# Transcritical R744 Supermarket Refrigeration System Integrated with a Heat-Driven Ejector Chiller
## 2 Materials and Methods
### Mathematical Modeling and Simulation  
The set of equations used to model the proposed system are presented in Equations (1)-(15). The following assumptions were taken into consideration in the present study [16]:  
* Steady state operations;
* Drop in pressure in the heat exchangers and the pipelines were neglected;
* Constant pressure in the mixing chamber of the ejector;
* Heat losses/gains from the components were neglected;
* Kinetic energy of the fluid stream at the inlet of the nozzle and outlet of the diffuser of the ejector were neglected. The refrigeration load in the MT evaporators was estimated using Equation (1) [11] based on the data obtained from the field measurements reported in [23]. The load had a linear variation from 100 kW to 190 kW with the rise in the outdoor air temperature from 10 \({}^{\circ}\)C to 35 \({}^{\circ}\)C, respectively. However, the refrigeration load was considered to be fixed at 100 kW at temperatures below 10 \({}^{\circ}\)C [23]. The LT refrigeration load (\(\dot{Q}_{LT}\)) was considered to be fixed at 35 kW [16]. \[\dot{Q}_{MT}=3.6T_{amb}+64 \tag{1}\]  
where, \(T_{amb}\) was the outdoor air temperature. The refrigerant flow rates through the LT and MT evaporators were calculated using Equation (2) and Equation (3), respectively. \[\dot{m}_{1}=\frac{\dot{Q}_{LT}}{(h_{2}-h_{1})}=\frac{\dot{Q}_{LT}}{(h_{2}-h_{ 7})} \tag{2}\]  
\[\dot{m}_{2}=\frac{\dot{Q}_{MT}}{(h_{5}-h_{4})}=\frac{\dot{Q}_{MT}}{(h_{5}-h_{ 7})} \tag{3}\]  
The refrigerant flow rate handled by the parallel compressor (PC) was calculated using Equation (4). \[\dot{m}_{4}=\frac{\dot{m}_{6}}{(1-x_{15})}-\dot{m}_{6} \tag{4}\]  
where, \(\dot{m}_{6}\big{(}\dot{m}_{6}=\dot{m}_{1}+\dot{m}_{2}\big{)}\) was the refrigerant flow rate through the MTC compressor, and \(x_{15}\) was the vapor fraction of the two-phase R744 after expansion in the HPEV (Figure 1). The global efficiencies of the R744 compressors were calculated based on the correlations [24] summarized in Table 1. \(P_{suc}\), and \(P_{dis}\) were the suction and discharge pressures across the R744 compressors. \begin{table}
\begin{tabular}{c c c} \hline
**Compressor** & **Operating Mode** & **Global Efficiency**[24] \\ \hline \hline \multirow{2}{*}{LTC} & \multirow{2}{*}{Subcritical} & \(\eta_{LTC}=-0.0012\Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}^{2}-0.0087\Big{(} \frac{P_{dis}}{P_{suc}}\Big{)}+0.6992\) \\ \hline \hline \multirow{2}{*}{MTC} & \multirow{2}{*}{Subcritical} & \((\eta_{MTC})_{sub}=-0.1155\Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}^{2}+0.5762 \Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}-0.0404\) \\ \hline \hline \multirow{2}{*}{MTC} & \multirow{2}{*}{Transcritical} & \((\eta_{PC})_{sub}=-0.172\Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}^{2}+0.7095\Big{(} \frac{P_{dis}}{P_{suc}}\Big{)}-0.0373\) \\ \hline \hline \multirow{2}{*}{PC} & \multirow{2}{*}{Subcritical} & \((\eta_{MTC})_{trans}=-0.0021\Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}^{2}-0.0155 \Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}+0.7325\) \\ \hline \hline \multirow{2}{*}{PC} & \multirow{2}{*}{Transcritical} & \((\eta_{PC})_{trans}=-0.0788\Big{(}\frac{P_{dis}}{P_{suc}}\Big{)}^{2}+0.3708\Big{(} \frac{P_{dis}}{P_{suc}}\Big{)}+0.2729\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Global efficiencies of the R744 compressors. The simplified layout of the generator is shown in Figure 2. The generator section of the HEC unit was divided into two sections, S1 and S2, in order to calculate the specific enthalpy of the refrigerants at the section '11 pp' and '17 pp'. In the generator, the pinch-point temperature (pp) was considered to be 3 K [25]. At state 11 pp, the specific enthalpy of R744 was estimated by energy balance in the S2 section of the generator (Equation (5)). \[h_{11 pp}=h_{11}-\left(\frac{\dot{m}_{8}(h_{17}-h_{17 pp})}{\dot{m}_{5}}\right) \tag{5}\]  
At state 17 pp, the phase of the HEC fluid was considered as saturated liquid, while at the state 17, the phase was considered as saturated vapor.