## 3 Methodology and Assumptions
### Exergy Analysis  
An exergy analysis can be applied to investigate the real thermodynamic inefficiencies, which are not accessible through a conventional energy analysis. The methodology is well established and has been applied to thermal energy conversion systems as well as to chemical processes [36]. Ambient conditions are assumed to be 15 \({}^{\circ}\)C and 1.013 bar, using the model of Szargut [37] for the determination of the specific chemical exergy of a \(j\)-th stream. Taking into account steady-state conditions, the exergy destruction rate \(\dot{E}_{D\dot{k}}\) within the \(k\)-th component is calculated as the difference between the exergy rate of fuel \(\dot{E}_{F,k}\) and exergy rate of product \(\dot{E}_{P,k}\)  
\[\dot{E}_{D,k}=\dot{E}_{F,k}-\dot{E}_{P,k} \tag{5}\]  
The exergy rates of fuel and product for different components are defined according to the SPECO approach [38]. Hereby, the exergy destruction rate \(\dot{E}_{D,k}\) quantifies the thermodynamic irreversibilities within the \(k\)-th component. Applying the exergy rate definitions, the exergetic efficiency \(\varepsilon_{k}\) for the \(k\)-th component can be calculated as  
\[\varepsilon_{tot}=\frac{\dot{E}_{P}}{\dot{E}_{F}}=1-\frac{\dot{E}_{D}}{\dot{E} _{F}} \tag{6}\]  
The exergetic efficiency is a unique indicator which allows for a characterization of a system's component in terms of performance and comparability. Another important parameter is the exergy destruction ratio, identifying the contribution of a single component to the reduction in exergetic efficiency of the overall system.  
\[y_{D,k}=\frac{\dot{E}_{D,k}}{\dot{E}_{F,k}} \tag{7}\]  
The exergy analysis provides information which is not available by a conventional thermodynamic analysis, identifies the main sources of thermodynamic inefficiencies, and develops first suggestions of possible improvements. However, a thermodynamic improvement which aims at minimizing the thermodynamic inefficiencies only represents a subcase of the general case of design improvement. Additionally, an economic analysis is required to consider the total investment cost in the design  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Component/System** & **Unit** & **Case I** & **Case II** & **Case III** \\ \hline \multicolumn{5}{l}{_Methanol Synthesis Unit (sub-system III)_} \\ \hline Length & m & 18 & 18 & 18 \\ Diameter & m & 0.037 & 0.037 & 0.037 \\ Heat Transfer Coefficient & kW/m\({}^{2}\)K & 0.28 & 0.28 & 0.28 \\ Purge Ratio & \% & 5 & 10 & 20 \\ \hline \multicolumn{5}{l}{_Purification (sub-system IV)_} \\ \hline \multicolumn{5}{l}{_Topping Column_} \\ \multicolumn{5}{l}{Reflux Ratio} & - & 0.6 & 0.6 & 0.6 \\ Distillate to Feed Mole Ratio & - & 0.3 & 0.1 & 0.1 \\ \multicolumn{5}{l}{_Pressure Column_} \\ \multicolumn{5}{l}{Reflux Ratio} & - & 0.75 & 0.6 & 0.75 \\ Distillate to Feed Mole Ratio & - & 0.9 & 0.95 & 0.95 \\ \multicolumn{5}{l}{_Atmospheric Column_} \\ \multicolumn{5}{l}{Reflux Ratio} & - & 0.9 & 0.8 & 0.8 \\ Distillate to Feed Mole Ratio & - & 0.77 & 0.95 & 0.95 \\ \hline \multicolumn{5}{l}{_Steam Cycle (sub-system V)_} \\ \hline \multicolumn{5}{l}{_steam turbine (driver) isentropic efficiency_} & - & 0.92 & 0.92 & 0.92 \\ \multicolumn{5}{l}{_condenser pressure_} & bar & 0.05 & 0.05 & 0.05 \\ \hline \hline \end{tabular}
\end{table}
Table 2: _Cont._improvement. Finally, an exergoeconomic analysis serves to assign monetary values to the thermodynamic inefficiencies, occurring within the system components.