The experimental plan was a full-factorial design for the five factors shown in Table 7. The full factorial design consisted of all possible combinations of levels of the five factors, so the total number of runs is 2700 (=6x5x6x5x3). Obviously, for this study, the primary factor was the solar reflectance of the roof. The other factors in Table 7 were, statistically speaking, considered to be "nuisance" factors.
Factors and Levels
In Table 7, the cities were ranked from lowest to highest by heating degree days, base 18 °C (see Table 2). This order is used hereafter in all plots and test results. As described above, these geographic locations were selected to cover a wide range of climatic conditions for the contiguous United States (Figures 1 and 2).
Five levels were selected for solar reflectance. The lower and upper values of 0.10 and 0.80 were based on measured extremes of building roofing products published in the literature (Berdahl 1994, Yarbrough 1994). A value of 0.10 corresponded to a dark or black-color shingle and a value of 0.80 corresponded to a highly reflective exterior radiation coating. In actuality, these values are somewhat extreme (particularly the upper limit), therefore, several intermediate values were also selected. Intermediate values of p were representative of either different color shingles (Reagan and Acklam 1979), or indicative of a "dirty or aged" reflective radiation coating (Bretz and Akbari 1994). In the TARP simulations, values for the solar absorptance (a) of the roof were computed from Equation (3).
p + a = 1 (3)
The levels for the ceiling thermal resistance (R-value) were based, in part, on criteria specified in ASHRAE 90.2 (ASHRAE 1993) and the DOE Insulation Fact Sheet (Department of Energy 1987). These publications typically recommended ceiling insulation values ranging from R-3.3 m2.K/W to R-8.6 m2.K/W (R-19 h.ft2-°F/Btu to R-49 h.ft2-°F/Btu) depending on geographic location. The minimum level of "none" was based on information compiled in Housing Characteristics 1990 (DOE 1992) which stated that, in 1990, a remarkable 19 % of single-family housing units in the United States had no ceiling insulation.
The attic ventilation rates were fixed at one of five levels for wind speeds ranging from 0.4 m/s to 8.9 m/s (1 mph to 20 mph). Table 8 summarizes wind speeds and attic ventilation rates for the five levels. The low level represented a poorly ventilated attic and the upper level a well-ventilated attic. An effective air leakage area (ELA) ratio of 300 for the floor-to-vent area was assumed. The volume of the attic was 106.4 m3. Using empirical data compiled for the Canada Mortgage and Housing Corporation (Buchan, Lawton, Parent Ltd. 1991), a relation for ventilation rate ( Va ) as a function of the wind speed was developed by Burch (1996):
I. 7 a = ELA (b v2)1/2 (4)
b = empirically derived regression coefficient (m2); and,
v = wind speed (m/s).
In the analysis, the mass framing area of the attic trusses was assumed to respond slowly to temperature changes and was modeled as an internal surface. The construction for the attic mass was modeled as wood and the surface area was set to one of three levels (Table 7). The mid-value for surface area was determined for 22 trusses, 0.6 m (24 in.) on center. Upper and lower values were computed using ± 20 % of the mid-value.
Simulation Runs for Building Loads
As before, the term full factorial means all possible combinations of settings for the five factors given in Table 7. There were 2700 (=6x5x6x5x3) combinations and hence 2700 simulation runs. The advantage of such a design is that all factors may be treated equivalently in the analysis, thus yielding conclusions that were truly global in nature, i.e., more robust. The disadvantage is that analysis of optimal settings for one factor, for example solar roof reflectance, would require analysis of subsets of data. Table 9 illustrates the development of the full factorial design undertaken for the study of building loads.
As observed in Table 9, the full-factorial design was developed systematically by varying each factor at all possible settings; in this case, starting with x5 (attic mass framing area) and proceeding to x4, x3, x2, and xl. Therefore, for each geographic location, 450 (5x6x5x3) separate simulation runs were conducted. For example, run #4 (11121) is for Miami FL with the following levels for the other factors: a solar roof reflectance of 0.1; ceiling thermal resistance of "none"; attic ventilation rate of 2.3 11-1; and, attic mass framing area of 31.0 m2. The last run for Miami is #450 (15653) which would have a solar roof reflectance of 0.8; ceiling thermal resistance of R-8.6 m2.KJW; attic ventilation rate of 9.2 -1; and, attic mass framing area of 46.5 m2. Completing this approach, run #2700 (65653) is for Bismarck ND with a solar roof reflectance of 0.8; ceiling thermal resistance of R-8.6 m2.K/W attic ventilation rate of 9.2 h '; and, attic mass framing area of 46.5 m2.
The 2700 input building description files for TARP were prepared as follows. Initially, a single file for Miami, Florida was prepared (see Appendix B for sample). The building description files for the other cities were prepared from the Miami input file by essentially modifying the prescriptive criteria of the building envelope in accordance with values given in Table 4. A BASIC program was subsequently used to generate the 2700 separate building description files needed for the analysis. Using a DOS batch file, all 2700 simulations were executed in 31 hours on a personal computer with a 486 DX CPU operating at 66 megahertz.
The output file of each TARP simulation run provided the following heating and cooling requirements for the conditioned living space (Zone 2): 1) the predicted annual heating and sensible cooling loads in MJ (or kBtu) (i.e., cumulative); and, 2) the predicted hourly peak heating and sensible cooling loads in kW. Hereafter, unless described otherwise, the term cooling load is taken to mean sensible cooling load. Latent cooling loads are treated later for the economic analysis. Note that only Zone 2 is conditioned in the model. Therefore, the heating and cooling requirements for Zone 2 (i.e., living space) are the same requirements for the building itself. It is also important to note that the results obtained for hourly peak loads are not to be confused with design loads that are calculated under specific design conditions. Using a (post-processor) BASIC program, the 2700 values of predicted annual heating and cooling loads and hourly peak heating and cooling loads were compiled into a single data file and subsequently analyzed with the NIST statistical plotting package, Dataplot (Filliben 1977).
This section presents the building load results for the 2700 TARP simulation runs plotted versus the input factors. Annual heating (cooling) loads and peak heating (cooling) loads are first ranked on a global basis, that is, collectively for all five factors. Next, the results are ranked locally for each geographic location comparing only three factors - roof solar reflectance, ceiling thermal resistance, and attic ventilation. Finally, a subset of the annual heating (cooling) loads and peak heating (cooling) loads is plotted as a function of roof solar reflectance for different levels of ceiling thermal resistance.
Global Ranking of the Factors
The five factors - city, roof solar reflectance, ceiling thermal resistance (R-value in the plots), attic ventilation, and attic mass framing area - were ranked (largest effect to smallest) using the design of experiment (dex) mean plot. This plot determines the mean value of the response variable for each level of each factor. The analysis is valid because the experimental design is balanced (see Experimental Design). That is, any particular value of the response variables has an equal number of runs for each independent factor. The advantage of this analysis is that general conclusions can be deduced quickly for the best settings (levels) of each factor. The disadvantage is that each point contains variations (or contamination) due to the other four factors. Therefore, optimal local settings of each factor require analysis of a subset of data. Figures 7 and 8 illustrate the dex-mean plot for annual heating (cooling) loads and peak heating (cooling) loads, respectively.
In Figure 7a, mean values for annual heating load (in megajoules [M.1]) are plotted on the y-axis versus the five input factors on the x-axis. For each factor on the x-axis, the data are plotted in the order shown in Table 7 from left to right. For example, the left point for city represents the average heating load of 450 simulation runs for Miami and the right point represents the average of 450 simulation runs for Bismarck. The plot shows a monotonic trend for each factor. The global ranking (i.e., largest effect to smallest) for each factor is as follows: city; ceiling R-value; roof solar reflectance or attic ventilation; and, mass framing area which was inconsequential. For all locations, the largest reduction in annual heating load was due to ceiling R-value. Moreover, the lowest annual heating load was at the highest level of ceiling insulation. Not surprisingly, the greatest reduction was due to adding the first level of ceiling insulation to an uninsulated attic. Thereafter, the reductions diminished for higher levels of ceiling R-value. Higher levels of solar reflectance and attic ventilation increased the annual heating load (sometimes referred to as heating penalty).
Figure 7b plots mean values for annual cooling load on the y-axis versus the five input factors on the x-axis. In this case, since the cities are ranked by heating degree days, the trend for city is not monotonic. The annual cooling loads required for Phoenix (level 2) and Portland ME (level 5) were the highest and lowest, respectively. The global ranking for each factor from highest to lowest effect is city, ceiling R-value, roof solar reflectance, ventilation, and mass framing area which was again inconsequential. As observed above in Figure 7a, a large reduction in annual cooling resulted by adding thermal insulation to an uninsulated attic with diminishing effects thereafter. However,
contrary to the annual heating results, the roof solar reflectance significantly reduced the annual cooling load. In fact, on average, the lowest annual cooling load for each and all locations and for each and all R-values occurred at the highest level of solar reflectance (Figure 7b). The reduction in annual cooling appears to be linearly related to roof solar reflectance. In general, higher levels of attic ventilation also decreased the annual cooling load, although to a lesser extent.
Figure 8a plots mean values for the hourly peak heating load (in kilowatts [kW]) on the y-axis versus the input factors on the x-axis. The results are similar to annual heating (Figure 7a). The global ranking for each factor is city; ceiling R-value, and, ventilation. The effects of roof solar reflectance and attic framing mass were both inconsequential (Figure 8a). Typically, peak heating for a residence occurs at night and, therefore, changes in roof solar reflectance would have no effect.
Figure 8b plots mean values for hourly peak cooling load on the y-axis versus the input factors on the x-axis. Here, the results are similar to annual cooling (Figure 7b), except for two findings. Ceiling R-value caused the greatest effect and the lowest peak cooling load was for Miami. Like annual cooling, the hourly peak cooling load was linearly related to the roof solar reflectance.
The global rank for the factors in Figures 7 and 8 are summarized in Table 10. In general, the geographic location of the city had the greatest effect, that is, the greatest overall change on building loads, in all cases except for hourly peak cooling. The next factor was ceiling R-value which was always beneficial, meaning higher levels of the ceiling R-value decreased both heating and cooling loads. In all cases, increasing the attic insulation from "none," (uninsulated case) to the first level of R-1.9 m2.K/W (R-11 h-ft2•°FBtu) resulted in the greatest load reduction. Thereafter, the incremental benefits diminished for each subsequent level of attic insulation. The roof solar reflectance was beneficial for cooling loads (both annual and hourly peak), detrimental for annual heating, and unimportant for hourly peak heating. The beneficial effect was linearly related to the roof solar reflectance. Likewise, attic ventilation was beneficial for cooling, although to a lesser extent, and detrimental for heating. The effect of the attic mass was inconsequential for all cases and was neglected in further analysis. Recall that one of the advantages of this analysis is that general conclusions can be deduced quickly for the best settings (levels) of each factor. Unfortunately, in this case, the effect of geographic location overwhelmed the other factors and therefore was removed from subsequent analyses by evaluating the other building factors on a local level.
Ranking the Factors by City
At the global level described above, the factors of interest were ranked in order of effect for annual heating (cooling), and peak heating (cooling). As noted in Table 10, geographic location was the primary factor and, in fact, overwhelmed the effect of the other factors. Ranking the factors by city allows further insight in determining the optimum levels of factors at each geographic location. Further, a local analysis allows us to verify (and strengthen) that the above global conclusions were in fact valid for all geographic locations. In order to rank the factors by city, the dex-mean plot was applied locally; that is, on a case-by-case basis to each individual city for three factors - solar reflectance, ceiling R-value, and attic ventilation rate.
Figures 9, 10, 11, and 12 illustrate the application of the dex-mean plot for the annual heating load, annual cooling load, peak heating load, and peak cooling load, respectively. Each figure contains the same sequence of six plots for the cities: Miami, Phoenix, Birmingham, Washington, D.C., Portland ME, and Bismarck. The factors of interest - solar reflectance, ceiling R-value, and attic ventilation rate - are plotted on the x-axis. Again, the levels for each factor are shown from left to right following the order in Table 7.
Examination of the data in Figures 9 to 12 reveals that the trends for each city were, for the most part, very similar to the global results presented in Figures 7 and 8. In Figure 9, the trends in the data for each factor at each geographic location changed monotonically and were in agreement with the global data in Figure 7a. At all locations, higher levels in ceiling insulation achieved the greatest reduction in annual heating. Moreover, the lowest annual heating load for each city was obtained with the highest level of attic thermal insulation. Again, the greatest reduction in annual heating load was observed by adding insulation to an uninsulated attic. Further reductions due to ceiling thermal resistance were incrementally smaller for each level of thermal resistance. Increased levels of solar reflectance and attic ventilation rates resulted in small but undesired increases in the required annual heating load, as noted previously in Figure 7a.
In Figure 10, the trends in the data for each factor at each geographic location changed monotonically and were in agreement with the global data in Figure 7b. At all locations, except Portland ME, ceiling insulation provided the greatest reduction in annual cooling. The solar reflectance of the roof, however, reduced annual cooling significantly. In fact, at all locations except Phoenix, the lowest annual cooling load was achieved at the highest value of solar reflectance. The reduction in annual cooling appears to be a linear effect of roof solar reflectance. It is interesting to note that if one neglects the uninsulated case, then the effect of solar reflectance provides the largest reduction in annual cooling loads. Increased levels of attic ventilation also decreased the annual cooling load, although to a lesser extent.
Figure 10 depicts somewhat peculiar results for Portland ME at high levels of ceiling R-value. At levels above R-3.3 m2•K/W (R-19 ft2.°F-h/Btu), the annual cooling load increased slightly; meaning that external cooling (from the HVAC system) was required to remove the heat gain to the conditioned space. This seemingly unusual result was an artifact of the input parameters for the building and is discussed further in Building Loads Discussion. It should be re-stated that the model did not account for the (beneficial) effects of natural ventilation. Specifically, the analysis did not include a schedule for opening and closing of the windows. The author acknowledges that ordinarily the occupants, if home, would more than likely (although not necessarily) take advantage of natural ventilation for space cooling.
Hourly peak heating and cooling loads are illustrated in Figures 11 and 12, respectively. For peak heating (Figure 11), trends in the data for each city were very similar to the global trends illustrated in Figure 8a. In order of importance, the largest benefit resulted from increased levels of attic insulation. The effect of solar reflectance was inconsequential and increased levels of ventilation were detrimental (i.e., small increase in peak heating). For peak cooling (Figure 12), trends in the data for each city were again very similar to the global trends illustrated in Figure 8b. As observed before in Figure 8b, the factors affecting peak cooling in order of benefit were R-value, solar reflectance, and ventilation. The effects were beneficial, that is, higher levels reduced peak cooling requirements. Again, if one neglects the uninsulated case, then solar reflectance provides the largest reduction in peak cooling.
Table 11 summarize the optimum settings (levels) for each factor by city. A precautionary note is required so that the results are not misinterpreted. The dex-mean plot is an averaging tool, so ranking of optimum settings is only valid within a factor, not between factors. Generalizing that a specific level of one factor is better than a level of the other factors should not be attempted.
The results in Table 11 are consistent with engineering expectations, as well as previous results published in the literature (Griggs and Courville 1989, Parker et al. 1998). In general, the best settings for heating (annual and peak) are low solar reflectance, high ceiling thermal resistance, and low ventilation rates for the attic'. For cooling, the best settings are high solar reflectance, high ceiling thermal resistance, and high ventilation rates for the attic. What is particularly powerful is that these settings are (globally) consistent, as shown above, across a wide range of geographical locations (and therefore climatic conditions) in the contiguous U.S. The effects for solar reflectance (as well as ventilation) are diametrically opposite for heating and cooling. Thus, for heating, a low-reflectance (i.e., black) roof is best, and for cooling a high-reflectance (i.e., white) roof is best. Since current technologies at present do not permit a roof with both levels, what is required is an analysis of economic trade-offs (i.e., cooling benefit versus heating penalty) covered in Economic Discussion.
Effect of Solar Reflectance on Predicted Annual Loads (Heating and Cooling)
In order to determine the effect of roof solar reflectance and ceiling R-value on the annual heating and cooling loads, a subset of the data presented in Figures 9 to 12 was examined. The attic ventilation rate was fixed at the lowest setting (0.5 air changes per hour) and the attic mass framing area fixed at the mid-value of 38.7 m2 (Table 7). For each location, the annual load in MJ was plotted on the y-axis and solar reflectance on the x-axis. The effect of ceiling R-value was treated parametrically and varied from "none" (i.e., uninsulated) to R-8.6 m2.K/W (R-49 ft2•°F-h/Btu) following the levels described in Table 7. Figures 13 to 18 illustrate the annual heating loads and annual cooling loads for Miami, Phoenix, Birmingham, Washington, D.C., Portland ME, and Bismarck.
A discussion of Figure 13 for Miami will serve as a useful example for the other cities (Figures 14 to 18). Figures 13a and 13b plot the annual heating and cooling loads, respectively, versus the roof solar reflectance over the range 0.10 to 0.80 at the levels shown in Table 7. Parametric values for ceiling R-values ranged from "none" (uninsulated case) to R-8.6 m2-K/W (R-49 ft2•°F•h/Btu) and are identified by line-type in the plot. In Figure 13a, the annual heating load increased nonlinearly as the solar reflectance increased. For a fixed level of ceiling R-value, the largest increase in annual heating was for an uninsulated attic. As before, a substantial reduction in the annual heating and cooling loads was noted between the uninsulated case and the level of R-1.9 m2-K/W (R-11 ft2-°F•h/Btu). In Figure 13b, the annual cooling load decreased linearly as the solar reflectance increased. In fact, for a particular level of ceiling R-value, the annual cooling load requirement was minimized at the highest level of the roof solar reflectance. The slope for the line, however, was less pronounced for higher levels of ceiling R-value. The greatest reduction in annual cooling load was for an uninsulated attic. It is interesting to note that the lines seem to converge at higher levels of solar reflectance (Figure 13b).
'Note that factors other than energy considerations such as moisture control are generally more important in determining the proper or appropriate levels of attic ventilation. The analysis presented here is strictly based on energy efficiency.
An examination of the other cities (Figures 14 to 18) revealed results consistent with the trends observed for Miami, and the global plots above. Note that for Portland ME and Bismarck (Figures 17 and 18, respectively), anomalous annual cooling loads were obtained at high levels of roof solar reflectance and ceiling R-values due to the removal of internal heat gains. As mentioned, these results are largely an artifact of the input parameters for the model and are discussed further in Building Loads Discussion. In general, however, for any given level of attic insulation, higher levels of solar reflectance caused an undesired increase in the annual heating load and a beneficial decrease in the annual cooling load. Of particular interest is whether the benefits from lower annual cooling outweigh the detrimental increase in annual heating. This will be discussed further in Economic Discussion.
Effect of Solar Reflectance on Predicted Hourly Peak Loads (Heating and Cooling)
Figures 19 to 24 show the hourly peak heating and cooling loads for each city as a function of solar reflectance and different levels of ceiling R-value. Like the previous plots for annual loads, the attic ventilation rate was fixed at the lowest level (see Table 7), the attic mass framing area at Level 2, and the solar reflectance and ceiling R-value ranged from their respective minimum to maximum levels. In general, the results are consistent with the global plots for peak heating (cooling) loads presented in Figures 11 and 12, respectively. For peak heating loads, increasing levels of solar reflectance were essentially inconsequential (Figures 19a to 24a). For peak cooling loads, however, the roof solar reflectance resulted in a linear reduction of peak cooling loads (Figures 19b to 24b). Thus, for any given level of ceiling R-value, the hourly peak cooling load requirement was minimized at the highest level of the roof solar reflectance. The slope of the line, however, was less pronounced for higher levels of ceiling R-value. The greatest reduction in the hourly peak cooling load was for the case of an uninsulated attic. Again, it is interesting to note that the lines tend to converge at high levels of roof solar reflectance. From a demand point-of-view, it would seem that high levels of roof solar reflectance could be quite beneficial for a residence.