Selecting T and ΔT, and a quick estimate for R
The temperatures for the R# calculation depend on the materials present in the section of the building envelope containing the enclosed reflective air space, as well as the temperatures of the exterior and interior surfaces of the section. Under steady-state conditions, the T and DT can both be calculated as follows:
- Rext = the sum of R-values on the exterior side of the air space;
- Rint = the sum of R-values on the interior side of the air space; and
- R# calculated for DT = 5 °F and 75 °F—reasonable starting temperatures in many cases.
Rmaterial can be included as either exterior or interior thermal resistance, depending on its location.
Figure 1 contains a diagram showing the locations of the above thermal resistances. If the exterior surface temperature is 38 C (100 F), and the interior surface temperature is 21 C (70 F), then:
- T (exterior surface of air space) = 100 − 30·Rext/(Rext + R# + Rint); and
- T (interior side of air space) =70 + 30·Rint /(Rext + R# + Rint).
The temperature difference across the air space is the absolute value of the temperature difference, a positive number:
I T(exterior surface of air space) – T(interior side of air space) I
The average temperature for the air space is:
[T(exterior surface of air space) + T(interior side of air space)]/2.
The average temperature and the ΔT can be used to calculate a new value for the R# of the enclosed reflective air space. If a precise value for R# is the objective, then the above process can be repeated until the sequence of estimated R#-values becomes constant. This calculation can be done using a spreadsheet program such as Excel.
As mentioned, building insulations in the United States are labeled with R-values determined at 75 F. It seems reasonable, therefore, to make a reflective air space estimate at that mean temperature. Calculated R-values for the air space with DT of 5°F and a second calculation with DT equal to 10 F will provide a range of likely values for the air-space thermal resistance in many cases. Figure 2 contains estimates for R# for air spaces from 1⁄2 to 2 in., with a low-emittance surface on one side.
How good is the estimate?
The enclosed air-space R#-values in the American Society of Heating, Refrigerating, and Air-conditioning Engineers’ ASHRAE Handbook of Fundamentals include selected results obtained from a large number of hot-box tests. (See the 2014 ASHRAE Handbook of Fundamentals, “Effective Thermal Resistances of Plane Air Spaces,” Chapter 26, Table 3.) R#-values estimated using ISO 6946 compare favorably with the ASHRAE values.
A few comparisons of air-space R#-values from the two sources are shown in Figure 3.
The comparison of R#-values from two sources is quite good in the absence of convection, as should be the case for heat-flow down. The two methods for obtaining R# use different approaches for the convective contribution, and the results differ by as much as 25 percent in the case of heat-flow up, as shown in Figure 3.
The R#s calculated with the ISO method are intended to be used for quick estimates of the potential benefits of using a reflective insulation in the building envelope. The heat-flow is taken to be one-dimensional and steady-state. This estimation provides a way to obtain the reflective air-space R#-value estimates for conditions not in the ASHRAE tabulations. It is important to know these estimates do not include multidimensional effect, which, in most cases, tends to reduce the R# for the enclosed air space. Nevertheless, the ISO model provides useful information for exploring several potential advantages of adding a reflective component to the building envelope.
This article shows the procedure for calculating the thermal resistance of a planar enclosed air space, as published in an international standard formatted to provide results with IP units. A limited comparison with hot-box results represented by R#-values published by ASHRAE shows agreement for heat-flow down and heat-flow horizontal. The small R-values provided when the heat-flow direction is up differ by as much as 0.5 or 25 percent in a few of the cases listed in Figure 3.
The R#-values in this discussion are for single air spaces enclosed and oriented as indicated by the heat-flow direction. ISO 6946 allows use of an average value for ‘d’ when an air space is tapered. The model is useful for exploring the performance for different climate regions, with thermal resistance increasing as temperature is reduced. However, these calculations are not intended to replace or satisfy any legal requirements for labeling and advertising.
David W. Yarbrough, PhD, PE, is vice-president of R&D Services Inc., an accredited, independent laboratory specializing in thermal insulations and related building materials. Yarbrough is an emeritus professor of chemical engineering at Tennessee Technological University, and a registered engineer in the states of both Tennessee and Florida. He is also a member of the Building Enclosure Technology and Environment Council (BETEC), an active member of ASTM Committee C 16 (Thermal Insulation), the International Thermal Conductivity Conference, and the American Society of Heating, Refrigerating, and Air-conditioning Engineers (ASHRAE). Yarbrough can be contacted via e-mail at firstname.lastname@example.org.