Theory for the Thermal Capacitor and Thermal Mass Components
This section presents the underlying theory of the Thermal Capacitor two-port component and the Thermal Mass one-port component.
Thermal Capacitance
The thermal capacitance C (SI unit: J/K) is a measure of how much heat a body can store. It is defined as:
with V the volume (SI unit: m3), ρ the density (SI unit: kg/m3), Cp the heat capacity at constant volume (SI unit: J/(kg·K)), and m the mass (SI unit: kg).
Lumped Thermal Capacitance Model
Under certain conditions, the temperature T of a solid body subjected to convective heat transfer on its boundaries may be considered as homogeneous, depending only on time, using the thermal capacitance:
where:
T0 is the initial body temperature
h is the convective heat transfer coefficient (SI unit: W/(m²·K))
A is the surface of convective heat transfer (SI unit: m²)
This approximation, referred as the lumped thermal capacitance model (see Ref. 40), holds when h, C, and A are constant and the gradients of temperature within the body are expected to be smaller than the gradients of temperature between the body and the surrounding. It happens for example when the thermal contact between the solid and the fluid is poor, or when the solid is a good thermal conductor.
This is characterized by the Biot number Bi (dimensionless):
where L is a relevant length scale of the body, and k is its thermal conductivity.
The lumped thermal capacitance model is usually assumed to be valid when Bi<0.1.
Regarding the network representation of thermal systems, the Biot number may be used to determine how many nodes should be included, assuming an homogeneous temperature distribution in the corresponding domains.