Analytical Modeling of Internal Thermal Mass: Transient Heat Conduction in a Sphere under Constant, Exponential, and Periodic Ambient Temperatures

Internal thermal mass, such as furniture and partitions, plays a crucial role in enhancing building energy efficiency and indoor thermal comfort by passively regulating temperature fluctuations. However, the irregular geometry of these elements poses a significant challenge for accurate modeling in building energy simulations. This study addresses this gap by developing a rigorous analytical model that idealizes internal thermal mass as a sphere, thereby capturing multi-directional heat conduction effects that are neglected in simpler one-dimensional slab models. The transient heat conduction within the sphere is solved analytically using Duhamel’s theorem for three representative indoor air temperature scenarios: (1) constant, simulating a space with active HVAC; (2) exponentially decaying, representing a free-floating space after HVAC shutdown; and (3) periodically varying, corresponding to a naturally ventilated environment. Closed-form solutions are derived for the sphere’s internal temperature field, surface heat flux, and cumulative heat absorbed. The results demonstrate that a material’s Biot number governs its transient thermal response, with high-Biot-number materials (e.g., plywood) exhibiting a faster surface temperature rise but a steeper internal temperature gradient compared to low-Biot-number materials (e.g., concrete). The analysis of exponentially decaying and periodic scenarios reveals that sphere radius is the dominant factor determining total heat storage capacity; larger spheres absorb and release significantly more energy per cycle, despite having a lower heat flux density. Furthermore, a quantitative comparison of the decrement factor and time lag shows that while different materials may similarly dampen temperature amplitudes, a material with lower thermal diffusivity (like reinforced concrete) provides a substantially longer time lag, making it more effective for shifting thermal loads. This work provides a versatile and physically insightful analytical framework that advances the modeling accuracy of internal thermal mass beyond existing lumped-parameter methods.