## 2 Thermodynamic Theory and Modeling
### Thermodynamic Model
Additionally, the net work may be written as \(\dot{W}_{\text{n}}=\dot{Q}-\dot{Q}_{c}\), \(\dot{Q}_{c}\) being the amount of extracted heat in the condenser. The amount of heat can be calculated by the enthalpy differences between two points of the cycle. For instance, for the heater stage, the energy change is given by  
\[\dot{Q}_{\text{h}}=m\left(\dot{H}_{3l}-\dot{H}_{2}\right)=-\dot{m}_{\text{s}} \big{(}\dot{H}_{o,s}-\dot{H}_{c,s}\big{)} \tag{2}\]  
Figure 2: Schematic illustrations of a rudimentary ORC in the temperature vs. entropy projection. Illustrations also depict the thermal source and thermal energy recovery temperature profiles. (**a**) ORC without the TES system. (**b**) ORC with two TES configurations. where the subscripts represent the same nomenclature shown in Figure 2. The ORC uses R134a, one of the most common, widespread, and cost-effective working fluids [29, 30]. Furthermore, R134 has low flammability, an acceptable GWP, and high availability. The thermophysical properties were obtained using the multiparametric equation of state (EOS) proposed by [PERSON] and [PERSON] [31]. The Tillner-Roth EOS is regarded as equivalent to experimental data, demonstrating remarkable accuracy: deviations for vapor pressure are within \(\pm\)0.05% at temperatures above 220 K and within \(\pm\)20 Pa at lower temperatures. Liquid density predictions are highly precise, with deviations below \(\pm\)0.01%, while isochoric heat capacity estimates maintain errors around \(\pm\)1.00%. Consequently, thermal properties such as enthalpy and entropy also exhibit deviations of less than \(\pm\)1.00%. All calculations were performed using Wolfram Mathematica (12.0.0.0) [32]. The flow-specific exergy can be defined as [33]  
\[\psi_{k}=\tilde{H}_{k}-\tilde{H}_{0}-T_{0}\big{(}\tilde{S}_{k}-\tilde{S}_{0} \big{)} \tag{3}\]  
where the subscript \(k\) concerns the cycle's flows, while the subscript 0 represents the dead-point condition, chosen at a temperature of 293.15 K.  
In a simple and adiabatic tube-and-shell heat exchanger, the supplied exergy is obtained through the decrease in the hot-flow exergy. Additionally, the recovered exergy is the increase in the cold-flow exergy, as long as it is not at a lower temperature than the surroundings [20, 33]. Therefore, the energetic efficiency between the heater-boiler and the source yields  
\[\eta_{\rm II}=\eta_{\rm ILs}=-\frac{\dot{m}(\psi_{3}-\psi_{2})}{\dot{m}_{s}( \psi_{o,s}-\psi_{i,s})} \tag{4}\]  
In the same way, which is evaluated as heater-boilerergetic efficiency, the heater and the boiler can be separately assessed. This study is centred on the evaluation of heater exergetic efficiency. The latter can be written as  
\[\eta_{\rm II,h}=-\frac{\dot{m}(\psi_{3l}-\psi_{2})}{\dot{m}_{s}(\psi_{m,s}- \psi_{o,s})} \tag{5}\]  
Adding the TES between the thermal source and the working fluid changes Equation (4). Considering C1 and C2, the exergetic efficiency for both configurations can be expressed as  
\[\eta_{\rm IL,C1}=\eta_{\rm IL,C2}=-\frac{\dot{m}(\psi_{3l}-\psi_{2})}{\dot{m} _{o}(\psi_{o,o}-\psi_{o,i})} \tag{6}\]  
where the differences in the TES exergies for both configurations are given by  
Figure 1:0  
where \(T_{0}\) is the dead-point temperature, while \(T_{1}\) and \(T_{2}\) are the operational temperatures. Additionally, the isobaric heat capacity of the thermal oil, \(C_{P,o}\), may be function of the temperature. The selection process was conducted among 28 thermal oils, based on an initial analysis of their density, viscosity, heat capacity, thermal conductivity, and operational temperature range. The density was required to be lower than 1000 kg m\({}^{-3}\) (\(A\)), while the viscosity needed to remain under 20 cSt (\(B\)) to ensure low energy consumption during pumping. The heat capacity and thermal conductivity were prioritized to be as high as possible to enhance heat absorption efficiency.