## 2 Thermodynamic Theory and Modeling  
### Thermodynamic Model  
Figure 1a shows a basic ORC fed by a thermal source, while Figure 1b displays the same layout connected to a TES system that operates with a thermal oil. In the standard ORC, depicted in Figure 1a, a saturated liquid (1) is compressed in a pump, obtaining a subcooled liquid (2). The liquid is heated until a saturated liquid at high pressure (3l) and then vaporized (3). These isobaric processes (2 to 3) obtain energy from exhaust gases. Finally, the saturated vapor (3) is expanded until the condenser pressure is reached (4). In contrast, in Figure 1b, the thermal source energizes the boiler while the remaining energy is used to heat a thermal oil. The oil preheats the fluid from the subcooled liquid to saturation. The rest of the cycle is identical to the previously described. The configuration of Figure 1 are represented in the temperature vs. entropy projection in Figure 2. Figure 2 represents the thermal source (exhaust gases) in crimson lines, while the TES unit is depicted in blue. In both cases, the points of the cycle are illustrated in green. The latter profile shows three fundamental points for analyzing the source. The thermal source enters the boiler or heat exchanger with a temperature \(T_{\text{i,s}}\). After the phase change of the working fluid, the temperature of the thermal source is \(T_{\text{m,s}}\), constituting an intermediated temperature of the source. The remaining energy from the thermal source is then delivered to the working fluid to preheat the saturated liquid after the compression in the pump, yielding a final temperature of \(T_{\text{o,s}}\). Figure 1: Schematic representation of an ORC composed of a pump, a heater, a heater or heat exchanger, a turbine, and a condenser. (**a**) The ORC is powered by exhaust. (**b**) The ORC is powered by a TES unit fed by exhaust gases. Figure 2b displays two different TES configurations representing the limiting conditions of the system. The C1 is set to have a thermal oil with a temperature near to the thermal source with a pinch point, \(T_{\text{PP,C1}}\), as a fixed parameter. The second configuration, C2, is defined as having a pinch point \(T_{\text{PP,C2}}\) as a function of the outlet temperature of the condenser, as shown in the blue lines. In both cases, configurations can be set using the design of the heat exchanger, the temperature of the thermal source, and the mass flow of oil or the heated fluid. The entropy scales in Figure 2 are merely illustrative. It is important to remark that the pinch-point temperature is defined differently depending on the configuration, as shown in Figure 2b. Although adding an extra heat exchanger and integrating thermal oil as an intermediate medium enhances the thermal and exergetic efficiencies of the ORC system, it is important to acknowledge the economic implications associated with this configuration. ORCs typically operate within a low power range, where the cost-effectiveness of additional components must be carefully evaluated. The increase in efficiency provided by the thermal oil loop does not necessarily translate into proportional net power gains, which may affect the financial viability of the system, particularly in small- to medium-scale applications. Consequently, this configuration is better suited for larger-scale installations where the efficiency improvements are more likely to offset the additional capital and operational costs. The thermal efficiency of a power cycle is determined by the ratio between the net power output, \(\dot{W}_{\text{n}}\), to the energy used in the process, \(\dot{Q}_{\text{n}}\). In a single-stage ORC, where each piece of equipment has the same mass flow, the thermal efficiency, \(\eta_{\text{I}}\), is given by  
\[\eta_{\text{I}}=\frac{\dot{W}_{\text{n}}}{Q} \tag{1}\]  
The added heat is the sum of the amount of energy of the heater and the boiler, \(\dot{Q}=\dot{Q}_{\text{n}}+\dot{Q}_{\text{b}}\).