## 2 Thermodynamic Theory and Modeling
### Heat Exchanger Sizing Model  
In order to size the shell-and-tube heat exchanger used for the heat transfer between the thermal oil and the refrigerant, the mass flow oil that passes through the heat exchanger is the key parameter. The mass flow refrigerant is obtained using compressor load [63; 64], which is given by  
\[\dot{m}=V_{d}R_{v}\rho_{4} \tag{9}\]  
where \(V_{d}\) is the displaced volume in m\({}^{3}\) h\({}^{-1}\), \(R_{v}\) the volumetric performance, and \(\rho_{4}\) the working fluid density in the turbine outlet. This density is related to the maximum value that may take the vapor density in an ORC. Therefore, it is also related to the maximum value of mass flow working fluid that may be leveraged in the turbine. Further, \(R_{v}=1-0.03r_{c}\) is a function of the compression relation, \(r_{c}=P_{3}/P_{1}\).  
The oil mass flow in the heat exchanger varies depending on the energy the working fluid needs to reach the saturated vapor state. Therefore, their values also depend on the  
\begin{table}
\begin{tabular}{l l c c c c} \hline \hline  & **Commercial Name** & \(\alpha_{1}\) & \(\alpha_{2}\cdot 10^{2}\) & \(\alpha_{3}\cdot 10^{6}\) & **Ref.** \\  & & & \(\mathbf{K^{-1}}\) & \(\mathbf{K^{-2}}\) & **Ref.** \\ \hline Oil 1 & Therminol LT & 12.8067 & 5.5078 & 2.4950 & [45] \\ Oil 2 & Paratherm HR & 43.8256 & 2.8091 & 46.8136 & [51] \\ Oil 3 & Paratherm NF & 7.7397 & 23.0976 & \(-15.0726\) & [52] \\ \hline  & & \(71\) & \(72\) & \(73\cdot 10^{4}\) & \\  & & \(\mathbf{kg\ m^{-3}}\) & \(\mathbf{kg\ m^{-3}K^{-1}}\) & \(\mathbf{kg\ m^{-3}K^{-2}}\) & \\ \hline Oil 1 & Therminol LT & 1027.42 & \(-0.3530\) & \(-6.8775\) & [45] \\ Oil 2 & Paratherm HR & 1188.77 & \(-0.7852\) & 0.1843 & [51] \\ Oil 3 & Paratherm NF & 1081.53 & \(-0.6787\) & 0.1990 & [52] \\ \hline \hline \end{tabular}
\end{table}
Table 2: Isobaric heat capacity parameters for the selected thermal oils, valid in the temperature range of 313.15 to 493.15 K. All fittings exhibit an R\({}^{2}\) greater than 0.998.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Value** & **Range** \\ \hline  & & **ORC** \\ \hline Turbine displaced volume & 120.00 cm\({}^{3}\) & - \\ Turbine revolutions per minute & 2900.00 rpm & - \\ Boiler saturation temperature & - & 330.15 to 368.15 K \\ Condenser temperature & 303.15 K & - \\ Expander isentropic efficiency & 1.0 & - \\ Pump isentropic efficiency & 1.0 & - \\ \hline  & & Energy source and TES \\ \hline Exhaust gases temperature & - & 493.15 K to 373.15 K \\ Pinch point temperature & - & 5, 10, and 15 K \\ \hline \hline \end{tabular}
\end{table}
Table 3: Input parameters and boundary conditions for ORC thermally powered by thermal energy recovery.  
mass flow of working fluid. To compute the volumetric flow, \(\dot{V}=\dot{m}/\rho\), the density as a function of the temperature is obtained by a fitted second-order polynomial. Table 2 shows the constants, \(\gamma_{i}\), for  
\[\rho=\gamma_{1}+\gamma_{2}T+\gamma_{3}T^{2} \tag{10}\]  
The velocity in the heat exchanger is a critical variable. Shell-side velocity lies between 0.6 to 1.5 m/s for water and similar liquids, and the number of tubes is chosen so that the tube-side velocity for water and similar liquids lies between 0.9 to 2.4 m/s [65,66]. In this work, the velocity is used as a function of the volumetric flow and piping diameter concerning the selection and sizing of the piping, fitting, valves, and pumps in hydraulic systems [67]. The velocity can calculated as  
\[v=353.678\frac{\dot{V}}{D^{2}} \tag{11}\]  
where \(v\) is the velocity in m/s, \(\dot{V}\) is the volumetric flow in m\({}^{3}\) h\({}^{-1}\), and \(D\) is the interior tube diameter in mm, limited to the commercial diameter (DN or NPS). The heat exchanger's base case comprises six tubes, as in [PERSON]'s work [65].