# Modeling the Transient Response of Thermal Circuits
## 1 Introduction  
The analysis of heat transfer phenomena is one of the most widely discussed topic in engineering literature [1]. Precise models have been achieved for most situations, with the exception of systems involving fluids, especially when turbulence plays a role. The majority of studies in literature use numerical methods to solve the heat equation [2]  
\[\rho c\frac{\partial T}{\partial t}=\
abla\cdot(k\
abla T), \tag{1}\]  
where \(T\) is the temperature, \(t\) the time, \(k\) the thermal conductivity, \(\rho\) the density, and \(c\) the specific heat. The heat equation can be solved analytically but for very strict conditions, which is out of the scope of the great majority of the actual conditions. An alternative technique is the lumped method where thermal elements are considered at 0 D but can only be applied for Biot numbers much below 1:  
\[Bi=\frac{hL}{k}\ll 1, \tag{2}\]  
where \(h\) is the heat transfer coefficient between the element and the environment, \(L\) is the characteristic length of the thermal element and \(k\) is the thermal conductivity of the thermal element.  
A special case occurs when changes do not occur in time. Stationary thermal circuits have been used extensively to calculate temperatures and heat transfer phenomena in steady state situations [3, 4]. It is one of the most simple and precise ways to study thermal processes. Nevertheless, the usual approach does not include the possibility to calculate transient responses. In fact, unlike electric circuits where the velocity of transmission of electric currents is dictated by the speed of light (300,000 ms\({}^{-1}\)), which is large enough to avoid collateral effects, the velocity of change in thermal circuits is extremely low, and, thus, collateral effect occur already for small characteristic lengths. As an example, thediffusion coefficient of copper, which can be seen as the equivalent to the speed of light in electric circuits, is only \(\sim\)10\({}^{-4}\) m\({}^{2}\)s\({}^{-1}\). This makes it difficult to calculate transient responses, e.g., within the context of thermal circuits that only use thermal resistances connected to reservoirs at constant temperatures and connected between them.  
The addition of heat capacitors to existing thermal circuit models was able to approximate transient responses for some applications. [PERSON] et al. proposed a resistance-capacity network model to investigate the energy performance of buildings [5]. [PERSON] et al. developed a time-series model using thermal capacitors for real-time thermal load estimation [6]. [PERSON] et al. used a lumped model together with RC circuits to study a doubled skin natural and mechanical ventilated test cell [7]. More recently, [PERSON] and [PERSON] et al. considered RC circuits to analyze the thermal performance of building wall and ventilation flow [8; 9], and [PERSON] et al. modeled cascaded caloric refrigeration systems based on thermal diodes or switches using heat capacitances [10]. In all situations, the thermal capacitors were connected to the ground temperature and the respective values were guessed for the specifics of each system. In this context, current tools to analyze general thermal processes lie within the scope of numerical methods, which are problematic to implement in situations where computational resources are limited.  
In this work, a new method to analyze heat transient processes is proposed within the scope of thermal circuits by extending thermal elements from thermal resistances to thermal resistances and heat capacitors in parallel. A model to generalize the calculation of heat capacitor values was developed and validated for three simple thermal circuits using numerical calculations from the Python package _heatrapy_. The method was also validated for a more complex thermal system model using experimental data of a thermal switch from literature.