## 1 Introduction  
The global push of the energy transition is leading to a shift away from conventional thermal power plants towards innovative concepts that supply climate-neutral energy in the form of electricity, heat, or cooling by efficiently utilizing renewable electricity and the integration of thermal energy storage systems. Carnot batteries are one of these innovative concepts that have been the subject of increased scientific research in this context for several years, with the majority of the literature focusing on pure electricity storage systems. \"Carnot Batteries are an emerging technology for the inexpensive and site-independent storage of electric energy at medium to large scale. Also referred to as \"Pumped Thermal Electricity Storage\" (PTES) or \"Pumped Heat Storage\" (PHES), a Carnot Battery transforms electricity into thermal energy, stores the thermal energy in inexpensive storage media such as water or molten salt and transforms the thermal energy back to electricity as required.\" [1]. One possible implementation of a Carnot battery uses the Joule or Brayton process, which operates with gaseous working fluids. This system, known as a Brayton battery, comprises two sets of turbomachinery (a turbine and a compressor) and two thermal storage systems at different temperature levels. These elements work together in a coupled cycle, as illustrated in the flow diagram in Figure 0(a). During the charging process (left part of the flow diagram), a motor drives the compression of the working fluid in a turbo compressor or blower, increasing its temperature, as shown in the T-s diagram in Figure 0(b) (counterclockwise in blue). The heated working fluid then transfers thermal energy to the high-temperature thermal energy storage (HT-TES) system. Afterward, the cooled working fluid moves to the turbine, where it expands and cools down while generating energy through a generator. The fluid then heats up again in the low-temperature thermal energy storage (LT-TES) system. In the discharging process (right part of the flow diagram), the cycle reverses, as depicted in the T-s diagram (clockwise in red). Electrical energy is supplied during the charging phase and discharged during the discharging phase. In theory, Brayton batteries can achieve very high round trip efficiencies (RTEs) if ideal components (component efficiencies = 100%) or very high process temperatures (>>1000 \({}^{\circ}\)C) are assumed. However, this is not practically achievable with current technology. RTEs in the range of 40 to 50% are more realistically achievable. During the charging and discharging cycles, a considerable amount of energy is converted into heat, which either cannot be utilized at all or only partially, especially when the system is used solely for electricity generation. This heat needs to be dissipated from the process, which is usually provided for during charging before the turbine or during discharging after the compressor. In principle, this can be achieved at different points, but it has not been systematically investigated so far. Generally, there are numerous theoretical studies on Brayton batteries in the literature, which are summarized in the following section. This summary focuses on subcritical processes. Essentially, this summary is based on the review papers by [PERSON] et al. [2], [PERSON] et al. [3], and [PERSON] et al. [4], supplemented by other important publications. Most concepts envisage a closed-loop system, but there are exceptions. [PERSON] [5], for example, proposed an open-loop system using air as the working fluid in a patent in 1979, where air is compressed to 800-900 \({}^{\circ}\)C, and the heat is stored in ceramic spheres in steel pipes. In the years 2008-2009, SAIPEM [6] and Isentropic Ltd. [7] each presented concepts based on a closed-loop argon cycle. SAIPEM proposed using maximum temperatures between 1000 \({}^{\circ}\)C and 1500 \({}^{\circ}\)C and storing thermal energy in bricks made of refractory material.