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A general approach for predicting the thermal performance of metal building fiberglass insulation assemblies.

INTRODUCTION

In an earlier paper (Choudhary and Kasprzak 2010), a system-based approach was developed for predicting the thermal performance (the U-factors) of metal building single layer fiberglass (FG) batt insulation assemblies. The approach consisted of calculating an overall or system thermal resistance parameter, [R.sub.insul-sys] that combined the thermal resistances (the R-values) of the insulation drape on both sides of the structural element (referred to as purlin for the roof assemblies and girt for the wall assemblies) with the thermal resistance of the insulation on the structural element itself. The overall heat transfer coefficient (U-factor) for the insulation assembly could then be predicted using the following correlation.

1 / U = 0.8672[R.sub.insul.sys] + 1-132 ([R.sup.2] = 0.998) (1)

In the above equation, U is in Btu/[ft.sup.2] x h x [degrees]F and R is in [ft.sup.2] x h x [degrees]F/Btu (1 Btu/[ft.sup.2] x h x [degrees]F = 5.6782 W/[m.sup.2] x K and 1 [ft.sup.2] x h x [degrees]F/Btu = 0.17611 [m.sup.2] x K/W).

The correlation (hereafter referred to as the Choudhary correlation) based approach was adopted by the ASHRAE Standing Standard Project Committee (SSPC) 90.1 (ASHRAE 2013a) to set U-factors for single- and double-layer draped FG insulation assemblies. The fact that the Choudhary correlation, derived from three-dimensional numerical modeling for single-layer FG insulation assemblies, could be found applicable to double-layer assemblies was in itself quite remarkable. The present work, performed at the request of ASHRAE SSPC 90.1, extends the application of the Choudhary correlation to assemblies where insulation is used in a manner designed to fill up the space (the cavity) between two consecutive structural elements. These assemblies involve either single or multiple insulation layers and the cavity is filled to various extents. Some assemblies may succeed in filling the cavity with the insulation substantially with small air gaps while the others may have relatively large air gaps between the insulation and the structural element. It is noted at the outset that the Choudhary correlation was derived for the purlin/girt spacing of 5 ft (1.524 m) and length of 8 in. (0.2 m). The coefficients (intercept and the slope) may differ from the above values (1.132 and 0.8672, respectively) for different spacing and length.

In the following, we first consider a general FG-based insulation assembly consisting of (1) insulation layers, (2) air gaps, and (3) structural elements. We consider the network of thermal resistances present in the general assembly, including the thermal bridges, and derive expressions for them. These thermal resistances are then combined analogously to electrical circuits consisting of electric resistances in series and parallel. Finally, we investigate the applicability of the Choudhary correlation to a number of insulation assemblies very different from the draped insulation assemblies considered earlier. We do so by comparing the U-factors calculated using the Choudhary correlation with those calculated using numerical simulation.

SOME FIBERGLASS-BASED INSULATION ASSEMBLIES FOR METAL BUILDINGS

In the earlier publication (Choudhary and Kasprzak 2010) we had considered the purlin single-layer FG insulation assemblies. The use of a parabolic profile allowed us to derive analytical expressions for the R-value of the insulation. In the present work we consider more thermally efficient insulation assemblies that attempt to fill up the cavity between two consecutive structural units. One such assembly is shown in Figure 1a. It involves two insulation layers, and the bottom layer essentially completely fills up the cavity except for some air gaps. Depending on installation practices, one may get some variations as shown in Figure 1b where the bottom insulation layer extends below the purlin to fully enclose the purlin flange. Yet another variation in the insulation profile is shown in Figure 1c. In both Figures 1b and 1c we have shown insulation in the immediate vicinity of the purlin. For convenience, we will be using the term purlin in this paper. The analysis, however, also applies to the wall assemblies and girts.

The thermal resistance analysis approach used in this work will be illustrated by using a generic FG metal building insulation assembly shown in Figure 2. Figure 2 represents a section of the assembly around a purlin with [L.sub.f] being the length of the purlin flange, [L.sub.1] + 0.5[L.sub.f] half the center-to-center spacing between two consecutive purlins, and [t.sub.p] is the thickness of the purlin. The three thermal zones present in the assembly are listed below.

[FIGURE 1 OMITTED]

1. Under the purlin. It consists of a portion of the air gap ([R.sub.AUP]), a portion of the bottom layer of the insulation, ([R.sub.2UP]), and the straight section of the purlin ([R.sub.PUP]). Thermally, [R.sub.2UP] and [R.sub.AUP] are in series, and the two combined are in parallel to [R.sub.PUP]. As shown in Figure 2, [R.sub.2UP] is in total contact with the purlin. The approach shown here may easily be modified if there is an air gap between the insulation and the purlin so long as the shape of the air gap or the insulation profile can be defined.

2. Over the purlin. This zone consists of the fiberglass and the foam insulation layers ([R.sub.1OP], [R.sub.FOP]), respectively. It should be noted that the technical approach is independent of the insulation materials. These two are in series. The purlin flange next to or in series with [R.sub.1OP] represents negligible thermal resistance and need not be included in the calculation.

3. Beyond the purlin. The constituents of this zone are R1BP, which represents the top insulation layer and any air gaps around it, and R2BP, which is the bottom insulation layer. R1BP and R2BP are in series. Figure 2 is a generalized representation of the top layer. Insulation exiting the purlin flange may expand in a way so as to allow air gaps both above and below the expanding section. The top insulation layer may be treated as a composite of the fiberglass insulation and the two air gaps shown in Figure 2.

[FIGURE 2 OMITTED]

In the analysis presented below we will treat the air layers ([R.sub.AUP] and those around [R.sub.1OP]) as stagnant. If there is convection in the air gap and/or radiation across it, then their effects on heat transfer will have to be estimated, for example, through correlations or compilations for heat transfer in the air enclosures (ASHRAE 2013b).

CALCULATION OF THERMAL RESISTANCES

In the analysis presented below, we will assume the heat flow to be primarily in the vertical or they direction. There will undoubtedly be some heat flow in the x and z directions, especially in the vicinity of contact between the purlin and the insulation. However, the overall dimensions in the x and the z directions are typically much larger than that in the y direction, and the small heat flow in the longitudinal (x) and the width (z) directions will, in a relatively short distance from the contact zones, be directed into the x direction. As mentioned above, we will neglect convection in and/or radiation across the air gaps. These effects, if significant, may impact the overall U-factor. In such cases, one may use empirically or numerically estimated effective or apparent thermal conductivity, larger than the molecular thermal conductivity of air, to calculate the thermal resistance of the air gaps. The thermal resistance equations for the air gaps may still be used by replacing the molecular thermal conductivity by the apparent conductivity.

Let us consider a zone of surface area S through which heat flows and in which the two boundaries in the heat flow direction are separated by a distance [delta]. The total heat flow, Q when the thermal conductivity is K and the temperature difference is [DELTA] T is given by the following:

Q = S K/[delta] [DELTA] T (2)

Let us define the thermal resistance R of this configuration as shown below:

R = [delta]/K (3b)

Q = S [DELTA]T/R (3a)

As shown in Figure 2, some of the thermal zones (e.g., R1BP, R2BP, and [R.sub.2UP]) do not have a constant thickness for the insulation. Further, the thermal conductivity of the FG insulation depends on its thickness (e.g., Choudhary et al. 2010). For such a situation, we need to generalize the calculation of R as explained below.

The effective thermal conductivity, K, of the FG insulation is given by the following equation (Wilkes 1979).

K = A + B[rho] + C/[rho] (4)

where A, B, C are empirically determined coefficients and [rho] is the fiberglass density.

At any location (x, z) the insulation thickness S and the density p are related by mass conservations, as shown below.

[rho][delta] = [[rho].sub.o] [[delta].sub.o] (5)

where [[rho].sub.o] is the reference density of the insulation at the reference thickness [[delta].sub.o].

For an FG insulation with an arbitrary profile, using Equations 4 and 5, we can rewrite Equation 3a as follows.

R = 1/XZ [[delta].sub.o][integral] [integral][delta]*/A + B([[rho].sub.o]/[delta]*) + C([delta]*/[[rho].sub.o]) dxdz (6)

where X and Z are the relevant overall dimensions in the x and z directions for the thermal zone and [delta]* is the nondimensional thickness, [delta]/[[delta].sub.o].

Thus, if the thickness profile [delta] (x, z) is known, one can, through analytical or numerical integration, calculate R. It is unlikely, however, to have such detailed information available. In that case, if the minimum and the maximum insulation thicknesses are available, one may assume the profile to be parabolic and calculate a mean insulation thickness at which the K-value and hence the R-value may be calculated.

Let us illustrate the calculation of R with respect to an insulation profile sketched in Figure 3. We will ignore insulation profile in the z direction (perpendicular to the plane of the paper) and use unit length in this direction.

R = 1/[X.sub.m] [integral] y/K dx (7)

where K is thermal conductivity of FG insulation corresponding to the local insulation thickness, y.

For a parabolic profile, the local insulation thickness is given as follows.

y - [Y.sub.[o]]/[Y.sub.m] - [Y.sub.o] = x/[X.sub.m] (2 - x/[X.sub.m]) (8)

where [Y.sub.o] and [Y.sub.m] represent the insulation thickness at x = 0 and x = [X.sub.m], respectively.

[FIGURE 3 OMITTED]

As mentioned earlier, at any location x, the insulation thickness, y, and the insulation density, [rho], are related by Equation 5, which is reproduced below (with [delta] replaced by y):

[rho]y = [[rho].sub.o] [[delta].sub.o] (9)

where, as explained earlier, [[rho].sub.o] is the reference density of the insulation at the reference thickness of [[delta].sub.o].

The equation for the insulation thermal conductivity, Equation 4, may now be expressed in terms of the local thickness.

K = A + B [[rho].sub.o] [[delta].sub.o]/y + C y/[[rho].sub.o] [[delta].sub.o] (10)

Using Equations 8 and 10, we may numerically integrate Equation 7 to calculate R.

An approximate and quick approach will be to calculate K and R at the average thickness of the profile. For the parabolic profile, represented by Equation 8, the value will be as given below:

R = 1/[K.sub.ave](2/3 [Y.sub.m] + 1/3 [Y.sub.o]) (11)

where [K.sub.avg] is thermal conductivity corresponding to the insulation of thickness (2[Y.sub.m]/3+[Y.sub.o]/3).

Figure 4 shows another configuration consisting of an FG insulation and an air gap. For simplicity we will only consider one air gap, say between the top and the bottom insulation layers.

The thermal resistance, R, of the FG/air composite insulation is given as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

where [K.sub.a] is the thermal conductivity of air. The first or the integral term on the right-hand side represents the combined or the composite R of air and insulation in 0 < x < [X.sub.a]. The second term is the R of the insulation of thickness [Y.sub.m].

[FIGURE 4 OMITTED]

Since the air conductivity does not depend on its thickness, we may simplify Equation 12 by using the mean thickness of the air gap, [Y.sub.a].

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (13)

One can take a step-by-step approach as outlined above to calculate the R of zones that have varying insulation thicknesses with or without the air gaps.

Thermal Resistance beyond the Purlin

With reference to Figure 2, the thermal resistance of the zone beyond the purlin is the sum of R1BP and R2BP calculated using the integration approaches described earlier. Let us call this combined resistance, [R.sub.BP].

[R.sub.BP] = R1 BP + R2 BP (14)

To the insulation R given above, we will add the air film resistance at the top, [R.sub.AT], and bottom, [R.sub.AB], defined below:

[R.sub.AB] = 1/[h.sub.AB] (15a)

[R.sub.AT] = 1/[h.sub.AT] (15b)

where [h.sub.AB] and [h.sub.AT] are the air film heat transfer coefficients at the top and the bottom, respectively.

The combined or effective R-value of insulation and air film beyond the purlin, [R.sub.BP + air], is

[R.sub.BP + air] = [R.sub.BP] + [R.sub.AB] + [R.sub.AT] (16)

Thermal Resistance under the Purlin

As described earlier and with reference to Figure 2, [R.sub.AUP] and [R.sub.2UP] are in series and their combined value is parallel to [R.sub.PUP]. Assuming no convection and/or radiation, [R.sub.AUP] is given by the following expression:

[R.sub.AUP] = [H.sub.3]/[K.sub.a] (17)

where, as mentioned earlier, [K.sub.a] is the thermal conductivity of the air and [H.sub.3] is the thickness of the air layer (see Figure 2).

[R.sub.2UP] represents a configuration like the one shown in Figure 3. If the insulation profile [i.e., v(x)] is known, [R.sub.2UP] may be calculated using Equation 7 with the help of Equation 10. If the profile is parabolic, then v(x) is defined by Equation 8. If the detailed information on the insulation profile is not available and the assumption of a parabolic profile is reasonable, then one may, as an approximation, use Equation 11.

The thermal resistance of the straight section of the purlin, [R.sub.PUP] is given by the following expression:

RPUP = [H.sub.3] + [H.sub.4]/[K.sub.p] (18)

where [K.sub.p] is the thermal conductivity of the purlin and, as seen in Figure 2, ([H.sub.3] + [H.sub.4]) is the length of the straight section of the purlin.

As explained earlier, the net thermal resistance under the purlin, RUP, may then be calculated by treating ([R.sub.AUP] + [R.sub.2UP]) in series and the sum in parallel to [R.sub.PUP].

[L.sub.f]/RUP = [L.sub.f] - [t.sub.p]/[R.sub.AUP] + R2UP' + [t.sub.p]/RPUP

From Equation 19, one may write the following expression for RUP:

RUP = (RAUP + R2UP)RPUP/([L.sub.f] - [t.sub.p]) RPUP + [t.sub.p] (RAUP + R2UP) [L.sub.f] (20)

The thickness of the purlin [t.sub.p] will typically be much smaller than the flange length [L.sub.f] and we may use [L.sub.f] instead of ([L.sub.f] - [t.sub.p]) in the denominator of Equation 20.

Thermal Resistance above the Purlin

For the general case, we have considered two different insulations on the top flange of the purlin: the foam ([R.sub.FOP]), block of thickness [H.sub.1] and thermal conductivity [K.sub.F], and the insulation ([R.sub.1OP]), of thickness [H.sub.2] and thermal conductivity [K.sub.I]. The foam insulation on top of the purlin is usually referred to as the thermal block. The two insulations are in series and the combined thermal resistance, [R.sub.OPI], is given by the following equation:

[R.sub.OPI] = [H.sub.1][K.sub.F] + [H.sub.2]/[K.sub.I] (21)

Following Johannesson and Vinberg (1986), the thermal bridging adjusted R of the insulation layers, [R.sub.OP], on the top flange is given as follows.

[R.sub.OP] = [R.sub.OPI] 1/(1 + 2[pi] [H.sub.T]/[L.sub.f]) (22)

where [H.sub.T] is the combined height of the two insulation layers on purlin and equals [H.sub.1] + [H.sub.2]. The effect of including the thermal resistance attenuation factor (i.e., the denominator in the above equation) is described subsequently.

Total Thermal Resistance under and above Purlin

RUP and ROP are in series and can be added to give the combined thermal resistance [R.sub.TP]:

[R.sub.TP] = RUP + ROP (23)

As done earlier for the thermal resistance beyond the purlin, we will add the bottom and the top air film resistances to give the total thermal resistance under and above the purlin, [R.sub.TP + air].

[R.sub.TP + air] = [R.sub.TP] + [R.sub.AB] + [R.sub.ATP] (24)

EFFECTIVE THERMAL RESISTANCE OF THE INSULATION SYSTEM

We will now combine the two thermal resistance calculated above, namely [R.sub.BP + air], and [R.sub.TP] + air] into a single thermal resistance of the insulation assembly, [R.sub.insul-sys]. It should be noted that, if the insulation profiles on the two sides of the purlin are significantly different then the thermal resistances on the two sides of the purlin ([R.sub.BP + air] on the left and [R.sub.BP + air] on the right) may also be different. In that case one would calculate the two thermal resistances separately. However, all the results discussed below are for the cases where the two are assumed equal as was the case also in the previous work (Choudhary and Kasprzak 2010).

The overall or the insulation system thermal resistance, [R.sub.insul-sys] is obtained by combining [R.sub.BP + air] and [R.sub.TP + air] as shown previously (Choudhary and Kasprzak 2010).

[R.sub.insul-sys] = [[(2[L.sub.1] + [L.sub.f]) [R.sub.BP + air] [R.sub.TP + air]]/ [2[L.sub.1] [R.sub.TP + air] + [L.sub.f] [R.sub.BP + air]]] (25)

The parameter [R.sub.insul-sys] calculated above may then be used to predict the overall U for a metal building insulation assembly by using the Choudhary correlation, given earlier by Equation 1. As mentioned earlier, this correlation was derived for a 5 ft (1.524 m) center-to-center spacing between the purlins/girts. All the results presented here and previously (Choudhary and Kasprzak 2010) were for this spacing. Similar correlations may be developed by carrying out limited numerical modeling for other purlin/girt spacing.

INSULATION ASSEMBLIES NUMERICALLY MODELLED

A total of thirteen insulation assemblies were investigated. Out of these twelve were roof assemblies and one was a wall assembly. As mentioned earlier, the approach described and used in this paper is applicable to both roof and wall assemblies. What matters is the overall or the insulation system resistance, [R.sub.insul-sys], that results from the insulation types and configurations used in any assembly. Nine assemblies were modeled by the author using a numerical solution procedure described in detail in a publication (Choudhary et al. 2010). The remaining four were modeled by Engrana, LLC for the North American Manufacturers Association (NAIMA) (Beard and Wangard 2013). Two types of roof assemblies, through fastened (TFR) and standing seam (SSR), were studied. Figures 5 and 6 show the general schematics of the TFR and SSR assemblies.

Table 1 summarizes key features of the 13 assemblies. Two types of roof assemblies, through fastened (TFR) and standing seam (SSR), were studied. Roof assemblies 1 to 8 and the wall assembly 13 were modeled by the author, while roof assemblies 9 through 12 were modeled by Bear and Wangard (2013).

The air pocket thickness in assemblies 9 through 12 was 0.8 in. (0.02 m) thick, 8 in. (0.20 m)long, and 40 in. (1.02 m) wide. The value given in the table is averaged or normalized for the surface area 28.33 x 60 in. (0.72 x 1.52 m) used in modeling. Here 28.33 in. (0.72 m) represents the length of the roof panel beyond the purlin flange, and 60 in. (1.52 m) is the width of the roof panel.

In assemblies 1 to 8, the insulation thickness varied only along the length (x direction in Figure 2). In assemblies 9 through 13, the thickness varied both along the length and the width (z direction in Figure 2) directions. The thickness profile details on assemblies 9 through 12 are described by Beard and Wangard (2013), and the profile details on assembly 13 are given in Choudhary et al. (2012). In all 13 cases, the approach to quantify the insulation thickness profile was similar and is described below.

The insulation profiles used in assemblies 1 to 7 are shown schematically in Figure 7. Assemblies 2, 4, and 5 have no air gap and that simply means that the air gap in Figure 7 will become zero and the bottom of the top insulation will coincide with the top of the bottom insulation. Figure 7 may also be used to represent the insulation thickness profile of assemblies 9 through 13. Since these assemblies had insulation thickness variation both in the length (x) and width (z) directions, the thickness depicted in Figure 7 will be the width averaged value.

[FIGURE 5 OMITTED]

[FIGURE 6 OMITTED]

[FIGURE 7 OMITTED]

The bottom (i.e., the thickness) of the top insulation and the bottom of the bottom insulation (i.e., the combined thickness of both the insulation layers and any air in between) are assumed to be parabolic with equations given below. In these equations, as shown in Figure 7, Y is the distance from the inside of the roof panel and X is the distance away from the edge of the purlin flange.

Equation for the bottom of the top insulation, [Y.sub.t]:

[[[Y.sub.t] - [h.sub.to]/[h.sub.tl] - [h.sub.to]]/ [[h.sub.tl] + [h.sub.bl] - H]] = X/[L.sub.1](2 - X/[L.sub.1]) (26)

Equation for the bottom of the bottom insulation, [Y.sub.b]:

[[[Y.sub.b] - H]/[[h.sub.tl] + [h.sub.bl] - H]] = X/[L.sub.1](2 - X/[L.sub.1]) (27)

where, as seen in Figure 7, His the combined depths of the top insulation ([h.sub.to]), the air gap ([h.sub.ao]), and the bottom insulation ([h.sub.bo]) at X = 0. The top and the bottom insulation thicknesses at X = [L.sub.1] are [h.sub.tl] and [h.sub.bl], respectively. As there is no air gap between the top insulation and the metal panel, the thickness of the top insulation is given by [Y.sub.t], calculated from Equation 27. Over 0 < X < [L.sub.a], the bottom insulation thickness remains constant at [h.sub.bo]. Over [L.sub.a] < X < [L.sub.1], the bottom insulation thickness is given by [Y.sub.b], calculated using Equation 28 minus [Y.sub.t].

The air gap thickness between the two insulation layers is [h.sub.ao] at X = 0 and it extends to X = [L.sub.a]. Over this domain, the air gap thickness at any location, X is given by [Y.sub.b] minus the combined thickness of the two insulation layers.

Assembly 8 is shown schematically in Figure 8. This assembly has an air pocket between the top insulation and the inside of the roof panel. The air pocket begins at X = [L.sub.1] - [L.sub.at] and extends to the end of the insulation, i.e. X = [L.sub.1].

[FIGURE 8 OMITTED]

With the symbols used in Figure 8, the equations describing the profiles of the bottom of the top insulation and the bottom of the bottom insulation are given below.

Equation for the bottom of the top insulation, [Y.sub.t]:

[Y.sub.t] - [h.sub.to]/[h.sub.tl] + [h.sub.apl] - [h.sub.to] = X/[L.sub.1] (2 - X/[L.sub.1]) (28)

Equation for the bottom of the bottom insulation, [Y.sub.b]:

[[[Y.sub.b] - H]/[[h.sub.tl] + [h.sub.apl] + [h.sub.bl] - H]] = X/[L.sub.1] (2 - X/[L.sub.1]) (29)

where the symbols have the same meaning as before except that we have an extra item, [h.sub.apl], which is the maximum thickness of the air pocket between the top insulation and the metal panel.

For 0 < X < ([L.sub.1] - [L.sub.at]), the thickness is given by [Y.sub.t] calculated from Equation 26.

For ([L.sub.1] - [L.sub.at]) < X < [L.sub.1], the thickness remains constant at the value at [h.sub.tl].

The bottom insulation thickness is constant at [h.sub.bo] for all values of X.

The air gap thickness between the two insulation layers is given by [Y.sub.b] minus the sum of [Y.sub.t] and the bottom insulation thickness. The thickness of the air pocket between the top insulation and the metal panel is [Y.sub.t] - [h.sub.tl].

Figures 7 and 8 show the bottom insulation drape at X = 0 to drape below the purlin bottom flange and, for ease in drawing, no thickness profile in the -[L.sub.f] < X < 0. These need not be the case, and the approach described is applicable to other situations, as well.

Table 2 summarizes key parameters relevant to the insulation profiles in assemblies 1 to 13 described above and in Figures 7 and 8. As noted in Table 1, assemblies 5 to 12 contain foam blocks.

The flange length [L.sub.f] was 2.5 in. (0.064 m) and the insulation length beyond the flange [L.sub.1] was 28.75 in. (0.73 m) in assemblies 1 to 8, as well as in assembly 13. The flange length was 3.34 in. (0.085 m) and the insulation length was 28.33 in. in assemblies 9 to 12. These and other design parameters and the air film heat transfer coefficients are summarized in Table 3.

MATERIAL PROPERTIES AND INSULATION K VALUES

As mentioned earlier, the effective thermal conductivity, K, of fiberglass insulation at any given thickness, y, is given by Equation 10, which is reproduced below with the specified values of coefficients, A, B, and C.

K = 0.179 + 0.00525275 [[rho].sub.o] ([[delta].sub.o]/y) + 0.0682766/[[rho].sub.o](y/[[delta].sub.o]) (30)

where K is in Btu x in./[ft.sup.2] x h x [degrees]F, the reference density [[rho].sub.o] is in lb/[ft.sup.3], and the local thickness y and the reference thickness [[delta].sub.o] are in inches (1 Btu x in/[ft.sup.2] x h x [degrees]F = 0.144 W/[m.sup.2] x K, 1 lb/[ft.sup.3] = 16.018 Kg/[m.sup.3], 1 in. = 0.0254 m).

The reference values for density [p.sub.o], thickness [[delta].sub.o], and thermal conductivity [K.sub.o] for the fiberglass insulation used in 13 assemblies are given in Table 4. Insulations used in assemblies 1 to 5 are designated with an X (e.g., R-11X, R-19X). These have lower reference densities than their counterparts in assemblies 6 to 13, which are commercial insulations used in metal building assemblies and are indicated simply as R-11, R-19, etc.

RESULTS

The calculated thermal resistance results (units: [ft.sup.2] x h x [degrees]F/Btu) for all the 13 assemblies hare summarized in Table 5.

Table 6 shows the U-factors calculated using the Choudhary correlation and by three dimensional numerical modeling. For assemblies 10 to 13, U-factors measured using calibrated hot-box testing are available and listed in Table 6.

As seen above in Table 6, the U-factors calculated using the Choudhary correlation are in 93.5% to 100% agreement with those calculated using three-dimensional numerical modeling. Out of a total of 13 assemblies, two U-factors are in 93% to 94% agreement with their modeling counterpart, the other eleven are in 97% to 100% agreement. The modeling-based U-factors are in 93% to 97% agreement and the correlation based U-factor are in 91% to 100% agreement with the measured values for the four cases (assemblies 10 to 13) for which the hot box data are available.

Finally, the effect of including the attenuation of the thermal resistance above the purlin (i.e., the denominator of Equation 22) was examined by comparing the U-factors calculated using with and without this correction. The two sets of results were in 95% to 100% agreement. The 95% figure is for assembly 4. For the remaining 12 assemblies, the U-factors calculated with and without this correction agreed within 97% to 100%. Since the attenuation reduces the thermal resistance above the purlin, the U-factors calculated with it were larger than their counterparts calculated without it. So the use of the attenuation factor for thermal resistance above the purlin, ROP, may not be necessary.

CONCLUSION

The paper has demonstrated that the Choudhary correlation-based approach developed earlier and adopted by the AHRAE SSPC 90.1 for calculating the overall heat transfer coefficients (the U-factors) of single and double layered fiberglass metal building (MB) insulation assemblies can also be used to calculate U-factors for assemblies where insulation is used in a manner designed to fill up the space (the cavity) between the two consecutive structural elements (purlins or girts). These assemblies may involve either single or multiple insulation layers and the cavity may be filled to various extents. The paper described a general approach to calculate the thermal resistances of various regions in an insulation assembly, namely regions beyond, underneath, and above the structural units. These zonal thermal resistances were then combined into a single, overall or system thermal resistance, [R.sub.insul-sys] using series and parallel combinations while also allowing for the thermal short circuiting or preferential heat flow through the metallic structural elements and the thermal resistances of the ambient air. The Choudhary correlation is given below.

1/U =0.8672 [R.sub.insul-sys] + 1.132

In the above equation U is in Btu/[ft.sup.2] x h x [degrees]F and R is in [ft.sup.2] x h x [degrees]F/Btu (1 Btu/[ft.sup.2] x h x [degrees]F = 5.6782 W/[m.sup.2] x K and 1 [ft.sup.2] x h x [degrees]F/Btu = 0.17611 [m.sup.2] x K/W).

A total of 13 metal building insulation assemblies were modeled using three-dimensional numerical approaches, 9 by the author and 4 by Beard and Wangard (2013). The 13 model calculated U-factors were compared with their counterparts calculated using the Choudhary correlation. The two sets of U-factors were in 93.5% to 100% agreement. Hot-box measured U-factors were available for 4 of the assemblies. The numerically modeling-based U-factors were in 93% to 97% agreement with the measured values. The correlation based U-factors were in 91% to 100% agreement with the measured values.

The Choudhary correlation and the approach for calculating and combining zonal thermal resistances into an overall or system thermal resistance provide a simpler, faster, and economical way of calculating the U-factors for a wide range of metal building fiberglass batt insulation assemblies than the three-dimensional mathematical modeling or hot box measurement based approaches.

An important caveat to remember is that the Choudhary correlation was obtained using results for assemblies with a fixed spacing of 5 ft (1.52 m) between and a fixed 8 in. (0.20 m) length of Z-type purlins/girts. The 13 assemblies considered in this paper satisfied those conditions. For the fixed design configurations used in these assemblies, the technical approach described here can be used to estimate U-factors for the relevant metal building insulation assemblies. For significantly different design scenarios (e.g., purlin/girt spacing, design of structural units) one would expect different coefficients in the correlation. This may be achieved through the numerical modeling for a limited number of relevant design scenarios. The technical approach described in this paper can then be used to calculate a system or overall thermal resistance and from it predict the U-factor using the modified correlation.

REFERENCE

ASHRAE. 2013a. ANSI/ASHRAE/IES Standard 90.1-2013, Energy Standard for Buildings Except Low-Rise Residential Buildings. Atlanta: ASHRAE.

ASHRAE 2013b. ASHRAE Handbook-Fundamentals, I-P ed., Chapter 26, p. 26-14. 2013. Atlanta: ASHRAE.

Beard, J.G., and W. Wangard. 2013. Thermal analysis of wall and roof metal building insulation assemblies. Report prepared for North American Insulation Manufacturing Association. Alexandria, Virginia: North American Insulation Manufacturing Association.

Choudhary, M.K., and C. Kasprzak. 2010. ASHRAE Standard 90.1, Metal Building U-Factors- Part 2: A Systems Based Approach for Predicting the Thermal Performance of Single Layer Fiberglass Batt insulation Assemblies. ASHRAE Transactions 116(1):169-76.

Choudhary, M.K, C. Kasprzak, R. Larson, and R. Venuturumilli. 2010. ASHRAE Standard 90.1, metal building U-factors part 1: mathematical modeling and validation by calibrated hot box measurements. ASHRAE Transactions 116(1):157-68.

Choudhary, M.K, C. Kasprzak, D. Musick, M. Henry, and N. D. Fast. 2012. ASHRAE Standard 90.1, metal building U-Factors- part 5: mathematical modeling of wall assemblies and validation by calibrated hot box measurements. ASHRAE Transactions 118(2):400-08.

Johannesson, G., and H.A. Vinberg. 1986. Thermal bridges in sheet metal construction. Reports of the Working Commissions, International Association for Bridge and Structural Engineering 49:409-14.

Wilkes, K.E. 1979. Thermophysical properties data base activities at Owens Corning fiberglass. Proceedings: 1979 ASHRAE/DOE-ORNL Conference on the Thermal Performance of the Exterior Envelope of Buildings, ASHRAE. 662.

M.K. Choudhary, PhD, PE

M.K. Choudhary is part of the senior technical staff at Owens Corning Science & Technology, Granville, OH.
Table 1. Insulation Assemblies Studied (1 [ft.sup.2] x h x
[degrees]F/Btu = 0.1761 [m.sup.2] x K/W; 1 in. = 0.0254 m)

Assembly   Assembly   Insulation   Insulation        Insulation Type
Number     Type       Type         Thickness Range   and Thickness
                                   (Flange           above or Girt,
                                   --Center), in.    in.

1          TFR        R-10+R-19    R-10: 0.375-3.4   R-10: 0.375
                                   R-19: 6.3-5.975

2          TFR        R-19+R-30    R-19: 0.375-6.3   R-19: 0.375
                                   R-30: 8.0-3.075

3          SSR        R-10+R-19    R-10: 0.25-3.4    R-10: 0.25
                                   R-19: 6.3-5.8

4          SSR        R-19+R-30    R-19: 1.75-6.3    R-19: 1.75
                                   R-30: 8.0-4.5

5          SSR        R-19+R-30    R-19: 1.75-6.3    R-19: 0.75
                                   R-30: 8.0-4.5     Foam: 1.0

6          SSR        R-11+R-19    R-11: 1.0-3.7     R-11: 0.25
                                   R-19: 6.3-5.3     Foam: 0.75

7          SSR        R-11+R-19    R-11: 1.0-3.7     R-11: 0.25
                                   R-19: 6.3-6.3     Foam: 0.75

8          SSR        R-11+R-19    R-11: 1.0-3.7     R-11: 0.25
                                   R-19: 6.3-6.3     Foam: 0.75

9 *        SSR        R-10+R-19    R-10: 1.31-2.95   R-11: 0.56
                                   R-19: 5.45-6.25   Foam: 0.75

10 *       SSR        R-11+R-19    R-11: 1.31-3.20   R-11: 0.56
                                   R-19: 5.45-6.25   Foam: 0.75

11 *       SSR        R-11+R-25    R-11: 1.31-3.20   R-11: 0.56
                                   R-25 6.51 -8.40   Foam: 0.75

12 *       SSR        R-11+R-30    R-11: 1.31-3.20   R-11: 0.56
                                   R-30: 7.13-9.75   Foam: 0.75

13         Wall       R-13+R-30    R-13: 0.25-3.88   R-13: 0.25
                                   R-30: 8.0-9.18

Assembly   Assembly   Thickness of      Air Pocket between
Number     Type       Air Gap between   Insulation and
                      Insulation        Metal Panel, in.
                      Layers, in.

1          TFR        1.7-0              --

2          TFR        --                 --

3          SSR        1.7-0              --

4          SSR        --                 --

5          SSR        --                 --

6          SSR        1.7-0              --

7          SSR        1.7-0              --

8          SSR        1.7-0              0-1.0

9 *        SSR        1.64-0             0.15

10 *       SSR        --                 0.15

11 *       SSR        --                 0.15

12 *       SSR        --                 0.15

13         Wall       --                 --

* Numerically modeled by Beard and Wangard (2013).

Table 2. Insulation and Air Gap Thickness Profiles (1 in. = 0.0254 m)

Assembly      Air Gap Length,     Air Pocket          Top Insulation
              [L.sub.a], in.                          Thickness at
                                                      X = 0,
                                                      [h.sub.to] in.

1. R10+R19    18                  NO                  0.375
2. R19+R-30   0                   NO                  0.375
3. R10+R19    16                  NO                  0.25
4. R19+R30    0                   NO                  1.75
5. R19+R30    0                   NO                  1.75
6. R11+R19    12                  NO                  1.0
7. R11+R19    28.75               NO                  1.0
8. R11+R19    28.75               YES *               1.0
9. R10+R19    2                   YES **              1.313
10. R11+R19   2                   YES **              1.313
11. R11+R25   2                   YES **              1.313
12. R11+R30   2                   YES **              1.313
13. R13+R30   0                   NO                  0.25

Assembly      Top Insulation      Air Gap Thickness   Bottom
              Thickness at        at X = 0,           Insulation
              X = 0, [L.sub.1],   [h.sub.ao], in.     Thickness at
              [h.sub.tl], in.                         X = 0,
                                                      [h.sub.bo] in.

1. R10+R19    3.40                1.70                6.3
2. R19+R-30   6.3                 0                   8.0
3. R10+R19    3.40                1.70                6.3
4. R19+R30    6.3                 0                   8.0
5. R19+R30    6.3                 0                   8.0
6. R11+R19    3.7                 1.70                6.3
7. R11+R19    3.7                 1.70                6.3
8. R11+R19    3.7                 1.70                6.3
9. R10+R19    2.95                1.64                5.58
10. R11+R19   3.2                 1.89                5.58
11. R11+R25   3.2                 1.89                6.81
12. R11+R30   3.2                 1.89                7.62
13. R13+R30   3.54                0                   8.0

Assembly      Bottom              Total
              Insulation          Depth at
              Thickness at        X = 0, H, in
              X = 0, [L.sub.1],
              [h.sub.tl], in.

1. R10+R19    5.975                8.375
2. R19+R-30   3.08                 8.375
3. R10+R19    5.85                 8.25
4. R19+R30    4.5                  9.75
5. R19+R30    4.5                  9.75
6. R11+R19    5.3                  9.0
7. R11+R19    6.3                  9.0
8. R11+R19    6.3                  9.0
9. R10+R19    6.12                 8.53
10. R11+R19   6.12                 8.78
11. R11+R25   8.03                10.01
12. R11+R30   9.58                10.82
13. R13+R30   8.82                 8.25

* The length, [L.sub.at] = 15.75 in. (0.40 m) and the maximum
depth, [h.sub.apl] = 1 in. (0.025 m).

** An air pocket of average thickness 0.15 in. (0.4 m) exists
 over the 28.33 in. (0.72 m) insulation length.

Table 3. Geometry Parameters and Air Film Heat Transfer Coefficients
(1 in. = 0.025 m, 1 Btu/[ft.sup.2] x h x [degrees]F = 5.6782 W/
/[m.sup.2] x K

Item                                           Value

Purlin/girt spacing, in.                       60

Purlin/girt length, in.                        8

Purlin/girt flange length, in.                 2.5 (1 to 8, 13),
                                               3.34 (9 to 12)

Purlin/girt thickness, in.                     0.06 (1 to 8),
                                               0.082 (9 to 13)

Interior air film heat transfer coefficient,   1.63 (1 to 8),
Btu/[ft.sup.2] x h x [degrees]F                1.31 (9 to 12),
                                               1.04 (13)

Exterior heat transfer coefficient,            1.63 (1 to 8), 1.29
Btu/[ft.sup.2] x h x [degrees]F                (9 to 12), 1.47 (13)

Table 4. K-Values of Insulation Used in Assemblies (1 in. = 0.0254
m, 1 lb/[ft.sup.3] = 16.018 Kg/[m.sup.3], 1 Btu x in./[ft.sup.2] x
h x [degrees]F = 0.144 W/m x K)

Fiberglass   Assembly    Reference    Reference    K at Reference
Insulation               Thickness,   Density,     Density,
                         in.          lb/          Btu x in./
                                      [ft.sup.3]   [ft.sup.2 x
                                                   h x [degrees]F

R-10X        1, 3        3.4           0.45        0.333

R-19X        1 to 5      6.3           0.455       0.331

R-30X        2,4, 5,     9.0           0.58        0.300
             13

R-10         9           3.4           0.525       0.312

R-11         6 to 8,     3.7           0.545       0.307
             10 to 12

R-13         13          4.3           0.555       0.305

R-19         6-10        6.3           0.565       0.303

R-25         11          8.0           0.64        0.289

R-30         12          9.25          0.675       0.284

Foam         5, 9 to                               0.2; 0.26
             12;
             6 to 8

Steel        1 to 8,                               313.8; 309
             13;
             9 to 12

Air          1 to 13,                              0.180
             Air
             pocket
             in 9                                  1.096 *
             through
             12 *

* This is the apparent thermal conductivity of the air pockets
between the top insulation and roof panel in assemblies 9
through 12. Beard and Wangard (2013) used a thermal
resistance of 0.73 [ft.sup.2] x [degrees]F/Btu (0.13
[m.sup.2] x K/W) for the 0.8 in. (0.02 m) thick air
pockets in these assemblies.

Table 5. Calculated Results on Thermal Resistances
(1 [ft.sup.2] x h x [degrees]F/Btu = 0.1761
[m.sup.2] x K/W)

Assembly    R1BP          R2BP           RBP           [R.sub.AUP]

1           10.370        18.859         29.230        9.428
2           14.444        18.782         33.226        0
3           10.058        18.772         28.829        9.428
4           15.802        21.713         37.515        0
5           15.802        21.713         37.515        0
6           11.571        19.942         31.513        9.428
7           13.017        20.805         33.822        9.428
8           15.694        20.805         36.499        9.428
9           10.260        20.287         30.547        9.095
10          11.274        20.287         31.561        10.482
11          11.274        27.116         38.391        10.482
12          11.274        32.162         43.436        10.482
13          9.236         27.435         36.671        0

Assembly    [R.sub.2UP]   [R.sub.1OP]    [R.sub.FOP]   RTP

1           19.0          1.727          0             2.601
2           27.868        1.644          0             2.525
3           19.0          1.125          0             2.082
4           27.868        7.632          0             6.304
5           27.868        3.457          5             6.874
6           20.805        1.088          2.89          4.198
7           20.805        1.088          2.89          4.198
8           20.805        1.088          2.89          4.198
9           19.144        2.590          3.75          6.089
10          19.144        2.594          3.75          6.093
11          24.510        2.594          3.75          6.099
12          27.936        2.594          3.75          6.101
13          28.576        1.058          0             1.752

Assembly    [R.sub.BP + air]   [R.sub.TP + air]   [R.sub.insul-sys]

1           30.457              3.828             23.613
2           34.453              3.752             25.693
3           30.056              3.309             22.484
4           38.742              7.531             33.037
5           38.742              8.102             33.468
6           32.740              5.425             27.062
7           35.049              5.425             28.552
8           37.726              5.425             30.226
9           32.085              7.627             27.225
10          33.100              7.632             27.915
11          39.929              7.637             32.322
12          44.975              7.640             35.357
13          38.315              3.396             26.824

Table 6. U-Factors (Units: Btu/[ft.sup.2] x h x [degrees]F) from
Choudhary Correlation, Numerical Modeling, and Hot-Box Measurements
(1 Btu/[ft.sup.2] x h x [degrees]F = 5.6782 W/[m.sup.2] x K)

Assembly    U from          U from         U from
            Correlation     Modeling       Hot Box

1           0.046           0.049          NA
2           0.043           0.043          NA
3           0.048           0.048          NA
4           0.034           0.035          NA
5           0.033           0.035          NA
6           0.041           0.040          NA
7           0.039           0.038          NA
8           0.037           0.037          NA
9           0.040           0.039          NA
10          0.039           0.038          0.037, 0.039 *
11          0.034           0.033          0.031
12          0.031           0.031          0.029
13          0.041           0.042          0.044

Assembly    % Difference    % Difference
            Correlation     Correlation
            and Modeling    and Measured

1           -6.5             NA
2            0.0             NA
3            0.0             NA
4           -2.9             NA
5           -6.1             NA
6            2.4             NA
7            2.6             NA
8            0.0             NA
9            2.5             NA
10           2.6             5.1, 0
11           2.9             8.8
12           0.0             6.5
13          -2.4            -7.3

* Two separate values are quoted by Beard and Wangard (2013).
The lower value is for a long tab banded system and the higher
one is for a liner system. The mathematical model gives the
same value for the two cases.
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Publication:ASHRAE Transactions
Article Type:Report
Date:Jan 1, 2016
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