# Transcritical R744 Supermarket Refrigeration System Integrated with a Heat-Driven Ejector Chiller
## 2 Materials and Methods
### Mathematical Modeling and Simulation
\(\dot{m}_{8}\) in Equation (5) was the mass flow rate of the working fluid in the HEC cycle through the generator, while \(\dot{m}_{5}\) was the mass flow rate of R744 through the generator. At state 12, the specific enthalpy of R744 was estimated by energy balance in the S1 section of the generator (Equation (6)). \[h_{12}=h_{11 pp}-\left(\frac{\dot{m}_{8}(h_{17 pp}-h_{16})}{\dot{m}_{5}}\right) \tag{6}\]  
The heat removed in the subcooler by the HEC fluid (Equation (7)) was determined iteratively by adjusting a set of variables, which included the GC pressure, degree of subcooling, and the generator temperature [16]. These variables were adjusted in such a way that the heat extracted by the HEC fluid exactly matched the heat rejected by the R744 in the subcooler (Equation (8)) [16]. The pinch-point temperature in the subcooler was assumed to be 3 K [25]. \[\left(\dot{Q}_{R744}\right)_{sub}=\dot{m}_{5}(h_{13}(T_{amb}+app,P_{GC})-h_{14 }(T_{13}-DOS,P_{GC})) \tag{7}\]  
\[\left(\dot{Q}_{HEC}\right)_{sub}=\dot{m}_{7}(h_{18}-h_{21}(=h_{20})) \tag{8}\]  
where, \(\dot{m}_{7}\) was the flow rate of the HEC fluid in the subcooler, \(app\) was the approach temperature at the gas-cooler (GC) outlet, \(DOS\) was the subcooling degree, and \(P_{GC}\) was the pressure in the gas-cooler. The phase of the HEC fluid at the state 20 (Figure 1) was considered as saturated liquid. The schematic of the ejector section in the HEC cycle is shown in Figure 3. A 1D flow approach similar to that reported in [26] was used to model the ejector. Figure 2: Simplified layout of the generator. The velocity of the HEC fluid at state b was calculated from Equation (9) [26]. The velocity of the HEC fluid at state 17 was neglected. \[u_{b}=\sqrt{2\eta_{noz}(h_{17}-h_{bs}(P_{mix},s_{17}))} \tag{9}\]  
where, \(h_{bs}\) was the specific enthalpy of the HEC fluid expanding via a reversible adiabatic process, and \(P_{mix}\) was the mixing chamber pressure. The actual specific enthalpy of the motive fluid at the nozzle outlet (section b) was determined by taking into consideration the various losses inside the motive nozzle (\(\eta_{noz}\)) using Equation (10) [26]. \[\eta_{noz}=\frac{h_{17}-h_{b}}{h_{17}-h_{bs}} \tag{10}\]  
The velocity of the HEC fluid at the nozzle outlet was calculated using Equation (11) [16; 26]. Since the increase in entropy during the expansion of the suction stream was very small, the process was assumed to be isentropic [26]. Figure 2.0  
The specific enthalpy of the refrigerant at section \(x\) (Figure 3) was determined by taking into account the losses in the mixing chamber (\(\eta_{mix}\)) using Equation (12) [26]. Figure 2.1  
The specific enthalpy of the refrigerant stream at the outlet of the diffuser (State 19 (Figure 3)) was estimated using energy balance across the diffuser section (Equation (13)) [26]. The losses in the diffuser section were taken into account in terms of the efficiency of the diffuser (\(\eta_{diff}\)). Figure 2.2  
The entrainment ratio of the ejector defined as the ratio of the suction to the motive flow rate, was computed using Equation (14) [26]. Figure 2.3  
The algorithm employed for solving the set of equations governing the functioning of the ejector of the HEC system was obtained from [16; 26]. The efficiencies of the nozzle,  
Figure 2.4 Sectional view of the ejector. mixing chamber, and the diffuser section of the ejector were taken as 90%, 80%, and 90% respectively [25]. The efficiency of the proposed system was evaluated in terms of COP, which was estimated using Equation (15). Figure 2.5  
where, the power required by the LT (\(\hat{W}_{LTC}\)), MT (\(\hat{W}_{MTC}\)), and PC (\(\hat{W}_{PC}\)) compressors were estimated using the correlations for the global efficiencies summarized in Figure 2.6 The power needed by the gas-cooler and condenser fans (\(\hat{W}_{fan}\)) was assumed to account for about 3% of the total heat rejected in the respective heat exchangers [23].