An Analytical Solution for Dynamic Thermal Transmission Loads

Thermal mass in building envelopes reduces the magnitude of diurnal conductive loads and, in some cases, reduces energy use for heating and cooling. In general, dynamic simulation is required to estimate the magnitude of these effects. This paper seeks to derive closed-form expressions of thermal time lag and amplitude dampening through building envelope structures, to aid in the design process before detailed simulation is performed, and to improve intuition about the effects of thermal mass in buildings. An analytical solution for the temperature distribution through an infinite wall subjected to a sinusoidal temperature boundary condition is derived. The solution is verified by comparison with a finite element solution. Next, it is hypothesized that the analytical solution for an infinite wall could also describe the temperature variation on the inside surface of a finite wall. The hypothesis is tested by comparing the temperature distribution predicted by the analytical solution to the temperature distribution predicted by a finite element model of a finite wall. The results confirm that analytical solution adequately described the temperature variation in a finite wall. Based on the analytical solution, separate closed-form expressions of time delay and amplitude dampening for thermal load transmission through building envelopes are derived. The use of these expressions is demonstrated for light frame and concrete walls.