchshornMember ASHRAEJonathan Ochshorn is an associate professor in the Department of Architecture, Cornell University, Ithaca, NY.LV-11-0192011. American Society of Heating, Refrigerating and Air-Conditioning Engineers, Inc. (www.ashrae.o
14、rg). Published in ASHRAE Transactions, Volume 117, 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.860 ASHRAE Transactionsdeck, interior finishes, other insulation, air fil
15、ms). Addingmaterial with constant R-value to material with variable R-value (e.g., to a piece of tapered insulation) changes the ratioof overall thickness (or R-value) upon which the efficiency ofthe tapered panel is based. Therefore, it is necessary toconsider the entire roof assembly when analyzin
16、g any piece oftapered insulation. Where the terms thickness or R-value areused in the following discussion, they always refer to the totalR-value of the entire roof assembly, and not just the portionconsisting of the actual tapered insulation. In general, insula-tion thickness may be substituted for
17、 R-value when only asingle material with uniform R-value is used; otherwise, thetotal R-value must be explicitly computed.There are two main strategies for determining the effi-ciency of tapered insulation. One strategy is to compare theheat flux of the tapered panel to that of an untapered (flat)pa
18、nel having the same volume. This method gives a truemeasure of the efficiency of the tapered panel, t, but may becumbersome to use in practice, since it requires the calculationof a true average thickness for the tapered panels. A more prac-tical strategy is to find the efficiency, a, based on an as
19、sumed“average” panel thickness. Letting H equal the thickness (orR-value) at the high point of tapered insulation and L equal thethickness (or R-value) at the low point of tapered insulation,this assumed average thickness (or R-value) is (H + L)/2.For a one- or two-way taper, there is no differenceb
20、etween these two strategies since the assumed average andtrue average thicknesses are the same; however, using anassumed average thickness of (H + L)/2 is much easier for allother taper geometries, since information about high- and low-point thicknesses is readily available. Derivation of the effi-c
21、iencies aand tfor various taper geometries follows.EFFICIENCY OF TAPERED INSULATIONThe following tapered forms will be considered, based onthe geometries illustrated in Figure 1: one-way taper, four-waytaper (sloping both to interior drain and to exterior drain); andtriangular cricket. In these deri
22、vations, let:A = length of tapered panel measured horizontally, in the direction of taper slopeB = width of tapered panel at high pointC = width of tapered panel at low pointE = heat flux at thickness, LH = thickness (or R-value) at high point of tapered insulation or cricketL = thickness (or R-valu
23、e) at low point of tapered insulation or cricketM = thickness (or R-value) at intermediate height of crickete = heat flux at any pointf = ratio of high- to low-point thickness (R-value) of taper, H/Lg = ratio of intermediate- to low-point thickness (R-value) of cricket, M/Lz = ratio of panel width a
24、t top and bottom of taper, B/CExcept for the cricket, which is analyzed separately, theefficiency of all these geometric forms can be derived fromconsideration of a single trapezoidal solid, shown in Figure 2a.By altering the ratio of high- and low-point sides, z = B/C, wecan find the efficiencies o
25、f one- and two-way slopes (Figure2b, where z = 1); one-way slopes converging downward to apoint or four-way slopes with internal drainage (Figure 2c,where z = ; and one-way slopes converging upward to a pointor four-way slopes with external drainage (Figure 2d, where z = 0).Figure 1 Common tapered i
26、nsulation forms include (a)one-way and two-way slopes, (b) four-way slopewith internal drainage, (c) four-way slope withexternal drainage, and (d) crickets.Figure 2 All commonly encountered taper geometries canbe derived from the consideration of thetrapezoidal solid (a). For the ratio of sides B/C
27、=z = 1 (b), we get a one-way or two-way slope; forz = (c), we get a one-way slope, convergingdownward to a point or a four-way slope withinternal drainage; and for z = 0 (d), we get a one-way slope converging upward to a point or a four-way slope with external drainage.2011 ASHRAE 861One-Way Trapezo
28、idal Taper with Parallel High- and Low-Point Sides1. From Figure 3, the thickness, t, of the trapezoidal solid atany distance, x, is the height of the line connecting (0, L)and (A, H):(1)2. The heat flux, e, at any point, inversely proportional to theR-value or thickness of the material at that poin
29、t (seeFigure 4), is,(2)where E is the heat flux at thickness equal to L.3. Substituting t from Equation 1 into Equation 2, we get.(3)4. Multiplying by the insulation panel width, (B C)(x/A) +C, and integrating Equation 3 over length A to find thetotal heat flux V of the panel, we get,(4)where f = H/
30、L and z = B/C.5. The solution to Equation 4 is.(5)6. The total heat flux, W, for a trapezoidal untapered panelwith constant thickness of (H + L)/2, sides B and C, andlength, A, is.(6)7. The assumed efficiency, a, of the trapezoidal taperedinsulationcompared with an un-tapered block of thick-ness (H
31、+ L)/2is found by dividing Equation 6 by Equa-tion 5, and multiplying by 100 to express the followingefficiency as a percentage:(7)8. The actual average thickness, t, for the one-way trapezoi-dal panel, computed by dividing the volume by the arearather than using (H + L)/2, is (8)and the true effici
32、ency, t, becomes(9)The efficiencies of one-way slopes, four-way slopes withinternal drains, and four-way slopes with external drains areall variations of Equations 7 and 9, found by setting z = 1,z = , and z = 0, respectively. The resulting efficiency expres-sions are summarized in Table 1. Typical
33、efficiency values forthese taper geometries are found in Table 2.It can be seen that tapered insulation efficiency dependsonly on the variable f, which, in turn, depends only on the ratioof the thicknesses (or R-values) at the high- and low-points ofthe tapered panel. However, there are two caveats.
34、 First of all,the panel cross section cannot be triangular (with the thicknessat one end equal to zero)as this would lead to the impossibletHLA-xL+=Figure 3 Geometry of one-way tapered insulation withtrapezoidal base.eELt-=eELHL()A- xL+-EHL()LA- x 1+-=VECA-z 1()xA+f 1()A- x 1+-xd0A=VEACfz()f() z 1()
35、f 1()+lnf 1()2-=wEAC z 1+()f 1+()-=a100 z 1+()f 1()2f 1+()fz()f() z 1()f 1()+ln-=t 2L()f 1()z 1()3 zf+()/2+z 1+()-=t=25 z 1+()2f 1()2f 1()z 1()3 zf+()/2+fz()f() z 1()f 1+()+ln-Figure 4 Heat flux, e, at any point along tapered insulation,where E is the heat flux at thickness L.862 ASHRAE Transactions
36、condition of “infinite” heat flux. Secondly, the slope of thetapered panel cannot be too great; in that case, the basicassumption underlying this methodthat the heat flux at eachpoint is inversely proportional to its vertical thickness at thatpointbecomes problematic, as the pattern of heat lossthro
37、ugh the sloping surface would be more complex than whatwas assumed. This concern is addressed in more detail below.Neither caveat affects typical tapered insulation panels. Inthe first case, the total R-value through any point on the insu-lation cannot be zero, since the R-value of the insulation do
38、esnot exist by itself, but must be measured together with the R-value of all other adjacent material layers, including air filmsand any substrate or interior finishes. In the second case, theslopes of tapered insulation are typically in the range of 1/8 in.per foot (1:96) up to 1 inch per foot (1:12
39、), values that areessentially flat for the purposes of this discussion.CricketCrickets are peculiar instances of tapered insulation typi-cally placed on one- or two-way slopes to direct water to drains(see Figures 1d and 5).While the actual cricket may literally rest on the slopingsurface(s) formed
40、by other pieces of tapered insulation,narrowing to zero thickness at its low point, our cricket isshown as if it extended downward to the horizontal roof deck,so that a triangular cricket element has different, non-zerothicknesses at each vertex. For example, triangle a-b-c(Figure 5) has a low-point
41、 thickness, L, and an intermediatethickness, M, at the perimeter; as well as a high-point, H, at thecenter. Crickets may be configured with one, two, or four ofthese triangular pieces, but, owing to their symmetry, it issufficient to examine a single triangle.Each triangle is analyzed by dividing it
42、 into two partsbounded by the line b-f (Figure 5) so that each of the triangularsolids thus obtained has a rectangular (and not a trapezoidal)side, as shown in Figure 6. The efficiencies computed for eachpart, based on an assumed average thickness of (H + L)/2, areas follows:(10)(11)where f = H/L an
43、d g = M/L.The efficiency of the whole cricket, derived from thesepartial values, is shown in Table 1. Values for cricket efficiencyat different thickness (R-value) ratios are shown in Table 3.What, then, is the loss of efficiency for tapered insulationassemblies in typical conditions? Considering ta
44、pered insula-tion with a one-way slope of 1/4 in. per foot (1:48) over a panellength of 40 ft (12.2 m), the change in thickness is 10 in.(250 mm). Where the slope and panel length are fixed as in thisexample, efficiency is a function only of the high- or low-pointthickness (either ignoring or includ
45、ing any additional constantR-values within the roof assembly). From Table 2, for a low-point thickness of 2 in. (50 mm) and, therefore, a high-pointthickness of 12 in. (300 mm), the ratio of thicknesses is f = 6,and the assumed efficiency, a, is 80%. If the low-point thick-ness is doubled to 4 in. (
46、100 mm), with the high-point thick-ness equal to 14 in. (350 mm), the ratio of thicknesses is f = 3.5,and aincreases to 89%. For a low-point thickness of 8 in.Table 1. Efficiency of Tapered Insulation1: Compared with Flat Panel Having Assumed Average Thickness (R-Value)of (H + L)/2: Compared with Fl
47、at Panel Having True Average Thickness (R-Value) of Volume/AreaOne-way slope (or two-way slope)Four-way slope with interior drain (or one-way slope converging down to a point)Four-way slope with exterior drain (or one-way slope converging up to a point)Triangular cricket21. Efficiencies based on f = H/L and (for crickets only) g = M/L, where H, L, and M are thicknesses (R-values) at high, low, and intermediate points, respectively.2. For cricket efficiency parameters, 1and 2, see Equations 10 and 11.ata200 f 1()f 1+()f()ln-= t200 f 1()f 1+()f()ln-
