## 3 Mathematical Model and Thermodynamics Analysis  
The utilization of solar energy for the desalination of water, air conditioning, electricity generation, and hot water generation is being investigated and deployed in multi-generation systems intended for domestic applications. The multi-generation system is split into three subsystems, ORC, HDH, and DCS, to determine each subsystem's thermo  
Figure 2: Psychometric cycles of proposed multi-generation: (a) MCS-I; (b) MCS-II. dynamic characteristics and the entire system's performance. During the simulation model phase, the first and second laws of thermodynamics are applied to each system component. In the current model, the following assumptions have been considered:  
* All system processes are assumed to be in a steady state. * Air/water leakage in system components is ignored. * The kinetic and gravitational energies are not considered. * The air wet-bulb temperature and the blowdown water temperature leaving the humidifier are the same. * Process air, return air, and water has the same mass flow rate. * The cold outlet streams from ORC condenser are assumed to be the same (t\({}_{\text{s1}}\) = t\({}_{\text{w3}}\)) to distribute the condenser capacity on HDH and DCS cycles. * At the dehumidifier's exit, the fresh water and the air wet-bulb temperatures are the same. * The ORC fluid's states at the turbine inlet are dry-saturated and superheated based on the different values of t\({}_{\text{evap}}\) and degree of superheating. * Based on the pressure of the condenser, the ORC fluid state at the pump inlet is saturated liquid. * In DCS and HDH systems, the special power of auxiliary components (power consumed by fans) is ignored. * Organic fluid (n-Octane) is selected to carry out the comparative study of the proposed systems due to its performance and thermodynamic properties [24]. * The studied and operating parameters values, ranges, and relevant efficiencies for the system's components are given in Tables 1 and 2. * Superheat degree at ORC turbine inlet and ORC evaporation temperature were selected based on the maximum temperature (200 \({}^{\circ}\)C) that can be obtained from the evacuated tube solar collectors and critical temperature of organic working fluids. * ORC condensation temperature was selected based on the temperature of refrigerant in the condenser that should be greater than the heated air (t\({}_{\text{a7}}\)) and sea water (t\({}_{\text{w2}}\)) condensation temperature where the heat transfer process occurs in the right direction
* Mass flow rate ratio (0-0.4) was selected based on previously published work for HDH systems
* Ambient air inlet temperature, ambient air inlet humidity, seawater inlet temperature and average solar intensity were selected based on average climatic conditions of the Jeddah city, Saudi Arabia
* Conditioned space air temperature, and humidity were selected based on average inside design human comfort conditions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value / Range** \\ \hline Superheat degree at turbine inlet, \(\Delta\)t\({}_{\text{sup}}\) & 0–45 \({}^{\circ}\)C \\ ORC evaporation temperature, t\({}_{\text{evap}}\) & 150 \({}^{\circ}\)C \\ ORC condensation temperature, t\({}_{\text{f1}}\) & 40–60 \({}^{\circ}\)C \\ ORC fluid mass flow rate, m\({}^{\bullet}\)ORC & 1 kJ/s \\ Mass flow rate ratio, MR & 0.1–0.4 \\ Ambient air inlet temperature, t\({}_{\text{a1}}\) & 35 \({}^{\circ}\)C \\ Ambient air inlet humidity, w\({}_{\text{a1}}\) & 15 g\({}_{\text{v}}\)/kg\({}_{\text{a}}\) \\ Seawater inlet temperature, t\({}_{\text{w1}}\) & 15–25 \({}^{\circ}\)C \\ Conditioned space air temperature, t\({}_{\text{a5}}\) & 25 \({}^{\circ}\)C \\ Conditioned space air humidity, w\({}_{\text{a5}}\) & 12 g\({}_{\text{v}}\)/kg\({}_{\text{a}}\) \\ Average solar intensity, I\({}_{\text{avg}}\) & 0.8 kW/m\({}^{2}\) (Jeddah city) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Studied and operating parameters; values and ranges.