# Fun with math models

## Greg Kallio assesses heat gain of various walls

Sustainable Space columnists Lori Brown and Greg Kallio are professors in the College of Engineering, Computer Science and Construction Management at Chico State University.

Super model
Engineers love to use mathematical models (i.e. a set of equations describing some physical event or system). To be trusted, the models must include all the correct physics and be verified by experiments. A robust mathematical model is powerful—the engineer or scientist can then use a computer to simulate complicated processes. The model that I’ve formulated is pretty simple: predicting the heat gain through a house wall due to varying outside temperature and solar irradiation.

Test subjects
The motivation to do this study comes from my interest in thermal mass—namely the effect of heat-storing building components on the summertime cooling load of a home. I have simulated the heat transfer through seven different wall systems: 1) a fictitious, no-mass wall; 2) a 2-by-6 wood-framed insulated wall; 3) a structural insulated panel (SIP) wall; 4) an insulated concrete form (ICF) wall; 5) a Thermomass wall (insulation sandwiched between two concrete layers); 6) an exterior-concrete, interior-insulation wall; and 7) an exterior-insulation, interior-concrete wall.

The four concrete walls have identical masses but differ in placement. All seven have an R-value of 23.4 and produce identical heat gains when subjected to steady outside and inside air temperatures and a constant solar irradiation. However, in reality, the outside conditions are constantly changing and this gives rise to different heat gains due to the thermal-mass effect.

Heat wave
The environmental conditions I chose correspond to the maximum and minimum temperatures recorded at my house on July 27 (105 degrees and 66 degrees, respectively). The simulation used an unshaded south-facing wall, and the interior air temperature was maintained at 77 degrees.

The accompanying graph shows the heat gains of each wall through the course of the day. You might expect the majority of heat gain to occur in the afternoon when the outside temperature is greatest. This is true for the low-mass walls (2-by-6 framed and SIP), but not for the four concrete walls.

These massive walls are able to store heat and distribute the heat gain over the entire day. This presents two advantages: 1) it allows the walls to discharge some heat to the outdoors during the cooler night time, thereby reducing the overall cooling load, and 2) it shifts the heat-gain peak to the evening when natural ventilation can be used to purge the heat instead of running the air conditioner. This also lessens the load on our electric utility grid and is particularly advantageous for those who have photovoltaic systems with time-of-use net metering.

In summation …
When the heat gain numbers are summed up (see graph legend), all four concrete walls have similar cooling loads over the day and yield about 10 percent to 12 percent improvement over the low-mass walls. The Thermomass wall yields the lowest cooling load during the 12-6 p.m. peak electricity period, while the ICF wall shows the greatest “buffering” effect.

Does this mean that 2-by-6 framed and SIP walls are poor choices for hot climates? Not necessarily. The concrete walls are typically more costly and their thermal-mass benefit relies on the amount of night-time cooling. Furthermore, thermal mass can be added to wood-framed and SIP walls by finishing with cement wallboard and siding.